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1 Consider the relationship between the savings rate and the steady level of per capita consumption that results from the Solow model

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1 Consider the relationship between the savings rate and the steady level of per capita consumption that results from the Solow model. Which level of the savings rate maximizes steady state per capita consumption? What does this optimal level of savings imply for respective steady state level of the interest rate and the fraction of national income that is used for consumption?

2 Assume the Solow growth model with the production function

y(t) = A(t)k(t)α without any population growth (i.e. n = 0).

a) Derive a closed form solution for the capital intensity k(t).

b) Assume now that A(t) = 0.5(sin(t)+2). Solve for the time path of

the capital intensity and explain your solution.
3 Consider two countries A,B and assume that income dynamics in

both countries are governed by the process y?i = b −ayi for i = A,B

with b = 10 and a = 0.02. Assume that at t = 0 the incomes in the

countries are given by y(0)A = 250 and y(0)B = 125, respectively.

a) How long does it take until country B reaches the initial income level of country A?

b) During the time period identified in a) country A’s income will have increased further. How long will it take to for country B to catch up with this income gap?

c) Compute the time it takes until the initial income difference between the two countries will have halved?

4 Assume that consumption in a Solow model is given by c(t) = c? +(1−s)f(k(t)) with c? > 0. Discuss the implications of this assumption for existence and stability of a balanced growth path.
5 Consider the Solow–Stiglitz model with a savings function

according to which the savings rate out of capital income sp differs

from the savings rate out of labor income sw , where

1 ≥ sp > sw ≥ 0. How does this change the results regarding the

long run distribution of wealth? What happens if sw = 0?
7 Assume that a production function Y = F(X?,L) = AX?α L1−α, with

A > 0. Here X? is a factor of production (e.g. land), which is in fixed

supply and that can not be accumulated. Thus, all output is used

for consumption.

a) Assume that L?

L = n > 0. What does this imply for the evolution of

per capital income y = Y/L?

b) Now assume that L?

L = n(y) = n? +ln

 

y

y+m

with n? > 0 and m > 0.

b.1) Discuss the properties of the assumed relationship between per capita income and population growth.

b.2) Derive the steady state level of per capita income and the steady level of the population size. Is this steady state solution stable?

b.3) Assume an exogenous increase in total factor productivity A. What

are the consequences for the steady level of per capital income and the population size? Describe the now resulting steady state as well as the transition towards this steady state from the initial situation. Discuss your result. Why is this a dismal result?
1 How does the solution of the infinite horizon cake eating problem

change if the utility function is given by V =

R ∞

0 e−ρt c1−θ

1−θ with

θ 6= 1?
1 Consider the Ramsey model with a Cobb Douglas production

function f(k) = kα and assume that the policy function

c(t) = g(k(t)) is given by c(t) = γk(t).

a) Use the Keynes Ramsey rule and the fact that the proposed policy

function implies c? = γk? to identify restrictions over the parameters

of the model that allow for such a linear policy function. What is the

implied value for γ?

b) Solve the differential equation k? = f(k)−δk −c using this policy

function.

2 Discuss based on (CF) how the intertemporal elasticity of

substitution 1

σ affects the reaction of the propensity to consume

out of the present value of income to changes of the interest rate

(Assume for simplicity that r(t) is constant, i.e. r(t) = r?).
3 Consider a Ramsey model with consumption externalities as

given by u(c(t),c?(t)) = (c(t)c?(t)−η)1−σ

1−σ with σ > 0 and 0 < η < 1 and

where c?(t) represents average consumption. Assume a standard

neoclassical production function f(k(t)) and the resource

constraint k?(t) = f(k(t))−δk(t)−c?(t) Show that the decentralized

solution is efficient.

u(c(t),c?(t)) =

 

c(t)1−η

 

c(t)

c?(t)

η1−σ

1−σ

=

(c(t)c?(t)−η)1−σ

1−σ

, 0 < η < 1

where the individual utility function is
5 Consider the Ramsey model with production externalities as

described

Assume that the government taxes capital incomes at the rate τ

while rebating all the revenues back to the households via lump

sum transfers. Derive the tax rate τ∗ which makes the

decentralized solution identical to the optimal solution.

f(k(t),k?(t)) = A(t)k(t)α

, A(t) = k?(t)α

−γ

0 < γ < 1

In a decentralized equilibrium, individual firms take k?(t) as given, while a central planner a priori assumes that k(t) = k?(t).

4 Consider the same model as before and assume that the utility

function takes the form u(c(t),c?(t)) = u(c(t))+s(c(t)/c?(t)), where u and s are both strictly concave functions.

a) Show that the steady states of the decentralized and the optimal

solution coincide.

b) Show that the decentralized solution is inefficient.
1 Consider an endogenous growth model with productive government expenditures and a production function

Y(t) = K(t)α (LG(t))1−α

. Show that the welfare maximizing

government share is given by G/Y = 1−α irrespective of whether government expenditures are financed via distorting taxes or lump sum taxes.
1 Consider the endogenous growth model with an expanding variety of intermediate goods.

a) Formulate the current value Hamiltonian which must be solved in order to derive the Pareto efficient growth equilibrium.

b) Solve the respective maximization problem and derive the optimal interest rate, the optimal growth rate as well as the optimal input level of the intermediate goods.

c) In what respects does this optimal solution differ from the decentralized solution?
1 Consider the Ramsey model with an exhaustible resource and assume that S(t) represents carbon budget that remains given a specific upper bound on global warming. Assume that anthropogenic carbon emissions are proportional R(t) and that there is some kind of natural decay rate µ of the atmospheric carbon carbon concentration such that S?(t) = µS(t)−φR(t).

How does this change the results of the model? What is the minimum level of growth required in order to obtain a non negative growth rate for consumption?