John Hassler

June 16, 1997

**EXAM
Macroeconomics I
5 points**

Choose 5 out of the following 6. Preferably, you should not use more than 1 page per. Worth 10 points each.

- Evaluated the following statement "In a closed economy, a big national debt is no problem because the members of society owes the debt to themselves."
- In a standard Solow model, let
*Y=K**a**N*1-a with a =0.5. Let population growth rate be 0.04, the savings rate = 0.10 and depreciation 0.10. Compute the steady state consumption per capita. - Consider an OLG model which has reached a dynamically inefficient steady state. Assume the government makes a transfers to the currently old and finance it by borrowing from the young. The government claims it never has to raise taxes but can roll over the debt forever. Could they?
- Is the following statement true, false or uncertain?
"If an individual’s savings behavior
*is*affected by risk one knows he*cannot*be risk-neutral." - Evaluated the following statement. "The government should try to smooth labor taxes, i.e., keep them constant over time."
- Evaluated the following statement. "Evidence from the growth literature suggest that the share of income that goes to factors of production that can be accumulated is larger than the share that goes to physical capital, recorded in the national accounts."

Choose 1 out of the following 2. Provide a strict formal argument. Worth 25 points

In this problem we are going to use the Lucas Asset Pricing
model to price an asset (the tree in the original Lucas model).
The economy consists of (infinitely) many individuals, indexed by
the real numbers between 0 and 1. There is a competitive market
for equally many trees, each producing a non-storable and
stochastic output *d**t*. In a
particular period each tree produces the same amount, so there is
no idiosyncratic uncertainty. By buying the tree in period *t *at
price *p**t** *the holder
gets next periods output *d**t+*1 and
can sell the tree at *p**t*+1.
The gross stochastic return is thus (*p**t*+1+*d**t+*1)/* p**t*.
The representative agent’s consumption–investment
problem in any period *t* can be represented by the value
functions

where *A**t** *is
financial assets and *c**t** *is
consumption.

1. Set up the Bellman equation for the consumption problem in . Make sure the notation is exact.

2. Derive the Euler equation.

We now assume, as in Lucas (1978) that humans and apple trees
live for ever. Assume, furthermore, that *U*(*c*)=(1-*a*)-1 *c*1-*a**
*. Since there is no way to transfer real resources between
periods it must be that *c**t*=*d**t*.

3. Use the assumptions above and Euler equation to derive an expression for the price of the Lucas asset.

Now assume that dividends, *dt*, follows

4. Given it is reasonable to guess that the asset price is
linear in the current dividend. So that we can write *p**t*=*pd**t*
with *p*, the price–dividend ratio,* *being
constant for all *t. *Use this guess together with the Euler
equation to get an explicit solution for *p* in terms of
expectations of *d**t*+1*/d**t*.

5. Use the result that if e is *N*(0*,**s** *) then

to express *p *as a function of problem parameters. What
is the price–divident ratio when *a *equals
unity and how does it change when *a*
increases/decreases from unity?

(Hint; For any constant *k, *. Use this and to show that is a normal with mean
zero and variance *k*2. What is the
appropriate expression for *k *in this case?)

We are now going to analyze recent demographic changes in a
closed economy Ramsey model where people work for a finite time
and then retire. We will analyze the case where fertility
unexpectedly falls permanently, which will decrease the support
ratio, i.e., the share of workers in population. Let *P**t** *and *Nt* be* *population
and number of workers alive at time *t.* Define the support
ratio *A**t**
*as *Nt*/*P**t* and the
growth rates of *A**t, P, *and
*N *by *a*, *p, *and *n.* Note that *a=n-p.*

Assume there is a CRS production function *F*(*K**t*,*N**t*)
where *K *is non-depreciating capital. Let *c**t** *be consumption per capita (*C/P*)*
*and *k**t** *be the
capital to labor ratio (*K*/*N*). Let *f*(*k*)=*F*(*k,*1).

1. Show that the capital accumulation equation can be written

3

Let there be a social planner who maximizes the sum of discounted individual utility. He thus solves

4

2. State the Hamiltonian for the planning problem prior to the shock, i.e., for constant growth rates. Show that the necessary conditions for an optimum imply that

5

3. Draw the phase diagram for the planning problem in the *c,k
*space *prior* to the shock with arrows that show the
movement everywhere in the diagram and that include potential
saddle paths.

4. Now analyze what happens after a sudden fall in *n *at
time *t*0*. *Note that while *n
*falls immediately to its new steady state it is going to take
time for the support ratio to reach its new lower steady state.
Draw one phase diagram for the initial phase, just after the fall
in *n, *and one for the long run case. Assume that long run
effects are dominated by the fall in *A*.
Describe how consumption per capita and capital stocks per worker
reacts dynamically to the shock. To simplify, assume that the
dynamics are given by the phase diagram for the initial phase
between *t*0* *and some future
date *t*1. From *t*1
and onwards, the dynamics are given by the long run phase
diagram.

Choose 1 out of the following 2. Emphasize discussion rather than formalism. Worth 25 points.

Discuss what the modern theory of economic growth predicts about convergence of the income per capita of different countries or regions! Do the predictions of different theories square with the available empirical evidence?

Discuss the effects of fully funded versus Pay-as-you-go pension systems in the OLG and the Ramsey model. Assume, at least in the primary discussion, that the economy is closed and has a perfect capital market.

A pass will require 60 points (out of 100) and a pass with distinction 80 points.

**Good Luck!**