Topics in Dynamic Public Finance

John Hassler Stockholm University November 25, 2008

1 Optimal unemployment insurance (UI)

There is a large literature of optimal unemployment insurance. The basic issue is how to provide the most efficient unemployment insurance when there is a moral hazard problem. This is arising from an assumption that unemployed individuals can affect the probability they find (and accept) a job offer. However, it is costly for the worker to increase this probability, e.g., because of effort costs, reduced reservation wages or opportunity costs of time.

1.1 The semi-static approach to optimal UI

The basic idea in Baily and Chetty is to simplify the dynamic problem into a static one. This makes the model simple and tractable also when savings is allowed. An important lesson is that when savings is allowed, we can use the drop in consumption at unemployment as a measure of the welfare loss associated with unemployment. In a dynamic model, this does not work when there is no market for savings. Why? The trade-off faced by the planner is to balance the loss of welfare associated with unemployment against the negative effect on search induced by UI.

1.1.1 The simplest model following Baily

• In the first period, the individual works and chooses how much to consume of the income, normalized to unity, and how much to save.

• In the beginning of the second period, the individual becomes unemployed with probability 1 − α and otherwise keeps his job.

• During the second period, the individual can determine how long it takes to find a job by choosing the reservation wage y_{n} and costly search effort c. A share β = β (c, y_{n}) of the second period is spent working in the new job.

• While unemployed, the individual gets UI-benefits b.These are paid by taxes on workers.

• Agents have access to a market for precautionary (buffer stock) savings.

• Both the unemployment duration and the wage upon rehiring is non-stochastic.

Total disposable income in second period if laid off is therefore the non-stochastic value

(1 − β) (b − c) + βy_{n} (1 − τ) ≡ y_{l}.

In first periods, individuals decide how much to save, s. Interest rate and subjective discount rate is normalized to zero. If an individual gets laid off, he consumes his resources, i.e., his disposable income plus savings.

Welfare is

V = u (1 − τ − s) + αu (1 − τ + s) + (1 − α) (u (y_{l} + s)) .

Government budget constraint is

(1 + α + (1 − α) βy_{n}) t = (1 − α) (1 − β) b.

⟹ b = \frac{(1 + α + (1 − α) βy_{n})}{(1 − α) (1 − β)}τ ≡ μτ

Denoting the endogenous total income by Y ≡ 1 + α + (1 − α) βy_{n}, this implies

b = \frac{Y}{(1 − α) (1 − β)}τ
≡ μτ,

where we note that μ is not a constant, but depends on individual choices of y_{n} and c and thus indirectly on taxes and benefits. Given the budget constraint and individual choices, we can therefore write μ = μ (τ) (provided there is a solution, which is not necessarily true for all τ.Explain!)

Note that in first best, c should be chosen to satisfy

(y_{n} + c) β_{c} = 1 − β

since social income is

(1 − β (y_{n}, c)) (b − c) + βy_{n}

implying that the marginal gain of a marginal unit of effort is β_{c} (y_{n} + c) and the cost is 1 − β.

The individual instead gains,

y_{n} (1 − τ) + c − b

so the private value of search is lower. Similarly, an increase in y_{n} has benefits β and costs − (y_{n} + c) β_{y_{n}}. While private benefits are (1 − τ) β and private costs − (y_{n} (1 − τ) + c − b) β_{y_{n}}.

The wedges between private and social costs/benefits imply that both choices will be distorted in second best.

We can now write

V = u (1 − τ − s) + αu (1 − τ + s) + (1 − α) (u ((1 − β) (μ (τ) τ − c) + βy_{n} (1 − τ) + s))
V = V (c, y_{n}, s, μ, τ)

The optimal UI system maximizes solves

max_{τ} V (c, y_{n}, s, μ (τ) , τ)

Although, c, y_{n}, s are affected by τ, these effects need not be taken into account since by individual optimality,

V_{c} = V_{Y_{n}} = V_{s} = 0.

This is the envelope theorem. Therefore, the first order condition for maximizing V by choosing τ is

\frac{dV}{dτ} = V_{μ} \frac{dμ}{dτ} + V_{τ} = 0,

where

V_{μ} = (1 − α) u′ (c_{u}) (1 − β) τ
V_{τ} = −u′ (c_{1}) − au′ (c_{2}) − (1 − α) u′ (c_{u}) βy_{n} + (1 − α) u′ (c_{u}) (1 − β) μ,

where c_{1} = 1 − τ − s is first period consumption, c_{2} = 1 − τ + s is second period consumption if the job is retained and c_{u} = (1 − β) (μτ − c)+βy_{n} (1 − τ)+s is second period consumption if the individual lost his job.

Note that by individual savings optimization (the Euler equation)

u′ (c_{1}) = au′ (c_{2}) + (1 − α) u′ (c_{u})
u′ (c_{1}) − (1 − α) u′ (c_{u}) = au′ (c_{2})

implying

V_{τ} = −u′ (c_{1}) − (u′ (c_{1}) − (1 − α) u′ (c_{u})) − (1 − α) u′ (c_{u}) βy_{n} + (1 − α) u′ (c_{u}) (1 − β) μ

= −2u′ (c_{1}) + (1 − α) (1 − βy_{n} + (1 − β) μ) u′ (c_{u}) .

Approximating

u′ (c_{1}) ≈ u′ (c_{u}) + u′′ (c_{u}) Δc

where Δc ≡ c_{1} − c_{u} is the fall in consumption if becoming unemployed. The first order condition is then

0 = (1 − α) u′ (c_{u}) (1 − β) τ \frac{dμ}{dτ} − 2 (u′ (c_{u}) + u′′ (c_{u}) Δc)
+ (1 − α) (1 − βy_{n} + (1 − β) μ) u′ (c_{u})
2 (1 + \frac{u′′}{u′} Δc) = (1 − α) (1 − β) τ \frac{dμ}{dτ} + (1 − α) (1 − βy_{n} + (1 − β) μ)
2 (1 + \frac{u′′}{u′} Δc) = (1 − α) (1 − β) τ \frac{dμ}{dτ} + (1 − α) (1 − βy_{n} + (1 − β) \frac{Y}{(1 − α) (1 − β)})
2 (1 + \frac{u′′}{u′} Δc) = (1 − α) (1 − β) τ \frac{dμ}{dτ} + (1 − α) (1 − βy_{n} + \frac{Y}{1 − α})
2 (1 + \frac{u′′}{u′} Δc) = (1 − α) (1 − β) τ \frac{dμ}{dτ} + (1 − α) (1 − βy_{n} + \frac{1 + α + (1 − α) βy_{n}}{(1 − α)} )
2 (1 + \frac{u′′}{u′} Δc) = (1 − α) (1 − β) τ \frac{dμ}{dτ} + 2
\frac{u′′}{u′} Δc = (1 − α) (1 − β) τ \frac{dμ}{dτ}

Using the definition

μ ≡ \frac{b}{τ} = \frac{Y}{(1 − α) (1 − β)}

we get

\frac{u′′}{u′} \frac{Δc}{Y} = \frac{τ}{μ} \frac{dμ}{dτ}
−R_{r} \frac{Δc}{c} = E_{μ,t}Y

where E_{μ,t} is the elasticity of μ with respect to taxes and R_{r} the relative risk aversion coefficient. Recall that μ is the ratio between benefits and taxes should be interpreted as the ratio between employment and unemployment.

Note that we should not interpret Y as the aggregate level of income since we have normalized the pre-unemployment income to unity. Instead, it is a measure of employment. Setting y_{n} ≈ 1, Y ≈ 1 + α + (1 − α) β which is the time people work. In this simple model, this is value is overstated since no unemplyment occur in the first period. More realistically, it should be close to one, giving

R_{r} \frac{Δc}{c} = −E_{μ,t}

The interpretation is that the welfare loss (the LHS) should optimally be given by how elastic the ratio of employment to unemployment is with respect to taxes.

Without moral hazard, \frac{dμ}{dτ} = 0 = E_{μ,t}, in which case optimality requires Δc = 0. With moral hazard, higher taxes tends to reduce μ since the employment to unemployment falls in in taxes, i.e., \frac{τ}{μ} \frac{dμ}{dτ} = E_{μ,t} is negative. Therefore, \frac{Δc}{c} > 0. We see that \frac{Δc}{c} increases if \frac{τ}{μ} \frac{dμ}{dτ} is large in absolute terms and falls if risk aversion is large. Baily claims that E_{μ,t} is in the order 0.15 − 0.4. With log utility, this is also how much consumption should fall on entering unemployment.

This approach has been generalized by Chetty showing that we can have repeated spells of unemployment, uncertain spells of unemployment, value of leisure, private insurance and borrowing constraints. The model can therefore be extended to evaluate UI reforms. With a more dynamic model, and in particular if capital markets are imperfect, it should be noted that one needs to know how the whole consumption profile is affected by unemployment. The drop at entering unemployment may not be enough. Shimer and Werning (2007), shows that the reservation wage can be used as a summary measure of how bad unemployment is.

In any case, this the model is not suitable to analyze

1. General equilibrium effects like impacts on wages, search spillovers and job creation.

2. Interaction with other taxes-fiscal spillovers.

3. Time varying benefits.

1.2 The dynamic approach with observable savings

The seminal paper by Shavel & Weiss (1979) focuses on the optimal time profile of benefits. It is a simple infinite horizon discrete time model where the aim is to maximize utility of a representative unemployed subject to a government budget constraint. Utility is given by

∑_{t=0}^{∞} (\frac{1}{1 + r})^{t} (u (c_{t}) − e_{t})

where c_{t} is period t consumption and e_{t} is a privately chosen unobservable effort associated with job search. The subjective discount rate is r, which is assumed to coincide with an exogenous interest rate.

It is assumed that the individual has no access to capital markets so c_{t} = b_{t} when the individual is unemployed. After regaining employment, the wage is w forever.

When the individual becomes employed he stays employed for ever for simplicity. Agents have no access to credit markets (or equivalently, savings is perfectly monitored and benefits can be made contingent on them) so the planner can perfectly control the consumption of the individual. The moral hazard problem is that individuals can affect the probability of finding a job. As in Baily (1978), the individual controls both the search effort (here called e_{t}) and the reservation wage (here w_{t}^{∗} ).

Given an effort level e_{t}, the individual receives one job offer per period with an associated wage drawn from a distribution with a time invariant probability density f (w_{t}, e_{t}) . The probability of finding an acceptable job in period t is thus

p (w_{t}^{∗}, e_{t}) = ∫_{w_{t}^{∗}}^{∞} f (w_{t}, e_{t}) dw_{t}

with

p_{w} (w_{t}^{∗}, e_{t}) = −f (w_{t}, e_{t}) ≤ 0 and
p_{e} (w_{t}^{∗}, e_{t}) > 0

where the latter is by assumption.

Let E_{t} be the expected utility of an unemployed individual that choose optimally a sequence {e_{t+s}, w_{t+s}^{∗}}_{s=0}^{∞} . Define

u_{t} = ũ (w_{t}^{∗}, e_{t}) ≡ \frac{1 + r}{r} ∫_{w_{t}^{∗}}^{∞} u (w_{t}) \frac{f (w_{t}, e_{t})}{p (w_{t}^{∗}, e_{t})}dw_{t}

This is the expected utility from next period, conditional on finding a job this period, which starts next period. We note that

ũ_{w} (w_{t}^{∗}, e_{t}) ≥ 0
ũ_{e} (w_{t}^{∗}, e_{t}) ≥ 0.

The first inequality follows from the fact that conditional on finding a job, wages are higher for higher reservation wages. The second inequality is by assumption, higher search effort leads to no worse distribution of acceptable job offers.

E_{t} satisfies the standard Bellman equation

E_{t} = max_{e_{t},w_{t}^{∗}} u (b_{t}) − e_{t} + \frac{1}{1 + r} (p (w_{t}^{∗}, e_{t}) ũ (w_{t}^{∗}, e_{t}) + (1 − p (w_{t}^{∗}, e_{t})) E_{t+1})

The first-order conditions are

e_{t}; \frac{1}{1 + r} (p_{e} (w_{t}^{∗}, e_{t}) (ũ (w_{t}^{∗}, e_{t}) − E_{t+1}) + p (w_{t}^{∗}, e_{t}) ũ_{e} (w_{t}^{∗}, e_{t})) = 1
w_{t}^{∗}; −p_{w} (w_{t}^{∗}, e_{t}) (ũ (w_{t}^{∗}, e_{t}) − E_{t+1}) = p (w_{t}^{∗}, e_{t}) ũ_{w} (w_{t}^{∗}, e_{t}) .

In the first equation, the LHS is the marginal benefit of higher search effort, coming from a higher probability of finding a job and better jobs if found. These balances the cost which is 1. In the second equation, the LHS is the marginal cost of higher reservation wages, coming from a lower probability of finding a job. The RHS is the gain, coming from better jobs if accepted.

By the envelope theorem

\frac{dE_{t}}{dE_{t+1}} = \frac{∂E_{t}}{∂E_{t+1}} = \frac{1 − p (w_{t}^{∗}, e_{t})}{1 + r}

Now, we will show the important results that anything that reduces next periods unemployment value E_{t+1} will reduce 1 − p (w_{t}^{∗}, e_{t}) , i.e., make hiring more likely. To see this, note that if E_{t+1} falls,

p_{e} (w_{t}^{∗}, e_{t}) (ũ (w_{t}^{∗}, e_{t}) − E_{t+1}) + p (w_{t}^{∗}, e_{t}) ũ_{e} (w_{t}^{∗}, e_{t}) , and
− p_{w} (w_{t}^{∗}, e_{t}) (u (w_{t}^{∗}, e_{t}) − E_{t+1})

both becomes larger if choices are unchanged. In words, the marginal benefit of searching harder and the marginal cost of setting higher reservation wages both increase. Thus, a reduction in E_{t+1} increase search effort and reduce the reservation wage increasing p.

Now, we can use this to show the key result that benefits should have a decreasing profile. Proof: Suppose contrary that b_{t} = b_{t+1}. Then consider an infinitessimal increase in b_{t} financed

by an actuarially fair reduction in b_{t+1}, that is

db_{t} = −\frac{1 − p}{1 + r}db_{t+1} > 0

where p (w_{t}^{∗}, e_{t}) is calculated at the initial (constant) benefit levels. The direct effect on felicitity levels (period utilities) is

u′ (b_{t}) db_{t} + \frac{1 − p}{1 + r}u′ (b_{t+1}) db_{t+1}

− u′ (b_{t}) \frac{1 − p}{1 + r}db_{t+1} + \frac{1 − p}{1 + r}u′ (b_{t+1}) db_{t+1}

= 0

since u′ (b_{t}) = u′ (b_{t+1}) .By the envelope theorem, we need not take into account changes in endogenous variables when calculating welfare. Therefore, E_{t} is unchanged. Since u (b_{t}) has increased, E_{t+1} must have fallen. When calculating the budgetary effects we need to into account the endogenous changes on p.

Let

B_{t} = b_{t} + \frac{1 − p}{1 + r}b_{t+1}

Then,

dB_{t} = db_{t} + \frac{1 − p}{1 + r}db_{t+1} − \frac{dp}{1 + r}b_{t+1}

= −\frac{dp}{1 + r}b_{t+1}

Since E_{t+1} has fallen, dp > 0. Thus dB_{t} < 0. I.e., the cost of providing utility E_{t} has fallen. Equivalently, the insurance is more efficient than the starting point b_{t} = b_{t+1}.

1.2.1 Extensions Hopenhayn and Nicolini extend the model by Shavel & Weiss in an important dimension – it enriches the policy space of the government by allowing taxation of workers to be contingent on their unemployment history. It is shown that the government should use this extra way of "punishing" unemployment. The intuition is that relative to the first best, which is a constant unemployment benefit, the government must "punish" unemployment. Doing this by only reducing unemployment benefits is suboptimal, by spreading the punishment of unsuccessful search over the entire future of the individual, a more efficient insurance can be achieved. I.e., lower cost of providing a given utility level. It is shown that this may be quantitatively important. Another contribution is to show that the problem can be formulated in a recursive way with the promised utility as state variable.

Using H&N’s notation, we assume that individuals can choose an unobservable effort level a_{t} that positively affects the hiring probability. In H&N 1997, it is assumed that p (a_{t}) is an concave and increasing function and hiring is an absorbing state with a wage w forever. In H&N 2005, it is instead assumed that spells are repeated, with an exogenous separation probability s and

p (a) = \{ \begin{matrix}p if a = 1 \\ 0 otherwise \end{matrix}

which is the assumption we make here.

The individual has a utility function

E ∑_{t=0}^{∞} (\frac{1}{1 + r})^{t} (u (c_{t}) − a_{t}) .

Let θ_{t} ∈ {0, 1} be the employment status of the individual in period t, where θ_{t} = 1 represents employment. Let θ^{t} = (θ_{0}, θ_{1}, ...θ_{t}) be the history of the agent up until period t. The history of a person that is unemployed in period t is therefore θ^{t−1}×0 = (θ_{0}, θ_{1}, ...θ_{t}, 0) ≡ θ_{u}^{t}, and similarly, θ^{t−1} × 1 ≡ θ_{e}^{t}).

An allocation is now defined as a rule that assigns consumption and effort as a function of θ^{t} at every point in time and for every possible history, c_{t} = c (θ^{t}) . We focus on allocations where a_{t} = 1. Individuals must be induced to volontarily choose a_{t} = 1. Allocations that satisfies this are called incentive compatible allocations.

Given an allocation we can compute the expected discouted utililty at every point in time for every possible history, V_{t} = V (θ^{t}) . The problem is now to choose the allocation that minimizes the cost of giving some fixed initial utility level to the representative individual. This problem can be written in a recursive way. In period zero, the planner gives a consumption level c_{0}, prescribes an effort level a_{0} (=1) and promised continuation utilities V_{1}^{e} ≡ V (θ_{e}^{1}) and V_{1}^{u} = V (θ_{u}^{1}) . The problem of the planner in period zero is to minimize costs of providing a given expected utility level V_{0} subject to the incentive constraint the individual voluntarily chooses a_{0}. The problem is recursive and at any node, costs of providing promised utilities are minimized given incentive constraints

The problem of the unemployed individual is also recursive. – as unemployed, maximized utility is (the agent only controls a_{t})

V (θ_{u}^{t}) = u (c_{t}) − 1 + \frac{1}{1 + r} (pV (θ_{u}^{t} × 1) + (1 − p) V (θ_{u}^{t} × 0))

with the incentive constraint

\frac{1}{1 + r}p (V (θ_{e}^{t+1}) − V (θ_{u}^{t+1})) ≥ 1.

Define W (V_{t}) as the minimum cost for the planner to provide a given amount of utility V_{t} to an employed. Similarly, let C (V_{t}) denote the minimal cost of providing utility V to an unemployed (are these function changing over time?). W satisfies

W (V_{t}) = min_{c_{t},V_{t+1}^{e},V_{t+1}^{u}} c_{t} − w + \frac{1}{1 + r} ((1 − s) W (V_{t+1}^{e}) + sC (V_{t+1}^{u}))
s.t.V_{t} = u (c_{t}) + \frac{1}{1 + r} ((1 − s) V_{t+1}^{e} + sV_{t+1}^{u}) ,

whereV_{t} = V (θ_{e}^{t}) , c_{t} = c (θ_{e}^{t}) , V_{t+1}^{e} = V (θ_{e}^{t} × 1) and V_{t+1}^{u} = V (θ_{e}^{t} × 0) .

The constraint can be called promise keeping constraint and has a Lagrange multiplier δ_{t}^{e}.

C satisfies

C (V_{t}) = min_{c_{t},V_{t+1}^{e},V_{t+1}^{u}} c_{t} + \frac{1}{1 + r} (pW (V_{t+1}^{e}) + (1 − p) C (V_{t+1}^{u}))

s.t. \frac{1}{1 + r}p (V_{t+1}^{e} − V_{t+1}^{u}) ≥ 1,

V_{t} = u (c_{t}) − 1 + \frac{1}{1 + r} (pV_{t+1}^{e} + (1 − p) V_{t+1}^{u}) .

where V_{t} = V (θ_{u}^{t}) , c_{t} = c (θ_{u}^{t}) , V_{t+1}^{e} = V (θ_{u}^{t} × 1) and V_{t+1}^{u} = V (θ_{u}^{t} × 0) .

The first constraint is the incentive constraint, with an associated Lagrange multiplier γ_{t} and the second is the promised utility with Lagrange multiplier δ_{t}^{u}.^{1} Given that u (c_{t}) is concave and u^{−1} (V_{t}) therefore is convex, it is straightforward to show that C and W are convex functions.

First order conditions when the agent is employed are

1 = δ_{t}^{e}u′ (c_{t})  (1)
W′ (V_{t+1}^{e}) = δ_{t}^{e}
C′ (V_{t+1}^{u}) = δ_{t}^{e}.

The envelope condition is

W′ (V_{t}) = δ_{t}^{e} = \frac{1}{u′ (c_{t})} = W′ (V_{t+1}^{e}) = C′ (V_{t+1}^{u}) .

The fact that W′ (V_{t}) = W′ (V_{t+1}^{e}) implies that nothing change for the employed individual as long as his remains employed. Since W′ (V_{t}) = C′ (V_{t+1}^{u}) ,marginal marginal utility does not change if the person becomes unemployed, i.e., consumption does not change upon loosing his job either. This is due to the fact that there is no moral hazard problem on the job and full insurance is therefore optimal.^{2} When the agent is unemployed, the FOC and envelope conditions are

1Note that the Lagrange multipliers depends on the history θ_{t}.

2From now, I will mostly skip writing out the explicit dependence on history, hopefully without creating confusion.

1 = δ_{t}^{u}u′ (c_{t})
W′ (V_{t+1}^{e}) = γ_{t} + δ_{t}^{u}
(1 − p) C′ (V_{t+1}^{u}) = −γ_{t}p + δ_{t}^{u} (1 − p)
C′ (V_{t}) = δ_{t}^{u}.

Giving

C′ (V_{t}) = \frac{1}{u′ (c_{t})}  (2)
W′ (V_{t+1}^{e}) = \frac{1}{u′ (c_{t})} + γ_{t}
C′ (V_{t+1}^{u}) = \frac{1}{u′ (c_{t})} − γ_{t} \frac{p}{1 − p}

Results Since the incentive constraint will bind^{3}, γ_{t} > 0 and therefore

W′ (V_{t+1}^{e}) > C′ (V_{t}) > C′ (V_{t+1}^{u}) ,
\frac{1}{u′ (c (θ_{u}^{t} × 1))} > \frac{1}{u′ (c (θ_{u}^{t}))} > \frac{1}{u′ (c (θ_{u}^{t} × 1))}
c (θ_{u}^{t} × 1) > c (θ_{u}^{t}) > c (θ_{u}^{t} × 1)

The result C′ (V_{t}) > C′ (V_{t+1}^{u}) and the convexity of C implies that the unemployed should be made successively worse off (V_{t+1}^{u} < V_{t}) as long as he remains unemployed. Since C′ (V_{t}) = \frac{1}{u′(c_{t})} this means that consumption must fall. Furthermore, as the IC-constraint \frac{1}{1+r}p (V_{t+1}^{e} − V_{t+1}^{u}) ≥ 1 binds, if V_{t+1}^{u} keeps falling as long as the unemployed remains unemployed, so must V_{t+1}^{e} implying that consumption when becoming employed is lower the lower the agent has been unemployed.

3Prove that it must by assuming that it doesn’t and derive the implications of that.

1.2.2 The inverse Euler equation. Multiplying the second line of (2) by p and the third by (1 − p) and adding them yields,

\frac{1}{u′ (c_{t})} = pW′ (V_{t+1}^{e}) + (1 − p) C′ (V_{t+1}^{u}) .  (3)

Recall that V_{t+1}^{e} is the utility next period if the agent becomes employed, in which case, by (1), W′ (V_{t+1}^{e}) = \frac{1}{u′(c_{t+1})}, where c_{t+1} = c (θ_{t+1}^{e}) denotes consumption in period t + 1 conditional on the getting a job in t + 1 (and the history that led to consumption in t being c_{t} = c (θ_{t})). Similarly, V_{t+1}^{u} is next periods utility if the agent remains unemployed. By (2), C′ (V_{t+1}^{e}) = _{\frac{1}{u′(c_{t+1}|_{θ_{t+1}=9})}}, where c_{t+1}|_{θ_{t+1}=0} denotes consumption if the agent remains unemployed. Equation (3) can therefore be written

\frac{1}{u′ (c (θ_{u}^{t}))} = p\frac{1}{u′ (c (θ_{u}^{t} × 1))} + (1 − p) \frac{1}{u′ (c_{t+1}|_{θ_{t+1}=0})}
\frac{1}{u′ (c_{t})} = E_{t} \frac{1}{u′ (c_{t+1})}.

This is the famous "Inverse Euler Equation" (Rogerson, -85 Econometrica)^{4}. Note the difference between this and the standard Euler equation.

u′ (c_{t}) = E_{t}u′ (c_{t+1}) .

The inverse Euler equation has an important implication. To see this, first note that Jensen’s inequality,

E_{t} \frac{1}{u′ (c_{t+1})} > \frac{1}{E_{t}u′ (c_{t+1})} ⇒ \frac{1}{E_{t} \frac{1}{u′(c_{t+1})}} < E_{t}u′ (c_{t+1})

since the inverse function is convex. Using this with the Inverse Euler equation gives,

u′ (c_{t}) = \frac{1}{E_{t} \frac{1}{u′(c_{t+1})}} < E_{t}u′ (c_{t+1}) .

The fact that u′ (c_{t}) < E_{t}u′ (c_{t+1}) in the optimal allocation means that the agent would like to save more, i.e., he is savings constrained. The incentive constraint implies that it is optimal to prevent the individual to save as much as he would like to. Suppose, for example, that utility is logarithmic, then we have

\frac{1}{c_{t}} = \frac{1}{E_{t}c_{t+1}} ⇒ c_{t} = E_{t}c_{t+1},

\frac{1}{u′ (c_{t})} \frac{1 + r}{1 + ρ} = E_{t} \frac{1}{u′ (c_{t+1})}.

4With a difference between subjective and market discount rates (ρ and r, respectively), we would get

while the Euler equation, guiding private preferences, implies the privately optimal consumption c_{t}^{∗} given future consumption is

c_{t}^{∗} = \frac{1}{E_{t} (\frac{1}{c_{t+1}} )} < E_{t}c_{t+1}.

The intuition is that with more wealth and higher consumption, it is more costly to implement the incentive constraint. Thus, the benevolent planner want to prevent some wealth accumulation. The standard interpretation of this is that when there are incentive constraints, it may be optimal to tax the returns to savings. However, it may turn out that this tax is nevertheless zero in expectation, thus not creating any revenue for the planner/government (Kocherlakota 2005, Econometrica). How can such a tax discourage savings? Hint: risk premium depends on covariance with marginal utility. Explain!

In the logarithmic example, suppose individuals can save and borrow a gross interest rate r. Consider a marginal tax rate that depends on employment status and last period individual asset holdings, τ_{t+1}^{e} = τ^{e} (a_{t}) and τ_{t+1}^{u} = τ^{u} (a_{t}) . Then, to have the individual Euler equation satisfied, we need

u′ (c_{t}) = βE_{t}u′ (c_{t+1}) (1 + r) (1 − τ (a_{t}))  (4)
\frac{1}{c_{t}} = (p\frac{1}{c_{t+1}^{e}} (1 − τ_{t+1}^{e}) + (1 − p) \frac{1}{c_{t+1}^{u}} (1 − τ_{t+1}^{u}))

The inverse Euler equation requires

c_{t} = pc_{t+1}^{e} + (1 − p) c_{t+1}^{u}  (5)

Suppose we consider a zero expected tax rate, i.e., pτ_{t+1}^{e} = − (1 − p) τ_{t+1}^{u}.Then,

τ_{t+1}^{e} = \frac{− (1 − p)}{p}τ_{t+1}^{u}.  (6)

Using (5) to replace c_{t} in (4) together with (6) yields

τ_{t+1}^{u} = \frac{p (c_{t+1}^{e} − c_{t+1}^{u})}{pc_{t+1}^{e} + c_{t+1}^{u} (1 − p)} = \frac{pΔc_{t+1}}{E_{t}c_{t+1}}
τ_{t+1}^{e} = −\frac{(1 − p) (c_{t+1}^{e} − c_{t+1}^{u})}{pc_{t+1}^{e} + c_{t+1}^{u} (1 − p)} = −\frac{(1 − p) Δc_{t+1}}{E_{t}c_{t+1}}

These tax rates leads to both the Euler and the inverse Euler equation being satisfied. Note that the tax is negative in case the agent becomes employed, while positive if he remains unemployed. That is, it creates a net return that is negatively correlated with marginal utility.

Result: Rendahl (2007)

Consider the repeated H&N economy but where individuals have access to a safe observable bond. It turns out that a tax/transfer that only depends on last period asset holdings and employment status can implement the second-best allocation as the private choices of individuals. Unemployment benefits falls in the asset position of the agent. Over an unemployment spell, unemployment benefits increase but consumption falls.

From Pontus Rendahl 2007 (Job market paper)

1.3 The Dynamic approach with unobservable saving

An key assumption in the previous subsection was that the planner can control the consumption level of the individual at all times, the only unobservable is search effort. In reality this assumption seems questionable, given the existence of alternative means of income, capital markets, insurance within an extended family and durable goods.

In this subsection, we assume that the planner cannot control the consumption of the individual – she has access to a perfect market for lending and borrowing at a fixed interest rate and her wealth is unobservable. Of course, this extreme is perhaps equally unrealistic and the truth might be somewhere in between.

An immediate problem is that seach decisions in this setting might depend on the unobservable wealth level. Making sure that there is always an incentive to search might then be unfeasible in general. In one special case, the seach decision is not dependent on wealth, when individuals have CARA utility. This is the way we go here. Furthermore, we simplify by assuming that seach is either one or zero.

Individuals maximize their intertemporal utility, given by

E ∫_{0}^{∞} e^{−rt}U (c_{t}) dt,

where

U (c_{t}) ≡ −e^{−γc_{t}}.

The purpose of the planner is to maximize time zero welfare of an employed agent subject to

1. budget balance expressed as actuarial fairness, i.e., that the expected discounted value of tax payments equals that of benefits (note that this is not the same as a budget balance in a pay-as-you-go system) and to

2. the constraint that agents volontarily search.

Without loss of generality, we let individuals pay lump-sum taxes, denoted τ, implying that

Ȧ_{t} = rA_{t} + y − c_{t} − τ,  (7)

where y = w if the individual is employed, y = b − s, if the individual is unemployed and search and y = b if the individual is unemployed without searching. An individual who searches, finds a job with an exogenous instantaneous probability h and a person with a job loses it with probability q. Define the average discounted probabilities (ADP’s) of being unemployed (in state 2) as

Π_{2} ≡ r ∫_{0}^{∞} e^{−rt}μ_{2,t}dt

It is straightforward to calculate that

Π_{2} ≡ \frac{q}{r + h + q} .

: \frac{q}{r+h+q}where μ_{2,t} is the probabilities of being unemployed at time t, respectively, conditional on being employed at time zero, provided that unemployed search for a job.

Actuarial fairness the UI system is now a simple linear function of the benefits

τ = Π_{2}b  (8)

Under constant absolute risk aversion and stationary income uncertainty, the value functions for the two states j ∈ {1, 2} can be separated

V (A_{t}, j) = W (A_{t}) Ṽ_{j} (τ, b) ,  (9)

where

W (A_{t}) ≡ \frac{e^{−γA_{t}}}{r}  (10)
Ṽ_{j} ≡ −e^{−γc_{j}},

and σ_{j} are state-dependent consumption constants such that the state dependent consumption functions are

c_{j} (A_{t}) = rA_{t} + σ_{j}.  (11)

The consumption constants σ_{j} are nonlinear functions of income in all states and thus, depend on the planner choice variables τ, and b. The constants are found as the unique solutions to the Bellman equations for each state:

σ_{1} = w − τ − \frac{q (e^{γΔ_{2}} − 1)}{γr},  (12)
σ_{2} = b − s − τ + \frac{h (1 − e^{−γΔ_{2}})}{γr},

where q is the exogeneous hiring rate, h is the hiring rate if the agent search actively and

Δ_{2} ≡ σ_{1} − σ_{2}.

Let us derive these results;

Conjecturing that the value functions are −\frac{1}{r}e^{−γ(rA_{t}+σ_{j})}, we can write the Bellman equations for the enployed as

−\frac{1}{r}e^{−γ(rA_{t}+σ_{1})} = max_{σ} −e^{−γ(rA_{t}+σ)}dt − (1 − rdt) (1 − qdt) \frac{1}{r}e^{−γ(rA_{t+dt}+σ_{1})}

− (1 − rdt) qdt\frac{1}{r} [e^{−γ(rA_{t+dt}+σ_{2})}] .

Using the budget constraint, A_{t+dt} = A_{t} + r (w − τ − σ) dt, and dividing by e^{−γrA_{t}}, this becomes

−\frac{1}{r}e^{−γσ_{1}} = max_{σ} −e^{−γσ}dt − (1 − rdt) (1 − qdt) \frac{1}{r}e^{−γ(r(w−τ−σ)dt+σ_{1})}

− (1 − rdt) qdt\frac{1}{r} [e^{−γ(r(w−τ−σ)dt+σ_{2})}] .

Using the first-order linear approximation, e^{−γ(r(w−τ−σ)dt+σ_{1})} ≈ e^{−γσ_{1}}−γr (w − τ − σ) dte^{−γσ_{1}}, adding \frac{1}{r}e^{−γσ_{1}} to both sides, dividing by dt and letting dt approach zero, yields

0 = max_{σ} {−re^{−γ(σ−σ_{1})} + r + γr (w − τ − σ)}  (13)
+ q (1 − e^{−γ(σ_{2}−σ_{1})})

Similarly, for the unemployed, we obtain

0 = max_{σ} {−re^{−γ(σ−σ_{2})} + γr (b_{2} − s − τ − σ)}  (14)
+ r + h − he^{−γ(σ_{1}−σ_{2})}

The right hand sides of (13) and (14) are maximized at σ = σ_{j}, implying that these values maximize the RHS’s of the Bellman equations.

Substituting σ_{1} and σ_{2} respectively for σ in (13) and (14) solves the maxima. Finally, solving for gives the σ_{j}^{s} gives (12), which by construction then solves the Bellman equations.

Clearly, the objective of the planner is now to maximize σ_{1}, from which also follows time consistency – the welfare of employed at all times is maximized.

The first step is now to derive an expression for σ_{1} in terms of Δ_{2} where the budget constraint (8) is used to replace the tax rate. For this purpose, we subtract the second line of (12) from the first and solve for b. Then, we use this expression in the budget constraint τ = Π_{2}b and substitute for τ in the first line of (12). This yields

σ_{1} = κ + Π_{2} (Δ_{2} − \frac{he^{−γΔ_{2}}}{γr} ) − (1 − Π_{2}) q \frac{e^{γΔ_{2}}}{γr} ,  (15)

where κ is a constant, independent of the choice variables. Straightforward calculus shows that (15) defines σ_{1} as a concave function of Δ_{2} with a unique maximum at 0. The reason for σ_{1} being maximized at Δ_{2} = 0 is obvious – when actuarial insurance is available, full insurance maximizes utility. However, Δ_{2} = 0 is not incentive compatible. Searching moving will not occur voluntarily. Now, as in Baily approach, we can use the consumption fall upon separation, Δ_{2}, to evaluate the gain by finding employment.

If the unemployed agent shirks she is unemployed for ever, getting an income b− τ and a utility

−\frac{1}{r}e^{−γrA_{t}}e^{−γ(b−τ)}.

The utility if the individual instead searches is

−\frac{1}{r}e^{−γrA_{t}}e^{−γσ_{2}}.

To induce search, we clearly need

σ_{2} ≥ b − τ.

Note that the consumption of the unemployed who search is rA_{t} + σ_{2}. Furthermore, her total income net of search costs is rA_{t} + b − τ − s. Therefore, the search condition implies consumption to be strictly higher than income. Over time, the unemployed depletes her assets and consumption therefore falls, despite the benefits being constant. The celebrated result by Shawell-Wais and Hopenhayn-Nicolini that consumption should optimally fall over the unemployment spell when the insurer can fully control consumption (no hidden savings) is therefore mimicked in this case, where hidden savings are allowed.

The final part is now to express the search constraint in terms of the consumption difference Δ_{2}. Using the second line of (12), the search constraint can be written

Δ_{2} ≥ −\frac{ln (1 − γr\frac{s}{h})}{γ},  (16)

which we label the IC2-condition. We depict this in Figure 1,

A higher r and s and lower h reduce the value of searching, and shifts the constraint to the right.

30

Finally, we can solve for the value of b that makes the IC2 condition bind exactly. Take the difference between the equations in (12) set it to −\frac{ln(1−γr \frac{s}{h})}{γ} and solve for b, which gives

−\frac{ln (1 − γr\frac{s}{h})}{γ} = w − τ − \frac{q (e^{− ln(1−γr \frac{s}{h})} − 1)}{γr} − (b − s − τ + \frac{h (1 − e^{ln(1−γr \frac{s}{h})})}{γr})
b = w + s + \frac{ln (1 − γr\frac{s}{h})}{γ} − \frac{q (−\frac{h}{−h+γrs} − 1)}{γr} − \frac{h (1 − −\frac{−h+γrs}{h})}{γr}

= w + \frac{ln (1 − γr\frac{s}{h})}{γ} − \frac{sq}{h − γrs}

In Hassler&Rodriguez (2008), we extend this model and show that it is useful to analyze multiple incentive constraints. It is immediate to show that benefits should optimally be constant over time. This since the incentive constraint does not change over time. We also introduce multiple incentive constraints, showing that if there is also a need to induce some individuals to move to find a job, this is optimally done with an initial period of low benefits.

2 Optimal taxation – the Ramsey approach

2.1 Optimal taxation under commitment – the Ramsey problem

To provide some intuition already before providing the famous Chamley & Judd result consider the following simple model.

Preferences

The representative agent has an additively separable utility function in consumption and leisure,

U = ∑_{t=0}^{∞} β^{t}u (c_{t}, 1 − n_{t}) .  (17)

satisfying the usual Inada conditions so first-order conditions are sufficient.

Technology

Output is produced by labor only on a competitive labor market. One unit of labor produces w units of the consumption good. Individuals have one unit of labor each period to split between work and leisure l = 1 − n. There is a perfect market for government bonds.

Budget constraints

The government needs to finance an exogenesous stream of consumption by tax revenues. For simplicity, we have already assumed that its consumption does not interfere with the individuals private problem. We will assumed that the government cannot finance its consumption by lump sum taxation. We do this without providing an explicit reason within the model. Instead, the government has at its disposal, a linear labor income tax τ_{n,t}, a consumption tax τ_{c,t} and a capital income tax, τ_{k}. Wages are exogeous at rate w_{t}.

The representative indivuals budget constraint period t is

c_{t} (1 + τ_{c,t}) + b_{t+1} = (1 − τ_{n,t}) w_{t}n_{t} + (1 − τ_{k,t}) (1 + r_{t}) b_{t}

where b isgovernment bonds.

Substituting forward yields,

∑_{t=0}^{∞} c_{t}∏_{s=0}^{t} (\frac{1}{1 + r_{t}} ) ∏_{s=0}^{t}W_{i,s} + lim_{t→∞}^{t} c_{t}∏_{s=0}^{t} (\frac{1}{1 + r_{t}} ) ∏_{s=0}^{t}W_{i,s}b_{t}  (18)

= ∑_{t=0}^{∞} w_{t}n_{t}W_{n,t}∏_{s=0}^{t} (\frac{1}{1 + r_{t}} ) ∏_{s=0}^{t}W_{i,s} + \frac{(1 − τ_{k,0})}{(1 + τ_{c,0})} (1 + r_{0}) b_{0}

where

W_{i,t} ≡ \frac{1 + τ_{c,t}}{(1 + τ_{c,t−1}) (1 − τ_{k,t})}
W_{i,0} = 1

and

W_{n,t} ≡ \frac{1 − τ_{n,t}}{1 + τ_{c,t}} .

We call W_{i,t} the intertemporal wedge between t − 1 and t and W_{n,t} the intratemporal wedge. In addition, there is an aggregate resource constraint,

g_{t} + c_{t} = w_{t}n_{t}.  (19)

We now make the following definitions:

Definition 1 A feasible allocation is a sequence {c_{t}, n_{t}, g_{t}}_{t=0}^{∞} that satisfies the aggregate budget constraint (22).

Definition 2 A price system is a sequence of interest rates {r_{t}}_{t=1}^{∞} that is bounded and non-negative.

Definition 3 A government policy is a sequence {τ_{c,t}, τ_{k,t}, τ_{n,t}, b_{t}}_{t=0}^{∞} .

Definition 4 A competitive equilibrium is a feasible allocation, a price system and a government policy such that

1. Given the price system and the government policy, the allocation solves the maximization problem of the individual.

2. The aggregate resource constraint is satisfied.

The problem of the consumer is to maximize (17) subject to (18).

First order conditions are

β^{t}u_{c} (c_{t}, 1 − n_{t}) = λ∏_{s=0}^{t} (\frac{1}{1 + r_{t}} ) ∏_{s=0}^{t}W_{i,s}
β^{t}u_{l} (c_{t}, 1 − n_{t}) = λw_{t}W_{n,t}∏_{s=0}^{t} (\frac{1}{1 + r_{t}} ) ∏_{s=0}^{t}W_{i,s}

which we can write

\frac{u_{l} (c_{t}, 1 − n_{t})}{u_{c} (c_{t}, 1 − n_{t}) w_{t}} = W_{n,t}  (20)
\frac{u_{c} (c_{t}, 1 − n_{t})}{u_{c} (c_{0}, 1 − n_{0})}β^{t}∏_{s=0}^{t} (1 + r_{t}) = ∏_{s=0}^{t}W_{i,s}

Result A sequence of consumption and labor supply satisfying (20), the budget constraint (18) and the resource constraint (19) is a competetive equilibrium.

Note: the governments budget constraint is redundant.

The Ramsey Problem

Maximizing (17) over the set of allocations that can be implemented as a competetive equilibrium is called the Ramsey problem.

Result

As we see, in the equations determining the competetive equilibrium, only the wedges and τ_{k,0} and τ_{c,0} appear. The latter two affects the value of the initial government debt. Therfore, the government has an over-supply of instruments in the sense that many sequences of taxes {τ_{c,t}, τ_{k,t}, τ_{n,t}}_{t=0}^{∞} can imply the same allocation.

Result Over any t + 1 periods starting from period 0, there are 3 (t + 1) independent tax rates but the budget constaint of individual is determined by 2t + 3 instruments given by t intertemporal wedges (W_{i,1}, ...W_{i,t}), t + 1 intratemporal wedges (W_{l,0}, ...W_{n,t}) and two initial tax-rates τ_{s,0} and τ_{c,0}.

Result Any sequence of taxes can be replicated using only labor and consumption taxes plus an initial capital income tax.

Proof: Using labor and income taxes gives 2t independent instruments that construct any sequence of wedges.

Result Consider a sequence of taxes such that consumption taxes are constant and capital income tax rates are constant at τ_{κ}. Given an initial consumption tax τ_{c,0},an identical intertemporal wedge can be constructed with zero capital income taxes and a sequence of consumption taxes satisfying

\frac{1 + τ_{c,1}}{1 + τ_{c,0}} = \frac{1}{1 − τ_{k}}
\frac{1 + τ_{c,t}}{1 + τ_{c,t−1}} = \frac{1}{1 − τ_{k}} ,

implying

1 + τ_{c,t} = \frac{1 + τ_{c,0}}{(1 − τ_{k})^{t}}

Note that if τ_{k} > 0, this sequence is increasing geometrically without bounds. It is perhaps intuitive that a sequence of consumption taxes that increases geometrically without bounds is suboptimal. Similarily, τ_{k} < 0,the consumption tax approaches -100%. That neither of this is optimal is really the Chamley-Judd result.

Before proceeding, we note that using the (20) in the private budget constraint, we get

∑_{t=0}^{∞} β^{t} (c_{t}u_{c} (c_{t}, 1 − n_{t}) − n_{t}u_{l} (c_{t}, 1 − n_{t})) = u_{c} (c_{0}, 1 − n_{0}) \frac{(1 − τ_{k,0})}{(1 + τ_{c,0})} (1 + r_{0}) b_{0}  (21)

An allocation that satisfies (21) satisfies the private budget constraint and is privately optimal for some sequences of taxes. If it also satisfies the aggregate budget constraint it is also implementable as a competetive equilibrium. Note, that there is no taxes or prices here except the two initial taxes on pre-existing capital.

Nevertheless we can reformulate the Ramsey problem as max

max U = ∑_{t=0}^{∞} β^{t}u (c_{t}, 1 − n_{t})
s.t. ∑_{t=0}^{∞} β^{t} (c_{t}u_{c} (c_{t}, 1 − n_{t}) − n_{t}u_{l} (c_{t}, 1 − n_{t})) = u_{c} (c_{0}, 1 − n_{0}) \frac{(1 − τ_{k,0})}{(1 + τ_{c,0})} (1 + r_{0}) b_{0}
g_{t} + c_{t} = w_{t}n_{t}

Sometimes, it is easier to work with this direct or primal approach. Here, it is then straighforward to construct the wedges and then taxes that implement the Ramsey optimal allocation.

2.2 The Chamley-Judd result

Now, we only add a production technology using capital. There is an infinitely lived representative agent with preferences

∑_{t=0}^{∞} β^{t}u (c_{t}, l_{t}) .

The household has one unit of labor per period, to be split between leisure l and work n.The aggregate budget constraint is

c_{t} + g_{t} + k_{t+1} = F (k_{t}, n_{t}) + (1 − δ) k_{t}  (22)

The production function is constant returns to scale and factor markets are competitive.

Profit maximization of the representative firm implies

w_{t} = F_{n} (k_{t}, n_{t})
r_{t} = F_{k} (k_{t}, n_{t})

The government needs to finance an exogenous stream of expenditures {g_{t}}_{t}^{∞} using taxes on labor and capital and can smooth taxes by using a bond. Following the literature, we let the interest rate on bonds be tax-free. Thus,

g_{t} + b_{t} = τ_{k,t}r_{t}k_{t} + τ_{n,t}w_{t}n_{t} + \frac{b_{t+1}}{R_{t}}

= F (k_{t}, n_{t}) − (1 − τ_{k,t}) r_{t}k_{t} − (1 − τ_{n,t}) w_{t}n_{t} + \frac{b_{t+1}}{R_{t}}

where b_{t} is government borrowing and R_{t} is the interest rate on government bonds. Households have budget constraints

c_{t} + k_{t+1} + \frac{b_{t+1}}{R_{t}} = (1 − τ_{n,t}) w_{t}n_{t} + (1 − τ_{k,t}) k_{t}r_{t} + (1 − δ) k_{t} + b_{t}

First order conditions are:

c_{t}; u_{c} (c_{t}, l_{t}) = λ_{t}
l_{t}; u_{l} (c_{t}, l_{t}) = λ_{t} (1 − τ_{n,t}) w_{t}
k_{t+1}; λ_{t} = βλ_{t+1} ((1 − τ_{k,t}) r_{t+1} + (1 − δ))
b_{t+1}; λ_{t} \frac{1}{R_{t}} = βλ_{t+1}

Clearly, the first three implies

\frac{u_{l} (c_{t}, l_{t})}{u_{c} (c_{t}, l_{t})} = (1 − τ_{n,t}) w_{t}
u_{c} (c_{t}, l_{t}) = βu_{c} (c_{t+1}, l_{t+1}) ((1 − τ_{k,t}) r_{t+1} + (1 − δ))

and the last two the no arbitrage condition

R_{t} = (1 − τ_{k,t}) r_{t+1} + (1 − δ)

Transversality conditions are

lim_{T →∞} (∏_{i=0}^{T −1}R_{i}^{−1}) k_{T +1} = 0
lim_{T →∞} (∏_{i=0}^{T −1}R_{i}^{−1}) \frac{b_{T +1}}{R_{T}} = 0

We can now make the following definitions: Definition 5 A feasible allocation is a sequence {k_{t}, c_{t}, l_{t}, g_{t}}_{t=0}^{∞} that satisfies the aggregate budget constraint (22). Definition 6 A price system is a sequence of prices {w_{t}, r_{t}, R_{t}}_{t=0}^{∞} that is bounded and nonnegative. Definition 7 A government policy is a sequence {τ_{n,t}, τ_{k,t}, b_{t}}_{t=0}^{∞} and perhaps {g_{t}}_{t=0}^{∞} if that can be chosen.

Definition 8 A competitive equilibrium is a feasible allocation, a price system and a government policy such that

1. Given the price system and the government policy, the allocation solves the maximization problem of the individual and of the firm.

2. The government budget constraints are satisfied.

Definition 9 The Ramsey problem is to choose a competitive equilibrium (i.e.,pick a particular government policy) that maximizes the welfare of the representative individual.

The Lagrangean of the Ramsey problem can be written

L = ∑_{t=0}^{∞} β^{t}{u (c_{t}, 1 − n_{t})
+ ψ_{t} (F (k_{t}, n_{t}) − (1 − τ_{k,t}) r_{t}k_{t} − (1 − τ_{n,t}) w_{t}n_{t} − b_{t} − g_{t} + b_{t+1}/R_{t})
+ θ_{t} (F (k_{t}, n_{t}) + (1 − δ) k_{t} − c_{t} − g_{t} − k_{t+1})
+ μ_{1,t} (u_{l} (c_{t}, l_{t}) − u_{c} (c_{t}, l_{t}) (1 − τ_{n,t}) w_{t})
+ μ_{2,t} (u_{c} (c_{t}, l_{t}) − βu_{c} (c_{t+1}, l_{t+1}) (1 − τ_{k,t}) r_{t+1} + (1 − δ))

Now, the first order condition for k_{t+1} is

θ_{t} = βψ_{t+1} (F_{k} (k_{t+1}, n_{t+1}) − (1 − τ_{k,t}) r_{t+1}) − θ_{t+1} (F_{k} (k_{t+1}, n_{t+1}) + (1 − δ))

and for c_{t}

u_{c} (c_{t}, 1 − n_{t}) = θ_{t}

giving

u_{c} (c_{t}, 1 − n_{t}) = βψ_{t+1} (F_{k} (k_{t+1}, n_{t+1}) − (1 − τ_{k,t}) r_{t+1})
+ βu_{c} (c_{t+1}, 1 − n_{t+1}) (F_{k} (k_{t+1}, n_{t+1}) + (1 − δ)) .

Suppose there is a steady state of the model, then

u_{c} = β (ψ (F_{k} − (1 − τ_{k,t}) F_{k}) + u_{c} (F_{k} + (1 − δ)))

= β (ψτ_{k,t}F_{k} + u_{c} (F_{k} + (1 − δ))) .

Private optimality (the Euler equation), implies in steady state

u_{c} = βu_{c} ((1 − τ_{k}) F_{k} + (1 − δ))
1 = β (F_{k} + (1 − δ) − τ_{k}F_{k})
\frac{1}{β} + τ_{k}F_{k} = F_{k} + (1 − δ)

giving

u_{c} = β (ψτ^{k}F_{k} + u_{c} (\frac{1}{β} + τ^{k}F_{k}))
= β ((ψ + u_{c}) τ^{k}F_{k}) + u_{c}
0 = β (ψ + u_{c}) τ^{k}F_{k}

requiring τ^{k} = 0.

2.3 Discussion

We have shown that also in this simple economy, tax smoothing implies that the intertemporal margin should not be distorted. We have also found an equivalence between constant consumption taxes and an investment tax. In an infinite horizon model, a positive investment tax in steady state has implications identical to ever increasing consumption taxes. This can thus provide some intuition for Chamley & Judd’s result that investment taxes should not be used in the long run. The result is quite robust. For example it extends to the case of heterogeneity, if the government wants to use it’s revenues to support some capital poor individuals, it should not tax capital accumulation in steady state. Here intuition could be that the capital stock in steady state is elastic enough to imply the tax incidence of capital taxes is on workers.

The result also extends to the stochastic case, in which case expected taxes should be zero and not distort savings.

An interesting case is if government spendings are stochastic. With complete markets, the government should then commit to a tax system that insures them against this (Chari et al. 1994). If spending needs are large, taxes on capital should be high and vice versa.

The zero capital income tax result does not go through in some cases:

1. If there are untaxed factors of production that generate profits and these factors are strict complements to capital. Then capital should be taxed (negatively if they are substitutes).

2. If market incompleteness makes people save too much for precautionary reasons.

In the short run, capital income taxes also collect revenue from sunk investments. Then, the tax is partly lump sum, which provides an argument for such taxes early in the planning horizon. But when is that zero? Has it already occurred a long time ago? In any case, we see a time consistency problem here.

Not also that the long-run maybe quite far out and people alive today might loose by a policy that maximizes the welfare of a constructed infinitely lived.

2.4 Time consistent taxation

Here we follow Klein and Rios-Rull 2003. Consider a stochastic economy productivity is z (s^{t}) and government consumption is g (s^{t}) where s^{t} is the history of a shock that in every period belongs to a finite element set S.The shock follows a Markov chain with transition maxtrix Γ. The representative individuals has utility given by

E ∑_{t}^{∞} β^{t}u (c_{t}, h_{t})

and the aggregate resource constraint is

F (K (s^{t−1}) , H (s^{t}) , z (s^{t})) + (1 − δ) K (s^{t−1}) = C (s^{t}) + K (s^{t}) + g (s^{t})

Individual budget constraints are

c_{t} + k_{t+1} = (1 − τ_{t}) w_{t}h_{t} + (1 + r_{t} (1 − θ_{t})) k_{t}

where lower case variables denoted individual and a balanced budget constraint is imposed on the government

θ_{t}k_{t}r_{t} + τ_{t}w_{t}h_{t} = g_{t}

If the government could set θ_{t} at t, this would be an ex-post lump-sum tax. Klein and Rios-Rull assume a limited commitment, i.e., that taxes are set for the next period. To find a time consistent solution, we require that the policy the government follows is of Markov type, i.e., it is a function of the set of state variables only. These are

{g, z, K, θ} ≡ x

Using budget balance, a policy rule is then

θ_{t+1} = ψ (x_{t})

We then define a recursive competitive equilibrium in the standard way, noting that the value function v depends on the policy rule

v (x, k; ψ) .

Assuming the government is benvolent, it assesses welfare according to

V (x; ψ) = v (x, k; ψ) .

We can also define the competitive equilibrium and its value function in case the government decides next periods tax to θ′ and following government follow ψ (x) . The value function is then

v̂ (x, θ′, k; ψ)

The associated welfare of the government is

V̂ (x, θ′; ψ)

Now define the current maximizing policy as

Ψ (x; ψ) = arg max_{θ′} V̂ (x, θ′; ψ)

A markov perfect optimal tax policy then satisfies the fixed-point requirement

Ψ (x; ψ (x)) = ψ (x) ,

i.e., if the government expects coming governmnts to use ψ it is optimal for itself to use ψ.

Klein and Rios-Rull use log utility and assume government consumption and productivity can each take on two different values respectively. They calibrate the model to US, data. Average g is 20%, varying 1.6% points up or down and an autocorrelation of .66. Productivity has a standard deviation of 2.4% with autocorrelation .88.

Comparing the commitment and no commitment they find that in commitment expected capital income tax rates are (almost) zero but with a standard deviation of 18%. having a strong positive correlation with g and a strong negative with z. Labor income taxes are 31% and almost fixed.

With 1 years commitment only, the average capital income tax rate is 65% with a standard deviation of 11%. It is positively correlated with g, but less than with full commitment. Labor income tax rates are 12% on average with a standard deviation of 3%. Output is 14% lower than under commitment and somewhat less volatile.

Also 4 years commitment produce high average tax rates on capital income 36%.

3 New Public Finance – the Mirrlees approach

Let us now consider the dynamic Mirrlees approach to optimal taxation. Here, individuals are assumed to be different. These differences can be either in their productivity or in their value of leisure. Such differences imply that there is differences between individuals in their trade-off between leisure and work. It is assumed that the government cannot directly observe this differences, only observe the individuals market choices. For example, governments observe income, but not the effort exerted to get this income.

Consider a simple two-period example from GTW.

Individual preferences are:

E (u (c_{1}) + v (n_{1}) + β (u (c_{2}) + v (n_{2})))

where c_{t} is consumption and n_{t} is labor supply/work effort. u is increasing and concave and v decreasing and concave. Individuals differ in their ability, denoted θ. It is assumed that there is a finite number i ∈ {1, 2, ..., N} of ability levels and ability might change over time. We will interchangeably use type and ability to denote θ. Output is produced in competitive firms using a linear technology where each individual i produces

y_{t} (i) = θ (i) n_{t} (i) .

There is a continuum of individuals of a unitary total mass. In the first period, individuals are given abilities by nature according to a probability function π_{1} (i). The ability can then change to the second period. Second period ability is denoted θ (i, j) and the transition probability is π_{2} (j|i) .

There is a storage technology with return R. Finally, the government needs to finance some spendings G_{1} and G_{2}. At first, we analyze the case of no aggregate uncertainty.

The aggregate resource constraint is

∑_{i} (y_{1} (i) − c_{1} (i) + ∑_{j} \frac{y_{2} (i, j) − c_{2} (i, j)}{R}π_{2} (j|i)) π_{1} (i) + K_{1} = G_{1} + \frac{G_{2}}{R}  (23)

where K_{1} is an aggregate initial endowment.

The problem is now to maximize the utilitarian welfare function subject to the resource constraints and the incentive constraints, i.e., that individuals themselves choose labor supply and savings. A way of finding the second best allocation is to let the planner provide consumption and work conditional on the ability an individual claims to have (and if relevant, the aggregate state). Here this is in the first period c_{1} (i) , y_{1} (i) and in the second, c_{2} (i, j) , y_{1} (i, j) . Individuals then report their abilities to the planner. The strategy of an individual is his first period report and then a reporting plan as a function of the realized period 2 ability. Let’s call the report i_{r} and j_{r} (j) , where the latter is the report as a function of the true ability. The incentive constraint is then that individuals voluntarily report their true ability. According to the revelation principle, this always yields the best incentive compatible allocation. The truth-telling constraint is then that

u (c_{1} (i)) + v (\frac{y_{1} (i)}{θ_{1} (i)}) + β∑_{j} (u (c_{2} (i, j)) + v (\frac{y_{2} (i, j)}{θ_{2} (i, j)})) π_{2} (j|i)  (24)

≥ u (c_{1} (i_{r})) + v (\frac{y_{1} (i_{r})}{θ_{1} (i)}) + β∑_{j} (u (c_{2} (i_{r}, j_{r} (j))) + v (\frac{y_{2} (i_{r}, j_{r} (j))}{θ_{2} (i, j)})) π_{2} (j|i)

for any possible reporting strategy i_{r}, j_{r} (j). Note that the θ_{s} are the true ones in both sides of the inequality. Note also that truth-telling implies that

u (c_{2} (i, j)) + v (\frac{y_{2} (i, j)}{θ_{2} (i, j)}) ≥ u (c_{2} (i_{r}, j_{r} (j))) + v (\frac{y_{2} (i_{r}, j_{r} (j))}{θ_{2} (i, j)}) ∀j,  (25)

otherwise utility could be increased by reporting j_{r} if the second period ability is j. The planning problem is to maximize

∑_{i} (u (c_{1} (i)) + v (\frac{y_{1} (i)}{θ_{1} (i)}) + β∑_{j} (u (c_{2} (i, j)) + v (\frac{y_{2} (i, j)}{θ_{2} (i, j)})) π_{2} (j|i)) π (i)

subject to (23) and (24).

Letting stars ^{∗},denote optimal allocations. We can now define three wedges (distortions) that the informational friction may cause. These are the consumption-leisure (intratemporal) wedges

τ_{y_{1}} (i) ≡ 1 + \frac{v′ (\frac{y_{1}^{∗}(i)}{θ_{1}(i)})}{θ_{1} (i) u′ (c_{1}^{∗} (i))},
τ_{y_{2}} (i, j) ≡ 1 + \frac{v′ (\frac{y_{2}^{∗}(i,j)}{θ_{2}(i,j)})}{θ_{2} (i, j) u′ (c_{2}^{∗} (i, j))},

and the intertemporal wedge

τ_{k} (i) ≡ 1 − \frac{u′ (c_{1}^{∗} (i))}{∑_{j}βRu′ (c_{2} (i, j)) π_{2} (j|i)}.

Clearly, in absence of government interventions, these wedges would be zero by perfect competition and the first-order conditions of private optimization.

3.1 The inverse Euler equation

We will now show that if individual productivities are not always constant over time, the intertemporal wedge will not be zero. The logic is as follows and similar to what we have done above. In an optimal allocation, the resource cost (expected present value of consumption) of providing the equilibrium utility to each type, must be minimized. Consider the following peturbation around the optimal allocation for a given first period ability type i. Increase utility by a marginal amount Δ for all possible second period types {i, j} the agent could become. To compensate, decrease utility by βΔ in the first period. Clearly, the objective function is not changed. What about the thruth-telling constraint?.

First, note that expected utility is not changed.

Second, since utility is changed in parallel for all ability levels the individual could have in the second period, there relative ranking cannot change. In other words, if we add Δ to both sides of (25) it must still be satisfied.

Thus, the incentive constraint is unchanged. However, the resource constraint is not necessarily invariant to this peturbation. Let

c̃_{1} (i; Δ) = u^{−1} (u (c_{1}^{∗} (i)) − βΔ) ,
c̃_{2} (i, j; Δ) = u^{−1} (u (c_{2}^{∗} (i, j)) + Δ)

denote the perturbed consumption levels. The resource expected resource cost of these are

c̃_{1} (i; Δ) + ∑_{j} \frac{1}{R}c̃_{2} (i, j; Δ) π_{2} (j|i)

= u^{−1} (u (c_{1}^{∗} (i)) − βΔ) + ∑_{j} \frac{1}{R}u^{−1} (u (c_{2}^{∗} (i, j)) + Δ) π_{2} (j|i) .

The first-order condition for minimizing the resource cost over Δ must be satisfied at Δ = 0, for the ^{∗} consumption levels to be optimal.

Thus,

0 =

= \frac{−β}{u′ (c_{1}^{∗} (i))} + ∑_{j} \frac{1}{R} \frac{1}{u′ (c_{2}^{∗} (i, j))}π_{2} (j|i)

⇒ \frac{1}{u′ (c_{1}^{∗} (i))} = E_{1} \frac{1}{βRu′ (c_{2}^{∗} (i, .))}

From Jensen’s inequality, we find that

u′ (c_{1}^{∗} (i)) < EβRu′ (c_{2}^{∗} (i, .))
⇒ τ_{k} (i) > 0,

if and only if there is some uncertainty in c_{2}^{∗}. Note that this uncertainty would come from second period ability being random and the allocation implying that second period consumption depends on the realization of ability. If second period ability is non-random, i.e., π_{2} (j|i) = 1 for some j, then τ_{k} (i) = 0.

3.2 A simple logarithmic example: insurance against low ability.

Suppose in the first period, ability is unity and in the second θ > 1 or \frac{1}{θ} with equal probability.Disregard government consumption – set G_{1} = G_{2} = 0, although non-zero spending is quite easily handled. The problem is therefore to provide a good insurance against a low-ability shock when this is not observed.

The first best allocation is the solution to

max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}} u (c_{1}) + v (y_{1}) + β (\frac{u (c_{h}) + v (\frac{y_{h}}{θ} )}{2} + \frac{u (c_{l}) + v (\frac{y_{l}}{\frac{1}{θ}} )}{2})

s.t.0 = y_{1} + \frac{y_{h} + y_{l}}{2R} − c_{1} − \frac{c_{h} + c_{l}}{2R}

First order conditions are

u′ (c_{1}) = λ
v′ (y_{1}) = −λ
βu′ (c_{h}) = \frac{λ}{R}
βu′ (c_{l}) = \frac{λ}{R}
βv′ (\frac{y_{h}}{θ} ) \frac{1}{θ} = −\frac{λ}{R}
βv′ (θy_{l}) θ = −\frac{λ}{R}

3.2.1 A simple example Suppose for example that u (c) = ln (c) and v (n) = −\frac{n^{2}}{2} and βR = 1. Then, we get

\frac{1}{c_{1}} = λ
\frac{1}{c_{h}} = λ
\frac{1}{c_{l}} = λ
y_{1} = λ
\frac{y_{h}}{θ^{2}} = λ
y_{l}θ^{2} = λ
c_{1} + \frac{c_{h} + c_{l}}{2} − y_{1} − \frac{y_{h} + y_{l}}{2} = 0

We see immediately that c_{1} = c_{h} = c_{l} while y_{h} = θ^{2}y_{1} and y_{l} = \frac{y_{1}}{θ^{2}} and y_{1} = \sqrt{\frac{2}{(1+ \frac{1}{2} (θ^{2}+θ^{−2}))}} = n_{1}. Therefore, n_{h} = \frac{y_{h}}{θ} = θ^{2}n_{1} and n_{l} = y_{l}θ = \frac{n_{1}}{θ} . Thus, if the individual becomes of high ability in the second period, he should work more but don’t get any higher consumption. Is this incentive compatible?

We conjecture that the binding incentive constraint is for the high ability type. High has to be given sufficient consumption to make him voluntarily choose not to report being low ability. If he misreports, he gets c_{l} and is asked to produce y_{l}. The constraint is therefore

u (c_{1}) + v (y_{1}) + β (\frac{u (c_{h}) + v (\frac{y_{h}}{θ} )}{2} + \frac{u (c_{l}) + v (θy_{l})}{2})

≥ u (c_{1}) + v (y_{1}) + β (\frac{u (c_{l}) + v (\frac{y_{l}}{θ} )}{2} + \frac{u (c_{l}) + v (θy_{l})}{2})

u (c_{h}) + v (\frac{y_{h}}{θ} ) ≥ u (c_{l}) + v (\frac{y_{l}}{θ} )
ln c_{h} − ln c_{l} ≥ \frac{y_{h}^{2} − y_{l}^{2}}{2θ^{2}}

We conjecture this is binding. The problem is then

max_{c_{1},y_{1},c_{h},c_{l},y_{h},y_{l}} ln (c_{1}) − \frac{y_{1}^{2}}{2} + (\frac{ln c_{h} − \frac{(\frac{y_{h}}{θ})^{2}}{2}}{2} + \frac{ln c_{l} − \frac{(θy_{l})^{2}}{2}}{2} )

s.t.0 = y_{1} + \frac{y_{h} + y_{l}}{2} − c_{1} − \frac{c_{h} + c_{l}}{2}
0 = ln c_{h} − ln c_{l} − \frac{y_{h}^{2} − y_{l}^{2}}{2θ^{2}} .

Denoting the shadow values by λ_{r} and λ_{I} the FOCs for the consumption levels are

c_{1} = \frac{1}{λ_{r}}
c_{h} = \frac{1 + 2λ_{I}}{λ_{r}}
c_{l} = \frac{1 − 2λ_{I}}{λ_{r}}

from which we see

\frac{c_{h}^{∗}}{c_{1}^{∗}} = 1 + 2λ_{I}, \frac{c_{l}^{∗}}{c_{1}^{∗}} = 1 − 2λ_{I}

and

τ_{k} (i) ≡ 1 − \frac{λ_{r}}{\frac{λ_{r}}{1+2λ_{I}} \frac{1}{2} + \frac{λ_{r}}{1−2λ_{I}} \frac{1}{2}} = (2λ_{I})^{2} ,

: implying a positive intertemporal wedge if the IC constraint binds.

The intratemporal wedges are found by analyzing the FOC’s for the labor supplies

y_{1}^{∗} = λ_{r}
y_{h}^{∗} = \frac{λ_{r}}{1 + 2λ_{I}} θ^{2} ⇒ y_{h}^{∗}c_{h}^{∗} = \frac{λ_{r}}{1 + 2λ_{I}} θ^{2} \frac{1 + 2λ_{I}}{λ_{r}} = θ^{2}
y_{l}^{∗} = \frac{λ_{r}}{θ^{4} − 2λ_{I}} θ^{2} ⇒ y_{l}^{∗}c_{l}^{∗} = \frac{λ_{r}}{θ^{4} − 2λ_{I}} θ^{2} \frac{1 − 2λ_{I}}{λ_{r}} = \frac{1 − 2λ_{I}}{θ^{2} (1 − 2λ_{I}θ^{−4})}

τ_{y_{1}} = 1 − \frac{y_{1}^{∗}}{\frac{1}{c_{1}^{∗}}} = 0,
τ_{y_{2}} (h) = 1 + \frac{v′ (\frac{y_{h}^{∗}}{θ} )}{θu′ (c_{h}^{∗})} = 1 + \frac{−\frac{y_{h}^{∗}}{θ}}{θ\frac{1}{c_{h}^{∗}}}

= 1 + \frac{−^{\frac{\frac{λr}{1+2λ_{I}} θ^{2}}{θ}}}{θ_{\frac{1}{\frac{1+2λ_{I}}{λr}}}} = 0

and

τ_{y_{2}} (l) = 1 + \frac{v′ (θy_{l}^{∗})}{\frac{1}{θ}u′ (c_{l}^{∗})} = 1 + \frac{−θy_{l}^{∗}}{\frac{1}{θ} \frac{1}{c_{h}^{∗}}}

= 1 + \frac{−θ\frac{λ_{r}}{θ^{4}−2λ_{I}} θ^{2}}{\frac{1}{θ} \frac{1}{\frac{1−2λ_{I}}{λr}}} = 2λ_{I} \frac{θ^{4} − 1}{θ^{4} − 2λ_{I}} > 0

As we see, the wedge for the high ability types is zero, but positive for the low ability type.^{5} For later use, we note that

5The wedge, asymptotes to infinity as λ_{I} approach \frac{θ^{4}}{2} . Can you explain?

y_{1}^{∗}c_{1}^{∗} = 1  (26)
y_{h}^{∗}c_{h}^{∗} = \frac{λ_{r}}{1 + 2λ_{I}} θ^{2} \frac{1 + 2λ_{I}}{λ_{r}} = θ^{2}
y_{l}^{∗}c_{l}^{∗} = \frac{λ_{r}}{θ^{4} − 2λ_{I}} θ^{2} \frac{1 − 2λ_{I}}{λ_{r}} = \frac{1 − 2λ_{I}}{θ^{2} (1 − 2λ_{I}θ^{−4})}

3.3 Implementation

It is tempting to interpret the wedges as taxes and subsidies. However, this is not entirely correct since the wedges in general are functions of all taxes. Furthermore, while there is typically a unique set of wedges this is generically not true for the taxes. As we have discussed above, many different tax systems might implement the optimal allocation. One example is the draconian, use 100% taxation for every choice except the optimal ones.

Only by putting additional restrictions is the implementing tax system found. Let us consider a combination if linear labor taxes and savings taxes that together with type specific transfers implement the allocation in the example. To do this, consider the individual problem,

max_{c_{1},y_{1},s,y_{h},y_{l},c_{h},c_{l}} ln (c_{1}) − \frac{y_{1}^{2}}{2} + (\frac{ln c_{h} − \frac{(\frac{y_{h}}{θ})^{2}}{2}}{2} + \frac{ln c_{l} − \frac{(θy_{l})^{2}}{2}}{2} )

s.t.0 = y_{1} (1 − τ_{1}) − c_{1} − s + T
0 = y_{h} (1 − τ_{h}) + s (1 − τ_{s,h}) − c_{h} + T_{h}
0 = y_{l} (1 − τ_{h}) + s (1 − τ_{s,l}) − c_{l} + T_{l}

with Lagrange multipliers λ_{1}, λ_{h} and λ_{r}. First order conditions for the indviduals are;

\frac{1}{c_{1}} = λ_{1}
y_{1} = λ_{1} (1 − τ_{1})
λ_{1} = λ_{h} (1 − τ_{s,h}) + λ_{l} (1 − τ_{l,h})
\frac{y_{h}}{2θ^{2}} = λ_{h} (1 − τ_{h})  (27)
\frac{θ^{2}y_{l}}{2} = λ_{l} (1 − τ_{l})
\frac{1}{2c_{h}} = λ_{h}
\frac{1}{2c_{l}} = λ_{l}

Using this, we see that

\frac{1}{c_{1}} = \frac{1}{2c_{h}} (1 − τ_{s,h}) + \frac{1}{2c_{l}} (1 − τ_{l,h})

Setting,

τ_{s,h} = −2λ_{I}
τ_{s,l} = 2λ_{I}.

this gives

\frac{1}{c_{1}} = \frac{1}{2c_{h}} (1 + 2λ_{I}) + \frac{1}{2c_{l}} (1 − 2λ_{I})

which is satisfied if we plug in the optimal allocation c_{h}^{∗} = c_{1}^{∗} (1 + 2λ_{I}) and c_{l}^{∗} = c_{1}^{∗} (1 − 2λ_{I})

\frac{1}{c_{1}^{∗}} = \frac{1 + 2λ_{I}}{2c_{1}^{∗} (1 + 2λ_{I})} + \frac{1 − 2λ_{I}}{2c_{1}^{∗}1 − 2λ_{I}}

Note that the expected capital income tax rate is zero, but it will make savings lower than without any taxes. Why?

Similarly, by noting from (26) that in the optimal second best allocation, we want

y_{1}c_{1} = y_{1}^{∗}c_{1}^{∗} = 1,

which is implemented by τ_{1} = 0. For the high ability type, the second best allocation in (26) is that y_{h}^{∗}c_{h}^{∗} = θ^{2}, which is implemented by τ_{h} = 0 since (27) implies that y_{h}c_{h} = θ^{2} (1 − τ_{h}) .

For the low ability type, we want y_{l}^{∗}c_{l}^{∗} = \frac{1−2λ_{I}}{θ^{2}(1−2λ_{I}θ^{−4})}. From (27), we know y_{l}c_{l} = \frac{1−τ_{l}}{θ^{2}} ,so we solve

\frac{1 − τ_{l}}{θ^{2}} = \frac{1 − 2λ_{I}}{θ^{2} (1 − 2λ_{I}θ^{−4})}

⇒ τ_{l} = 2λ_{I} \frac{θ^{4} − 1}{θ^{4} − 2λ_{I}} .

Note that if λ_{I} = \frac{1}{2}, τ_{l} = 1. I.e., the tax rate is 100%. There is no point going higher than that, so λ_{I} cannot be higher than \frac{1}{2}.

Finally, to find the complete allocation, we use the budget constraints of the private individual and the aggregate resource constraint. This will recover the transfers T, T_{h} and T_{l}. We should note that T_{l} > T_{h} is consistent with incentive compatibiity. Why? Because if you claim to be a low ability type you will have to may a high labor income tax which is bad if you are high ability and earn a high income. Thus, by taxing high income lower, we can have a transfer system that transfers more to the low ability types.

3.3.1 Third best – laizzes faire The allocation in without any government involvements is easily found by setting all taxes to zero.

\frac{1}{c_{1}} = λ_{1}
y_{1} = λ_{1}
λ_{1} = λ_{h} + λ_{l}
\frac{y_{h}}{2θ^{2}} = λ_{h}  (28)
\frac{θ^{2}y_{l}}{2} = λ_{l}
\frac{1}{2c_{h}} = λ_{h}
\frac{1}{2c_{l}} = λ_{l}

Using these and the budget constraints, we get

y_{1} = \frac{1}{c_{1}}
\frac{1}{c_{1}} = \frac{1}{2c_{h}} + \frac{1}{2c_{l}}
\frac{y_{h}}{2θ^{2}} = \frac{1}{2c_{h}}
\frac{θ^{2}y_{l}}{2} = \frac{1}{2c_{l}}
y_{1} = c_{1} + s
y_{h} + s = c_{h}
y_{l} + s = c_{l}

which implies

c_{1} + s = \frac{1}{c_{1}}
\frac{1}{c_{1}} = \frac{1}{2c_{h}} + \frac{1}{2c_{l}}
c_{h} = \frac{1}{2}s + \frac{1}{2}\sqrt{s^{2} + 4θ^{2}}
c_{l} = \frac{\frac{1}{2}sθ + \frac{1}{2} \sqrt{s^{2}θ^{2} + 4}}{θ}

I did not find an analytical solution to this, but setting θ = 1.1 I found the solution {c_{1} = 0.997 75, c_{h} = 1. 102 3, s = 4. 504 5 × 10^{−3}, c_{l} = 0.911 35} .

3.4 Time consistency

Under the Mirrlees approach, the government announces a menu of taxes or of consumption baskets. People then make choices that in equilibrium reveal their true types (abilities) to the government. Suppose the government could then re-optimize. Would it like to do this?

The problem is more severe in a dynamic setting provided abilities are persistent. Why? In a finite horizon economy, there might only be very bad equilibria (Roberts, 84). But

better equilibria might arise in infinite horizon.