\documentclass[12pt,a4paper]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsmath} \usepackage{amssymb} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2606} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %TCIDATA{Created=Tuesday, March 14, 2006 10:42:59} %TCIDATA{LastRevised=Tuesday, May 04, 2010 14:24:32} %TCIDATA{} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{CSTFile=40 LaTeX article.cst} %TCIDATA{ComputeDefs= %1$p_{h}=\beta \frac{\left( 1+0.018+0.036\right) ^{\alpha -1}}{2}$ %1$p_{l}=\beta \frac{\left( 1+0.018-0.036\right) ^{\alpha -1}}{2}$ %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \section{Overview} Let's first look at standard preferences -- additive expected utility and define some notation. Time is discrete, $t=0,1,...T.$ At each $t\geq 0$, an event $z_{t}$ is drawn from a set $\Gamma _{t}.$ A history, denoted $z^{t}$ is a collection of events up to an including $z_{t}$, i.e., e.g., $\left\{ z_{0},z_{1},...,z_{t}\right\} .$ The set of possible histories at $t$ is $% \Gamma ^{t}.$ Let $c\left( z^{t}\right)$ denote the vector of consumption goods (possibly including service flows and leisure) consumed in period $t$ if $z^{t}$ is realized$.$Then, the standard assumption is that for any $t,$ preferences over $\left\{ c\left( z^{t}\right) |z^{t}\in \Gamma ^{t}\right\} _{t=0}^{T}$ with associated probabilities $p\left( z^{t}\right)$ can be represented by the function% \begin{equation*} U\left( \left\{ c\left( z^{t}\right) |z^{t}\in \Gamma ^{t}\right\} _{t=0}^{T}\right) =\sum_{t=0}^{T}\beta ^{t}\sum_{z^{t}\in \Gamma ^{t}}p\left( z^{t}\right) u\left( c\left( z^{t}\right) \right) =E\sum_{t=0}^{T}\beta ^{t}u\left( c\left( z^{t}\right) \right) \end{equation*}% where $u$ is a smooth increasing concave function, called the per-period utility or felicity function. Several important features of these preferences will be relaxed during the course. \begin{enumerate} \item Curvature of $u$ determines elasticity of substitution both between states within a period (inverse of risk aversion) and between periods, intertemporal elasticity of substitution. Therefore, these intrinsically different concepts cannot be disentangled theoretically. \item No preference for time revelation -- it does not matter when we get to know our faith (unless, of course, if we can condition our actions on the information). \item No history effect; the utility \begin{equation*} E\sum_{t=s}^{T}\beta ^{t}c\left( z^{t}\right) \end{equation*}% is independent of consumption taking place before $s.$ (However, history of course matters by affecting the budget, i.e., the consumption possibility set, usually simply captured by a state variable like wealth. \item Time consistency; if \begin{equation*} U\left( c_{1},\left\{ c\left( z^{t}\right) \right\} _{t=2}^{T}\right) \geq U\left( c_{1},\left\{ c^{\prime }\left( z^{t}\right) \right\} _{t=2}^{T}\right) \end{equation*}% then% \begin{equation*} U\left( \left\{ c\left( z^{t}\right) \right\} _{t=2}^{T}\right) \geq U\left( \left\{ c^{\prime }\left( z^{t}\right) \right\} _{t=2}^{T}\right) , \end{equation*}% i.e., if the sequence of consumption $\left\{ c\left( z^{t}\right) \right\} _{t=2}^{T}$ is preferred over $\left\{ c^{\prime }\left( z^{t}\right) \right\} _{t=2}^{T}$ at $t=1,$ it is preferred also at $t=2.$ \item Risk matters only through second order effects. E.g., suppose in period $t,$ there is a lottery with payoff $\tilde{x}$ and $E\tilde{x}>0$. Then all individuals will prefer to hold a strictly positive amount of this lottery over having a certain consumption level $\bar{c}$. I.e., there is a $% \bar{k}>0$ such that for all $k\in \left( 0,\bar{k}\right)$ \begin{equation*} E\left( u\left( \bar{c}+k\tilde{x}\right) \right) >u\left( \bar{c}\right) . \end{equation*}% To see this, let's take the derivative of the left hand side w.r.t. $k$ and evaluate at $k=0.$% \begin{eqnarray*} \frac{\partial \left( E\left( u\left( \bar{c}+k\tilde{x}\right) \right) \right) }{\partial k}|_{k=0} &=&E\left( u^{\prime }\left( \bar{c}+k\tilde{x}% \right) \tilde{x}\right) |_{k=0} \\ &=&u^{\prime }\left( \bar{c}\right) E\tilde{x} \end{eqnarray*}% which is positive if $E\tilde{x}>0,$ regardless of riskaversion. \end{enumerate} Empirical findings, introspection and lab experiments have all shown that these implications are often invalidated. \subsection{Two important macro-puzzles.} \begin{enumerate} \item \textbf{Equity premium puzzle } \item \textbf{Too little insurance puzzle.} \end{enumerate} Most well-known example. Mehra-Prescott, "equity premium puzzle". Consider the following example (almost) in there original formulation and we will return to it later. Suppose consumption grows fast with $p=1/2$ and slowly otherwise. Starting from a consumption level $c_{t}$, history at $t+1$ is either $z^{t+1}=z_{h}$ in which case $c_{t+1}=c_{t}\left( 1+g+\sigma \right)$ or $z^{t+1}=z_{l}$ and $c_{t+1}=c_{t}\left( 1+g-\sigma \right) .$ We can now using the standard Euler condition for maximizing utility to price a bond that gives 1 unit of consumption in both cases and a share that gives $% 1+g+\sigma$ or $1+g-\sigma$. The share is thought to be a claim to the process that provides the consumption possibilities. Individuals eat only apples. Apples are grown on a tree (or a number of identical trees( which has exogenous but stochastic output that grows with a rate $g+\sigma$ or $% g-\sigma$. A share is ownership of the tree. The Euler equation is \begin{equation*} u^{\prime }\left( c_{t}\right) =\beta Eu^{\prime }\left( c_{t+1}\right) R_{t+1}. \end{equation*} Well known intuition, give up one unit of consumption "costs" $u^{\prime }\left( c_{t}\right) ,$ invest it and increasing consumption by $R_{t+1}$ next period gives an increase in utility given by $\beta Eu^{\prime }\left( c_{t+1}\right) R_{t+1}.$ We find asset prices by first defining state-contingent prices of Arrow-Debreu type. In period $t$ there is one lottery that gives 1 unit of consumption in $t+1$ if growth is high and zero otherwise. There is also a lottery that gives one unit of consumption in the low growth state. The price of these lotteries and $p_{h}$ and $p_{l}$ and the returns $\frac{% p\left( z_{h}\right) }{p_{h}}$ and $\frac{p\left( z_{l}\right) }{p_{l}}$ respectively. Therefore, \begin{eqnarray*} p_{h} &=&\beta \frac{u^{\prime }\left( c\left( z_{h}\right) \right) p\left( z_{h}\right) }{u^{\prime }\left( c_{t}\right) } \\ p_{l} &=&\beta \frac{u^{\prime }\left( c\left( z_{l}\right) \right) p\left( z_{l}\right) }{u^{\prime }\left( c_{t}\right) } \end{eqnarray*} Using the standard CRRA felicity function% \begin{eqnarray*} u\left( c\right) &=&\frac{1}{\alpha }c^{\alpha } \\ \alpha &\leq &1, \end{eqnarray*}% recalling that the coefficient of RRA is \begin{equation*} -c\frac{u^{\prime \prime }\left( c\right) }{u^{\prime }\left( c\right) }=-c% \frac{\left( \alpha -1\right) c^{\alpha -2}}{c^{\alpha -1}}=1-\alpha . \end{equation*} Setting $p\left( z_{h}\right) =p\left( z_{l}\right) =\frac{1}{2}$ we have% \begin{eqnarray*} p_{h} &=&\beta \frac{\left( 1+g+\sigma \right) ^{\alpha -1}}{2}, \\ p_{l} &=&\beta \frac{\left( 1+g-\sigma \right) ^{\alpha -1}}{2}. \end{eqnarray*} A portfolio consisting of one each of the lotteries mimics perfectly the safe one-period bond. The price of a bond is thus \begin{equation*} p_{b}=p_{h}+p_{l} \end{equation*}% with a return, \begin{equation*} r_{b}=\frac{1}{p_{h}+p_{l}}. \end{equation*} Let us now compute the return on a claim to next periods dividends -- a one-period share. A portfolio consisting of $\left( 1+g+\sigma \right)$ of the $h$ lottery and $\left( 1+g-\sigma \right)$ of the $l$ lottery exactly replicates such a one period share. The price of this risky portfolio is \begin{equation*} p_{r}=p_{h}\left( \left( 1+g+\sigma \right) \right) +p_{l}\left( 1+g-\sigma \right) \end{equation*}% and its expected return is therefore% \begin{eqnarray*} r_{r} &=&\frac{1+g}{p_{h}\left( \left( 1+g+\sigma \right) \right) +p_{l}\left( 1+g-\sigma \right) } \\ &=&\frac{1+g}{\beta \frac{\left( 1+g+\sigma \right) ^{\alpha -1}}{2}\left( 1+g+\sigma \right) +\beta \frac{\left( 1+g-\sigma \right) ^{\alpha -1}}{2}% \left( 1+g-\sigma \right) } \\ &=&\frac{2\left( 1+g\right) }{\beta \left( \left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }\right) } \end{eqnarray*} In this simple economy, the return on a more standard type of share, i.e., a one that gives rights to all future dividends is the same. Why? To see this, we recall that the price of the share with CARA utility will be proportional to current income/consumption. The price of the share will therefore be $% P_{r}c_{t}$. and the return \begin{equation*} \frac{P_{r}c_{t+1}+c_{t+1}}{P_{r}c_{t}}=\frac{\left( 1+P_{r}\right) c_{t+1}}{% P_{r}c_{t}}. \end{equation*}% From the Euler equation, \begin{eqnarray*} P_{r} &=&\beta E\frac{u^{\prime }\left( c_{t+1}\right) \left( 1+P_{r}\right) c_{t+1}}{u^{\prime }\left( c_{t}\right) c_{t}} \\ &=&\left( 1+P_{r}\right) \beta \frac{\frac{\left( c_{t}\left( 1+g+\sigma \right) \right) ^{\alpha -1}}{2}\left( c_{t}\left( 1+g+\sigma \right) \right) +\frac{\left( c_{t}\left( 1+g-\sigma \right) \right) ^{\alpha -1}}{2}% \left( c_{t}\left( 1+g-\sigma \right) \right) }{c_{t}^{\alpha -1}c_{t}} \\ &=&\left( 1+P_{r}\right) \beta \frac{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}{2}. \\ \frac{1+P_{r}}{P_{r}} &=&\frac{2}{\beta \left( \left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }\right) } \end{eqnarray*} Finally, calculate the expected return on the share:% \begin{eqnarray*} &&\frac{1+P_{r}}{P_{r}}E\frac{c_{t+1}}{c_{t}} \\ &=&\frac{1+P_{r}}{P_{r}}\left( 1+g\right) \\ &=&\frac{2\left( 1+g\right) }{\beta \left( \left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }\right) }, \end{eqnarray*}% as with the one-period share. Using (US) data $g$ is around 1.8\% per year and $\sigma$ is around 3.6\%. Stock market returns have averaged around 8\% per year and the risk-free rate around 1\% over the last 100 years or so. Therefore% \begin{eqnarray*} p_{h} &=&\beta \frac{\left( 1+0.018+0.036\right) ^{\alpha -1}}{2} \\ p_{l} &=&\beta \frac{\left( 1+0.018-0.036\right) ^{\alpha -1}}{2} \\ r_{b} &=&\frac{1}{p_{h}+p_{l}} \\ r_{r} &=&\frac{1+0.018}{p_{h}\left( 1+0.018+0.036\right) +p_{l}\left( 1+0.018-0.036\right) } \end{eqnarray*} Can we find $\alpha$ and $\beta$ to generate the observed values of $% r_{r}=1.08$ and $r_{b}=1.01?$ \FRAME{dtbpFUX}{4.8304in}{3.2197in}{0pt}{\Qcb{Combinations of $\protect% \alpha$ and $\protect\beta$ such that $r_{a}=1.08$ (red,dashed) and $% r_{b}=1.01$ (black,solid)}}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.8304in;height 3.2197in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "-20";xmax "0.9";ymin "0";ymax "1.5";xviewmin "-20.418";xviewmax "1.32636";yviewmin "0.924580501867469";yviewmax "1.12470972646838";plottype 12;num-x-gridlines 24;num-y-gridlines 24;plotstyle "patch";axesstyle "normal";xis \TEXUX{v58123};yis \TEXUX{v58124};var1name \TEXUX{$\alpha$};var2name \TEXUX{$\beta$};function \TEXUX{$\frac{1+0.018}{p_{h}\left( 1+0.018+0.036\right) +p_{l}\left( 1+0.018-0.036\right) }=1.08$};linecolor "blue";linestyle 3;pointstyle "point";linethickness 3;lineAttributes "Dots";var1range "-20,0.9";var2range "0,1.5";num-x-gridlines 24;num-y-gridlines 24;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"XY";function \TEXUX{$\frac{1}{p_{h}+p_{l}}=1.01$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "-20,0.9";var2range "0,1.5";num-x-gridlines 24;num-y-gridlines 24;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"XY";valid_file "T";tempfilename 'L1MPX700.wmf';tempfile-properties "XPR";}} If, for example, we set $\beta =0.98,$ riskaversion. $1-\alpha$ should be 3.5, to motivate 8\% stock return. But then the bond return should be 7.6\%, leaving a mere 0.4\% risk premium. In fact, it is very difficult to get the right risk premium. Let's look closer at the risk premium. Let's express it as the ratio of the price of the bond to the ratio of the price of the risky asset \begin{eqnarray*} \frac{p_{b}}{p_{r}} &=&\frac{p_{h}+p_{l}}{p_{h}\left( 1+g+\sigma \right) +p_{l}\left( 1+g-\sigma \right) } \\ &=&\frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }} \end{eqnarray*} In reality, this ratio is \begin{equation*} \frac{\frac{1}{1.01}}{\frac{1+g}{1.08}}\approx 1.05 \end{equation*} However, by plotting \begin{equation*} \left[ \frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}\right] _{g=0.018,\sigma =0.036} \end{equation*}% against $RRA=1-\alpha ,$\FRAME{dtbpFX}{4.4998in}{3.0004in}{0pt}{}{}{Plot}{% \special{language "Scientific Word";type "MAPLEPLOT";width 4.4998in;height 3.0004in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "100";xviewmin "-2";xviewmax "102.04";yviewmin "0.981599299107496";yviewmax "1.01900022318977";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{R};var1name \TEXUX{$R$};function \TEXUX{$\left[ \frac{\left( 1+g+\sigma \right) ^{-R}+\left( 1+g-\sigma \right) ^{-R}}{\left( 1+g+\sigma \right) ^{1-R}+\left( 1+g-\sigma \right) ^{1-R}}\right] _{g=0.018,\sigma =0.036}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,100";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'L01QYF07.wmf';tempfile-properties "XPR";}} we see that it is very difficult to get the right risk premium. In fact, in the realistic case where $\sigma >g,$ it is easy to bound the risk premium, \begin{eqnarray*} \lim_{\alpha \rightarrow -\infty }\left[ \frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}\right] _{g<\sigma } &=&\lim_{\alpha \rightarrow -\infty }\left[ \frac{\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g-\sigma \right) ^{\alpha }}\right] \\ &=&\frac{1}{1+g-\sigma }\approx 1.0183. \end{eqnarray*} The fist line comes from the fact that $\left( 1+g-\sigma \right) ^{\alpha }$ goes to infinity as $\alpha$ approach $-\infty ,$ while $\left( 1+g+\sigma \right) ^{\alpha }$ approach zero. Of course, with lower growth and higher risk, the risk premium can get larger, but we are stuck with data. 2. \textbf{Too little risk-sharing} In a complete markets equilibrium where individuals have homothetic preferences, e.g., CRRA, there should be full risk sharing. Consumption growth should be perfectly correlated between individuals and everyone should hold a share in a global portfolio of assets. This is not the case, obviously frictions and asymmetric information may be one explanation. But sometimes these explanations don't seem to suffice. An example is the home bias puzzle. All around the world local investors hold unbalanced portfolios with to much domestic assets. It is shown in the literature that expected returns could increase a lot, without increasing risk by having more balanced portfolios, containing more foreign assets. The explanation cannot be that information is superior. Then, domestic holders should sometimes have more negative information than foreign investors, in which case they should sell, moving to foreign assets, this we don't see. Conversely, they should sometimes go short abroad, having an investment share above unity at home, which we don't see either. \subsection{Lab puzzles} \textbf{Ambiguity aversion -- Ellsberg Paradox.} Consider following lottery. There are two urns, each with 100 balls. In urn 1, there are 50 red and 50 black. In urn 2, there are only red and black balls but the proportions are unknown. The subject is given a color and can pick one ball. If a ball with the given color comes up, the gain is 50\$, if not the gain is zero. The subject is asked to rank lotteries. Typically the following response comes up. \begin{enumerate} \item Red from urn 1$\sim $Black from urn 1. \item Red from urn 2$\sim $Black from urn 2. \item Red from urn 1$\succ $Red from urn 2. \item Black from urn 1$\succ $Black from urn 2. \end{enumerate} This contradicts expected utility since from 2, we expect subjective probability to imply that they believe$p\left( red\right) =1/2.$Then, 3 and 4 should be with indifference. \textbf{Time inconsistency}. In lab experiments, preference reversal occurs. Example. Suppose you can choose between 10CD's 1 year from now or 11 CD's 1 year and a week. Often the latter is preferred. However, if the same individual is asked after a year, 10 CD's today might be preferred over 11 CD's in a week. This is inconsistent with standard time-additive utility with geometric discounting. A week's discounting depends on how close in time it is. Other examples, people sometimes seems to pay to commit. They tend to over-consume during the year, and, for example, ask their employer to keep money for tax-payments at the end of the year or, say for big holidays. \textbf{First-order risk-aversion} With smooth preferences, people should as we have seen not care much about small gambles, ilke holding a balanced stockmarket protfolio. Big risks, on the other hand, are detrimental. In fact, with CRRA coefficient bigger than unity, sufficiently big losses can never be compensated since$U\left( c_{t}\right) $is bounded from above but not from below. For example, consider a lottery that gives a relative loss of$x$, forcing a consumption loss of$xc$with$p=\frac{1}{2}$and otherwise gives consumption$\left( 1+k\right) c$. For different values of$x,$how large must$k$be to compensate for so that% \begin{equation*} U\left( c\right) =\frac{1}{2}U\left( \left( 1-x\right) c\right) +\frac{1}{2}% U\left( (1+k)c\right) \end{equation*} Here, I plot this$k$as a function of$x$for$\alpha =-3,-4$and$-6$.% \FRAME{dtbpFX}{4.4998in}{3.0004in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4998in;height 3.0004in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "0.4";ymin "0";ymax "10";xviewmin "-0.004123333642983E0";xviewmax "0.210372482464997";yviewmin "-0.2";yviewmax "10.204";plottype 12;num-x-gridlines 24;num-y-gridlines 24;plotstyle "patch";axesstyle "normal";xis \TEXUX{x};yis \TEXUX{k};var1name \TEXUX{$x$};var2name \TEXUX{$k$};function \TEXUX{$\left[ \frac{1}{\alpha }=\frac{1}{2}\frac{\left( 1-x\right) ^{\alpha }}{\alpha }+\frac{1}{2}\frac{\left( 1+k\right) ^{\alpha }}{\alpha }\right] _{\alpha =-3}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,0.4";var2range "0,10";num-x-gridlines 49;num-y-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"XY";function \TEXUX{$\left[ \frac{1}{\alpha }=\frac{1}{2}\frac{\left( 1-x\right) ^{\alpha }}{\alpha }+\frac{1}{2}\frac{\left( 1+k\right) ^{\alpha }}{\alpha }\right] _{\alpha =-4}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,0.4";var2range "0,10";num-x-gridlines 49;num-y-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"Y";function \TEXUX{$\left[ \frac{1}{\alpha }=\frac{1}{2}\frac{\left( 1-x\right) ^{\alpha }}{\alpha }+\frac{1}{2}\frac{\left( 1+k\right) ^{\alpha }}{\alpha }\right] _{\alpha =-6}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,0.4";var2range "0,10";num-x-gridlines 49;num-y-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"Y";valid_file "T";tempfilename 'L01QY905.wmf';tempfile-properties "XPR";}} :To get people to behave like they do for small gambles,$\sigma $has to be so large as to give unreasonable predictions for large gambles. In fact, sometimes no upside can compensate for a sufficiently large but finite downside. This fact is due to that when CRRA coefficient larger than 1, i.e., when$\alpha <0,$utility is bounded, since \begin{equation*} \frac{c^{\alpha }}{\alpha }<0,\forall c,\alpha <0. \end{equation*} This means that solving \begin{eqnarray*} \frac{c^{\alpha }}{\alpha } &=&\frac{1}{2}\frac{\left( \left( 1-x\right) c\right) ^{\alpha }}{\alpha }+\frac{1}{2}0 \\ 1 &=&\frac{1}{2}\left( 1-x\right) ^{\alpha } \\ x &=&1-2^{\frac{1}{\alpha }} \end{eqnarray*}% we find an upper bound to the downside risk that could be compensated by any upside. In the graph, we see the maximum loss occuring with 50\% chance that could be compensated by any gain as a function of the level of riskaversion. \FRAME{dtbpFX}{4.5002in}{3.0001in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.5002in;height 3.0001in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "1";xmax "20";xviewmin "0.824813986787879";xviewmax "20.3835037202642";yviewmin "0.017174150272224E0";yviewmax "0.987339440142524";rangeset"X";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{R};var1name \TEXUX{$R$};function \TEXUX{$1-2^{\frac{1}{1-R}}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "1,20";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'L01QYB06.wmf';tempfile-properties "XPR";}} For$\alpha =-11,x$is as low as$6.1\%.$Would you refuse a 50/50 bet of loosing 25\% of your lifetime income vs. getting the fortune of Bill Gates? If, not, you cannot have absolute riskaversion above 3.4, which is quite low. \newpage \section{ Non-additive recursive preferences} \subsection{Aggregation over time} Now, disregard risk. In general, preferences can be described as a function that associates a particular level of overall utility to any sequence of consumption levels% \begin{equation*} U\left( c_{1},c_{2}.,,,c_{T}\right) \equiv U\left( \left\{ c_{t}\right\} _{0}^{T}\right) \end{equation*} MRS\ is defined \begin{equation*} MRS_{t,t+1}\equiv \frac{\frac{\partial U\left( c_{1},c_{2}.,,,c_{T}\right) }{% \partial c_{t+1}}}{\frac{\partial U\left( c_{1},c_{2}.,,,c_{T}\right) }{% \partial c_{t}}} \end{equation*} We define time preference as MRS along a path of constant consumption$c$% \begin{equation*} \beta \left( c\right) _{t,t+1}=\frac{\frac{\partial U\left( c_{1},c_{2}.,,,c_{T}\right) }{\partial c_{t+1}}}{\frac{\partial U\left( c_{1},c_{2}.,,,c_{T}\right) }{\partial c_{t}}}|_{c_{t}=c\forall t} \end{equation*}% noting that this may depend onf$c.$For the time additive utility with constant discounting, however, we have% \begin{equation*} U=\sum_{t=s}^{T}\beta ^{t}u\left( c_{t}\right) \end{equation*}% with \begin{equation*} \beta \left( c\right) _{t,t+1}=\beta \forall c. \end{equation*} \textbf{Koopman's time aggregator} Assume preferences \emph{at all dates }are represented by a time zero utility function, so preferences are time consistent. First notation,% \begin{equation*} _{t}c\equiv \left\{ c_{t},c_{t+1},c_{t+2},....c_{t+\infty }\right\} \end{equation*} Utility at time zero is% \begin{equation*} U\left( _{0}c\right) =U\left( c_{0^{,}1}c\right) \end{equation*} Assume \emph{history independence,} here marginal rate of substitution$% MRS_{t,t+1}$\emph{does not} depend on consumption prior to$t$(is this innocuous?) and if$c_{t}$is a vector, also the intra-temporal MRS between goods in$t,$is independent of prior consumption. Then, but not otherwise, we can write% \begin{equation*} U\left( _{0}c\right) =\tilde{V}\left[ c_{0},U_{1}\left( _{1}c\right) \right] \end{equation*}% for an aggregator function$V$and a function that gives the continuation utility$U_{1}\left( _{1}c\right) $Choices over$_{1}c_{1},$in particular, what maximizes$U_{1}$in some choice set, does not depend on$c_{0}.$But the choice set can, of course, be affected. Now also assume \emph{future independence} preferences over$c_{t}$does not depend on$_{t+1}c.$(Is this innocuous? Yes, clearly if$c_{0}$is a scalar, then more is just better, but if$c_{0}$is a vector this is a restriction. One could prefer chicken over fish if one plans to eat a lot of fish in the future. However, future independence seems like a less strong assumption than history independence). Now, we can write utility as \begin{equation*} U\left( _{0}c\right) =V\left[ u\left( c_{0}\right) ,U_{1}\left( _{1}c\right) % \right] \end{equation*}$V$aggregates utility coming from current consumption, and future consumption. It is \emph{not} restricted to simply add them like standard preferences. Finally, assume \emph{stationarity}, then for all$t$, \begin{equation*} U\left( _{t}c\right) =V\left[ u\left( c_{t}\right) ,U\left( _{t+1}c\right) % \right] , \end{equation*} and recursivity is implied% \begin{equation*} U\left( _{t}c\right) =V\left[ u\left( c_{t}\right) ,V\left[ u\left( c_{t+1}\right) ,U\left( _{t+2}c\right) \right] \right] \end{equation*} MRS$_{t,t+1}$is \begin{equation*} \frac{\frac{\partial U\left( c_{1},c_{2}.,,,c_{T}\right) }{\partial c_{t+1}}% }{\frac{\partial U\left( c_{1},c_{2}.,,,c_{T}\right) }{\partial c_{t}}}=% \frac{V_{2}\left[ u\left( c_{t}\right) ,U\left( _{t+1}c\right) \right] V_{1}% \left[ u\left( c_{t+1}\right) ,U\left( _{t+2}c\right) \right] u^{\prime }\left( c_{t+1}\right) }{V_{1}\left[ u\left( c_{t}\right) ,U\left( _{t+1}c\right) \right] u^{\prime }\left( c_{t}\right) } \end{equation*} As we know, time preference is MRS evaluated at a constant consumption path, where by stationarity, also$u\left( c_{t}\right) $and$U\left( _{t}c\right) $is constant at$u\left( c\right) $and$U\left( c\right) $(excuse the notation, I am here letting$c$denote a path of constant levels of consumption). Then, \begin{equation*} \beta \left( u\left( c\right) \right) =V_{2}\left[ u\left( c\right) ,U\left( c\right) \right] \end{equation*}% which can depend on$c$unless$V$is a linear aggregator (standard). The Uzawa simplification is a particular example of the Koopmans aggregator. \begin{equation*} U\left( _{t}c\right) =u\left( c_{t}\right) +\beta \left( u\left( c_{t}\right) \right) U\left( _{t+1}c\right) \end{equation*} First order condition: \begin{equation*} \frac{\partial u\left( c_{t}\right) }{\partial c_{t,i}}\left( 1+\beta ^{\prime }\left( u\left( c_{t}\right) \right) U\left( _{t+1}c\right) \right) +\beta \left( u\left( c_{t}\right) \right) \frac{\partial U\left( _{t+1}c\right) }{\partial c_{t,i}}=0 \end{equation*} The expression$\frac{\partial U\left( _{t+1}c\right) }{\partial c_{t,i}}$is a a (sloppy) way of denoting the effect (via the budget constraint) of consumption today on future utility. A bit more general, by not imposing future independence. \begin{equation*} U\left( _{t}c\right) =u\left( c_{t}\right) +\beta \left( c_{t}\right) U\left( _{t+1}c\right) \end{equation*} Note that here, preference over elements in$c_{t}$may depend on$U\left( _{t+1}c\right) $, which matters if$c_{t}$is a vector. First order condition:% \begin{equation*} \frac{\partial u\left( c_{t}\right) }{\partial c_{t,i}}+\frac{\partial \beta \left( c_{t}\right) }{\partial c_{t,i}}U\left( _{t+1}c\right) +\beta \left( c_{t}\right) \frac{\partial U\left( _{t+1}c\right) }{\partial c_{t,i}}=0 \end{equation*} \textbf{Examples:} Growth and fiscal policy (Dolmas and Wynne (1998)). Using Usawa% \begin{eqnarray*} \max U\left( _{t}c\right) &=&u\left( c_{t}\right) +\beta \left( u\left( c_{t}\right) \right) U\left( _{t+1}c\right) \\ s.t.c_{t} &=&f\left( k_{t}\right) -k_{t+1}-g_{t} \end{eqnarray*} We can derive a Bellman equation:% \begin{equation*} J\left( k\right) =\max_{k_{t+1}}u\left( f\left( k_{t}\right) -k_{t+1}-g_{t}\right) +\beta \left( u\left( f\left( k_{t}\right) -k_{t+1}-g_{t}\right) \right) J\left( k_{t+1}\right) . \end{equation*} Only non-standard is endogenous discounting. FOC:% \begin{equation*} u^{\prime }\left( c_{t}\right) \left( 1+\beta ^{\prime }\left( u\left( c_{t}\right) \right) J\left( k_{t+1}\right) \right) =\beta \left( u\left( c_{t}\right) \right) J^{\prime }\left( k_{t+1}\right) \end{equation*} Envelope: \begin{eqnarray*} J^{\prime }\left( k_{t}\right) &=&u^{\prime }\left( c_{t}\right) f^{\prime }\left( k_{t}\right) +\beta ^{\prime }\left( u\left( c_{t}\right) \right) u^{\prime }\left( c_{t}\right) f^{\prime }\left( k_{t}\right) J\left( k_{t+1}\right) \\ &=&u^{\prime }\left( c_{t}\right) f^{\prime }\left( k_{t}\right) \left( 1+\beta ^{\prime }\left( u\left( c_{t}\right) \right) J\left( k_{t+1}\right) \right) . \end{eqnarray*}% Giving \begin{equation*} J^{\prime }\left( k_{t}\right) =\beta \left( u\left( c_{t}\right) \right) J^{\prime }\left( k_{t+1}\right) f^{\prime }\left( k_{t}\right) \end{equation*} In a steady state \begin{equation*} 1=\beta \left( u\left( c_{t}\right) \right) f^{\prime }\left( k_{t}\right) \end{equation*} Compare this to the standard case% \begin{equation*} 1=\beta f^{\prime }\left( k_{t}\right) \end{equation*}% being independent of fiscal policy. In particular, an increase in$g,$must reduce$c$one-for-one, since$c_{ss}=f\left( k_{ss}\right) -k_{ss}-g_{t}$. Now changes in$g$can affect the steady state. To see this, consider \begin{equation*} 1=\beta \left( u\left( f\left( k_{ss}\right) -k_{ss}-g\right) \right) f^{\prime }\left( k_{ss}\right) \end{equation*} An increase in$g$reduces$u$, suppose this makes people more patient, i.e.,$\beta ^{\prime }<0.$Then, the increase in$g$makes$\beta \left( u\right) f^{\prime }>1$. This will lead to \emph{more saving} and a growing capital stock. Crowding out of consumption more than one-for-one. In this case, since both$\beta ^{\prime }$and$f^{\prime \prime }$are negative, there is a unique steady state. If, instead$\beta ^{^{\prime }}>0,$there may be multiple solutions to \begin{equation*} 1=\beta \left( u\left( f\left( k_{ss}\right) -k_{ss}-g\right) \right) f^{\prime }\left( k_{ss}\right) . \end{equation*} In this case, some steady states are unstable - a higher level of$k$increases$u,$and$\beta $more than the fall in$f^{\prime }.$Therefore,$% \beta >f\prime $and individuals accumulates capital. In the graph, I have specified \begin{eqnarray*} \beta \left( u\right) &=&\left[ 1-e^{\gamma u}\right] _{\gamma =-0.1} \\ u\left( c\right) &=&\ln c \\ f\left( k\right) &=&\left[ \theta k^{\alpha }\right] _{\alpha =\frac{1}{2}% ,\theta =10.} \end{eqnarray*} I plot$f^{\prime }\left( k\right) $(the fat black line) and$\left( \beta \left( u\left( f\left( k\right) -k-g\right) \right) \right) ^{-1}$for$g=0$and$g=2.$A decrease in$g$shifts the latter curve down and steady state$% k $increases. \FRAME{dtbpFX}{4.4998in}{3.0004in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4998in;height 3.0004in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0.4";xmax "2";xviewmin "0.368";xviewmax "2.03264";yviewmin "3.44813070104297";yviewmax "7.99484541940851";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{k};var1name \TEXUX{$k$};function \TEXUX{$\left( \left[ 1-e^{\gamma \left( \left( \ln \left( \theta k^{\alpha }-k-g\right) \right) \right) }\right] _{\theta =10,\alpha =.5,g=1,\gamma =-.1}\right) ^{-1}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0.4,2";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left[ \alpha \theta k^{\alpha -1}\right] _{\alpha =0.5,\theta =10}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0.4,2";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left( \left[ 1-e^{\gamma \left( \left( \ln \left( \theta k^{\alpha }-k-g\right) \right) \right) }\right] _{\theta =10,\alpha =.5,g=0,\gamma =-.1}\right) ^{-1}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0.4,2";num-x-gridlines 100;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JZ262302.wmf';tempfile-properties "PR";}} Other examples is small open economies with a fixed net interest rate,$r.$The steady state is \begin{equation*} 1=\beta \left( u\left( f\left( k_{ss}\right) -k_{ss}-g\right) \right) \left( 1+r\right) \end{equation*} With standard preferences no steady state exists generically. With$\beta ^{\prime }<0$a unique stable one exists. Intuition, with low capital and low consumption,$\beta $is higher than$r$and people save a lot. As capital and consumption increases, the discount factor falls. \subsection{Aggregation over states} As in the intro, consider states (within a period) to be$z\in \Gamma ,$with an associated probability measure$p\left( z\right) .$Utility is now \begin{equation*} U\left( \left\{ c\left( z\right) \right\} _{z\in \Gamma }\right) \end{equation*} We can solve for the certainty equivalent consumption level$\mu $of$% \left\{ c\left( z\right) \right\} _{z\in \Gamma }$from \begin{equation*} U\left( \left\{ c\left( z\right) \right\} _{z\in \Gamma }\right) =U\left( \left\{ \mu \right\} \right) \end{equation*} Standard theory says \begin{equation*} U\left( \left\{ c\left( z\right) \right\} \right) =\sum_{z\in \Gamma }p\left( z\right) u\left( \left( z\right) \right) \end{equation*} and since \begin{equation*} u\left( \mu \right) =\sum_{z\in \Gamma }p\left( z\right) u\left( c\left( z\right) \right) \end{equation*}% we have \begin{equation*} \mu \left( \left\{ c\left( z\right) \right\} \right) =u^{-1}\left( \sum_{z\in \Gamma }p\left( z\right) u\left( c\left( z\right) \right) \right) . \end{equation*} For example: Suppose preferences are CRRA, then \begin{equation*} u\left( c\right) =\frac{1}{\alpha }c^{\alpha } \end{equation*} So \begin{eqnarray*} \frac{1}{\alpha }\mu ^{\alpha } &=&\sum_{z\in \Gamma }p\left( z\right) \frac{% 1}{\alpha }c\left( z\right) ^{\alpha } \\ \mu &=&\left( \alpha \sum_{z\in \Gamma }p\left( z\right) \frac{1}{\alpha }% c\left( z\right) ^{\alpha }\right) ^{\frac{1}{\alpha }} \end{eqnarray*} Note the linear homogeneity in this case. For a constant$k>0.$% \begin{equation*} \mu \left( k\left\{ c\left( z\right) \right\} \right) =k\mu \left( \left\{ c\left( z\right) \right\} \right) . \end{equation*} Chew and Dekel generalizes this by allowing the certainty equivalent of$% \left\{ c\left( z\right) \right\} $to be a more general function \emph{% while maintaining }first order conditions that are linear in probabilities by implicitly defining% \begin{equation*} \mu \left( \left\{ c\left( z\right) \right\} \right) =\sum_{z\in \Gamma }p\left( z\right) M\left[ c\left( z\right) ,\mu \right] . \end{equation*} As we see, this generalizes standard utility by implying that the marginal value of consumption in state$z$depends on consumption in other states through their effect on$\mu .$An example of this is that people might care more (or less) about consumption in states that provide less consumption than the certainty equivalence (disappointment aversion). Notice the relative comparison here. With concave utility, marginal utility in states with low consumption is high, but \emph{independent of consumption in other states.} This is not necessarily the case here since consumption in state$% z, $relative to$\mu $depends on consumption in all other states since they affect$\mu .$We assume that$M\left( \mu ,\mu \right) =\mu $(Why?),$M_{1}>0,M_{11}<0$(first order stochastic dominance and riskaversion). Often we want to maintain the linear homogeneity of preferences like in CRRA. \begin{equation*} M\left( kc,k\mu \right) =kM\left( c,\mu \right) \end{equation*} Examples: To show that the Chew-Dekel generalizes and includes e.g., CRRA preferences: Note that if we set% \begin{equation} M\left( c,\mu \right) =\frac{c^{\alpha }\mu ^{1-\alpha }}{\alpha }-\frac{\mu }{\alpha }+\mu \label{eq_ChewDekelCRRA} \end{equation} we get \begin{eqnarray*} \mu &=&\sum_{z\in \Gamma }p\left( z\right) \left( \frac{c\left( z\right) ^{\alpha }\mu ^{1-\alpha }}{\alpha }-\frac{\mu }{\alpha }+\mu \right) \\ 0 &=&\sum_{z\in \Gamma }p\left( z\right) \left( c\left( z\right) ^{\alpha }\mu ^{1-\alpha }-\mu \right) \\ \mu &=&\mu ^{1-\alpha }\sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\alpha } \\ \mu ^{\alpha } &=&\sum_{z\in \Gamma }p\left( z\right) \left( c\left( z\right) ^{\alpha }\right) \\ \mu &=&\left( \sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\alpha }\right) ^{\frac{1}{\alpha }} \end{eqnarray*}% which is the CRRA certainty equivalence. Examples: "Weighted expected utility" Let \begin{equation*} M=\left( \left( \frac{c}{\mu }\right) ^{\gamma }\left( \frac{c^{\alpha }\mu ^{1-\alpha }}{\alpha }-\frac{\mu }{\alpha }\right) +\mu \right) \end{equation*} Compare to (\ref{eq_ChewDekelCRRA}). Now, we have% \begin{eqnarray*} \mu &=&\sum_{z\in \Gamma }p\left( z\right) \left( \frac{c\left( z\right) ^{\gamma }}{\mu ^{\gamma }}\left( \frac{c\left( z\right) ^{\alpha }\mu ^{1-\alpha }}{\alpha }\right) -\frac{c\left( z\right) ^{\gamma }}{\mu ^{\gamma }}\frac{\mu }{\alpha }+\mu \right) \\ 0 &=&\sum_{z\in \Gamma }p\left( z\right) \left( \frac{c\left( z\right) ^{\gamma }}{\mu ^{\gamma }}\left( \frac{c\left( z\right) ^{\alpha }\mu ^{1-\alpha }}{\alpha }\right) -\frac{c\left( z\right) ^{\gamma }}{\mu ^{\gamma }}\frac{\mu }{\alpha }\right) \\ \mu \sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\gamma } &=&\mu ^{1-\alpha }\sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\gamma }c\left( z\right) ^{\alpha } \\ \mu ^{\alpha } &=&\frac{\sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\gamma }c\left( z\right) ^{\alpha }}{\sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\gamma }} \end{eqnarray*} Here, we can interpret \begin{equation*} \frac{p\left( z\right) c\left( z\right) ^{\gamma }}{\sum_{z\in \Gamma }p\left( z\right) c\left( z\right) ^{\gamma }}\equiv \hat{p}\left( z\right) \end{equation*}% as a weighted probability. If$\gamma <0,$these weights decrease in$c,$i.e., bad outcomes are weighted higher than otherwise. Suppose we have two outcomes$c_{1}$and$c_{2}$with equal probabilities$\frac{1}{2}$each. The weighted probabilities are then \begin{eqnarray*} \hat{p}\left( z_{1}\right) &=&\frac{\frac{1}{2}c_{1}^{\gamma }}{\frac{1}{2}% c_{1}^{\gamma }+\frac{1}{2}c_{1}^{\gamma }}=\frac{c_{1}^{\gamma }}{% c_{1}^{\gamma }+c_{1}^{\gamma }} \\ \hat{p}\left( z_{2}\right) &=&\frac{\frac{1}{2}c_{2}^{\gamma }}{\frac{1}{2}% c_{1}^{\gamma }+\frac{1}{2}c_{1}^{\gamma }}=\frac{c_{2}^{\gamma }}{% c_{1}^{\gamma }+c_{1}^{\gamma }} \end{eqnarray*} Consequently, \begin{equation*} \mu ^{\alpha }=\frac{c_{1}^{\gamma }}{c_{1}^{\gamma }+c_{1}^{\gamma }}% c_{1}^{\alpha }+\frac{c_{2}^{\gamma }}{c_{1}^{\gamma }+c_{1}^{\gamma }}% c_{2}^{\alpha } \end{equation*} For any constant$k,$indifference curves then satisfy% \begin{equation*} k=\frac{c_{1}^{\alpha +\gamma }}{c_{1}^{\gamma }+c_{2}^{\gamma }}+\frac{% c_{2}^{\alpha +\gamma }}{c_{1}^{\gamma }+c_{2}^{\gamma }} \end{equation*}% Plotting one together with standard utility we find these preferences produce a higher level of riskaversion. \FRAME{dtbpFX}{6.4274in}{4.2858in}{% 0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 6.4274in;height 4.2858in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "3";ymin "0";ymax "3";xviewmin "0";xviewmax "3.04736182710055";yviewmin "0";yviewmax "3.04736182710055";plottype 12;num-x-gridlines 24;num-y-gridlines 24;plotstyle "patch";axesstyle "normal";xis \TEXUX{cQSUB1ESUB};yis \TEXUX{cQSUB2ESUB};var1name \TEXUX{$c_{1}$};var2name \TEXUX{$c_{2}$};function \TEXUX{$\left[ 1=\frac{c_{1}^{1+\gamma -\sigma }}{c_{1}^{\gamma }+c_{2}^{\gamma }}+\frac{c_{2}^{1+\gamma -\sigma }}{c_{1}^{\gamma }+c_{2}^{\gamma }}\right] _{\sigma =1.5,\gamma =-1}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,3";var2range "0,3";num-x-gridlines 24;num-y-gridlines 24;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"XY";function \TEXUX{$\left[ 1=\frac{c_{1}^{1+\gamma -\sigma }}{c_{1}^{\gamma }+c_{2}^{\gamma }}+\frac{c_{2}^{1+\gamma -\sigma }}{c_{1}^{\gamma }+c_{2}^{\gamma }}\right] _{\sigma =2.5,\gamma =0}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,3";var2range "0,3";num-x-gridlines 24;num-y-gridlines 24;curveColor "[flat::RGB:0000000000]";curveStyle "Line";valid_file "T";tempfilename 'JA7XOY08.wmf';tempfile-properties "PR";}} We can also have Faruk Gul's disappointment aversion. Then \begin{equation*} M\left( c,\mu \right) =\left\{ \begin{array}{c} \left( 1+\delta \right) \left( \frac{c^{\alpha }\mu ^{1-\alpha }}{\alpha }-% \frac{\mu }{\alpha }\right) +\mu \text{ if }c\left( z\right) <\mu \\ \frac{c^{\alpha }\mu ^{1-\alpha }}{\alpha }-\frac{\mu }{\alpha }+\mu \text{ else}% \end{array}% \right. \end{equation*} This means that all outcomes worse than the certainty equivalence get scaled up.% \begin{eqnarray*} \mu &=&\sum_{z\in \Gamma }p\left( z\right) \left( \left( 1+\delta I\left[ c\left( z\right) <\mu \right] \right) \left( \frac{c\left( z\right) ^{\alpha }\mu ^{1-\alpha }}{\alpha }-\frac{\mu }{\alpha }\right) +\mu \right) \\ \mu \sum_{z\in \Gamma }p\left( z\right) \left( 1+\delta I\left[ c\left( z\right) <\mu \right] \right) &=&\mu ^{1-\alpha }\sum_{z\in \Gamma }p\left( z\right) \left( 1+\delta I\left[ c\left( z\right) <\mu \right] \right) c\left( z\right) ^{\alpha } \\ \mu ^{\alpha } &=&\frac{\sum_{z\in \Gamma }p\left( z\right) \left( 1+\delta I% \left[ c\left( z\right) <\mu \right] \right) c\left( z\right) ^{\alpha }}{% \sum_{z\in \Gamma }p\left( z\right) \left( 1+\delta I\left[ c\left( z\right) <\mu \right] \right) } \end{eqnarray*} The weighted probability here is \begin{equation*} \frac{p\left( z\right) \left( 1+\delta I\left[ c\left( z\right) <\mu \right] \right) }{\sum_{z\in \Gamma }p\left( z\right) \left( \left( 1+\delta I\left[ c\left( z\right) <\mu \right] \right) \right) } \end{equation*} For any constant$k.$indifference curves can be written as \begin{equation*} k=\left\{ \begin{array}{c} \frac{\left( 1+\delta \right) \frac{1}{2}c_{1}^{\alpha }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}+\frac{\frac{1}{2}c_{2}^{\alpha }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}\text{if }c_{1}c_{2}% \end{array}% \right. \end{equation*} \FRAME{dtbpFX}{6.9566in}{4.638in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 6.9566in;height 4.638in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "2";ymin "0";ymax "2";xviewmin "0";xviewmax "2.03787430473557";yviewmin "0";yviewmax "2.03787430473557";plottype 12;num-x-gridlines 24;num-y-gridlines 24;plotstyle "patch";axesstyle "normal";xis \TEXUX{cQSUB1ESUB};yis \TEXUX{cQSUB2ESUB};var1name \TEXUX{$c_{1}$};var2name \TEXUX{$c_{2}$};function \TEXUX{$\left[ 1=\frac{\left( 1+\delta \right) \frac{1}{2}c_{1}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}+\frac{\frac{1}{2}c_{2}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}\right] _{\sigma =.5,\delta =.5}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,2";var2range "0,2";num-x-gridlines 99;num-y-gridlines 99;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"XY";function \TEXUX{$\left[ 1=\frac{\frac{1}{2}c_{1}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}+\frac{\left( 1+\delta \right) \frac{1}{2}c_{2}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}\right] _{\sigma =.5,\delta =.5}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,2";var2range "0,2";num-x-gridlines 99;num-y-gridlines 99;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"XY";function \TEXUX{$\left[ 1=\frac{\left( 1+\delta \right) \frac{1}{2}c_{1}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}+\frac{\frac{1}{2}c_{2}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}\right] _{\sigma =.5,\delta =.5}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,1";var2range "0,2";num-x-gridlines 24;num-y-gridlines 24;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"XY";function \TEXUX{$\left[ 1=\frac{\frac{1}{2}c_{1}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}+\frac{\left( 1+\delta \right) \frac{1}{2}c_{2}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}\right] _{\sigma =.5,\delta =.5}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 3;lineAttributes "Solid";var1range "0,2";var2range "0,1";num-x-gridlines 99;num-y-gridlines 99;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"XY";function \TEXUX{$\left[ 1=\frac{\left( 1+\delta \right) \frac{1}{2}c_{1}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}+\frac{\frac{1}{2}c_{2}^{1-\sigma }}{\left( 1+\delta \right) \frac{1}{2}+\frac{1}{2}}\right] _{\sigma =1.5,\delta =0}$};linecolor "blue";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,2";var2range "0,2";num-x-gridlines 99;num-y-gridlines 99;curveColor "[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"XY";valid_file "T";tempfilename 'JA7XOY09.wmf';tempfile-properties "PR";}} Here, it is important to notice the kink around unity. This is first order risk-aversion. Individuals will be more averse to small risks than under standard preferences. \textbf{Risk premia under Chew-Dekel Preferences. } Define the risk-premium$R$from \begin{equation*} R=Ec\left( z\right) -\mu \left( \left\{ c\left( z\right) \right\} \right) \end{equation*} Let us consider two state case above. In the two cases of CRRA E utility and weighted utility. Suppose$c=c_{1}=1-\sigma $with prob$\frac{1}{2}$and$% c=c_{2}=1+\sigma $with prob.$\frac{1}{2}.$The shock is thus$\sigma $and$\sigma $is the standard deviation of$c.$\begin{eqnarray*} R_{s\tan d} &=&1-\left( \frac{\left( 1-\sigma \right) ^{\alpha }+\left( 1+\sigma \right) ^{\alpha }}{2}\right) ^{\frac{1}{\alpha }}\equiv R_{s}\left( \sigma \right) \\ &\approx &R_{s}\left( 0\right) +R_{s}^{\prime }\left( 0\right) \sigma +\frac{% 1}{2}R_{s}^{\prime \prime }\left( 0\right) \sigma ^{2} \\ &=&\left[ \frac{\partial \left( 1-\left( \frac{\left( 1-\sigma \right) ^{\alpha }+\left( 1+\sigma \right) ^{\alpha }}{2}\right) ^{\frac{1}{\alpha }% }\right) }{\partial \sigma }\right] _{\sigma =0}\sigma \\ &&+\frac{1}{2}\left[ \frac{\partial ^{2}\left( 1-\left( \frac{\left( 1-\sigma \right) ^{\alpha }+\left( 1+\sigma \right) ^{\alpha }}{2}\right) ^{% \frac{1}{\alpha }}\right) }{\partial \sigma ^{2}}\right] _{\sigma =0}\sigma ^{2} \\ &=&\frac{1}{2}\left( 1-\alpha \right) \sigma ^{2} \end{eqnarray*} In the weighted expected utility, we have \begin{eqnarray*} R_{wheigt} &=&\left( 1-\left( \frac{\left( 1-\sigma \right) ^{\alpha +\gamma }+\left( 1+\sigma \right) ^{\alpha +\gamma }}{\left( 1-\sigma \right) ^{\gamma }+\left( 1+\sigma \right) ^{\gamma }}\right) ^{\frac{1}{\alpha }% }\right) \equiv R_{w}(\sigma ) \\ &\approx &R_{w}(0)+R_{w}^{\prime }(0)\sigma +\frac{1}{2}R_{w}^{\prime }(0)\sigma ^{2} \\ &=&\left[ \frac{\partial \left( 1-\left( \frac{\left( 1-\sigma \right) ^{\alpha +\gamma }+\left( 1+\sigma \right) ^{\alpha +\gamma }}{\left( 1-\sigma \right) ^{\gamma }+\left( 1+\sigma \right) ^{\gamma }}\right) ^{% \frac{1}{\alpha }}\right) }{\partial \sigma }\right] _{\sigma =0}\sigma \\ &&+\frac{1}{2}\left[ \frac{\partial ^{2}\left( 1-\left( \frac{\left( 1-\sigma \right) ^{\alpha +\gamma }+\left( 1+\sigma \right) ^{\alpha +\gamma }}{\left( 1-\sigma \right) ^{\gamma }+\left( 1+\sigma \right) ^{\gamma }}% \right) ^{\frac{1}{\alpha }}\right) }{\partial \sigma ^{2}}\right] _{\sigma =0}\sigma ^{2} \\ &=&\frac{1}{2}\left( 1-\alpha -2\gamma \right) \sigma ^{2} \end{eqnarray*} In the disappointment aversion case, the first derivative is not going to be zero, people are not locally riskneutral. Therefore, \begin{eqnarray*} R_{disapp} &=&\left( 1-\left( \frac{\left( 1+\delta \right) \left( 1-\sigma \right) ^{\alpha }+\left( 1+\sigma \right) ^{\alpha }}{2+\delta }\right) ^{% \frac{1}{\alpha }}\right) \equiv R_{d}\left( \sigma \right) \\ &\approx &R_{d}\left( 0\right) +R_{d}^{\prime }\left( 0\right) \sigma +\frac{% 1}{2}R_{d}^{\prime \prime }\left( 0\right) \sigma ^{2} \\ &=&\left[ \frac{\partial \left( 1-\left( \frac{\left( 1+\delta \right) \left( 1-\sigma \right) ^{\alpha }+\left( 1+\sigma \right) ^{\alpha }}{% 2+\delta }\right) ^{\frac{1}{\alpha }}\right) }{\partial \sigma }\right] _{\sigma =0}\sigma \\ &&+\frac{1}{2}\left[ \frac{\partial ^{2}\left( 1-\left( \frac{\left( 1+\delta \right) \left( 1-\sigma \right) ^{\alpha }+\left( 1+\sigma \right) ^{\alpha }}{2+\delta }\right) ^{\frac{1}{\alpha }}\right) }{\partial \sigma ^{2}}\right] _{\sigma =0}\sigma ^{2} \\ &=&\frac{\delta }{2+\delta }\sigma +2\left( 1-\alpha \right) \frac{1+\delta }{\left( 2+\delta \right) ^{2}}\sigma ^{2} \end{eqnarray*} The key here is the linear term. \subsection{Time and Risk} Let us now combine the aggregation over time and over states so we can define preferences over both states and time, allowing for rates of substitution to differ depending on whether we are talking about states or time periods. Generalize the notation so that \begin{equation*} _{t}c \end{equation*}% denotes a stochastic consumption stream starting from period$t,$i.e.,$% c_{t},c\left( z_{t+1}\right) ,...,$and denote% \begin{equation*} U\left( _{t}c\right) \equiv U_{t} \end{equation*} The aggregator can be written \begin{equation*} U_{t}=V\left( c_{t},\mu \left( U_{t+1}\right) \right) \end{equation*}% or by imposing \emph{future independence}% \begin{equation} U_{t}=V\left( u\left( c_{t}\right) ,\mu \left( U_{t+1}\right) \right) \label{eq_KrepsPorteus} \end{equation} Note that we are now aggregating the direct utility of$c_{t}$and the certainty equivalent of the \emph{stochastic continuation payoff }from$t+1$and onwards. Thus, since at$t,U_{t+1}$is stochastic (depending on the realization of$z_{t+1}$), we construct the period$t+1$certainty equivalent of this stochastic utility, i.e.,$\mu \left( U_{t+1}\right) $. If we use a standard expected utility certainty equivalent$\mu $in (\ref% {eq_KrepsPorteus}), we get what is typically called \emph{Kreps-Porteus }% utility (some times also Epstein-Zin, although they often are associated with more general specification of state aggregation, \emph{a la} Chew-Dekel). The key here is that we can now separate risk-aversion, which is determined by the properties of$\mu ,$from intertemporal substitution, determined by$V.$Let us use the CRRA specification for$\mu ,$so \begin{equation*} \mu \left( U\right) =\left[ EU^{\alpha }\right] ^{\frac{1}{\alpha }} \end{equation*}% and a constant elasticity aggregator \begin{equation*} V\left( u\left( c_{t}\right) ,\mu \left( U_{t+1}\right) \right) =\left[ \left( 1-\beta \right) u^{\rho }+\beta \mu \left( U\right) ^{\rho }\right] ^{% \frac{1}{\rho }}. \end{equation*} If we let$u$be linear,$u=c,$so we let all curvature be taken care of by$% \mu $and$V,$we can now interpret$1-\alpha $a s the degree of risk-aversion and$\frac{1}{1-\rho }$as the elasticity of intertemporal substitution. To see this simply, look at a two period example where consumption is stochastic in the second period. In the second period, stochastic utility is \begin{equation*} U_{2}=c\left( z_{2}\right) \end{equation*} Suppose first that second period consumption is non-stochastic at$c_{2}.$Then,% \begin{equation*} \mu \left( U_{2}\right) =\left[ EU^{\alpha }\right] ^{\frac{1}{\alpha }% }=c_{2} \end{equation*} Then, \begin{equation*} U_{1}=\left[ \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{2}^{\rho }\right] ^{\frac{1}{\rho }} \end{equation*}% MRS is now, \begin{eqnarray*} MRS_{c_{1},c_{2}} &\equiv &\frac{\partial U_{1}}{\partial c_{2}}/\frac{% \partial U_{1}}{\partial c_{1}} \\ &=&\frac{\frac{1}{\rho }\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{2}^{\rho }\right) ^{\frac{1}{\rho }-1}\rho \beta c_{2}^{\rho -1}}{\frac{1% }{\rho }\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{2}^{\rho }\right) ^{\frac{1}{\rho }-1}\rho \left( 1-\beta \right) c_{1}^{\rho -1}} \\ &=&\frac{\beta }{1-\beta }\left( \frac{c_{2}}{c_{1}}\right) ^{\rho -1} \end{eqnarray*} as usual and% \begin{equation*} \frac{-\partial MRS_{c_{1},c_{2}}}{\partial \left( \frac{c_{1}}{c_{2}}% \right) }\frac{\frac{c_{1}}{c_{2}}}{MRS\left( \frac{c_{1}}{c_{2}}\right) }% =IES^{-1}=1-\rho . \end{equation*} Note that we have imposed that$U$is linearly homogeneous by having the power$\frac{1}{\rho },$which is with no consequence. To see this, consider \begin{equation*} \tilde{U}_{1}=\left( 1-\beta \right) c_{1}^{\rho }+\beta c_{2}^{\rho } \end{equation*}% and calculate \begin{equation*} \frac{\partial \tilde{U}_{1}}{\partial c_{2}}/\frac{\partial \tilde{U}_{1}}{% \partial c_{1}}=\frac{\rho \beta c_{2}^{\rho -1}}{\rho \left( 1-\beta \right) c_{1}^{\rho -1}}=\frac{\beta }{1-\beta }\left( \frac{c_{2}}{c_{1}}% \right) ^{\rho -1} \end{equation*} It should be clear from the aggregator \begin{equation*} \mu \left( c_{2}\left( z_{2}\right) \right) =\left[ Ec\left( z_{2}\right) ^{\alpha }\right] ^{\frac{1}{\alpha }} \end{equation*}% that we can think of$1-\alpha $as measuring risk aversion. A key characteristic of Kreps-Porteus preferences is that they give rise to preference for when risk is revealed. A basic intuition is that early revelation implies that some risk is \emph{converted }into intertemporal substitution. To see an example of this, consider the example in the introduction, where second period consumption is high or low ($z_{2}=z_{h}$or$z_{l}).$Suppose that this is reveled in the first period. Denote$% \tilde{U}_{1}(z)$first period utility given a realized value of$z.$Then, \begin{eqnarray*} \tilde{U}_{1}\left( z_{l}\right) &=&\left[ \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{l}^{\rho }\right] ^{\frac{1}{\rho }} \\ \tilde{U}_{1}\left( z_{h}\right) &=&\left[ \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right] ^{\frac{1}{\rho }} \end{eqnarray*}% and \begin{eqnarray*} U_{1}^{early} &=&\mu \left( U_{1}\right) \\ &=&\left( \frac{1}{2}\left( \left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right) ^{\frac{1}{\rho }}\right) ^{\alpha }+\frac{1}{2}% \left( \left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right) ^{\frac{1}{\rho }}\right) ^{\alpha }\right) ^{\frac{1}{\alpha }} \\ &=&\left( \frac{1}{2}\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right) ^{\frac{\alpha }{\rho }}+\frac{1}{2}\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right) ^{\frac{\alpha }{% \rho }}\right) ^{\frac{1}{\alpha }} \end{eqnarray*} Compare this to the late resolution case. Then,% \begin{equation*} \mu \left( U_{2}\right) =\left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}% \right) ^{\frac{1}{\alpha }} \end{equation*}% and \begin{equation*} U_{1}^{late}=\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta \left( \frac{% c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}\right) ^{% \frac{1}{\rho }} \end{equation*}% Comparing these, we see \begin{eqnarray*} &&U_{1}^{early}-U_{1}^{late} \\ &=&\left[ \left( \frac{1}{2}\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right) ^{\frac{\alpha }{\rho }}+\frac{1}{2}\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{l}^{\rho }\right) ^{\frac{% \alpha }{\rho }}\right) ^{\frac{1}{\alpha }}-\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}\right) ^{\frac{1}{\rho }}\right] _{c_{1}=1,c_{h}=1.1,c_{l}=0.9,\rho =.5,\beta =.45} \end{eqnarray*}% \FRAME{dtbpFX}{4.4998in}{3.0004in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4998in;height 3.0004in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "1";xviewmin "-0.009220316484848E0";xviewmax "1.0201844063297";yviewmin "-0.000645274641323E0";yviewmax "0.000633420184066E0";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{v58123};var1name \TEXUX{$\alpha $};function \TEXUX{$\left[ \left( \frac{1}{2}\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{h}^{\rho }\right) ^{\frac{\alpha }{\rho }}+\frac{1}{2}\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta c_{l}^{\rho }\right) ^{\frac{\alpha }{\rho }}\right) ^{\frac{1}{\alpha }}-\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}\right) ^{\frac{1}{\rho }}\right] _{c_{1}=1,c_{h}=1.1,c_{l}=0.9,\rho =.5,\beta =.45}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,1";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY0A.wmf';tempfile-properties "PR";}} Here I plotted the difference for$\rho =\frac{1}{2}$and for$\alpha <\rho , $we see that early resolution is preferred. Note that when$\alpha <\rho $% , risk aversion$(1-\alpha )$is larger than the inverse of intertemporal elasticity of substitution, so in a sense, it is less costly to substitute between time periods than between states of nature. Is this important? With Kreps-Porteus preferences we can clearly get low interest rates if individuals are facing little risk and high if they are facing large. But can we get what we want when we have a representative household that prices \emph{both }risky and non-risky assets while having to bear reasonable amounts of risk himself. \textbf{An example}. Look at the case in the introduction; given$c_{1}$% \begin{equation*} c_{2}=\left\{ \begin{array}{c} c_{h}=c_{1}\left( 1+g+\sigma \right) \text{ with prob }\frac{1}{2} \\ c_{l}=c_{1}\left( 1+g-\sigma \right) \text{ with prob }\frac{1}{2}% \end{array}% \right. \end{equation*} First, we note that Arrow-Debreu prices now have the property that the price of an asset that pays out in state$z=z_{h},$depends on consumption also in the other state. This is clear from the expression of utility:% \begin{equation*} U_{1}=\left( \left( 1-\beta \right) c_{1}^{\rho }+\beta \left( \frac{% c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}\right) ^{% \frac{1}{\rho }} \end{equation*}% where we see that the marginal contribution to utility$U_{1}$of an asset that pays out in one of the states, say in$z=z_{h}$depends on consumption also in the other state. The simple calculation of Arrow-Debreu securities is lost. What is now the price of a safe bond$p_{b}$and a risky asset (the apple tree)$p_{r}$that pays dividends$\left( 1+g+\sigma \right) $or$\left( 1+g-\sigma \right) $per share depending on which state occurs? The prices have to be such that if a consumer buys one marginal unit of the assets, utility is unchanged. Take the safe bond, it has to satisfy \begin{eqnarray*} 0 &=&-p_{b}\left( 1-\beta \right) \frac{\partial U_{1}}{\partial c_{1}}% +\beta E\left( \frac{\partial U_{1}}{\partial c_{2}}\right) \\ 0 &=&-p_{b}\left( 1-\beta \right) \frac{\partial c_{1}^{\rho }}{\partial c_{1}}+\beta \left( \frac{\partial \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}}{\partial c_{h}}+\frac{% \partial \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{% \rho }{\alpha }}}{\partial c_{l}}\right) \\ p_{b} &=&\left[ \frac{\beta \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}% \right) ^{\frac{\rho -\alpha }{\alpha }}\left( c_{h}^{\alpha -1}+c_{l}^{\alpha -1}\right) }{2\left( 1-\beta \right) c_{1}^{\rho -1}}% \right] _{c_{h}=c_{1}\left( 1+g+\sigma \right) ,c_{l}=c_{1}\left( 1+g-\sigma \right) } \\ &=&\frac{\beta }{2\left( 1-\beta \right) }\left( \frac{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}{2}\right) ^{\frac{% \rho -\alpha }{\alpha }}\left( \left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}\right) \end{eqnarray*} With the risky asset we have \begin{eqnarray*} 0 &=&-p_{r}\left( 1-\beta \right) \frac{\partial c_{1}^{\rho }}{\partial c_{1}}+\beta \left( \frac{\partial \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}\left( 1+g+\sigma \right) }{\partial c_{h}}+\frac{\partial \left( \frac{c_{h}^{\alpha }+c_{l}^{\alpha }}{2}\right) ^{\frac{\rho }{\alpha }}\left( 1+g-\sigma \right) }{\partial c_{l}}\right) \\ &\Rightarrow &p_{r}=\frac{\beta }{2\left( 1-\beta \right) }\left( \frac{% \left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}{2}% \right) ^{\frac{\rho -\alpha }{\alpha }}\left( \left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }\right) \end{eqnarray*}% Note that the ratio% \begin{equation*} \frac{p_{b}}{p_{r}}=\frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }} \end{equation*}% is determined by risk aversion and \emph{exactly the same} as in the standard case. Therefore it seems as we have got little help from Kreps and Porteus. Not entirely right. We have learnt that it is difficult to get the risk premia by assuming that stock market returns are fully determined by consumption growth. Maybe this is not the way to go. Suppose there is some other more stable income so that the share dividend is more volatile than consumption. Say that the standard deviation of consumption remains at$\sigma $while the standard deviation of dividends is$x\sigma .$Then% \begin{equation*} \frac{p_{b}}{p_{r}}=\frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha -1}\left( 1+g+x\sigma \right) +\left( 1+g-\sigma \right) ^{\alpha -1}\left( 1+g-\sigma x\right) } \end{equation*} Setting$g=0.018,\sigma =0.036$and$x=4.5,$we get the following graph. \FRAME{dtbpFX}{6.5726in}{4.382in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 6.5726in;height 4.382in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "-50";xmax "1";xviewmin "-51.02";xviewmax "2.0404";yviewmin "0.978831101282206";yviewmax "1.16023367623609";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{v58123};var1name \TEXUX{$\alpha $};function \TEXUX{$\left[ \frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha -1}\left( 1+g+x\sigma \right) +\left( 1+g-\sigma \right) ^{\alpha -1}\left( 1+g-\sigma x\right) }\right] _{g=0.018,\sigma =0.036,x=4.5}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-50,1";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY0B.wmf';tempfile-properties "PR";}} Targeting$\left[ \frac{\left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}}{\left( 1+g+\sigma \right) ^{\alpha -1}\left( 1+g+x\sigma \right) +\left( 1+g-\sigma \right) ^{\alpha -1}\left( 1+g-\sigma x\right) }\right] _{g=0.018,\sigma =0.036,x=4.5}=1.05$yields$% \alpha =-11.1,$i.e., a CRRA coefficient of 10. To also get the riskfree rate right, we need \begin{equation*} p_{b}=\frac{1}{1.01} \end{equation*}% with a reasonable value of$\beta .$Setting \begin{equation*} \frac{\beta }{1-\beta }=.99\Rightarrow \beta =0.497\,49 \end{equation*}% and using$\alpha =-11.1,$we solve \begin{equation*} \left[ \frac{\beta }{2\left( 1-\beta \right) }\left( \frac{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}{2}\right) ^{\frac{% \rho -\alpha }{\alpha }}\left( \left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}\right) \right] _{g=0.018,\sigma =0.036,\beta =0.497\,49,\alpha =-11.1}=\frac{1}{1.01} \end{equation*}% we find$\rho =0.34,$so the intertemporal elasticity of substitution is$% \frac{1}{1-0.34}=1.515.$This is quite far away from the EU case where we would have IES=$\frac{1}{1-(-11.1)}=0.08.$Producing an interest rate of% \begin{equation*} \left( \left[ \frac{\beta }{2\left( 1-\beta \right) }\left( \frac{\left( 1+g+\sigma \right) ^{\alpha }+\left( 1+g-\sigma \right) ^{\alpha }}{2}% \right) ^{\frac{\rho -\alpha }{\alpha }}\left( \left( 1+g+\sigma \right) ^{\alpha -1}+\left( 1+g-\sigma \right) ^{\alpha -1}\right) \right] _{g=0.018,\sigma =0.036,\beta =0.497\,49,\alpha =-11.1,\rho =-11.1}\right) ^{-1}-1 \end{equation*}% which equals$13.8\%.$Recent paper by Bansal and Yaron use Kreps-Porteus specification with CRRA=10 and IES 1.5, i.e., a large deviation from standard EU. They use data to show that dividends are more volatile than consumption, 4.5 times higher. As you can see, these assumptions seems able to account for the risk-premia. \subsection{Closed form value function in partial equilibrium} Consider an individual with no labor income who has access to a market for investments with an exogeneous stochastic i.i.d. return$\tilde{R}.$His budget constraint is thus% \begin{equation*} A_{t+1}=\left( A_{t}-c_{t}\right) \tilde{R}_{t+1} \end{equation*} We can now write a Bellman equation \begin{equation*} W\left( A_{t}\right) =\max_{c_{t}}V\left( u\left( c_{t}\right) ,\mu \left( W\left( \left( A_{t}-c_{t}\right) \tilde{R}_{t+1}\right) \right) \right) \end{equation*} where as above \begin{eqnarray*} u\left( c\right) &=&c \\ \mu \left( W\right) &=&\left[ EW^{\alpha }\right] ^{\frac{1}{\alpha }} \end{eqnarray*}% and \begin{equation} V\left( u,\mu \right) =\left[ \left( 1-\beta \right) u^{\rho }+\beta \mu ^{\rho }\right] ^{\frac{1}{\rho }}. \label{eq_KP} \end{equation} Conjecture that \begin{equation*} W\left( A\right) =\psi _{1}A \end{equation*}% and \begin{equation*} c=\psi _{2}A \end{equation*}% for yet undetermined coefficients$\psi _{1}$and$\psi _{2}.$Using this, we find that \begin{eqnarray*} W\left( A_{t+1}\right) &=&W\left( \left( A_{t}-c_{t}\right) \tilde{R}% _{t+1}\right) =\psi _{1}\left( A_{t}-c_{t}\right) \tilde{R}_{t+1}=\psi _{1}\left( 1-\psi _{2}\right) A_{t}\tilde{R}_{t+1} \\ \mu \left( W\left( A_{t+1}\right) \right) &=&\left( E\left( \psi _{1}\left( 1-\psi _{2}\right) A_{t}\tilde{R}_{t+1}\right) ^{\alpha }\right) ^{\frac{1}{% \alpha }}=\psi _{1}\left( 1-\psi _{2}\right) A_{t}\left( E\left( \tilde{R}% _{t+1}\right) ^{\alpha }\right) ^{\frac{1}{\alpha }} \end{eqnarray*} Using this in the Bellman equation; \begin{eqnarray*} \psi _{1}A_{t} &=&\left( \left( 1-\beta \right) \left( \psi _{2}A_{t}\right) ^{\rho }+\beta \left( \psi _{1}\left( 1-\psi _{2}\right) A_{t}\left( E\left( \tilde{R}_{t+1}\right) ^{\alpha }\right) ^{\frac{1}{\alpha }}\right) ^{\rho }\right) ^{\frac{1}{\rho }} \\ &=&\left( \left( 1-\beta \right) \left( \psi _{2}\right) ^{\rho }+\beta \left( \psi _{1}\left( 1-\psi _{2}\right) \left( E\left( \tilde{R}% _{t+1}\right) ^{\alpha }\right) ^{\frac{1}{\alpha }}\right) ^{\rho }\right) ^{\frac{1}{\rho }}A_{t} \end{eqnarray*}% which is satisfied, provided \begin{equation} \psi _{1}=\left( \left( 1-\beta \right) \left( \psi _{2}\right) ^{\rho }+\beta \left( \psi _{1}\left( 1-\psi _{2}\right) \left( E\left( \tilde{R}% _{t+1}\right) ^{\alpha }\right) ^{\frac{1}{\alpha }}\right) ^{\rho }\right) ^{\frac{1}{\rho }}. \label{eq_psi1} \end{equation} The first-order condition for$c_{t}$is the same as choosing$\psi _{2}$optimally,% \begin{equation} \frac{\partial \left( \left( \left( 1-\beta \right) \left( \psi _{2}\right) ^{\rho }+\beta \left( \psi _{1}\left( 1-\psi _{2}\right) \left( E\left( \tilde{R}_{t+1}\right) ^{\alpha }\right) ^{\frac{1}{\alpha }}\right) ^{\rho }\right) \right) }{\partial \psi _{2}}=0 \label{eq_psi2} \end{equation}% Solving these equations gives us the solution to the problem. Assuming a specific form for the distribution of$\tilde{R},$makes it possible to calculate$\left( E\left( \tilde{R}\right) ^{\alpha }\right) ^{\frac{1}{% \alpha }}.$If, for example, if$\ln \left( \tilde{R}\right) \equiv r$is normal with mean$m$and standard deviation$\sigma ,$i.e.,$\ln R\thicksim n\left( R;m,\sigma \right) $we have \begin{eqnarray*} E\left( \tilde{R}^{\alpha }\right) &=&\frac{1}{\sigma \sqrt{2\pi }}% \dint\limits_{-\infty }^{\infty }\left( e^{r}\right) ^{\alpha }e^{-\frac{% \left( r-m\right) ^{2}}{2\sigma ^{2}}}dr \\ &=&e^{\alpha m+\frac{\sigma ^{2}\alpha ^{2}}{2}}, \\ E\left( \tilde{R}^{\alpha }\right) ^{\frac{1}{\alpha }} &=&e^{m+\frac{\sigma ^{2}\alpha }{2}} \end{eqnarray*} Note here that increasing$\sigma $has a direct effect on the expected return, so it is not a mean preserving spread, i.e., \begin{equation*} E\left( \tilde{R}\right) =e^{m+\frac{\sigma ^{2}}{2}}. \end{equation*} We can define a mean-preserving spread as increasing$\sigma $by letting$m=% \bar{m}-\frac{\sigma ^{2}}{2},$then \begin{equation*} E\left( \tilde{R}\right) =e^{\left( \bar{m}-\frac{\sigma ^{2}}{2}\right) +% \frac{\sigma ^{2}}{2}}. \end{equation*}% That is, if for any$\sigma $, the mean of$r$is$\bar{m}-\frac{\sigma ^{2}% }{2},$and the standard deviation$\sigma ,$an increase in$\sigma ,$is a mean preserving spread. In this case, \begin{equation*} E\left( \tilde{R}^{\alpha }\right) ^{\frac{1}{\alpha }}=e^{\bar{m}-\frac{% \sigma ^{2}}{2}+\frac{\sigma ^{2}\alpha }{2}}=e^{\bar{m}-\frac{\sigma ^{2}}{2% }\left( 1-\alpha \right) } \end{equation*} Clearly, we here see that an increase in$\sigma ,$while keeping the mean of the return constant, reduces the certainty equivalent, since$\alpha \leq 1$and more so the smaller is$\alpha .$Denoting$R\equiv \left( E\left( \tilde{R}\right) ^{\alpha }\right) ^{\frac{1% }{\alpha }},$the choice of$\psi _{2}$solves (\ref{eq_psi2}), implying that$\psi _{2}$is a root of \begin{equation*} \psi _{2}^{\rho }\left( 1-\psi _{2}^{-1}\right) \left( 1-\beta \right) +\beta \left( \psi _{1}\left( 1-\psi _{2}\right) R\right) ^{\rho }=0. \end{equation*} In the simplest case of unitary intertemporal elasticity of substitution, i.e.,$\rho =0,$this is \begin{equation*} \left( 1-\psi _{2}^{-1}\right) \left( 1-\beta \right) +\beta =0 \end{equation*}% with the solution$\psi _{2}=1-\beta .$We should then substitute our optimized value of$\psi _{2}$back into the Bellman equation and find$\psi _{2}$in (\ref{eq_psi1}). Note, however, that the formulation in (\ref{eq_KP}) is not valid if$\rho =0.$We can however, look at the limit as$\rho \rightarrow 0;$% \begin{eqnarray*} V\left( u,\mu \right) &=&\lim_{\rho \rightarrow 0}\left( \left( 1-\beta \right) u^{\rho }+\beta \mu ^{\rho }\right) ^{\frac{1}{\rho }} \\ &=&u^{1-\beta }\mu ^{\beta } \end{eqnarray*} Using this is formalution plus$u_{t}=c_{t}=\psi _{2}A_{t}$and$\mu \left( W\left( A_{t+1}\right) \right) =\psi _{1}\left( 1-\psi _{2}\right) A_{t}R$with$\psi _{2}=1-\beta $in the Bellman equation gives \begin{eqnarray*} \psi _{1}A_{t} &=&u_{t}^{1-\beta }\mu _{t}^{\beta } \\ &=&\left( \left( 1-\beta \right) A_{t}\right) ^{1-\beta }\left( \psi _{1}\beta A_{t}R\right) ^{\beta } \\ &=&A_{t}\left( \left( 1-\beta \right) \right) ^{1-\beta }\left( \psi _{1}\beta R\right) ^{\beta } \\ \psi _{1} &=&\psi _{1}^{\beta }\left( 1-\beta \right) ^{1-\beta }\left( \beta R\right) ^{\beta } \\ &=&\left( 1-\beta \right) \left( \beta R\right) ^{\frac{\beta }{1-\beta }}>0. \end{eqnarray*}% In which case we conclude that if$\tilde{R}$is log-normal \begin{equation*} W\left( A\right) =\left( 1-\beta \right) \left( \beta e^{m+\frac{\alpha \sigma ^{2}}{2}}\right) ^{\frac{\beta }{1-\beta }}A \end{equation*}% and \begin{equation*} c=\left( 1-\beta \right) A. \end{equation*} In this case, high riskaversion or higher risk reduces welfare by reducing the certainty equivalent return (high riskaversion means low$\alpha )$but saving is unaffected. What happens if$\rho >0?$First we note that higher risk reduces$R$and therefore tends to reduce$\psi _{1},$then, in (\ref% {eq_FOCKP}) a lower$R$and lower$\psi _{1}$reduces (increases) the term$% \beta \left( \psi _{1}\left( 1-\psi _{2}\right) R\right) ^{\rho }$if$\rho $\TEXTsymbol{>} (\TEXTsymbol{<}) 0. In optimum, this term must equal \begin{equation*} -\psi _{2}^{\rho }\left( 1-\psi _{2}^{-1}\right) \left( 1-\beta \right) \end{equation*}% which is downward sloping since$\rho <1$% \begin{equation*} \frac{d\left( -\psi _{2}^{\rho }\left( 1-\psi _{2}^{-1}\right) \left( 1-\beta \right) \right) }{d\psi _{2}}=-\left( 1-\beta \right) \psi _{2}^{\rho -1}\left( \frac{1-\rho }{\psi _{2}}+\rho \right) <0. \end{equation*}% \FRAME{dtbpFX}{4.5002in}{3.0001in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.5002in;height 3.0001in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0.3";xmax "0.9";xviewmin "0.288";xviewmax "0.91224";yviewmin "0.036666666666667E0";yviewmax "1.30826666666667";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{v58146QSUB2ESUB};var1name \TEXUX{$\psi _{2}$};function \TEXUX{$-\left[ \psi _{2}^{\rho }\left( 1-\psi _{2}^{-1}\right) \left( 1-\beta \right) \right] _{\beta =.45,\rho =0}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0.3,0.9";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$.45$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0.3,0.9";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'K6VBVZ00.wmf';tempfile-properties "PR";}} Therefore, an increase in risk leads to a higher (lower)$\psi _{2}$if$% \rho >(<)$0. I.e., consumption increases and savings decreases if the certainty equivalent return decreases iff the intertemporal elasticity of substitution is higher than unity. This is very much along the results with standard expected utility -- if the intertemporal elasticity of substitution is higher than unity, the substitution effect dominates after an decrease in interest rates so consumption increases. The converse is true if the elasticity is lower than unity. \newpage \section{Ambiguity} Aversion to unknown odds, as is demonstrated in labs, e.g., the Ellsberg paradox, is given axiomatic foundations by Gilboa and Schmeidler. They show that reasonable axioms capturing such ambiguity averse behavior can be represented by a sort of$max-min$preferences. Suppose it is known that there is a set of possible realizations of the state,$z\in \Gamma $with associated consumption levels$c\left( z\right) .$Then, assume that the individual does not know the probability distribution over these states, but he knows that this probability distribution belongs to a set$\Pi .$Call the elements of this set$p$being particular possible probability distributions. An element$p$is thus a vector of probabilities$p\left( z\right) .$Then, preferences are given by \begin{equation*} U\left( \left\{ c\left( z\right) \right\} \right) =\min_{p\in \Pi }\dsum\limits_{z\in \Gamma }p\left( z\right) c\left( z\right) =\min_{p\in \Pi }E_{p}c\left( z\right) . \end{equation*} As with standard preferences, it is not necessary to take this literally in the sense the individuals actually maximize these minimized preferences. It may, for example, not be possible to ask individuals to tell us directly about$\Pi .$A simple example, consider again the Ellsberg paradox. There are two urns each with 100 balls. In urn 1, there are 50 red and 50 black. In urn 2, there are only red and black balls but the proportions are unknown. The subject is given a color and can pick one ball. If a ball with the given color comes up, the gain is 50\$, if not the gain is zero. The subject is asked to rank lotteries. Typically the following response comes up. \begin{enumerate} \item Red from urn 1 $\sim$ Black from urn 1. \item Red from urn 2 $\sim$ Black from urn 2. \item Red from urn 1 $\succ$ Red from urn 2. \item Black from urn 1 $\succ$ Black from urn 2. \end{enumerate} Suppose we now ask individuals about the "ambiguity premium", i.e., we find value $r,$ such that if we let urn 1 contain $50-r$ red balls and $50+r$ black ones. \begin{itemize} \item Red from urn 1 $\sim$ Red from urn 2.\newline and hopefully if we let urn 1 contain $50+r$ red balls and $50-r$ black ones \item Black from urn 1 $\sim$ Black from urn 2. \end{itemize} In this case, the relevant $p^{\prime }s$ are the shares of red balls in urn 2. The value of $r,$ pins down $\Pi ,$being the interval $\left[ \frac{50-r}{% 100},\frac{50+r}{100}\right] .$ Given this, the value of urn 2 in a bet on red is \begin{eqnarray*} &&\min_{p\in \left[ \frac{50-r}{100},\frac{50+r}{100}\right] }pU\left( 50\right) +\left( 1-p\right) U(0) \\ &=&\frac{50-r}{100}U\left( 50\right) +\frac{50+r}{100}U(0) \end{eqnarray*} with the same value in a bet on black since in this case, we have \begin{eqnarray*} &&\min_{p\in \left[ \frac{50-r}{100},\frac{50+r}{100}\right] }pU\left( 0\right) +\left( 1-p\right) U(50) \\ &&\frac{50+r}{100}U\left( 0\right) +\frac{50-r}{100}U(0). \end{eqnarray*} Suppose now that there are two individuals, one owns an asset that gives 50 dollars with a probability $p$ he knows is 50\% but that he cannot verify this knowledge to the other person. Otherwise it pays zero. The other person is in the same position, he owns an asset that gives 50 dollars with probability of $p=.5$ that he can not credibly verify. Suppose individuals consume 1+the payoff from the asset and have log utility. Under standard assumption including that both individuals assign a 50\% probability to the other's urn. Individuals should share the risk and get a payoff of% \begin{equation*} \frac{1}{4}\ln \left( 51\right) +\frac{1}{2}\ln \left( 26\right) +\frac{1}{4}% \ln 1=2.612 \end{equation*} If instead individuals have ambiguity aversion the probability of winning in the "foreign" asset is $p^{\prime }\in \Pi .$ How big share $\omega$ should he choose to invest in the other persons asset? Given a $p\in \Pi ,$ the four states of the world, $\left\{ high,high\right\} ,\left\{ low,high\right\} ,\left\{ high,low\right\} ,\left\{ low,low\right\}$ happen with probabilities, $\frac{1}{2}p,\frac{1}{2}p,\frac{1}{2}\left( 1-p\right) ,$and $\frac{1}{2}\left( 1-p\right) .$ Given the symmetric nature of the game, we focus on the case when the price of the "foreign" asset in terms of the "domestic" is unity. Consumption, given $\omega$ is then $51,.1+\omega 50,1+\left( 1-\omega \right) 50,1.$ The utility of the optimal portfolio is therefore \begin{equation*} \min_{p\in P}\max_{\omega }\left( \frac{1}{2}p\ln \left( 1+50\right) +\frac{1% }{2}p\ln \left( 1+\omega 50\right) +\frac{1}{2}\left( 1-p\right) \ln \left( 1+\left( 1-\omega \right) 50\right) +\frac{1}{2}\left( 1-p\right) \ln 1\right) \end{equation*} The first-order condition for the maximization problem is% \begin{equation*} \frac{d\left( \frac{1}{2}p\ln \left( 1+51\right) +\frac{1}{2}p\ln \left( 1+\omega 50\right) +\frac{1}{2}\left( 1-p\right) \ln \left( 1+\left( 1-\omega \right) 50\right) \right) }{d\omega } \end{equation*} \begin{equation*} \omega =\frac{52p-1}{50} \end{equation*} Of course, this is increasing in $p.$ Note also that for $p=1/52,$ $\omega =0$ and for $p<1/52,\omega <0.$ For example, if $p=0.01,$ $\omega \approx -0.01,$i.e., a short position in the foreign asset. The short position implies that if the foreign asset pays 50, a payment from hom to abroad takes place. Now, we have to pick $p.$To do this, we consider the maximized value, i.e.,% \begin{eqnarray*} &&\max_{\omega }\left( \frac{1}{2}p\ln \left( 1+51\right) +\frac{1}{2}p\ln \left( 1+\omega 50\right) +\frac{1}{2}\left( 1-p\right) \ln \left( 1+\left( 1-\omega \right) 50\right) \right) \\ &=&\frac{1}{2}\left( p\ln \left( 51\right) +\ln \left( 52\right) +p\ln \left( p\right) +\left( 1-p\right) \ln \left( 1-p\right) \right) \end{eqnarray*} Let's plot this, what we see at the this is increasing in $p$ for $p$ close to 0.5. But, it is not monotone. \FRAME{itbpFX}{4.5002in}{3.0001in}{0in}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.5002in;height 3.0001in;depth 0in;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "0.5";xviewmin "-0.004610158242424";xviewmax "0.510092203164849";yviewmin "1.95300498614071";yviewmax "2.62518467101285";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{p};var1name \TEXUX{$p$};function \TEXUX{$\frac{1}{2}\left( p\ln \left( 51\right) +\ln \left( 52\right) +p\ln \left( p\right) +\left( 1-p\right) \ln \left( 1-p\right) \right)$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,0.5";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY0D.wmf';tempfile-properties "PR";}} \FRAME{itbpFX}{4.5002in}{3.0001in}{0in}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.5002in;height 3.0001in;depth 0in;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "0.1";xviewmin "-0.000922031648485";xviewmax "0.10201844063297";yviewmin "1.96503771761548";yviewmax "2.01056433296352";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{p};var1name \TEXUX{$p$};function \TEXUX{$\frac{1}{2}\left( p\ln \left( 51\right) +\ln \left( 52\right) +p\ln \left( p\right) +\left( 1-p\right) \ln \left( 1-p\right) \right)$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,0.1";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY0E.wmf';tempfile-properties "PR";}}. In fact, \begin{equation*} \arg \min_{p}\frac{1}{2}\left( p\ln \left( 51\right) +\ln \left( 52\right) +p\ln \left( p\right) +\left( 1-p\right) \ln \left( 1-p\right) \right) =% \frac{1}{52} \end{equation*}% which corresponds to a share $\omega =0.$ Why is this? The answer is that if the individual would be short in the foreign asset, what is a bad realization has flipped. It is now the state when the foreign asset pays 50, happening with probability $p.$ Therefore, in the range where $\omega <0,$ a higher $p$ means lower utility. In general, since trade can only increase utility, the worst $p,$ is the one that implies autarky. The situation would be quite different under asymmetric information, in which case we would expect to sometimes see agents with more information (domestic) go short in home securities. Let's take another example, suppose production in each country is 2 or 1, so the state space is $\left\{ 1,2,3,4\right\} ,$ implying output $\left\{ y_{1},y_{2}\right\}$ is $\left\{ 2,2\right\} ,\left\{ 2,1\right\} ,\left\{ 1,2\right\}$ or $\left\{ 1,1\right\} .$ Utility is \begin{equation*} U_{i}=\min_{p\in \Pi }\dsum p\left( z\right) \ln c_{i}\left( z\right) \end{equation*}% Suppose also individual the knows that the probability of there own production being high is 0.5, while the set of possible probabilities for the other countries production being high is $0.5-2\gamma ,$ for $\gamma \in % \left[ -a,a\right]$ and that these events are independent. The agent decides how much to invest abroad $\omega .$ Due to symmetry, the relative price of the two assets, foreign and domestic production should be one. The budget constraint is therefore for agent 1. \begin{equation*} c_{1}\left( 1\right) =2,c_{1}\left( 2\right) =\left( 1-\omega \right) 2+\omega ,c_{1}\left( 3\right) =\left( 1-\omega \right) +2\omega ,c_{1}\left( 4\right) =1 \end{equation*} Given $\gamma ,$ the maximization problem is \begin{eqnarray*} &&\max_{\omega }\dsum p\left( z\right) \ln c_{i}\left( z\right) \\ &=&\max_{\omega }\frac{1}{2}\left( \frac{1}{2}-2\gamma \right) \ln 2+\frac{1% }{2}\left( \frac{1}{2}+2\gamma \right) \ln \left( \left( 1-\omega \right) 2+\omega \right) \\ &&+\frac{1}{2}\left( \frac{1}{2}-2\gamma \right) \ln \left( \left( 1-\omega \right) +2\omega \right) +\frac{1}{2}\left( \frac{1}{2}+2\gamma \right) \ln 1 \end{eqnarray*} The first order condition is \begin{eqnarray*} \frac{d\left( \frac{1}{2}\left( \frac{1}{2}+2\gamma \right) \ln \left( \left( 1-\omega \right) 2+\omega \right) +\frac{1}{2}\left( \frac{1}{2}% -2\gamma \right) \ln \left( \left( 1-\omega \right) +2\omega \right) \right) }{d\omega } &=&0 \\ \omega &=&\frac{1}{2}-6\gamma \end{eqnarray*} Substituting this into the utility function, yields% \begin{eqnarray*} &&\max_{\omega }\dsum p\left( z\right) \ln c \\ &=&\left[ \frac{1}{2}\left( \frac{1}{2}-2\gamma \right) \ln 2+\frac{1}{2}% \left( \frac{1}{2}+2\gamma \right) \ln \left( \left( 1-\omega \right) 2+\omega \right) +\frac{1}{2}\left( \frac{1}{2}-2\gamma \right) \ln \left( \left( 1-\omega \right) +2\omega \right) \right] _{\omega =\frac{1}{2}% -6\gamma } \\ &=&\frac{-1}{4}\ln 2+\frac{1}{2}\ln 3-\gamma \left( \ln 2\right) +\left( \frac{1}{4}+\gamma \right) \ln \left( 1+4\gamma \right) +\left( \frac{1}{4}% -\gamma \right) \ln \left( 1-4\gamma \right) . \end{eqnarray*} In the following graph, this is plotted against $\gamma$\FRAME{dtbpFX}{% 4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "-0.2";xmax "0.2";xviewmin "-0.208";xviewmax "0.208160000000001";yviewmin "0.339532208195823";yviewmax "0.705863862504196";plottype 4;numpoints 100;plotstyle "patch";axesstyle "normal";xis \TEXUX{v58125};var1name \TEXUX{$\gamma$};function \TEXUX{$\left( \frac{-1}{4}\ln 2+\frac{1}{2}\ln 3-\gamma \left( \ln 2\right) +\left( \frac{1}{4}+\gamma \right) \ln \left( 1+4\gamma \right) +\left( \frac{1}{4}-\gamma \right) \ln \left( 1-4\gamma \right) \right)$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-0.2,0.2";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JZRM2G00.wmf';tempfile-properties "PR";}} For small $a,$i.e., $\gamma$ close to zero. The maximized value is falling in $\gamma$ in which case $\gamma$ is "picked" at the corner $a.$ However, for sufficiently large $a$, this is not true, since the maximized value is non-monotone in $\gamma .$ The first order condition for minimizing the maimized value over $\gamma$ \begin{equation*} \frac{d\left( \frac{-1}{4}\ln 2+\frac{1}{2}\ln 3-\gamma \left( \ln 2\right) +\left( \frac{1}{4}+\gamma \right) \ln \left( 1+4\gamma \right) +\left( \frac{1}{4}-\gamma \right) \ln \left( 1-4\gamma \right) \right) }{d\gamma }=0 \end{equation*}% giving $\gamma _{\min }=\frac{1}{12}.$ This means that $\gamma$ will never be chosen larger than $\frac{1}{12}$ and in particular that \begin{equation*} \omega _{\min }=\frac{1}{2}-6\gamma _{\min }=\frac{1}{2}-\frac{6}{12}=0. \end{equation*}% : This is implies the important result that shortsales cannot occur. This is in line with empirical evidence and is not the prediction of models with asymmetric information. Why is this? Note first that if $\omega =0,$ the probability of high production abroad has no effect on utility. If $\omega$ is negative, a reduction in $p$ actually increases welfare. Why, the stream of payment from the foreign asset is \begin{equation*} \omega \left( 2p+(1-p)\right) =\omega (1+p), \end{equation*}% which decreases in $p$ if $\omega$ is negative. In this case, high production abroad is the bad state!\newpage \section{Time-inconsistency and temptation} Lab evidence discussed in the introduction shows \emph{preference }reversal, quicker discounting in for close time periods than for distant. Also preference for commitment. People sometimes prefer to restrict their future behavior -- force themselves to save, hide the jar of cookies, not bring to much money to the bar, and so on. Evidence that hyperbolic discount factor represents time preference better than geometric. Two approaches: \begin{itemize} \item Quasi geometric preferences. \item Preferences over sets (menus), allows modelling of temptation, cost of employing self control and welfare analysis. \end{itemize} \subsection{Quasi-geometric preferences} Between current and next period, an "extra" discount factor $\delta ,$ is introduced (the $\beta \delta$ model).% \begin{eqnarray*} \text{Self 0} &&\ ~\quad \quad U_{0}\left( _{0}c\right) =u\left( c_{0}\right) +\delta \left( \beta u\left( c_{1}\right) +\beta ^{2}u\left( c_{2}\right) +\beta ^{3}u\left( c_{3}\right) +\cdots \right) \\ \text{Self} &\text{1}&\quad \quad \quad \quad \quad U\left( _{1}c\right) =u\left( c_{1}\right) +\delta \left( \beta u\left( c_{2}\right) +\beta ^{2}u\left( c_{3}\right) +\beta ^{3}u\left( c_{3}\right) +\cdots \right) \\ \text{Self} &\text{2}&\quad \quad \quad \quad \quad \quad \quad \quad \;\;\;\;U\left( _{2}c\right) =u\left( c_{2}\right) +\delta \left( \beta u\left( c_{3}\right) +\cdots \right) \end{eqnarray*} This implies that preferences are changing over time. \begin{eqnarray*} \text{Self 0 }MRS_{1,2} &=&\frac{\beta ^{2}u^{\prime }\left( c_{2}\right) }{% \beta u^{\prime }\left( c_{1}\right) }=\beta \frac{u^{\prime }\left( c_{2}\right) }{u^{\prime }\left( c_{1}\right) } \\ \text{Self 1 }MRS_{1,2} &=&\delta \beta \frac{u^{\prime }\left( c_{2}\right) }{u^{\prime }\left( c_{1}\right) } \end{eqnarray*} Self 1 cares relatively less of period 2 utility than self 1. \underline{Assumptions about behavior}: \begin{itemize} \item The consumer cannot commit to future actions. \item The consumer is \textquotedblleft sophisticated\textquotedblright : he realizes that his preferences will change and makes the current decision taking this into account. \item The decision-making process is viewed as a dynamic game, with the agent's current and future selves as players. (Alternative: \textquotedblleft naive\textquotedblright\ behavior. The agent thinks that he will not change preferences.) \item Focus is on Markov equilibria, but other equilibria with trigger strategies also exists. For example, a self "behaves" and does not overconsume as long as previous selves has behaved well. \item Markov equilibria can be strange or non-existent. Standard existence theorems not applicable. \end{itemize} \textbf{Example: Consumption and savings problem. } Suppose the agent has log period utility% \begin{equation*} U_{t}\left( _{t}c\right) =\ln \left( c_{t}\right) +\delta \sum_{s=1}^{\infty }\beta ^{s}\ln \left( c_{t+s}\right) \end{equation*} and face a constant return $r$. The budget constraint is \begin{equation*} a_{t+1}=r\left( a_{t}-c_{t}\right) \end{equation*} The state variable is $a_{t}$ and let us implicitly define a continuation value from \begin{equation} J\left( a_{t}\right) =\ln c_{t}^{\ast }+\beta J\left( r\left( a_{t}-c_{t}^{\ast }\right) \right) , \label{eq_recursion} \end{equation}% for some value $c_{t}^{\ast }.$ If $\delta =1,$ $J$ is also the value function if $c_{t}^{\ast }$ is the $% argmax$ to the RHS (the whole equation is then the Bellman equation). Under quasi geometric discounting, $c_{t}^{\ast }$ is NOT $\arg \max_{c}\ln c_{t}+\beta J\left( r\left( a_{t}-c_{t}\right) \right)$ instead \begin{equation*} c_{t}^{\ast }=\arg \max_{c_{t}}\ln c_{t}+\beta \delta J\left( r\left( a_{t}-c_{t}\right) \right) \end{equation*}% and the utility of self $t$ is% \begin{equation*} W\left( a_{t}\right) =\max_{c_{t}}\ln c_{t}+\beta \delta J\left( r\left( a_{t}-c_{t}\right) \right) \end{equation*} Note that the value of giving assets to the next self is depreciated by the fact that self 0 knows that self 1 is going to overconsume in the eyes of self 0. Now, we can guess that $J$ has the form \begin{equation*} J\left( a_{t}\right) =A+b\ln a_{t} \end{equation*} Given this, $c_{t}^{\ast }$ is the solution to the first order condition \begin{eqnarray*} 0 &=&\frac{1}{c_{t}^{\ast }}-\beta \delta J^{\prime }\left( r\left( a_{t}-c_{t}^{\ast }\right) \right) r \\ &=&\frac{1}{c_{t}^{\ast }}-\beta \delta \frac{b}{r\left( a_{t}-c_{t}^{\ast }\right) }r \\ &\Rightarrow &c_{t}^{\ast }=\frac{a_{t}}{1+\beta \delta b} \end{eqnarray*} Substituting this in (\ref{eq_recursion}) gives \begin{eqnarray*} A+b\ln a_{t} &=&\ln \frac{a_{t}}{1+\beta \delta b}+\beta \left( A+b\ln r\left( a_{t}-\frac{a_{t}}{1+\beta \delta b}\right) \right) \\ &=&\left( 1+\beta b\right) \ln a_{t}+\beta A+\beta b\ln \frac{r\beta \delta b% }{1+\beta \delta b}-\ln \left( 1+\beta \delta b\right) \end{eqnarray*}% This is satisfied for all $a_{t}$ iif% \begin{eqnarray*} \left( 1+\beta b\right) &=&b \\ &\Rightarrow &b=\frac{1}{1-\beta } \end{eqnarray*}% implying \begin{equation*} c_{t}^{\ast }=\frac{a_{t}}{1+\beta \delta \frac{1}{1-\beta }}=\frac{1-\beta }{1-\beta \left( 1-\delta \right) }a_{t} \end{equation*}% and $A$ is \begin{equation*} A=\frac{\frac{\beta }{1-\beta }\ln \frac{r\beta \delta }{1-\beta \left( 1-\delta \right) }+\ln \left( \frac{1-\beta }{1-\beta \left( 1-\delta \right) }\right) }{1-\beta }. \end{equation*} As we see, if $\delta <1,$ consumption is higher and savings are lower than in the time-consistent case, when the consumption rate is $1-\beta .$ Let's now find the commitment solution if self 0 determines all consumption values. In this case, we first calculate $J_{c}\left( a_{t}\right) ,$which is the continuation value when everything is determined by self 0. Note, however, that future selves will agree with self zero on this. The difference between the no-commitment case is that now the continuation value maximizes the standard Bellman equation without any $\delta .$ Thus, $J_{c}$% must satisfy, \begin{equation*} J_{c}\left( a_{t}\right) =\max_{c_{t}}\ln c_{t}+\beta J_{c}\left( r\left( a_{t}-c_{t}\right) \right) \end{equation*}% If we don't remember the solution to this, we do as usual. We take the first order condition% \begin{equation*} \frac{1}{c_{t}}=\beta rJ^{\prime }\left( r\left( a_{t}-c_{t}\right) \right) . \end{equation*}% Guessing% \begin{equation*} J_{c}\left( a_{t}\right) =A_{c}+b_{c}\ln a_{t} \end{equation*}% implies \begin{eqnarray*} \frac{1}{c_{t}} &=&\beta b_{c}r\frac{1}{r\left( a_{t}-c_{t}\right) } \\ c_{t} &=&\frac{a_{t}}{1+\beta b_{c}}. \end{eqnarray*}% Substituting, \begin{eqnarray*} A_{c}+b_{c}\ln a_{t} &=&\ln \frac{a_{t}}{1+\beta b_{c}}+\beta A_{c}+\beta b_{c}\ln ra_{t}\left( 1-\frac{1}{1+\beta b_{c}}\right) \\ A_{c}+b_{c}\ln a_{t} &=&\ln a_{t}+\ln \frac{1}{1+\beta b_{c}}+\beta A_{c}+\beta b_{c}\ln a_{t}+\beta b_{c}\ln r\left( \frac{\beta b_{c}}{1+\beta b_{c}}\right) \end{eqnarray*} Giving \begin{eqnarray*} &\Rightarrow &b_{c}=\left( 1+\beta b_{c}\right) ,b_{c}=\frac{1}{1-\beta } \\ c_{t} &=&\frac{a_{t}}{1+\beta b_{c}}=\left( 1-\beta \right) a_{t} \\ A_{c} &=&\ln \frac{1}{1+\beta \frac{1}{1-\beta }}+\beta A_{c}+\beta \frac{1}{% 1-\beta }\ln r\left( \frac{\beta \frac{1}{1-\beta }}{1+\beta \frac{1}{% 1-\beta }}\right) \\ &=&\ln \left( 1-\beta \right) +\beta A_{c}+\frac{\beta }{1-\beta }\ln r\beta \\ &\Rightarrow &A_{c}=\frac{\ln \left( 1-\beta \right) +\frac{\beta }{1-\beta }% \ln r\beta }{1-\beta } \end{eqnarray*} As we see, the coefficient on $\ln a_{t+1}$ is the same in both cases, commitment and no commitment. The difference between the constants under no commitment and commitment is negative, I think. The fact that the coefficient on $\ln a_{t+1}$ is the same in $J_{c}$ and $J,$ implies that the marginal value of leaving assets to self 1 is the same in both cases. Thus, consumption in period 0 is independent of whether there is commitment or not. Note that two forces here are affecting the results. On the one hand, giving assets to self 1 under no commitment has a lower value since he consumers too much in the eyes of self 0. This reduces the incentive to save for self 0. On the other hand, this leaves self 2, 3,... with too little consumption and the way only way self 0 can increase consumption of self 2,3,... is to save. This increases the value of saving. Apparently, this two effects cancel in the log utility case. From period 1 and onwards, savings is higher under commitment. \begin{equation*} \beta >\frac{\beta \delta }{1-\beta \left( 1-\delta \right) } \end{equation*} Commitment would of course increase welfare for self $0,$ it can never reduce it. What about later selves? In the no commitment case, self 1 gets \begin{eqnarray*} W_{nc} &=&\ln c_{1}^{\ast }+\beta \delta J\left( r\left( a_{1}-c_{t+1}^{\ast }\right) \right) \\ &=&\ln \frac{1-\beta }{1-\beta \left( 1-\delta \right) }a_{1} \\ &&+\beta \delta \left( A+\frac{1}{1-\beta }\ln ra_{1}\left( \frac{\beta \delta }{1-\beta \left( 1-\delta \right) }\right) \right) \\ &=&\frac{1-\beta \left( 1-\delta \right) }{1-\beta }\ln a_{1}+\ln \frac{% 1-\beta }{1-\beta \left( 1-\delta \right) } \\ &&+\beta \delta \left( A+\frac{1}{1-\beta }\ln r\left( \frac{\beta \delta }{% 1-\beta \left( 1-\delta \right) }\right) \right) \end{eqnarray*} With commitment, the continuation value is different since now self 0 determines everything. Self 1 gets under commitment \begin{eqnarray*} &&\ln \left( 1-\beta \right) a_{1}+\beta \delta J_{c}\left( r\beta a_{1}\right) \\ &=&\ln \left( 1-\beta \right) a_{1}+\beta \delta b_{c}\ln \left( r\beta a_{1}\right) +\beta \delta A_{c} \\ &=&\left( \frac{1-\beta \left( 1-\delta \right) }{1-\beta }\right) \ln a_{1}+\ln \left( 1-\beta \right) +\frac{\beta \delta }{1-\beta }\ln r\beta +\beta \delta A_{c} \end{eqnarray*} The difference between no comittment and commitment is.% \begin{equation*} \frac{1}{\left( 1-\beta \right) ^{2}}\left( \left( \ln \left( 1-\beta \left( 1-\delta \right) \right) \right) \left( \beta \left( 2-\beta \right) \left( 1-\delta \right) -1\right) +\beta \delta \ln \delta \right) \equiv D\left( \delta \right) \end{equation*} This expression is a little ugly, but we can evaluated it by a Taylor approximation around $\delta =1,$ using% \begin{eqnarray*} \left[ D\left( \delta \right) \right] _{\delta =1} &=&0 \\ \left[ D^{\prime }\left( \delta \right) \right] _{\delta =1} &=&0 \\ \left[ D^{\prime \prime }\left( \delta \right) \right] _{\delta =1} &=&\left( 1-2\beta \right) \frac{\beta }{1-\beta } \end{eqnarray*} which is negative if $\beta >\frac{1}{2}.$Therefore, a reduction in $\delta$ from unity makes the no commitment better (worse) if $\beta >\frac{1}{2}$($% \beta <\frac{1}{2}).$ Commitment gives extra value which is good also for self 1, but she cannot control her consumption which reduces the value. For self 1, commitment can therefore be better than no commitment, also if it is done by self 1. For later individuals, it may be even better with previous commitment since asset levels are higher. \subsection{Preferences over choice sets, Gul and Pesendorfer} An alternative approach. Does not assume multiple selves, no game thus no multiplicity. Also allows resistance to temptation and to model costs of resisting temptation. \begin{itemize} \item Two subperiods. \item Second subperiod preferences defined over ordered pairs $(A,x)$, where $A$ is a choice set and $x\in A$ is a choice (consumed). \item \underline{Definition}: $y$ tempts $x$ if $(\{x\},x)$ is preferred to $% (\{x,y\},x)$. That is, individuals are better of getting $x$ \emph{without }% having $y$ in the choice set. \item \underline{Assumptions}: \begin{enumerate} \item Eliminating temptations cannot make the consumer worse off. \item If $y$ tempts $x$, then $x$ does not tempt $y$. \item The utility of a fixed choice is affected by the choice set only through its most tempting element. \end{enumerate} \item Second-period preferences induce first-period preferences over choice sets themselves: $A\succeq B$ if and only if there is an $x\in A$ such that $% (A,x)$ is preferred to $(B,y)$ for all $y\in B$. \item The above assumptions imply what is labelled \emph{set betweenness}: \begin{equation*} A\succeq B\Rightarrow A\succeq A\cup B\succeq B. \end{equation*}% \emph{Choice sets cannot be compared simply by looking at their "best" or chosen elements.} Instead, the utility of a fixed choice depends on the choice set (through its most \textquotedblleft tempting\textquotedblright\ element). Note that this violates one of the axioms in standard theory. Removing a non-choosen element from a choice set cannot change utility or behavior (independence of irrelevant alternatives). Set betweenness allows three possibilities: \begin{enumerate} \item Standard decision maker: $A\sim A\cup B\succeq B$. \item \emph{Preference for commitment\/} and \emph{self-control\/}: $A\succ A\cup B\succ B$. \newline Interpretation: there is an element in $B$ that tempts me. Nevertheless, I choose the same element in $A\$and $A\cup B,$ but if faced with only $A,$ I don't have to take the effort of controlling myself. Thus $A\succ A\cup B.$ Furthermore, $A\cup B\succ B$ since the choice in $A\cup B$ provides higher utility,than the tempting choice. \item \emph{Preference for commitment\/} and \emph{succumbing to temptation\/% }: $A\succ A\cup B\sim B$.\newline Interpretation: there is an element in $B$ that tempts me. $A\succ A\cup B$ since it provides higher utility. Faced with the tempting choice, however. I cannot resist. I choose the same element in $A\cup B$ and $B$ and there is no cost of controlling myself$.$Thus, $A\cup B\sim B$ \end{enumerate} \end{itemize} \subsection{The representation theorem} The assumptions implies that preference over sets in the first period can be written% \begin{equation*} W(A)={\max }_{x\in A}\;\{U(x)+V(x)\}-{\max }_{\tilde{x}\in A}\;V(\tilde{x}) \end{equation*} Second period, preference are represented by \begin{equation*} W^{\ast }(A,x)=\;\{U(x)+V(x)\}-{\max }_{\tilde{x}\in A}\;V(\tilde{x}) \end{equation*} Interpretation: \begin{itemize} \item $U$ determines the \emph{commitment} ranking (i.e., the utility of singleton sets, no temptation). \item $V$ determines the \emph{temptation} ranking (i.e., $V$ gives higher values to more tempting elements). \item ${\arg \max }_{\tilde{x}\in A}\;V(\tilde{x})$ is the most tempting element in $A$. \item The second-period choice picks $x$ as \begin{equation*} \arg \max_{x\in A}\{U(x)+V(x)\} \end{equation*}% giving utility \begin{equation*} \max_{x\in A}\{U(x)+V(x)\}-{\max }_{\tilde{x}\in A}\;V(\tilde{x}). \end{equation*} \item If a person is given $x\in A$, \emph{without }anything else to choose from, there is no cost of self control. The utility is \begin{equation*} U(x)+V(x)-V(x)=U\left( x\right) . \end{equation*} \item If a person \emph{chooses }$x\in A,$ the disutility of self-control is $V(x)-{\max }_{\tilde{x}\in A}\;V(\tilde{x})\geq 0,$so utility is \begin{equation*} U(x)+V(x)-{\max }_{\tilde{x}\in A}\;V(\tilde{x}). \end{equation*} \item If a person \emph{chooses }$\tilde{x}={\arg \max }_{\tilde{x}\in A}\;V(% \tilde{x})$, he gives in to temptation, there is no cost of self-control, and the utility is \begin{equation*} U(\tilde{x})+{\max }_{\tilde{x}\in A}V(x)-{\max }_{\tilde{x}\in A}\;V(\tilde{% x})=U(\tilde{x}). \end{equation*} \end{itemize} \subsection{A 2-period consumption-savings model} \begin{itemize} \item Consumption today and tomorrow. \item Neoclassical production. \item Standard budget set (borrowing and lending at $r$). \item General equilibrium. \item With $U(c_{1},c_{2})$ playing the role of $U$ and $V(c_{1},c_{2})$ the role of $V$, let the temptation function $V$ have a stronger preference for present consumption. For example, let \begin{equation*} U(c_{1},c_{2})=u(c_{1})+\beta u(c_{2}) \end{equation*}% and \begin{equation*} V(c_{1},c_{2})=\gamma \left( u(c_{1})+\beta \delta u(c_{2})\right) , \end{equation*}% with $\delta ,\beta <1$. Aggregate resource constraint given by \begin{eqnarray*} c_{1}+k_{2} &=&f\left( k_{1}\right) \\ c_{2} &=&f\left( k_{2}\right) \end{eqnarray*} \item Strength of temptation determined by $\gamma .$ Standard model when $% \gamma =0.$ As $\gamma \rightarrow \infty ,$ Laibson model. In equilibrium choices are made to maximize \begin{equation*} U(c_{1},c_{2})+V(c_{1},c_{2}) \end{equation*} \item In competitive general equilibrium individuals take prices (here interest rate) as given, provides a linear budget set for the individual. \end{itemize} \FRAME{itbpF}{5.8825in}{4.4192in}{0in}{}{}{slide1.emf}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.8825in;height 4.4192in;depth 0in;original-width 10.0024in;original-height 7.5022in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../China/Slides/Slide1.EMF';file-properties "XNPEU";}} Policy implications. Command optimum can achieve $\arg \max U$, without any temptation cost. Nothing is better than this. A subsidy to investments (tax on first period consumption) can improve upon \emph{laissez faire. }Does so by reducing temptation. For example, let $u\left( x\right) =\ln \left( x\right)$. Then, choices are governed by solving \begin{eqnarray*} &&\max \left( \ln c_{1})+\beta \ln (c_{2})\right) +\gamma \left( \ln (c_{1})+\beta \delta \ln (c_{2})\right) \\ \text{s.t. }c_{1}+\left( \frac{c_{2}}{1+r}\right) &=&w \\ c_{1} &=&\arg \max \left( \left( 1+\gamma \right) \ln c_{1}+\beta \left( 1+\gamma \delta \right) \ln (\left( 1+r\right) \left( w-c_{1}\right) )\right) \\ &=&w\frac{1+\gamma }{1+\beta +\gamma \left( 1+\beta \delta \right) } \end{eqnarray*}% which increases in $\gamma .$ The maximum temptation is \begin{eqnarray*} c_{1}^{t} &=&\arg \max \gamma \left( \ln (c_{1})+\beta \delta \ln (\left( 1+r\right) \left( w-c_{1}\right) )\right) \\ &=&\frac{w}{1+\beta \delta } \end{eqnarray*} \textbf{Interesting implication} Compare autarky, i.e., each individual runs his own machine. Then the interest rate is not exogenous. \FRAME{itbpF}{4.5567in}{3.4255in}{0in}{}{}{slide6.emf}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 4.5567in;height 3.4255in;depth 0in;original-width 10.0024in;original-height 7.5022in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../China/Slides/Slide6.EMF';file-properties "XNPEU";}} \textbf{Result:} Autarky delivers the same allocation, but at higher welfare. Why? The choice sets shrinks, the temptation to overconsume is reduced and the cost of resisting temptation falls. \subsection{Macroeconomic applications} Krusell, Kuru\c{s}\c{c}u{\small , }Smith "Temptation and Taxation" \begin{itemize} \item Consider long horizons: the limit of the finite-horizon problems. \item Study competitive equilibrium under two kinds of parametric restrictions: \begin{enumerate} \item \emph{Logarithmic utility, Cobb-Douglas production, and full depreciation\/}: full analytical solution of recursive competitive equilibria. \item \emph{Iso-elastic utility and no restrictions on technology\/}: analytical characterization of steady state and computational analysis of dynamics. \end{enumerate} \end{itemize} \textbf{Analysis}: Vary $\delta$ and $\gamma$, while adjusting $\beta$ to keep the steady-state interest rate constant. \textbf{Results:} \begin{itemize} \item Almost observationally equivalence (like in Barro "Laibson meets Ramsey", when the utility is $\log$, the speed of adjustment to the steady state does not depend on $\delta$ (as in Barro). \item With more (less) curvature in utility, the speed of adjustment is decreasing (increasing) in $\delta$. \item The effects of $\delta$ on the speed of adjustment is quantitatively small: observational equivalence found in Barro \textquotedblleft almost\textquotedblright\ carries over. \item Savings should be subsidized. But not much, and the welfare gains are small (little reduction in temptation costs). \end{itemize} \newpage \section{Habits} We have previously assumed \emph{history independence,} meant to mean that the marginal rate of substitution, evaluated at $t$ between: \begin{itemize} \item goods consumed at $t$ and $t+1,$ and \item different goods consumed in the same time is independent of the consumption history prior to $t.$ \end{itemize} This is not necessarily a good assumption. There are cases in which we might want to relax history independence. \begin{enumerate} \item The case when previous consumption leads to higher aspiration. To live on a small budget may be easier if you are used to it than if your are used to the good life. \item The relative taste for some goods is affected by previous consumption of them, e.g., food, culture goods and drugs. Another case, perhaps conceptually different but analytically similar is when relative consumption matters for utility. "Catching up with the Joneses or "Poverty is more easily accepted if it is shared by everyone" Ernst Wigfors, Social Democratic Finance Minister 1932-49. \end{enumerate} In the literature, the first case and the case of "Catching up with the Joneses" have been used to try to explain asset market puzzles, i.e., why standard models have great problems explaining the co-movements of prices and consumption. The second case is used to explain, for example, cultural diversity. To simplify, in particular in order to be able to specify recursive preferences, utility is assumed to be \begin{equation*} U_{t}=\sum_{j=0}^{\infty }\beta ^{j}u\left( c_{t+j},\nu _{t+j}\right) \end{equation*}% where \begin{equation*} \nu _{t}=v\left( \tilde{c}_{t-1},\tilde{c}_{t-2},...\tilde{c}_{t-n}\right) , \end{equation*}% is denoted the habit. A couple of things to note, \begin{enumerate} \item We maintain time additivity here, although this should be straightforward to generalize to the constant elasticity aggregator. \item $\tilde{c}_{t-s}$ can denote the own previous consumption of the household, the consumption of some reference group or some combination. If $% \tilde{c}_{t-s}=c_{t-s}$ (own consumption), we have what is called "internal habits" while if it is the consumption of some reference group, it is called "external" habit. Abel uses a geometric average \begin{equation*} \nu _{t}\equiv \left( c_{t-1}^{D}C_{t-1}^{1-D}\right) ^{\gamma }, \end{equation*}% where $C_{t-1}$ is aggregate consumption. Here $\gamma =0$ gives the standard model, $\gamma \neq 0,$ and $D=1,$gives the internal habit case and $D=0$ the external case. \item History matters only though the habit function, i.e., through \begin{equation*} v\left( \tilde{c}_{t-1},\tilde{c}_{t-2},...\tilde{c}_{t-n}\right) \end{equation*} \item We assume either a finite value of $n$ or at least that \begin{equation*} \lim_{n\rightarrow \infty }\frac{\partial v\left( \tilde{c}_{t-1},\tilde{c}% _{t-2},...\tilde{c}_{t-n}\right) }{\partial \tilde{c}_{t-n}}=0. \end{equation*}% so that we can hope to find stationary decision rules. \end{enumerate} \subsection{Optimal consumption under external vs. internal habits.} Suppose the representative agent solves \begin{eqnarray*} &&\max \sum_{t=0}^{\infty }\beta ^{t}u\left( c_{t},\nu _{t}\right) \\ \text{s.t. }a_{t+1} &=&\left( a_{t}-c_{t}\right) r \\ v_{t} &=&\left( c_{t-1}^{D}C_{t-1}^{1-D}\right) ^{\gamma } \end{eqnarray*}% and a no-Ponzi condition. Let's look at the Bellman equation. Note that now, $v_{t}$ is a state variable. Furthermore, aggregate assets $A_{t}$ is a state variable, since aggregate assets tell us what aggregate consumption will be in the future. Therefore% \begin{eqnarray*} V\left( a_{t},A_{t},v_{t}\right) &=&\max_{c_{t}}u\left( c_{t},\nu _{t}\right) +\beta V\left( a_{t+1},A_{t+1},v_{t+1}\right) \\ \text{s.t. }A_{t+1} &=&\left( A_{t}-C_{t}\right) r, \\ v_{t} &=&\left( c_{t-1}^{D}C_{t-1}^{1-D}\right) ^{\gamma }, \\ a_{t+1} &=&\left( a_{t}-C_{t}\right) r \end{eqnarray*} The first order condition is% \begin{eqnarray*} u_{1}\left( c_{t},v_{t}\right) &=&\beta V_{1}\left( a_{t+1},A_{t+1},v_{t+1}\right) r-\beta V_{3}\left( a_{t+1},A_{t+1},v_{t+1}\right) \frac{\partial v_{t+1}}{\partial c_{t}} \\ &=&\beta V_{1}\left( a_{t+1},A_{t+1},v_{t+1}\right) r-\beta \gamma D\frac{% v_{t+1}}{c_{t}}V_{3}\left( a_{t+1},A_{t+1},v_{t+1}\right) \end{eqnarray*} Clearly, the first order condition is not affected if $\gamma D=0.$ Suppose instead $\gamma D>0.$ Then an increase in today's consumption increases the habit. Suppose this reduces utility, then this implies that there is a negative dynamic effect of consumption which will show up in a negative $% V_{3}.$ Therefore, the term $-\beta \gamma D\frac{v_{t+1}}{c_{t}}V_{3}\left( a_{t+1},A_{t+1},v_{t+1}\right)$ is positive and marginal utility of consumption should be set higher in period $t.$ Consider the case of external habits, $D=0$ . We have seen that in this case, the FOC is the same as under no habits. Suppose first for simplicity that \begin{equation*} u\left( c_{t},\nu _{t}\right) =\ln c_{t}-\ln v_{t} \end{equation*} Recall the solution strategy in the case when we expect that there can be an analytical solution. \begin{enumerate} \item Write Bellman equation. \item Guess a functional form of the value function with unknown parameters. From the Bellman equation we see that it has to be of the same functional form as the per-period utility function. \item Solve for the choice variable that maximizes the RHS\ of the Bellman equation given our guess on the value function. \item Substitute your optimal choice variable into the RHS of the Bellman equation to express the maximized RHS. \item Verify that the Bellman equation is satisfied for \emph{all} values of the state variables by finding the unknown parameters. \end{enumerate} If step 5 fails you have made an incorrect guess and must start with another. However, most problems do not admit closed form solutions for the value function in which case this approach is useless. Now, we guess that the value function is \begin{equation*} V\left( a_{t},A_{t},v_{t}\right) =B_{1}\ln a_{t}+B_{2}\ln A_{t}+B_{3}\ln v_{t}+B_{4} \end{equation*}% for the unknown coefficients $B_{1},B_{2},$ $B_{3}$ and $B_{4}.$ The FOC is% \begin{eqnarray*} \frac{1}{c_{t}} &=&\frac{\beta B_{1}r}{a_{t+1}} \\ \frac{1}{c_{t}} &=&\frac{\beta B_{1}}{a_{t}-c_{t}} \\ &\Rightarrow &c_{t}=\frac{1}{1+\beta B_{1}}a_{t}, \\ a_{t+1} &=&\left( a_{t}-c_{t}\right) r \\ &=&\frac{\beta B_{1}}{1+\beta B_{1}}ra_{t} \end{eqnarray*} Which we recognize well. Substituting this into our guess gives \begin{gather*} B_{1}\ln a_{t}+B_{2}\ln A_{t}+B_{3}\ln v_{t}+B_{4}=\ln \frac{a_{t}}{1+\beta B_{1}}-\ln v_{t} \\ +\beta \left( B_{1}\ln a_{t+1}+B_{2}\ln A_{t+1}+B_{3}\ln v_{t+1}+B_{4}\right) \\ =\ln \frac{a_{t}}{1+\beta B_{1}}-\ln v_{t}+\beta \left( B_{1}\ln \frac{\beta B_{1}}{1+\beta B_{1}}ra_{t}+B_{2}\ln A_{t+1}+B_{3}\ln v_{t+1}+B_{4}\right) \end{gather*} This cannot work unless we get rid of $v_{t+1}$ and $A_{t+1}$ in the RHS. To do this we note that \begin{equation*} v_{t+1}=C_{t}^{\gamma } \end{equation*}% and since everybody does the same thing, $C_{t}=c_{t}$ and therefore, \begin{eqnarray*} C_{t} &=&\frac{1}{1+\beta B_{1}}A_{t} \\ v_{t+1} &=&\left( \frac{1}{1+\beta B_{1}}A_{t}\right) ^{\gamma }. \end{eqnarray*} Furthermore, for the same reason $a_{t+1}=\frac{\beta B_{1}}{1+\beta B_{1}}% ra_{t}$ implies $A_{t+1}=\frac{\beta B_{1}}{1+\beta B_{1}}rA_{t}$ Therefore, \begin{eqnarray*} &&B_{1}\ln a_{t}+B_{2}\ln A_{t}+B_{3}\ln v_{t}+B_{4} \\ &=&\ln \frac{a_{t}}{1+\beta B_{1}}-\ln v_{t}+\beta \left( B_{1}\ln \frac{% \beta B_{1}}{1+\beta B_{1}}ra_{t}+B_{2}\ln \frac{\beta B_{1}}{1+\beta B_{1}}% rA_{t}+B_{3}\gamma \ln \frac{1}{1+\beta B_{1}}A_{t}+B_{4}\right) \end{eqnarray*} We solve this by equalizing the coefficients on the different terms% \begin{eqnarray*} B_{1} &=&\left( 1+\beta B_{1}\right) \\ B_{2} &=&\beta B_{2}+\beta \gamma B_{3} \\ B_{3} &=&-1 \\ B_{3} &=&\beta B_{1}\ln \beta B_{1}r-\left( 1+\beta B_{1}+\beta B_{2}+\beta \gamma B_{3}\right) \ln \left( 1+\beta B_{1}\right) +\beta B_{4} \end{eqnarray*} Giving% \begin{eqnarray*} B_{1} &=&\frac{1}{1-\beta } \\ B_{2} &=&-\frac{\beta \gamma }{1-\beta } \\ B_{3} &=&-1 \end{eqnarray*} Again using the FOC, we get \begin{eqnarray*} c_{t} &=&\frac{1}{1+\beta \frac{1}{1-\beta }}A_{t} \\ &=&\left( 1-\beta \right) A_{t} \end{eqnarray*} As we see, consumption is independent of $\gamma .$ This is due to the fact marginal utility is not affected by the habit. This is due to the log utility. Marginal utility of consumption is $\frac{1}{c_{t}}$regardless of $% v_{t}$ since $u$ is separable in $c$ and $v.$ Let us therefore consider a generalization. Instead of solving the full problem, we can at least characterize consumption dynamics. Suppose \begin{eqnarray*} u\left( c_{t},v_{t}\right) &=&\frac{1}{1-\alpha }\left( \frac{c_{t}}{v_{t}}% \right) ^{1-\alpha } \\ u_{1}\left( c_{t},v_{t}\right) &=&\frac{1}{c_{t}}\left( \frac{c_{t}}{v_{t}}% \right) ^{1-\alpha } \end{eqnarray*} The Euler condition under purely external habits is the usual \begin{equation*} 1=\beta \frac{u_{1}\left( c_{t+1},v_{t+1}\right) }{u_{1}\left( c_{t},v_{t}\right) }r \end{equation*} In general equilibrium $v_{t}=c_{t-1}^{\gamma }$, giving \begin{equation*} 1=\beta \frac{\frac{1}{c_{t+1}}\left( \frac{c_{t+1}}{v_{t+1}}\right) ^{1-\alpha }}{\frac{1}{c_{t}}\left( \frac{c_{t}}{v_{t}}\right) ^{1-\alpha }}% r=\beta \frac{\frac{1}{c_{t+1}}\left( \frac{c_{t+1}}{c_{t}^{\gamma }}\right) ^{1-\alpha }}{\frac{1}{c_{t}}\left( \frac{c_{t}}{c_{t-1}^{\gamma }}\right) ^{1-\alpha }}r=\beta c_{t+1}^{-\alpha }c_{t}^{\gamma \left( \alpha -1\right) +\alpha }c_{t-1}^{\gamma \left( 1-\alpha \right) }r \end{equation*} Taking logs% \begin{eqnarray*} 0 &=&\ln \beta r-\alpha \ln c_{t+1}+\left( \alpha -\gamma \left( 1-\alpha \right) \right) \ln c_{t}+\gamma \left( 1-\alpha \right) \ln c_{t-1} \\ \alpha \left( \ln c_{t+1}-\ln c_{t}\right) &=&\ln \beta r-\gamma \left( 1-\alpha \right) \left( \ln c_{t}-\ln c_{t-1}\right) \end{eqnarray*} In the case $\gamma =0,$the standard case, the growth rate of consumption is constant at% \begin{equation*} \ln c_{t+1}-\ln c_{t}=\frac{\ln \beta r}{\alpha }, \end{equation*}% as we should now from standard models. We have also seen that with $\alpha =1,$% \begin{equation*} \ln c_{t+1}-\ln c_{t}=\ln \beta r \end{equation*} Dynamics becomes interesting now under $\alpha \neq 1$ and $\gamma >0.$ Define \begin{equation*} \ln c_{t+1}-\ln c_{t}\equiv g_{t+1} \end{equation*}% Then, \begin{equation} g_{t+1}=\frac{\ln \beta r}{\alpha }-\gamma \frac{\left( 1-\alpha \right) }{% \alpha }g_{t} \label{eq_HabitEuler} \end{equation} When riskaversion is low (IES high), that is $a>0,$ we get oscillations! A low growth rate is followed by a high and vice versa. The oscillations may even be unstable if $\gamma \frac{\left( 1-\alpha \right) }{\alpha }>1.$ If instead $\alpha <0,$we get a monotone path, that is stable if $-\gamma \frac{% \left( 1-\alpha \right) }{\alpha }<1.$ Note that we have assumed a constant interest rate $r,$ this is quite easy to relax. With a varying interest rate, for example if we include capital accumulation, we still have \begin{equation*} g_{t+1}=\frac{\ln \beta r_{t+1}}{\alpha }-\gamma \frac{\left( 1-\alpha \right) }{\alpha }g_{t}. \end{equation*} \subsection{Adding stochastics} With stochastics, we have \begin{eqnarray*} \beta E_{t}\frac{1}{c_{t+1}}\left( \frac{c_{t+1}}{c_{t}^{\gamma }}\right) ^{1-\alpha }r_{t+1} &=&\frac{1}{c_{t}}\left( \frac{c_{t}}{c_{t-1}^{\gamma }}% \right) ^{1-\alpha } \\ \beta E_{t}c_{t+1}^{-\alpha }c_{t}^{\gamma \left( \alpha -1\right) }r_{t+1} &=&c_{t}^{-\alpha }c_{t-1}^{\gamma \left( \alpha -1\right) } \\ E_{t}\beta \left( \frac{c_{t+1}}{c_{t}}\right) ^{-\alpha }r_{t+1} &=&\left( \frac{c_{t}}{c_{t-1}}\right) ^{\gamma \left( 1-\alpha \right) } \\ E_{t}\beta \left( e^{g_{t+1}}\right) ^{-\alpha }r_{t+1} &=&\left( e^{g_{t}}\right) ^{\gamma \left( 1-\alpha \right) }. \end{eqnarray*} : : : This can be analyzed by linearization. Suppose \begin{equation*} r_{t+1}=e^{z_{t+1}}r \end{equation*}% where $\beta r=1,$ giving a steady state of the economy. Approximating around $g=0,z=0$, \begin{eqnarray*} \beta \left( e^{g_{t+1}}\right) ^{-\alpha }e^{z_{t+1}}r &\approx &\beta r-\alpha g_{t+1}\beta r+\beta rz_{t+1} \\ \left( e^{g_{t}}\right) ^{\gamma \left( 1-\alpha \right) } &\approx &1+\gamma \left( 1-\alpha \right) g_{t} \end{eqnarray*}% Giving% \begin{eqnarray*} E_{t}\left( \beta r-\alpha g_{t+1}\beta r+\beta rz_{t+1}\right) &=&1+\gamma \left( 1-\alpha \right) g_{t} \\ E_{t}g_{t+1} &=&-\frac{\gamma \left( 1-\alpha \right) }{\alpha }g_{t}+\frac{% E_{t}z_{t+1}}{\alpha }, \end{eqnarray*}% which can provide interesting dynamics. To understand the reason for the oscillatory behavior, it may be of help to note that individuals with habits might prefer variations over time. Clearly, when $\gamma =0,$ individuals are risk averse for $\alpha >0$ and also averse to variations over time. Consider instead the case when $\gamma =1.$ Assume that (individual and aggregate) consumption is \begin{equation*} c_{t}=\left\{ \begin{array}{c} c\left( 1+\varepsilon \right) \text{ if }t\text{ odd} \\ c\left( 1-\varepsilon \right) \text{ else.}% \end{array}% \right. \end{equation*} In this case, utility in even periods (multiplied by $\left( 1-\alpha \right)$ for convenience) is% \begin{equation*} u_{t}=\left( \frac{1+\varepsilon }{1-\varepsilon }\right) ^{1-\alpha } \end{equation*}% and in odd% \begin{equation*} u_{t+1}=\left( \frac{1-\varepsilon }{1+\varepsilon }\right) ^{1-\alpha } \end{equation*}% we have \begin{eqnarray*} \frac{u_{t}+u_{t+1}}{2} &=&\frac{\left( \frac{1-\varepsilon }{1+\varepsilon }% \right) ^{1-\alpha }+\left( \frac{1+\varepsilon }{1-\varepsilon }\right) ^{1-\alpha }}{2} \\ &=&\frac{\left( \frac{1-\varepsilon }{1+\varepsilon }\right) ^{1-\alpha }+\left( \frac{1+\varepsilon }{1-\varepsilon }\right) ^{1-\alpha }}{2} \\ &\approx &1+\varepsilon \left[ \frac{d\left( \frac{\left( \frac{% 1-\varepsilon }{1+\varepsilon }\right) ^{1-\alpha }+\left( \frac{% 1+\varepsilon }{1-\varepsilon }\right) ^{1-\alpha }}{2}\right) }{% d\varepsilon }\right] _{\varepsilon =0}+\frac{\varepsilon ^{2}}{2}\left[ \frac{d^{2}\left( \frac{\left( \frac{1-\varepsilon }{1+\varepsilon }\right) ^{1-\alpha }+\left( \frac{1+\varepsilon }{1-\varepsilon }\right) ^{1-\alpha }% }{2}\right) }{d\varepsilon ^{2}}\right] _{\varepsilon =0} \\ &=&1+2\left( 1-\alpha \right) ^{2}\varepsilon ^{2} \end{eqnarray*} which is increasing in $\varepsilon ^{2}.$ \subsection{Asset market implications.} Abel has shown that habits also may have an ability to explain asset market puzzles. Let us finally go over this. Using \begin{eqnarray*} u\left( c_{t},v_{t}\right) &=&\frac{1}{1-\alpha }\left( \frac{c_{t}}{v_{t}}% \right) ^{1-\alpha } \\ u_{1}\left( c_{t},v_{t}\right) &=&\frac{1}{c_{t}}\left( \frac{c_{t}}{v_{t}}% \right) ^{1-\alpha } \end{eqnarray*}% utility in period $t$ can be written \begin{eqnarray*} U_{t} &=&\frac{1}{1-\alpha }\frac{c_{t}}{v_{t}}^{1-\alpha }+\beta \left( \frac{1}{1-\alpha }\frac{c_{t+1}}{v_{t+1}}^{1-\alpha }+\beta V\left( A_{t+2},v_{t+2}\right) \right) \\ \text{s.t. }A_{t+1} &=&\left( A_{t}-c_{t}\right) r_{t+1},v_{t}=\left( c_{t-1}^{D}C_{t-1}^{1-D}\right) ^{\gamma } \\ A_{t+2} &=&\left( A_{t+1}-c_{t+1}\right) r_{t+2} \end{eqnarray*}% Using the definition of habits, we get \begin{eqnarray*} U_{t} &=&\frac{1}{1-\alpha }\left( \frac{c_{t}}{v_{t}}\right) ^{1-\alpha }+\beta \left( \frac{1}{1-\alpha }\left( \frac{c_{t+1}}{\left( c_{t-1}^{D}C_{t-1}^{1-D}\right) ^{\gamma }}\right) ^{1-\alpha }+\beta V\left( A_{t+2},v_{t+2}\right) \right) \\ && \end{eqnarray*}% Therefore \begin{eqnarray*} \frac{\partial U_{t}}{\partial c_{t}} &=&\frac{\partial \left( \frac{1}{% 1-\alpha }\left( \frac{c_{t}}{v_{t}}\right) ^{1-\alpha }+\beta \left( \frac{1% }{1-\alpha }\left( \frac{c_{t+1}}{\left( c_{t}^{D}C_{t}^{1-D}\right) ^{\gamma }}\right) ^{1-\alpha }\right) \right) }{\partial c_{t}} \\ &=&\frac{1}{c_{t}}\left( \left( \frac{c_{t}}{v_{t}}\right) ^{1-\alpha }-\beta \gamma D\left( \frac{c_{t+1}}{\left( c_{t}^{D}C_{t}^{\left( 1-D\right) }\right) ^{\gamma }}\right) ^{1-\alpha }\allowbreak \right) \\ &=&\frac{1}{c_{t}}\left( \left( \frac{c_{t}}{v_{t}}\right) ^{1-\alpha }-\beta \gamma D\left( \frac{c_{t+1}}{\nu _{t+1}}\right) ^{1-\alpha }\right) \\ &=&\frac{1}{c_{t}}\left( \left( \frac{c_{t}}{\nu _{t}}\right) ^{1-\alpha }-\beta \gamma D\left( \frac{c_{t+1}}{\nu _{t+1}}\right) ^{1-\alpha }\right) \\ &=&\frac{1}{c_{t}}\left( \left( \frac{c_{t}}{\nu _{t}}\right) ^{1-\alpha }-\beta \gamma D\left( \frac{c_{t+1}\nu _{t}}{c_{t}\nu _{t+1}}\right) ^{1-\alpha }\left( \frac{c_{t}}{\nu _{t}}\right) ^{1-\alpha }\right) \\ &=&\frac{1}{c_{t}}\left( \frac{c_{t}}{\nu _{t}}\right) ^{1-\alpha }\left( 1-\beta \gamma D\left( \frac{c_{t+1}}{c_{t}}\right) ^{1-\alpha }\left( \frac{% \nu _{t}}{\nu _{t+1}}\right) ^{1-\alpha }\right) \end{eqnarray*} Now, define gross output growth% \begin{equation*} x_{t+1}\equiv \frac{y_{t+1}}{y_{t}} \end{equation*} and since the economy is closed% \begin{equation*} x_{t+1}=\frac{c_{t+1}}{c_{t}}=\frac{C_{t+1}}{C_{t}} \end{equation*} and thus% \begin{equation*} \frac{\nu _{t+1}}{v_{t}}=\frac{\left( c_{t}^{D}C_{t}^{1-D}\right) ^{\gamma }% }{\left( c_{t-1}^{D}C_{t-1}^{1-D}\right) ^{\gamma }}=x_{t}^{\gamma }, \end{equation*} implying \begin{eqnarray} \frac{\partial U_{t}}{\partial c_{t}} &=&\frac{1}{c_{t}}\left( \frac{c_{t}}{% \nu _{t}}\right) ^{1-\alpha }\left( 1-\beta \gamma D\left( \frac{c_{t+1}}{% c_{t}}\right) ^{1-\alpha }\left( \frac{\nu _{t}}{\nu _{t+1}}\right) ^{1-\alpha }\right) \label{eq_MU1} \\ &=&c_{t}^{-\alpha }\nu _{t}^{\alpha -1}\left( 1-\beta \gamma D\left( \frac{% x_{t+1}}{x_{t}^{\gamma }}\right) ^{1-\alpha }\right) \notag \\ &=&c_{t}^{-\alpha }\nu _{t}^{\alpha -1}H_{t+1} \notag \end{eqnarray} where \begin{equation*} H_{t+1}\equiv 1-\beta \gamma D\left( \frac{x_{t+1}}{x_{t}^{\gamma }}\right) ^{1-\alpha } \end{equation*} Clearly the marginal utility of consumption in $t$ is increasing in $% H_{t+1}.$ Furthermore, $\frac{\partial H_{t+1}}{\partial x_{t+1}}$ is positive if $\alpha >1$, and negative otherwise. So if growth is expected to be high between $t$ and $t+1,$ and IES is small ($\alpha >1$), this boosts the marginal utility of consumption. In other words, an expectation of high growth has a \emph{negative }effect on savings. Note that this strngthens the standard smoothing results that if you expect high income in the future, the savings motive falls. Furthermore, in asset market equilibrium, this effect tends to \emph{reduce the }price of assets, i.e., increasing the expected return. The opposite is true if $\alpha <1,$ i.e., the intertemporal elasticity is high. Let us now first consider the case when $D=0.$ We will see that we can get some interesting results for bond and asset returns. When $D=0,$ $H_{t}=1\forall t$ and $\frac{\partial U_{t}}{\partial c_{t}}% =c_{t}^{-\alpha }\nu _{t}^{\alpha -1}.$The Euler equation implies as usual \begin{eqnarray*} \frac{\partial U_{t}}{\partial c_{t}} &=&E_{t}\beta r_{t+1}\frac{\partial U_{t+1}}{\partial c_{t+1}} \\ 1 &=&E_{t}\beta r_{t+1}\frac{c_{t+1}^{-\alpha }\nu _{t+1}^{\alpha -1}}{% c_{t}^{-\alpha }\nu _{t}^{\alpha -1}} \end{eqnarray*} Now consider the endowment economy, where $c_{t}=y_{t}$ and we define \begin{equation*} x_{t+1}\equiv \frac{y_{t+1}}{y_{t}}. \end{equation*}% Then, we have \begin{equation*} 1=E_{t}\beta r_{t+1}\frac{c_{t+1}^{-\alpha }\nu _{t+1}^{\alpha -1}}{% c_{t}^{-\alpha }\nu _{t}^{\alpha -1}}=E_{t}\beta r_{t+1}x_{t+1}^{-\alpha }x_{t}^{\gamma \left( \alpha -1\right) } \end{equation*} Now, consider a risky share that pays $y_{t}$ as dividend (the apple trees) with price $p_{r,t}.$ The price of this asset must satisfy% \begin{equation*} r_{r,t+1}=\frac{p_{r,t+1}+y_{t+1}}{p_{r,t}}. \end{equation*}% Denoting the price-dividend ratio \begin{equation*} w_{t}\equiv \frac{p_{r,t}}{y_{t}} \end{equation*}% we can write% \begin{eqnarray*} r_{r,t+1} &=&\frac{w_{t+1}y_{t+1}+y_{t+1}}{w_{t}y_{t}}, \\ &=&\frac{1+w_{t+1}}{w_{t}}x_{t+1}. \end{eqnarray*} Now, using this last expression for the return on stocks into the Euler equation yields% \begin{eqnarray*} 1 &=&E_{t}\beta r_{t+1}x_{t+1}^{-\alpha }x_{t}^{\gamma \left( \alpha -1\right) } \\ &=&E_{t}\beta \frac{1+w_{t+1}}{w_{t}}x_{t+1}x_{t+1}^{-\alpha }x_{t}^{\gamma \left( \alpha -1\right) } \\ w_{t} &=&x_{t}^{\gamma \left( \alpha -1\right) }E_{t}\beta \left( 1+w_{t+1}\right) x_{t+1}^{1-\alpha }, \end{eqnarray*}% since $w_{t}$ is known in $t.$ When growth rates are i.i.d., this can be calculated quite easily. In particular, we will show that the expression \begin{equation*} E_{t}\left( \beta \left( 1+w_{t+1}\right) x_{t+1}^{1-\alpha }\right) \end{equation*}% is constant at some value $A$, so that we can write% \begin{equation*} w_{t}=Ax_{t}^{\gamma \left( \alpha -1\right) } \end{equation*}% for some $A$ and verify that this satisfies the pricing equation (\ref% {eq_pricing}). Using our "guess", we have% \begin{eqnarray*} Ax_{t}^{\gamma \left( \alpha -1\right) } &=&x_{t}^{\gamma \left( \alpha -1\right) }E_{t}\beta \left( 1+Ax_{t+1}^{\gamma \left( \alpha -1\right) }\right) x_{t+1}^{1-\alpha }, \\ A &=&E_{t}\beta \left( 1+Ax_{t+1}^{\gamma \left( \alpha -1\right) }\right) x_{t+1}^{1-\alpha } \\ &=&\beta \left( E_{t}x_{t+1}^{1-\alpha }+E_{t}Ax_{t+1}^{\gamma \left( \alpha -1\right) +\left( 1-\alpha \right) }\right) \\ &=&\beta E_{t}x_{t+1}^{1-\alpha }+\beta E_{t}Ax_{t+1}^{\left( 1-\gamma \right) \left( 1-\alpha \right) } \end{eqnarray*} Clearly, the RHS\ is a constant if growth rates are i.i.d. and we have established that \begin{equation*} w_{t}=Ax_{t}^{\gamma \left( \alpha -1\right) } \end{equation*}% where \begin{eqnarray*} A &=&\beta Ex^{1-\alpha }+\beta EAx^{\left( 1-\gamma \right) \left( 1-\alpha \right) } \\ A &=&\frac{\beta Ex^{1-\alpha }}{1-\beta Ex^{\left( 1-\gamma \right) \left( 1-\alpha \right) }} \end{eqnarray*} To calculate the expected stock market return, we use% \begin{eqnarray*} r_{r,t+1} &=&\frac{1+w_{t+1}}{w_{t}}x_{t+1} \\ &=&\frac{1+Ax_{t+1}^{\gamma \left( \alpha -1\right) }}{Ax_{t}^{\gamma \left( \alpha -1\right) }}x_{t+1} \end{eqnarray*} and \begin{eqnarray*} E_{t}r_{r,t+1} &=&\frac{1+AEx^{\gamma \left( \alpha -1\right) }}{% Ax_{t}^{\gamma \left( \alpha -1\right) }}Ex \\ &=&\frac{1+AEx^{\gamma \left( \alpha -1\right) }}{Ax_{t}^{\gamma \left( \alpha -1\right) }}Ex \end{eqnarray*} As we see, the expected return is time dependent, despite the i.i.d. assumption, provided $\gamma \neq 0.$ Why? Furthermore, if $\gamma >0,$ and $\alpha <1,$ the denominator decreases in $% x_{t}$ and the expected return is thus higher when $x_{t}\equiv \frac{y_{t}}{% y_{t-1}}$ is high. If $\alpha >1,$ the opposite is true. We can easily calculate the unconditional return, i.e., the average return over time% \begin{equation*} Er_{r}=E\left[ \frac{1+AEx^{\gamma \left( \alpha -1\right) }}{Ax^{\gamma \left( \alpha -1\right) }}Ex\right] . \end{equation*} In a similar fashion, using the Euler equation \begin{equation*} 1=\beta r_{b}\frac{E_{t}\frac{\partial U_{t+1}}{\partial c_{t+1}}}{\frac{% \partial U_{t}}{\partial c_{t}}}=\beta r_{b}x_{t}^{\gamma \left( \alpha -1\right) }E_{t}x_{t+1}^{-\alpha } \end{equation*}% the unconditional return on bonds is% \begin{equation*} Er_{b}=\frac{Ex^{-\gamma \left( \alpha -1\right) }}{\beta \left( Ex^{-\alpha }\right) } \end{equation*} Consider the special case when output growth $x$ is lognormal, with mean $g$ and standard deviation $\sigma .$ Recall that then \begin{equation*} Ex=e^{g+\frac{1}{2}\sigma ^{2}}. \end{equation*} Furthermore, if $\ln x$ is normal, $\ln x^{\xi }=\xi \ln x$ is normal with mean $\xi g$ and standard deviation $\xi \sigma .$Thus, \begin{equation*} Ex^{\xi }=e^{\xi g+\frac{\xi ^{2}\sigma ^{2}}{2}}. \end{equation*} Now, let $\ln x$ be normally distributed with mean $g$ and standard deviation $\sigma$. Then,% \begin{eqnarray*} &&\beta Ex^{1-\alpha }+\beta AEx^{\left( 1-\gamma \right) \left( 1-\alpha \right) } \\ A &=&\frac{\beta Ex^{1-\alpha }}{1-\beta Ex^{\left( 1-\gamma \right) \left( 1-\alpha \right) }}=\frac{\beta e^{\left( 1-\alpha \right) g+\frac{\left( 1-\alpha \right) ^{2}\sigma ^{2}}{2}}}{1-\beta e^{\left( 1-\gamma \right) \left( 1-\alpha \right) g+\frac{\left( \left( 1-\gamma \right) \left( 1-\alpha \right) \right) ^{2}\sigma ^{2}}{2}}} \\ \ln Er_{r} &=&\ln \left( Ex^{-\gamma \left( \alpha -1\right) }\frac{% 1+AEx^{\gamma \left( \alpha -1\right) }}{AEx^{\gamma \left( \alpha -1\right) }}Ex\right) \\ &=&\ln \left( e^{-\gamma \left( \alpha -1\right) g+\left( \gamma \left( \alpha -1\right) \right) ^{2}\frac{\sigma ^{2}}{2}}\frac{e^{g+\frac{\sigma ^{2}}{2}}+Ae^{\left( 1+\gamma \left( \alpha -1\right) \right) g+\left( 1+\gamma \left( \alpha -1\right) \right) ^{2}\frac{\sigma ^{2}}{2}}}{A}% \right) \end{eqnarray*} and \begin{eqnarray*} \ln ER^{B} &\equiv &r_{B}=\ln \frac{Ex^{-\gamma \left( \alpha -1\right) }}{% \beta \left( Ex^{-\alpha }\right) } \\ &=&\ln \left( \frac{e^{-\gamma \left( \alpha -1\right) g+\frac{\left( \gamma \left( \alpha -1\right) \right) ^{2}\sigma ^{2}}{2}}}{\beta e^{-\alpha g+% \frac{\alpha ^{2}\sigma ^{2}}{2}}}\right) \\ &=&\left( \alpha -\gamma \left( \alpha -1\right) \right) g+\frac{\left( \left( \gamma \left( \alpha -1\right) \right) ^{2}-\alpha ^{2}\right) \sigma ^{2}}{2}-\ln \beta \end{eqnarray*} Setting, $\gamma =0,\sigma =0.036,g=0.018,\beta =.99$ and plotting against $% r_{s}$ and $r_{B}$ against $1-\alpha ,$ we have the no habit case.\FRAME{% dtbpFUX}{6.1585in}{4.1054in}{0pt}{\Qcb{Average stockmarket return $r_{S}$ (solid line) and safe return $r_{B}$ against risk aversion $(\protect\alpha % ).$ Standard utility ($\protect\gamma =0).$}}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 6.1585in;height 4.1054in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "12";xviewmin "-0.24";xviewmax "12.2448";yviewmin "0.007285531984002";yviewmax "0.151110428116986";plottype 4;labeloverrides 1;x-label "Relative Riskaversion";numpoints 49;plotstyle "patch";axesstyle "normal";xis \TEXUX{v945};var1name \TEXUX{$\alpha$};function \TEXUX{$-20.741\,913\,84-0.000\,648\,\alpha ^{2}+\allowbreak 0.019\,296\,\alpha +\ln \left( 1.\,\allowbreak 029\,114\,101\times 10^{9}-1.0\exp \left( 0.000\,072\,\left( \alpha -1.0\right) \left( 9.0\alpha -259.0\right) \right) \right)$};linecolor "black";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "0,12";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left[ \left( \alpha -\gamma \left( \alpha -1\right) \right) \mu +\frac{\left( \left( \gamma \left( \alpha -1\right) \right) ^{2}-\alpha ^{2}\right) \sigma ^{2}}{2}-\ln \beta \right] _{\gamma =0,\sigma =0.036,\mu =0.018,\beta =.99}$};linecolor "black";linestyle 3;pointstyle "point";linethickness 2;lineAttributes "Dots";var1range "0,12";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY0H.wmf';tempfile-properties "PR";}} Keeping the other parameters, but now introducing external habits by setting $\gamma =1,$ the returns are given in the second figure. \FRAME{dtbpFUX}{5.9508in}{3.9669in}{0pt}{\Qcb{Average stockmarket return $% r_{S}$ (solid line) and safe return $r_{B}$ against risk aversion $(\protect% \alpha ).$ Extrenal Habit ($\protect\gamma =1).$}}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 5.9508in;height 3.9669in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "12";xviewmin "-0.24";xviewmax "12.2448";yviewmin "0.009445328427039E0";yviewmax "0.201971734751614";plottype 4;labeloverrides 1;x-label "Relative riskaversion";numpoints 49;plotstyle "patch";axesstyle "normal";xis \TEXUX{v58123};var1name \TEXUX{$\alpha$};function \TEXUX{$\ln \left( 1.\,\allowbreak 029\,114\,101\times 10^{-2}+0.999\,999\,999\,\allowbreak 9\exp \left( -0.001\,296\,\alpha +0.018\,648\,+0.001\,296\,\alpha ^{2}\right) \right)$};linecolor "black";linestyle 1;pointstyle "point";linethickness 2;lineAttributes "Solid";var1range "0,12";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left[ \left( \alpha -\gamma \left( \alpha -1\right) \right) \mu +\frac{\left( \left( \gamma \left( \alpha -1\right) \right) ^{2}-\alpha ^{2}\right) \sigma ^{2}}{2}-\ln \beta \right] _{\gamma =1,\sigma =0.036,\mu =0.018,\beta =.99}$};linecolor "black";linestyle 3;pointstyle "point";linethickness 2;lineAttributes "Dots";var1range "0,12";num-x-gridlines 49;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY0I.wmf';tempfile-properties "PR";}} As we see, the stock market return quickly becomes very high as we reduce $% \alpha .$ \subsection{Appendix: The case when $D>0.$} The stochastic Euler equation \begin{equation*} E_{t}\frac{\partial U_{t}}{\partial c_{t}}=E_{t+1}\beta r_{t+1}\frac{% \partial U_{t+1}}{\partial c_{t+1}} \end{equation*} Note that is $\frac{\partial U_{t}}{\partial c_{t}}$ is actually not realized at $t,$ thus the expectations operator.% \begin{eqnarray} E_{t}\frac{\partial U_{t}}{\partial c_{t}} &=&\beta E_{t}r_{t+1}\frac{% \partial U_{t+1}}{\partial c_{t+1}} \notag \\ 1 &=&\frac{\beta E_{t}\left( r_{t+1}\frac{\partial U_{t+1}}{\partial c_{t+1}}% \right) }{E_{t}\frac{\partial U_{t}}{\partial c_{t}}}. \label{eq_Euler} \end{eqnarray} Now, it is convenient to find an expression for \begin{equation*} \frac{\frac{\partial U_{t+1}}{\partial c_{t+1}}}{E_{t}\left( \frac{\partial U_{t}}{\partial c_{t}}\right) } \end{equation*} Shifting (\ref{eq_MU1}) forward, yields% \begin{equation*} \frac{\partial U_{t+1}}{\partial c_{t+1}}=H_{t+2}\nu _{t+1}^{\alpha -1}c_{t+1}^{-\alpha }. \end{equation*} Using, $\frac{\nu _{t+1}}{\nu _{t}}=x_{t}^{\gamma }$and $\frac{c_{t+1}}{c_{t}% }=x_{t+1},$yields% \begin{eqnarray*} \frac{\frac{\partial U_{t+1}}{\partial c_{t+1}}}{E_{t}\left( \frac{\partial U_{t}}{\partial c_{t}}\right) } &=&\frac{H_{t+2}\nu _{t+1}^{\alpha -1}c_{t+1}^{-\alpha }}{E_{t}H_{t+1}\nu _{t}^{\alpha -1}c_{t}^{-\alpha }} \\ &=&\frac{H_{t+2}}{E_{t}H_{t+1}}\left( \frac{\nu _{t+1}}{\nu _{t}}\right) ^{\alpha -1}\left( \frac{c_{t+1}}{c_{t}}\right) ^{-\alpha } \\ &=&\frac{H_{t+2}}{E_{t}H_{t+1}}x_{t}^{\gamma \left( \alpha -1\right) }x_{t+1}^{-\alpha }. \end{eqnarray*} Now, consider a risky share that pays $y_{t}$ as dividend (the apple trees) with price $p_{r,t}.$ The price of this asset must satisfy% \begin{equation*} r_{r,t+1}=\frac{p_{r,t+1}+y_{t+1}}{p_{r,t}}. \end{equation*}% Denoting the price-dividend ratio \begin{equation*} w_{t}\equiv \frac{p_{r,t}}{y_{t}} \end{equation*}% we can write% \begin{eqnarray*} r_{r,t+1} &=&\frac{w_{t+1}y_{t+1}+y_{t+1}}{w_{t}y_{t}}, \\ &=&\frac{1+w_{t+1}}{w_{t}}x_{t+1}. \end{eqnarray*} Now, using this last expression for the return on stocks into the Euler equation yields% \begin{eqnarray} 1 &=&\frac{\beta E_{t}r_{r,t+1}\frac{\partial U_{t+1}}{\partial c_{t+1}}}{% E_{t}\frac{\partial U_{t}}{\partial c_{t}}}, \notag \\ w_{t} &=&\frac{\beta E_{t}\left( 1+w_{t+1}\right) x_{t+1}\frac{\partial U_{t+1}}{\partial c_{t+1}}}{E_{t}\frac{\partial U_{t}}{\partial c_{t}}}, \label{eq_Pdiv1} \end{eqnarray}% since $w_{t}$ is known in $t.$ From (\ref{eq_Pdiv1}), we have% \begin{eqnarray*} w_{t} &=&\frac{\beta E_{t}\left( \left( 1+w_{t+1}\right) x_{t+1}\frac{% \partial U_{t+1}}{\partial c_{t+1}}\right) }{E_{t}\left( \frac{\partial U_{t}% }{\partial c_{t}}\right) } \\ &=&\frac{\beta E_{t}\left( \left( 1+w_{t+1}\right) x_{t+1}\frac{\partial U_{t+1}}{\partial c_{t+1}}\right) }{E_{t}\left( \frac{\partial U_{t}}{% \partial c_{t}}\right) } \\ &=&\frac{\beta E_{t}\left( \left( 1+w_{t+1}\right) x_{t+1}c_{t+1}^{-\alpha }\nu _{t+1}^{\alpha -1}H_{t+2}\right) }{E_{t}\left( c_{t}^{-\alpha }\nu _{t}^{\alpha -1}H_{t+1}\right) } \\ &=&\frac{\beta E_{t}\left( \left( 1+w_{t+1}\right) x_{t+1}\left( \frac{% c_{t+1}}{c_{t}}\right) ^{-\alpha }\left( \frac{\nu _{t+1}}{\nu _{t}}\right) ^{\alpha -1}H_{t+2}\right) }{E_{t}\left( H_{t+1}\right) } \\ &=&\frac{\beta E_{t}\left( \left( 1+w_{t+1}\right) x_{t+1}x_{t+1}^{-\alpha }\left( x_{t}^{\gamma }\right) ^{\alpha -1}H_{t+2}\right) }{E_{t}\left( H_{t+1}\right) } \\ &=&\frac{\beta E_{t}\left( \left( 1+w_{t+1}\right) x_{t}^{\gamma \left( \alpha -1\right) }H_{t+2}x_{t+1}^{1-\alpha }\right) }{E_{t}H_{t+1}} \\ &=&\frac{\beta x_{t}^{\gamma \left( \alpha -1\right) }E_{t}\left( \left( 1+w_{t+1}\right) H_{t+2}x_{t+1}^{1-\alpha }\right) }{E_{t}H_{t+1}}. \end{eqnarray*} Using the law of iterated expectations% \begin{equation*} E_{t}\left( 1+w_{t+1})H_{t+2}x_{t+1}^{1-\alpha }\right) =E_{t}\left( \left( 1+w_{t+1}\right) x_{t+1}^{1-\alpha }E_{t+1}\left( H_{t+2}\right) \right) \end{equation*} Define% \begin{equation*} J_{t}\equiv E_{t}\left( H_{t+1}\right) =1-\beta \gamma Dx_{t}^{\gamma \left( \alpha -1\right) }E_{t}x_{t+1}^{1-\alpha }. \end{equation*} Then we have \begin{equation} w_{t}=\frac{\beta x_{t}^{\gamma \left( \alpha -1\right) }E_{t}\left( \left( 1+w_{t+1}\right) J_{t+1}x_{t+1}^{1-\alpha }\right) }{J_{t}}. \label{eq_pricing} \end{equation}% We now need to find $w_{t}$ as a function of state variables (which are they?) that satisfies (\ref{eq_pricing}). When growth rates are i.i.d., this can be calculated quite easily. In particular, we will show that the expression \begin{equation*} \beta E_{t}\left( \left( 1+w_{t+1}\right) J_{t+1}x_{t+1}^{1-\alpha }\right) \end{equation*}% is constant at some value $A$, so that we can write \begin{equation*} w_{t}=\frac{Ax_{t}^{\gamma \left( \alpha -1\right) }}{J_{t}} \end{equation*}% for some $A$ and verify that this satisfies the pricing equation (\ref% {eq_pricing}). Using our "guess", we have% \begin{eqnarray*} \frac{Ax_{t}^{\gamma \left( \alpha -1\right) }}{J_{t}} &=&\frac{\beta x_{t}^{\gamma \left( \alpha -1\right) }}{J_{t}}E_{t}\left( \left( 1+\frac{% Ax_{t+1}^{\gamma \left( \alpha -1\right) }}{J_{t+1}}\right) J_{t+1}x_{t+1}^{1-\alpha }\right) \\ A &=&\beta E_{t}\left( \left( 1+\frac{Ax_{t+1}^{\gamma \left( \alpha -1\right) }}{J_{t+1}}\right) J_{t+1}x_{t+1}^{1-\alpha }\right) \\ &=&\beta \left( E_{t}J_{t+1}x_{t+1}^{1-\alpha }+E_{t}Ax_{t+1}^{\gamma \left( \alpha -1\right) +\left( 1-\alpha \right) }\right) \\ &=&\beta E_{t}J_{t+1}x_{t+1}^{1-\alpha }+\beta E_{t}Ax_{t+1}^{\left( 1-\gamma \right) \left( 1-\alpha \right) } \end{eqnarray*} So% \begin{equation} A\left( 1-\beta E_{t}x_{t+1}^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) =\beta E_{t}J_{t+1}x_{t+1}^{1-\alpha } \label{eq_A7} \end{equation} Now, under the assumption of i.i.d. output (consumption) shocks, \begin{equation*} E_{t}x_{t+1}^{1-\alpha }=E_{t}x_{t+2}^{1-\alpha }=Ex^{1-\alpha } \end{equation*}% we have \begin{eqnarray*} J_{t} &=&E_{t}H_{t+1} \\ &=&1-\beta \gamma Dx_{t}^{\gamma \left( \alpha -1\right) }E_{t}x_{t+1}^{1-\alpha } \\ &=&1-\beta \gamma Dx_{t}^{\gamma \left( \alpha -1\right) }Ex^{1-\alpha } \end{eqnarray*}% so% \begin{eqnarray*} J_{t+1} &=&1-\beta \gamma Dx_{t+1}^{\gamma \left( \alpha -1\right) }Ex^{1-\alpha } \\ J_{t+1}x_{t+1}^{1-\alpha } &=&x_{t+1}^{1-\alpha }\left( 1-\beta \gamma Dx_{t+1}^{\gamma \left( \alpha -1\right) }Ex^{1-\alpha }\right) \end{eqnarray*} Take conditional expectation at $t$% \begin{eqnarray*} E_{t}\left( J_{t+1}x_{t+1}^{1-\alpha }\right) &=&E_{t}\left( x_{t+1}^{1-\alpha }\left( 1-\beta \gamma Dx_{t+1}^{\theta }Ex^{1-\alpha }\right) \right) \\ &=&E_{t}x_{t+1}^{1-\alpha }-\beta \gamma DEx^{1-\alpha }E_{t}\left( x_{t+1}^{1-\alpha }x_{t+1}^{\gamma \left( \alpha -1\right) }\right) \\ &=&Ex^{1-\alpha }-\beta \gamma DEx^{1-\alpha }E\left( x^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) \\ &=&Ex^{1-\alpha }\left( 1-\beta \gamma DE\left( x^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) \right) \end{eqnarray*} Finally, use this in (\ref{eq_A7}), and use the i.i.d. assumption to replace conditional expectations \begin{eqnarray*} A\left( 1-\beta E_{t}x_{t+1}^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) &=&\beta Ex^{1-\alpha }\left( 1-\beta \gamma DE\left( x^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) \right) \\ A &=&\frac{\beta Ex^{1-\alpha }\left( 1-\beta \gamma DE\left( x^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) \right) }{\left( 1-\beta Ex^{\left( 1-\gamma \right) \left( 1-\alpha \right) }\right) }, \end{eqnarray*}% which is clearly a constant under the i.i.d. assumption. To calculate the expected stock market return, we use \begin{eqnarray*} R_{t+1}^{S} &=&\frac{p_{t+1}^{S}+y_{t+1}}{p_{t+}^{S}} \\ &=&\frac{\left( 1+w_{t+1}\right) x_{t+1}}{w_{t}} \end{eqnarray*} and \begin{eqnarray*} E_{t}R_{t+1}^{S} &=&\frac{E_{t}\left( \left( 1+\frac{Ax_{t+1}^{\gamma \left( \alpha -1\right) }}{J_{t+1}}\right) x_{t+1}\right) }{w_{t}} \\ &=&\frac{Ex+E_{t}\frac{Ax_{t+1}^{1+\gamma \left( \alpha -1\right) }}{J_{t+1}}% }{\frac{Ax_{t}^{\gamma \left( \alpha -1\right) }}{J_{t}}} \end{eqnarray*} Which we at least can simulate.\newpage \section{Loss-Aversion} Substantial amounts of lab-evidence suggests that individuals behave like if the formed reference levels for consumption. Preferences over actual consumption then depend in a particular way of consumption relative to this reference level. Specifically, preferences are consistent with lab evidence if \begin{enumerate} \item utility is concave in consumption if consumption is above the reference level, \item utility is convex in consumption if consumption is below the reference level, and \item marginal utility is discretely larger below the reference level than what it is above. \end{enumerate} The utility function can then be depicted as follows: \FRAME{itbpF}{3.1981in}{2.4016in}{0in}{}{}{slide1.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.1981in;height 2.4016in;depth 0in;original-width 9.9998in;original-height 7.4996in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'Slide1.GIF';file-properties "XNPEU";}} Two important implications of this utility function that is supported by (at least) expermental evidence is that \begin{enumerate} \item Individuals have a strict distaste also for abitrarily small gambles (because of the discontinuity). \item Individuals are risk-lovers for losses. This means that they may prefer a 50/50 bet of loosing $x$ or nothing over a sure loss of $x/2.$ Kahneman \& Tversky proposes% \begin{equation*} u\left( c-r\right) =\left\{ \begin{array}{c} \left( c-r\right) ^{\alpha }\text{if }c\geq r \\ -\lambda \left( -\left( c-r\right) \right) ^{\beta }\text{ else}% \end{array}% \right. \end{equation*} \end{enumerate} and in lab-experiments finds that $\alpha =\beta =0.88,$ and $\lambda =2.25$% as in the following graph\footnote{% A better formulation, allowing $\alpha ,\beta <0,$ is $u\left( c-r\right) =\left\{ \begin{array}{c} \frac{\left( c-r\right) }{\alpha }^{\alpha }\text{if }c\geq r \\ -\frac{\lambda \left( -\left( c-r\right) \right) ^{\beta }}{\beta }\text{ else}% \end{array}% \right.$}\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "Maple";xmin "0";xmax "1";xviewmin "-1.04";xviewmax "1.0408";yviewmin "-2.315";yviewmax "1.0663";plottype 4;num-x-ticks 2;num-y-ticks 1;numpoints 100;plotstyle "patch";axesstyle "boxed";xis \TEXUX{x};var1name \TEXUX{$x$};function \TEXUX{$\left[ \left( x\right) ^{\alpha }\right] _{\alpha =.88}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "0,1";num-x-gridlines 129;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";function \TEXUX{$\left[ -\lambda \left( -x\right) ^{\beta }\right] _{\lambda =2.25,\beta =0.88}$};linecolor "black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range "-1,0";num-x-gridlines 156;curveColor "[flat::RGB:0000000000]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JA7XOY03.wmf';tempfile-properties "PR";}} In the paper by Bowman et al., this implication is expanded into a dynamic setting. One implication is then that consumption may respond asymmetrically to positive and negative news about future income. To get the intuition, consider a two period setting and suppose that income at the outset is expected to be $w$ in both periods. Suppose also that the reference point for consumption is $r=w.$ For simplicty suppose that the interest rate equals the subjective discount rate. Clearly, the optimal consumption is now $c=w$ in both periods. Consider now a positive but uncertain signal about period 2 income. Say that income is $w+2x$ with probability $1/2$ and $w$ with probability $1/2.$ Expected lifetime income is then $2w+x$ and unless there is some precautionary savings, consumption in period 1 will be $w+\frac{1}{2}x.$ This is the permanent income hypothesis. In any case, consumption will certainly increase when this positive signal comes. Behavior is "standard" for gains. Consider now instead a negative signal saying that second period income is $% w-2x$ with probability $\frac{1}{2}$ and $w$ otherwise. Now, it may very well pay for the household to continue to consume $w$ in period 1 and then with probability $1/2$ consume $w$ also in the final period and with probability $\frac{1}{2}$ consume $w-2x$, rather than consuming the permanent income $w-\frac{1}{2}x$ in the first period, in which case second period consumption is either $w+\frac{1}{2}x$ or $w-\frac{3}{2}x.$ Why? Let $r$ denote the reference level for consumption. Then, the expected utility of the first strategy is% \begin{eqnarray*} &&u\left( w-r\right) +\frac{1}{2}u\left( w-r\right) +\frac{1}{2}u\left( w-2x-r\right) \\ &&0+0-\frac{1}{2}\lambda \left( 2x\right) ^{\beta } \end{eqnarray*}% and for the second strategy is it \begin{eqnarray*} &&u\left( w-r-\frac{1}{2}x\right) +\frac{1}{2}u\left( w+\frac{1}{2}% x-r\right) +\frac{1}{2}u\left( w-\frac{3}{2}x-r\right) \\ &=&-\lambda \left( \frac{1}{2}x\right) ^{\beta }+\frac{1}{2}\left( \frac{1}{2% }x\right) ^{\alpha }-\lambda \left( \frac{3}{2}x\right) ^{\beta } \end{eqnarray*} The difference is \begin{eqnarray*} &&-\frac{1}{2}\lambda \left( 2x\right) ^{\beta }-\left( -\lambda \left( \frac{1}{2}x\right) ^{\beta }+\frac{1}{2}\left( \frac{1}{2}x\right) ^{\alpha }-\lambda \left( \frac{3}{2}x\right) ^{\beta }\right) \\ &=&\lambda \left( -\frac{1}{2}\left( 2\right) ^{\beta }+\left( \frac{1}{2}% \right) ^{\beta }+\left( \frac{3}{2}\right) ^{\beta }\right) x^{\beta }-\left( \frac{1}{2}\right) ^{1+\alpha }x^{\alpha } \end{eqnarray*} Under the assumption $a=\beta ,$ this is positive if \begin{eqnarray*} \lambda \left( -\frac{1}{2}\left( 2\right) ^{\beta }+\left( \frac{1}{2}% \right) ^{\beta }+\left( \frac{3}{2}\right) ^{\beta }\right) -\left( \frac{1% }{2}\right) ^{1+\beta } &>&0 \\ \lambda &>&\frac{2\left( \frac{1}{2}\right) ^{1+\beta }}{-2^{\beta }+2\left( \frac{1}{2}\right) ^{\beta }+2\left( \frac{3}{2}\right) ^{\beta }}\in \left[ \frac{1}{4},\frac{1}{2}\right] \end{eqnarray*} Due to the convexity of utility in losses, its better to take a chance that consumption might not need to be reduced below the reference point. In other words, there is a tendency that consumption does not fall "unless it is clear that it has to". Bowman et al documents such an asymmetry in U.S. consumption data. In a dynamic setting, a key issue is how reference points are formed. Unfortunately, not much empirical evidence is collected regarding this issue. Bowman et al assume $r_{1},$ the reference point for period 1, is given and that \begin{equation*} r_{2}=\left( 1-\alpha \right) r_{1}+\alpha c_{1} \end{equation*} If $\alpha =0,$ we have the case discussed above -- static reference points. For $\alpha =1,$ next periods reference points is completely determined by the previous periods consumption. Here, we could think both of the case when the agent internalize the effect her consumption has on the reference point and the case when she doesn't (due to naivite or external reference points). \subsection{Loss-aversion as commitment (Hassler\&Rodriguez Mora)} \begin{itemize} \item Two types of rational and non-altruistic individuals, (poor) \emph{% workers} and \emph{entrepreneurs}, living in a two period OLG-setup. \item The workers make no private choices, having a fixed wage normalized to zero, consuming in the second period of life, having high marginal utility since they are poor. \item Young entrepreneurs in $t$ choose investments $i_{t}$ at a utility cost $\frac{i_{t}^{2}}{2}$, returning $i_{t}\left( 1-\tau _{t+1}\right)$ in second period of life when consumption takes place and the capital fully depreciates. \item Old workers get a transfer $G_{t}$, financed by taxes on installed capital. \item Taxes, $\tau_t\in \left[0,1\right]$, are determined without commitment by probabilistic voting with equal weight on all living individuals. Alternative interpretation, a benevolent planner that cares equally of all living individuals. \item Without commitment, the only Markov equilibrium is one with 100 percent taxation since temptation to tax installed capital is too high. \item As K\"{o}szegi and Rabin, we consider the case when reference points for consumption are \emph{forward-looking}. We can refrase reference points for cosumption in terms of the corresponding tax-level, $\tau ^{r}.$ \item We require $\tau _{t+1}^{r}$ to be in the set of equilibrium tax rates for $t+1$. \item We allow politicians to affect reference points by making "promises" about the future. But remember that the promise is empty -- the politician does not remain in office nor runs again and he has no formal commitment power. \item The promise can affect the future if it is believed, in which case it becomes the the reference point. \item It is believed if it is done by the winning candidate and is in the set of equilibria for next period. If the promise is not an equilibrium, $% \tau _{t+1}^{r}$ is some element of the set of equilibrium tax rates. \item As a variation, we consider the opposite case of history dependence. Reference tax-levels are \emph{backward-looking}, $\tau _{t+1}^{r}=\tau _{t}$ \end{itemize} \subsubsection{Results} Under both backward and forward-looking reference points, there is a Markov equilibrium with limited amounts of taxation. Dynamics differ between the two cases. The level of taxation in equilibrium depends inversely on on the degree of loss-aversion. Intuition: If people have reference points, implying that they feel "entitled" to some return on their investments -- if becomes politically costly to go against this. If the entitlements are not too large, they will be satisfied in equilibrium. \end{document}