ECON3102 Macroeconomics 3

29/08/2022 by 7:45 PM

Solve the following three problems.
1. [1+2+2=5 points] Consider the standard economy we considered in the
class with
Y t = 4K 0.3
t
L 0.7
t
U = c 0.5
y
c 0.5
o
Total population in this country is N = 200 and the initial total (gross)
capital stock in the country is 300. Assume that no one in this economy
is unemployed (N = L).
(a) Calculate the steady state values of the following variables:
k t ;r t ,w t ;Y t
(b) Suppose the economy is at the steady state (see part (a)). Sud-
denly a series of earthquakes destroyed one quarter of the country’s
capital stock. Calculate the new steady state values of the above
variables mentioned in part (a).
(c) Compare the steady state values derived in (a) with (b) and the
long run after the earth quake. Explain any differences and simi-
larities. Draw a picture that supports your answer.

2. [3+1+2+4=10 points] Consider an overlapping generation model with
L t agents in period t. The net population growth rate is n = 2%. The
production function in this economy is given by
F(K,L) = AK β L 1−β
with A = 10 and β =
2
5 . Life time utility of an agent is given by
U(c) = log(c y ) + αlog(c o )
with α =
4
5 . At time t, each of the N t
agents are supplying labour 1.
(a) i. Specify the utility maximization problem of an agent.
ii. Derive the steady state capital stock under the competitive
equilibrium.
[Hint: modify the steady state formula from week 3 by the new
saving rate from week 4 and the population growth rate. Note
that L t = N 0 (1 + n) t ]
To continue with (c) the (general) solution is:
¯
k =
?
α
1 − α
A(1 − β)
1 + n
? 1
1−β
iii. Figure out the Golden rule capital stock for this economy. The
capital level maximizes the steady state consumption level c ∗ .
To continue with (b), (c) and (d) the (general) solution is:
k gr =
?
Aβ
1 + n
? 1
1−β
(b) Calculate the golden rule for capital k gr with the given parameters.
(c) Compare the golden rule level of capital stock with the competitive
level derived in part (a) ii.
(d) Suppose there is a mandatory saving (along with the routine sav-
ing) requirement imposed by the government. Government knows
that the agents are myopic and therefore it collects π proportion
of income w when the agents are young and when they are old in
the next period, the government returns the saving at a market
rate of interest.
i. Write down your new budget constraint
ii. Show the path for capital under the competitive equilibrium.
Hint: there are two cases
α
1+α
< π and
α
1+α
≥ π
iii. Calculate the steady state under the competitive equilibrium.
iv. Is there a change in Golden rule capital stock?
3. [3 points] Consider a standard Cobb-Douglas production function. Us-
ing variables rather than numbers show that under competitive equi-
librium, the pure profit (Total Revenue - Total Cost) is zero.