521/601 Adv. Macroeconomic Theory
Problem Set 1
Professor: Miguel A. Iraola
Fall 2020
Due Date: 9/23/2020
1. Finite Horizon: Consider the following Planner’s Problem:
max
{ct ,kt+1 }t=T
t=0
:
t=T
X
β t log (ct )
t=0
ct + kt+1 = Aktα , t = 0, 1, 2...
s.t.
ct , kt+1 ≥ 0, t = 0, 1, 2...
k0 > 0, given.
a) Suppose that A = 10, α = 0.3, β = 0.95, k0 = 0.05 y T = 1. Compute the optimal
capital level at t = 1.
b) (Voluntary) Provide a MATLAB program to compute the optimal sequence of
capital for a generic finite horizon T. Plot the optimal sequence of capital and the
steady state level of capital for T = 5, T = 10, T = 50 y T = 100. Guess the
optimal sequence of capital for T = +∞.
2. Analitic Solution: Consider the following Planner’s Problem:
max
{ct ,kt+1 }t=∞
t=0
s.t.
:
t=∞
X
β t [γlog (ct ) + (1 − γ)log(1 − nt )]
t=0
ct + kt+1 = Aktα nt1−α , t = 0, 1, 2...
ct , kt+1 ≥ 0, nt ∈ [0, 1] t = 0, 1, 2...
k0 > 0, given.
The notation is as follows: β ∈ (0, 1) is the subjective discount factor, γ ∈ (0, 1)
determines the relative weight of each good in the utility function, ct denotes per
capita consumption at t, kt is the stock of physical capital per capita at t, nt is the
time per capita devoted to work at t, α ∈ (0, 1) is the capital income share and A > 0
is the Total Factor Productivity. Assume that the population size is normalized to one.
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a) Provide the equations that characterize the solution.
b) Write the optimal sequence of capital kt as a function of k0 . (Hint: You may assume that the time devoted to work is constant through time and that consumption
is a constant fraction of output and then verify these assumptions).
c) Compute the steady state level of capital kss > 0. Is it stable?
3. Solow Problem: Consider an economy such that the per capita production level is
yt = ktα , with α = 0.4 where kt denotes the level of capital per capita. Assume that
investment is a constant fraction of production s = 0.1. Assume that the depreciation
of capital is δ = 0.08. Compute the steady state of this economy. Is it stable?
4. Changing Population: Consider the following optimization problem:
max
{Ct ,Kt+1 }t=∞
t=0
s.t.
:
t=∞
X
β t log (Ct /Nt )
t=0
Ct + Kt+1 − (1 − δ)Kt = AKtα Nt1−α , t = 0, 1, 2...
Ct , Kt+1 ≥ 0, t = 0, 1, 2...
K0 > 0, given.
Nt = η t N0 .
The notation is as follows: Ct consumption at t, Kt stock of capital, Nt population
size, δ ∈ (0, 1) depreciation rate, A > 0 total factor productivity η > 0 population
growth rate. Note that the endowment of time each period is equal to one, and hence,
the amount of time devoted to work is equal to the population size Nt .
a) We are here assuming that individuals allocate all their endowment of time to
work. Why?
b) Define a balanced growth path and provide the equations characterizing it.
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