Problem Set 1 (Growth Models) ECON 6002, Semester 2 2016

Problem 1
Consider the Solow model with production function
Yt = F(Kt
; Lt) = K
t
(ALt)
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with A = 1 and = 0:3. Yt
, Kt
, and Lt are the levels of output, physical capital, and labor
at time t, respectively. The savings rate is s = 0:30, with population growth rate and capital
depreciation rate equal to n = 0:03 and = 0:05, respectively.
(i) Derive the production function in per-capita term (yt = Yt=Lt) as function of capital
per capita (kt = Kt=Lt).
(ii) Derive and describe the equation for _kt (the change in capital per capita over time).
(iii) Sketch the Solow diagram in this economy and write the equations (actual and
break-even investments).
(iv) What are the steady-state growth rates of yt and Yt?
(v) Derive the steady state values (when kt 6= 0) of kt
, yt
. Further, assume that the
economy starts at t = 0 with initial stock of labor L0ó whatís the steady state value of Yt
(hint: it depends on L0 and is time varying)? Is the steady state stable (illustrate this using
the Solow diagram)?
(vi) Suppose a politician argues that the savings rate is too high (and consumption is too
low) and proposes that the savings rate should be permanently lowered to s = 0:20. Analyze,
using a diagram and in words, what should happen to kt over time under this proposal.
(vii) Another policitian disagrees with this policy change on the ground that the permanent
decrease in s would lead to lower level and growth rate of output per capita, both in the
short run and long run. In the context of Solow model, do you agree with this statement?
Why? Would your answer change if there is a positive technological progress (At grows at a
constant rate)?
Problem 2
Suppose an economy has a production function given by Yt = K0:5
t
(AtLt)
0:5
. Assume that
L_
t=Lt = n = 0:05 and A_
t=At = g = 0:07. Further, assume that the savings rate is constant
at s = 0:20 and the capital depreciates at the rate = 0:03.
(i) Derive output per e§ective labor, yt = Yt=AtLt
, as a function of capital per e§ective
labor, kt = Kt=AtLt
.
(ii) Find the steady state (balanced growth path) values of kt and yt
.
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(iii) What are the growth rates of kt
, yt
,
~kt = Kt=Lt (capital per capita), and y~t = Yt=Lt
(output per capita) in the steady state?
(iv) Assume that the economy starts at time t = 0 with an initial stock of labor L0.
Suppose at time t = j > 0, there is an extremely large number of refugees coming to this
country (legally) that makes the labor stock, L, to double immediately (the economy was
already at the steady state prior to this increase). After this sudden increase, the growth
rate of population (and labor) goes back to the previous level of n = 0:05. Describe what
happens to kt and yt both in the short-run and long-run (you can use a combination of
diagrams and equations). Do the same thing for ~kt and y~t
.