MU Problem Set Worksheet
Description
1 attachmentsSlide 1 of 1attachment_1attachment_1.slider-slide > img { width: 100%; display: block; }
.slider-slide > img:focus { margin: auto; }
Unformatted Attachment Preview
ECO 317 – Intermediate Macroeconomic Theory
Problem Set #3
David Lindequist
Spring 2022
This problem set covers material from the Solow Growth Model. The relevant chapters in
the textbook are chapter 4 and chapter 8. You can achieve a maximum of 100 points. The
problem set is due in class on Thursday, March 31. You are allowed to work in groups, but
please hand in your individual copy of the solution and indicate who you worked with.
Question 1 (15 points)
Chinas real GDP per capita has been growing rapidly in the last four decades. Its average
growth rate between 1979 and 2018 was 9.5%. At the same time, US real GDP per capita
grew an average of 1.7% per year. In 2019, real GDP per capita in the US was equal to
$65,118. Chinese real GDP per capita was equal to $16,784 (in $US PPP).
(a) Assuming the average growth rates from the last few decades persist into the future,
how long will it take for China to catch up with the US in terms of income per capita?
(b) Now assume that Chinas average GDP growth rate after the year 2019 will slow down
to 7% while the US growth rate remains at 1.7% per year. How long will it take for
China to catch up with the US in terms of income per capita?
(c) Now assume that Chinas average GDP growth rate after the year 2019 will slow down
to 3% while the US growth rate remains at 1.7% per year. How long will it take for
China to catch up with the US in terms of income per capita?
(d) Assume that the US growth rate remains at 1.7% per year. Calculate the average
annual growth rate that is needed for China to catch up with the US in 2040.
(e) Assume that the US growth rate remains at 1.7% for the next 10 years and increases
to 3% afterwards. The Chinese growth rate remains at 9.5% per year. How long will
it take for China to catch up with the US in terms of income per capita?
Question 2 (45 points)
Consider an overlapping generations model in which households live for two periods. Lifetime
utility for a household born at t is given by
U = ln cy,t + ? ln c0,t+1 , ? ? [0, 1]
1
(1)
where cy,t denotes consumption when young, and c0,t+1 denotes consumption when old.
The household faces two flow budget constraints. When young, the households inelastically
supplies one unit of labor and earns wage rate wt :
cy,t + st ? wt
(2)
When old, the household consumes his savings:
co,t+1 ? (1 + Rt+1 ) st
(3)
where Rt+1 is the gross return on capital in t + 1. Note that we assume here that capital
does not depreciate, i.e. ? = 0. Cohort size at t is given by Nt . Population grows at rate n.
(a) Derive expressions for the households optimal consumption when young (c?y,t ), optimal
consumption when old (c?o,t+1 ), and optimal savings s?t .
Assume that output in this economy is produced by a firm which lives forever and combines
labor Lt and capital Kt to produce output according to
Yt = At F (Kt , Lt ) = At Kt? L1??
t
(4)
The firm hires labor at wage rate wt and capital at interest rate Rt .
(b) Solve the firms profit maximization problem to derive its optimal labor demand LD
t
and capital demand KtD .
(c) Determine the equilibrium on the labor market. That is, derive L?t and wt? .
The capital stock evolves according to
Kt+1 = st Nt
(5)
(d) Determine the equilibrium on the capital market. That is, derive Kt? and Rt? .
(e) Show that in equilibrium, the goods market clears: Yt = Ct + It .
t+1
(f) Find an expression that links the future capital stock per worker kt+1 ? K
to the
Nt+1
Kt
current capital stock per worker kt ? Nt and the exogenous variables of the model
(?, ?, A, n).
(g) Solve for the capital stock per worker and output per worker in the steady state of the
economy.
(h) Determine what happens to output per worker in steady state if households become
more patient, i.e. if ? increases. Provide some intuition.
(i) Assume that you have two countries, A and B. Suppose that in steady state, income
per capita in country A is three times higher than income in country B. It holds that
?A = 0.7, ?B = 0.8, and ?A = ?B = 1/3. Further assume that nA = nB . Assuming
that AB = 1, determine the value of AA that rationalizes this income difference.
2
Question 3 (20 points)
This question is similar to Question 2 with the only difference that we assume a different
lifetime utility function for households. More specifically, lifetime utility for a household
born at t is now given by
U = (cy,t )0.5 + ?(c0,t+1 )0.5 , ? ? [0, 1]
(6)
As before, cy,t denotes consumption when young, and c0,t+1 denotes consumption when old.
The household faces two flow budget constraints. When young, the households inelastically
supplies one unit of labor and earns wage rate wt :
cy,t + st ? wt
(7)
When old, the household consumes his savings:
co,t+1 ? (1 + Rt+1 ) st
(8)
where Rt+1 is gross the return on capital in t + 1. Note that we assume here that capital
does not depreciate, i.e. ? = 0. Cohort size at t is given by Nt . Population grows at rate n.
(a) Derive an expression for the households optimal savings s?t . Note how it differs from
the expression for savings you obtained in part (a) of Question 2.
Assume that output in this economy is produced by a firm which lives forever and combines
labor Lt and capital Kt to produce output according to
Yt = At F (Kt , Lt ) = At Kt? L1??
t
(9)
The firm hires labor at wage rate wt and capital at interest rate Rt .
(b) Determine the equilibrium on the labor market. That is, derive L?t and wt? . Is the
labor market equilibrium different from the equilibrium you found in Question 2?
The capital stock evolves according to
Kt+1 = st Nt
(10)
(c) Graph the equilibrium on the capital market, i.e. provide a graph of capital supply and
capital demand in a space where capital Kt is on the x-axis and the interest rate Rt
is on the y-axis. [Note: You are not required to solve for the equilibrium analytically
here as this would be quite challenging algebraically.]
(d) Describe how the capital market equilibrium you depicted in part (c) is different from
the capital market equilibrium in Question 2.
(e) What happens to equilibrium capital when households become more patient, i.e. if
? increases? Is the effect on equilibrium capital smaller or greater compared to the
effect of an increase in ? on equilibrium capital in Question 2? Provide some intuition.
[Note: You do not need to provide any formal analysis here, intuition based on your
results in part (c) and (d) suffices.]
3
Question 4 (20 points)
The central equation governing the dynamics of capital per worker in the Solow model is
given by:
kt+1 =
?(1 ? ?)At ?
k
(1 + ?)(1 + n) t
(11)
yt = At kt?
(12)
Output per worker is given by
(a) Solve for expressions for steady state capital and output per worker as functions of At
and other parameters.
Consider a country with ? = 0.9, ? = 13 , n = 0 and At = 1.
(b) Calculate the steady state level of output per worker for this country.
(c) Now suppose that at some time t, At is permanently increased to At = 3. Calculate
and plot the responses of output per worker over the next 10 periods.
(d) Calculate the new steady state level of output per worker with At = 3. Do your results
from (c) confirm that the economy is indeed converging to this new steady state?
(e) Assume that the country doubles its level of total factor productivity At . Is output
per capita in steady state doubled in response?
4
Purchase answer to see full
attachment
Explanation & Answer:
400 words
Tags:
Economic
Equilibrium
problem set
User generated content is uploaded by users for the purposes of learning and should be used following FENTYESSAYS.COM ESSAY’s honor code & terms of service.
