ECO 317: Intermediate Macroeconomic Theory
Department of Economics
Miami University
Professor Kimberly Berg
Spring 2021
Problem Set 3
Instructions: This problem set is due in Canvas by 11:00am (Eastern) on March 1, 2021. Prob-
lem set solutions must be written in pen or typed and the figures associated with questions one
and three must be digitally produced. The problem set solutions must be submitted in order of
the questions (i.e. the solution for question one will come first, then the solution for the second
question, etc.). Also, your problem set solutions must be uploaded in Canvas as a pdf document
and your name must be written on each pdf document submitted in Canvas. Failure to follow these
instructions will result in a zero score.
Grading: Completion (3 points; no partial credit). Selected question (2 points; partial credit).
Economic Growth
1. This problem asks you to use Excel to plot and examine the Cobb-Douglas production function
Yt = F (Kt, Nt) = K
α
t N
1−α
t , with α =
1
3 .
(a) Create a table of numbers in an Excel worksheet. Let the rows represent a grid of values
for capital, Kt = {0, 1, 2, 3, 4, 5}, and the columns represent a grid of values for labor,
Nt = {0, 1, 2, 3, 4, 5}. Then, fill in the 6×6 grid with the values of output, Yt = Kαt N1−αt ,
using the value of Kt corresponding to that row and the value of Nt corresponding to that
column (you should use “=(Row value)^(1/3)*(Column value)^(2/3)”). Comment
on the properties of the production function
(
F (Kt, Nt) = K
α
t N
1−α
t
)
.
(b) Use Excel to create a plot of the production function. Use Excel’s “surface plot” (listed
under “other charts” in the chart section) to create a 3-D plot of the grid of numbers
created in (a). Label the x, y, and z axes and be sure that the grid marks on the three
axes correspond to the actual values of Kt, Nt, and Yt.
(c) Plot a “slice” (i.e. line plot) of the production function where Kt = 4. That is, for the
row associated with Kt = 4, use a “line plot” to plot the values of output for each value
of labor. Label figure. Comment on the figure.
2. Suppose that we have a standard Solow Model. There is no population or productivity growth.
(a) The firm problem is to maximize profits, where profits (Πt) are given by:
Πt = AK
α
t N
1−α
t − wtNt −RtKt.
The firm takes the factor prices (wt and Rt) as given and chooses capital (Kt) and labor
(Nt). Productivity (A) is exogenous, Rt is the real rental rate on capital, wt is the real
wage, and α (0 < α < 1) is a parameter. Find the first order conditions characterizing
optimal firm behavior. What is the economic interpretation of each first order condition?
(b) Use the first order conditions from (a) to show that wtNtYt = 1−α, where Yt = AKαt N1−αt .
(c) The household is endowed with labor (Nt), owns the capital stock (Kt), and leases the
capital stock to firms on a period-by-period basis. Firms remit any profits (Πt) back to
households. The household budget constraint is given by:
Ct + It = wtNt +RtKt + Πt.
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Ct is consumption and It is investment. Use your previous answers to show that the
right hand side of the equation reduces to Yt.
(d) The capital accumulation equation is standard:
Kt+1 = It + (1− δ)Kt.
It is investment and δ (parameter; 0 < δ < 1) is the depreciation rate. Kt is given
in period t. It is inherited from past decisions. Assume that the household consumes
a constant fraction of its income each period, 1 − s (0 ≤ s ≤ 1), and supplies labor
inelastically. Re-write the capital accumulation equation as a difference equation relating
kt+1 to kt (k represents capital per worker, kt =
Kt
Nt
) using exogenous variables and
parameters only.
(e) Create a figure plotting kt+1 against kt of the capital accumulation equation in per
worker terms. Label the figure (axes, intercepts, and curves). Explain in words why the
figure looks the way it does and also argue that there exists a steady state k∗ where
kt+1 = kt. Label k
∗ in the figure.
(f) Algebraically solve for steady state capital per worker (k∗), steady state output per
worker (y∗), steady state consumption per worker (c∗), and steady state investment per
worker (it).
(g) What value of s would maximize steady state output per worker (y∗)? Do you think the
household would like this saving rate? Why or why not?
(h) What value of s would maximize current consumption per worker (ct)? Do you think it
would be a good thing to have this saving rate?
(i) What value of s would maximize steady state consumption per worker (c∗)? Please
derive an analytical expression for s that makes c∗ as big as possible.
(j) A reasonable value for α is 13 . With a saving rate between 10-15 percent, would the U.S.
be near the “Golden Rule” saving rate you found in (i)? If not, would you necessarily
recommend that we increase the saving rate?
(k) Suppose that the economy begins in the steady state and then there is a surprise per-
manent increase in productivity (A). Graphically show the time path of kt, yt, ct, and
it. Label figures. If there is any ambiguity, please state why.
3. Suppose that we have a standard Solow model. It is the same setup as problem 2. The central
equation summarizing the model is given by:
kt+1 = sAk
α
t + (1− δ)kt.
Other variables can be expressed in terms of capital per worker as:
yt = Ak
α
t ,
ct = (1− s)yt,
wt = (1− α)Akαt ,
Rt = αAk
α−1
t .
(a) Solve for an analytic expression for the steady state capital per worker (k∗).
(b) Suppose that s = 0.2, A = 1, α = 13 , and δ = 0.05. What is the numeric value for k
∗
given these values?
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(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 (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 = sAk
α
0 + (1− δ)k0. Your value in time
period 2 would be k2 = sAk
α
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.
Label figures. Does the economy appear to converge toward k∗?
(d) Suppose that the initial capital per worker is k0 = k
∗, A = 1, α = 13 , and δ = 0.05.
Suppose that in time periods 0 through 8, s = 0.2. In time period 9, the saving rate
increases to s = 0.25 and 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
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. Label figures. Comment on the figures.
(e) Suppose that the initial capital per worker is k0 = k
∗, s = 0.2, α = 13 , and δ = 0.05.
Suppose that in periods 0 through 8, A = 1. In period 9, A increases from 1 to 1.1
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. Label figures. Comment on the figures.
4. Suppose we have a standard Solow model with a government. Each period, the government
consumes a constant fraction of output, sG (0 ≤ sG ≤ 1). The aggregate resource constraint
is:
Yt = Ct + It +Gt,
where Yt is output, Ct is consumption, It is investment, Gt is government spending, and
Gt = sGYt. Define private output as Y
p
t = Yt − Gt. Suppose that investment is a constant
fraction, s, of private output (consumption is then (1−s) times private output). The rest of the
model is standard. Labor is inelastically supplied and there is no population or productivity
growth.
Yt = AK
α
t N
1−α
t
Kt+1 = It + (1− δ)Kt
Kt is given in period t, 0 < α < 1, and 0 < δ < 1.
(a) Re-write the capital accumulation equation as a difference equation relating kt+1 to kt
(k represents capital per worker, kt =
Kt
Nt
) using exogenous variables and parameters
only.
(b) Algebraically solve for steady state capital per worker (k∗), steady state output per
worker (y∗), and steady state consumption per worker (c∗).
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5. Consider the Solow Model with population growth where Nt+1 = (1 + gn)Nt and labor
augmenting productivity growth where Zt+1 = (1+gz)Zt. gn (0 < gn < 1) and gz (0 < gz < 1)
are taken as given. The firm produces output (Yt) according to Yt = AK
α
t (ZtNt)
1−α where
A is the level of productivity, Kt is capital, Nt is labor, and 0 < α < 1. The firm takes factor
prices (wt and Rt) as given and chooses capital and labor to maximize profits (Πt) where
Πt = Yt − wtNt − RtKt. Rt is the real rental rate on capital and wt is the real wage. 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. It is investment, Kt is given in period t, and Ct is
consumption.
(a) Define kˆt =
Kt
ZtNt
(capital per effective worker). Re-write the capital accumulation equa-
tion relating kˆt+1 to kˆt using exogenous variables and parameters only.
(b) Algebraically solve for steady state capital per effective worker (kˆ∗), steady state output
per effective worker (yˆ∗), and steady state consumption per effective worker (cˆ∗).
(c) Suppose that the economy is in the steady state where kˆt+1 = kˆt = kˆ
∗. Define kt = KtNt
(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
)
, the growth rate of output
per worker
(
yt+1
yt
)
, 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
)
? Comment on how
some of these results compare with the economic growth time series stylized facts.
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