Economic Growth
1. Quantitative Solow Model Exercise. Suppose that we have a standard Solow model.
It is the same setup as question 2 in problem set 3. The central equation summarizing the
model is given by:
kt+1 = sAkα
t + (1 − δ)kt
.
Other variables can be expressed in terms of capital per worker as:
yt = Akα
t
,
ct = (1 − s)yt
,
wt = (1 − α)Akα
t
,
Rt = αAkα−1
t
.
(a) Solve for an analytic expression for the steady state capital per worker, k
∗
.
(b) Suppose that s = 0.3, A = 1, α =
1
3
, and δ = 0.1. What is the numeric value for k
∗
given
these values?
(c) Suppose that time begins in period t = 0. Create an Excel spreadsheet with rows corresponding
to time periods. Have time periods running from 0 to 100. I want you to do
three simulations of capital per worker considering three different values of the predetermined
capital stock in time period 0 where capital per worker is: (i) k0 = 0.5 × k
∗
, (ii)
k0 = 1.5 × k
∗
, and (iii) k0 = k
∗
. Given this initial capital per worker and the parameter
values given in part (b), in your Excel file you can produce subsequent values of capital
per worker. For example, for a given capital per worker in time period 0, your capital
per worker in time period 1 would be k1 = sAkα
0 + (1 − δ)k0. Your value in time period
2 would be k2 = sAkα
1 + (1 − δ)k1 and so on. Plot (i.e. line plot) out the time paths of
capital per worker starting from the three different assumed starting values. Does the
economy appear to converge toward k
∗
?
(d) Suppose that the initial capital per worker is k0 = k
∗
, A = 1, α =
1
3
, and δ = 0.1. Suppose
that in time periods 0 through 8, s = 0.3. In time period 9, the saving rate decreases to
s = 0.15 and is expected to remain at this lower value forever. Remember that capital is
predetermined. Use excel to compute values of capital per worker, output per worker,
1
consumption per worker, the real wage, and the rental rate on capital for time periods
0 to 100. Produce plots (i.e. line plots) of the time paths of capital per worker, output
per worker, consumption per worker, the real wage, and the rental rate of capital for
time periods 0 to 100. Comment on the figures.
(e) Suppose that the initial capital per worker is k0 = k
∗
, s = 0.3, α =
1
3
, and δ = 0.1.
Suppose that in periods 0 through 8, A = 1. In period 9, A increases from 1 to 1.2
and A is expected to remain at this higher value forever. Remember that capital is
predetermined. Use excel to compute values of capital per worker, output per worker,
consumption per worker, the real wage, and the rental rate on capital for periods 0 to
100. Produce plots (i.e. line plots) of the time paths of capital per worker, output per
worker, consumption per worker, the real wage, and the rental rate of capital for periods
0 to 100. Comment on the figures.
2. Solow Model with Growth. Consider the Solow Model with population growth where
Nt+1 = (1+gn)Nt and productivity growth where Zt+1 = (1+gz)Zt
. The firm produces output
according to Yt = AKα
t
(ZtNt)
1−α where 0 < α < 1. The firm’s objective is to choose capital
and labor to maximize profits where Πt = Yt − wtNt − RtKt
. The household consumes a
constant fraction, (1−s), of its income each period. The household invests the other fraction,
s, of its income in new capital, with capital accumulating according to Kt+1 = It + (1 − δ)Kt
where 0 < δ < 1.
(a) Define ˆkt =
Kt
ZtNt
(capital per effective worker). Re-write the capital accumulation equation
relating ˆkt+1 to ˆkt and exogenous variables and parameters.
(b) Create a graph plotting ˆkt+1 against ˆkt
. Argue that there exist a steady state, ˆk
∗
, at
which ˆkt+1 = ˆkt
.
(c) Algebraically solve for steady state capital per effective worker, ˆk
∗
, steady state output
per effective worker, ˆy
∗
and steady state consumption per effective worker, ˆc
∗
.
(d) Suppose that the economy begins in the steady state and then there is a surprise permanent
increase in s. Trace out the dynamic responses of ˆkt
, ˆyt and ˆct
. If there is any
ambiguity, please state why.
(e) Suppose that the economy is in the steady state where ˆkt+1 = ˆkt = ˆk
∗
. Define kt =
Kt
Nt
(capital per worker). In the steady state, what is the growth rate of capital per worker
(
kt+1
kt
), the growth rate of consumption per worker (
ct+1
ct
) and the growth rate of output
per worker (
yt+1
yt
). Comment on how some of these results compare with the economic
growth time series stylized facts.
(f) Suppose that the economy is in the steady state where ˆkt+1 = ˆkt = ˆk
∗
. In the steady
state, what is the growth rate of capital (
Kt+1
Kt
), the growth rate of the real rental rate
on capital (
Rt+1
Rt
) and the growth rate of the real wage (
wt+1
wt
). The real rental rate on
capital, Rt
, is equal to the marginal product of capital. The real wage, wt
, is equal to
the marginal product of labor. Comment on how some of these results compare with
the economic growth time series stylized facts.