# Exam 3 Key

```Question I Homage to the production function. (50 points, 10 points each part)
a) What is meant by a country’s production function?
A country’s production function is the relation between factor inputs and output.
b) What is meant by constant returns to scale?
Constant returns to scale means that if you double all inputs you’ll double output.
c) What is meant by decreasing returns to labor?
Decreasing returns to labor means that, holding the amount of capital constant, each
d) Why is it reasonable that a country’s production function will exhibit i) constant returns
to scale and ii) decreasing returns to labor? Do not simply repeat your answers to (b)
and (c) above!
i) It makes sense that a country’s production function will exhibit constant returns to scale
because if you have a way to produce things that you’re satisfied with you can simply
replicate the process.
ii) It makes sense that a country’s production function will exhibit decreasing returns to labor
because additional workers share the existing capital…capital per worker decreases as
more workers are employed.
e) Many countries that were desperately poor in the 1950s have grown rapidly in the last
half century. These formerly poor countries have narrowed the gaps between their per
capita incomes and those of countries like the United States that were rich in the 1950s
but have grown more slowly since then. How does the Solow growth model with
constant returns to scale, decreasing returns to capital, and decreasing returns to labor
explain this convergence of per capita incomes over the last half century?
The Solow growth model with constant returns to scale, decreasing returns to capital, and
decreasing returns to labor predicts convergence of per capita incomes across countries because
countries that are initially rich presumably have high capital:labor ratios while poor
countries countries have low ratios. With decreasing returns to capital, capital increments
in (initially) rich countries provide less bang-per-buck than similar increments in poor
countries, allowing poor country output per worker to catch up with that of (initially) rich
countries. With depreciation proportional to the amount of capital, countries (initially)
well-endowed with capital approach the steady state sooner than poorer countries as more
and more of what they save simply offsets the increased depreciation of their increased
capital stocks. Hence, capital per worker, outputs per worker and incomes per worker
converge.
Question II
Imagine an economy without population growth and without technological progress. Output (Y)
in this economy as a function of capital (K) and labor (N) inputs is given by
Y = F(K,N) = .5 K&frac12; N&frac12;
Existing capital stock depreciates at a rate of ten percent per year (δ = .1) and people save and
invest ten percent of their incomes (s = .1).
a) Express output per worker, y=Y/N, as a function of capital per worker, k =K/N. (5 points)
Y/N = .5 (K/N)&frac12; (N/N)&frac12; =.5k &frac12;
b) Sketch output per worker, saving per worker, and depreciation per worker as functions
of capital per worker. (10 points)
c) Determine the steady state values of capital per worker, k, and output per worker, y. Label
these numerical values on your sketch. (10 points)
Steady state condition: sy = .1(.5k &frac12;) = δk = .1k  k &frac12; = .5  k = (.5)2= .25
and y =.5k &frac12; = .5*.5 = .25
d) The “golden rule” saving rate for this economy is s = .5. What is special about this saving
rate? (10 points)
The golden rule saving rate is special because steady-state consumption per worker is
maximized at the golden rule rate of saving.
e) Suppose the economy described above has settled at its steady state with s = .1 when people
decide to live by the golden rule instead, i.e., to increase their saving rate from s=.1 to s =.5.
i) Based on your answer to (c) above, what is the value of consumption per worker when
the economy is in its initial steady state? Show your calculation. (5 points)
c* = (1-s)*y = .9*.25 = .225
ii) What is the value of consumption per worker in the first year that the saving rate increases
from s = .1 to s = .5? Show your calculation. (5 points)
c1 = (1-s)*y = .5*.25 = .125
iii) What is the value of consumption per worker when the economy settles in its new steady
state? Show your calculations. (5 points)
Steady state condition: sy = .5(.5k &frac12;) = δk = .1k  k &frac12; = 2.5  k = (2.5)2= 6.25
and y =.5k &frac12; = .5*2.5 =1 .25 ; c = (1-s)*y = .5*1.25 = .625
Question III
Imagine an economy with annual workforce growth of one percent (gN = .01), technological
progress of two percent (gA = .02), and a depreciation rate of ten percent (δ = .1). Output (Y) in
this economy is a constant-returns-to-scale/decreasing-returns-to-inputs function of capital (K)
and effective labor (AN) inputs,
Y = F(K,AN).
a) What is the annual growth rate of effective labor in this economy? Show your
calculation. (5 points)
gAN = gN + gA = .01 + .02 = .03
b) How much gross investment is needed to maintain a constant level of capital stock per
To keep K/N steady, we must offset depreciation (=.1K) and outfit each additional
worker with capital (=gN K = .01 K). In total, then, gross investment must equal
11% of the existing capital stock to keep K/N constant. Note, this does not keep
capital per effective worker steady. It will decline by gA =2% each year.
c) What is the steady state growth rate of output per worker, Y/N, in this economy? Show
If capital per effective worker were kept steady, which would require gross
investment equal to 13% of the capital stock each year, output per worker, Y/N,
would increase at the rate of technological progress or 2%. If only capital per
worker is kept steady, Y/N will increase at successively lower rates than 2% (there
are decreasing returns to effective labor) and would eventually level off.
d) Expectations of inflation track actual inflation pretty well but expectations of productivity
growth, gA, adjust only slowly. Suppose productivity growth accelerates to three percent
per year but workers and employers expect it to remain at the two percent rate they are
familiar with.
Recall, Wage Setting: W = Ae Pe F(u,z) and Price Setting: P = (1 + μ) W/A
i) At what annual rate will the real wage grow? Explain why. (10 points)
If gross investment keeps capital per effective worker steady, the real wage will grow at a
rate of 3 percent per year because technology is advancing at that rate. Otherwise, there will be
an initial spurt in the real wage to a new, higher level but eventually the real wage would level
off.
ii) How will the normal, (natural, or equilibrium) rate of unemployment respond to the increase
in productivity growth? Explain why. (10 points)
The equilibrium rate of unemployment in the economy will decrease because
it will take a lower rate to get wage setters to accept the higher real wage they will enjoy, a
real wage higher than they had bargained for.
iii) What will happen to the equilibrium rate of unemployment in this economy once
expectations of productivity growth catch up to the reality of accelerated productivity growth?
Explain why. (10 points)
The equilibrium rate of unemployment in the economy will return to its initial value once
wage setters anticipate and build the higher rate of productivity growth into their bargains.
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