Example Question – Solow Model and OLG
Output per worker in each period is a function of capital per worker:
yt = f (kt ) = Ak tα + Bk t ,
where A>1, 0<B<1, and 0<α<1.
Population growth rate is n and the depreciation rate is δ ∈ (0,1) .
i. The Solow Model
Consider the Solow model in which aggregate saving in the economy is a
fraction s of total output.
a. Find the dynamical system governing the evolution of capital per worker
over time: k t +1 = φ (k t )
(2 marks)
b. Find φ ' (kt )
(2 marks)
c. Find lim kt →0 φ ' ( k t ) and
lim kt →∞ φ ' (kt )
(2 marks)
d. Find a sufficiently large s such that the economy is characterized by
growth in the long run. (A non-trivial steady state doesn’t exist and
income is endlessly growing).
(2 marks)
e. Find a condition on the coefficient B such that a sufficiently large s, as
defined above, exists.
(2 marks)
ii. The OLG model
Consider now the overlapping generations model. Individuals live for two
periods. They work, and save in the first period of their lives and retire in the
second, consuming their entire wealth. Utility of an individual that joins the
economy in period t is
y 1−β
t
( ) (c )
ut = c
where
old.
o β
t +1
o
0 < β < 1, cty is consumption when young, and ct +1 is consumption when
(Production, population, and depreciation are as above)
f. Find the marginal product of labour, wt , as a function of k t .
(2 marks)
g. Solve the optimization problem of the young, and find their saving as a
function of wt .
(2 marks)
h. Find the dynamical system governing the evolution of capital per worker
over time: k t +1 = ψ (k t ) .
(2 marks)
i.
Find the steady state level of capital per worker k = ψ (k ) .
(2 marks)
j.
The coefficient B has no effect on the level of capital in the steady state.
The reason is (choose one of the following as an answer):
B has no effect on the wage rate and, under the specific utility function,
the rate of return on savings has no effect on the saving rate.
B has no effect on the wage rate and no effect on the rate of return on
savings
B has no effect on the accumulation of capital since the old consume
the entire stock of capital net of depreciation, after production takes
place
In the OLG model the rate of capital depreciation is one, and B is
practically a manipulation on the depreciation rate
1.
2.
3.
4.
(2 marks)
Suppose now that utility is:
ut = cty + cto+1 .
(Everything else is unchanged).
k. Under the assumption that B < δ , Find the level of capital in t+1 such that
individuals in t are indifferent regarding their savings.
(2 marks)
l.
Find the steady state level of capital per worker if B > δ .
(2 marks)
m. Find the dynamical system
(Bonus part)