ECN 101 SSI Modjtahedi An economy’s technology is described by the following

ECN 101 SSI Modjtahedi
Due: Tuesday July 05, 2016
There are three questions in the problem set
Question 1:
An economy’s technology is described by the following Cobb-Douglas production function.
Y = K 0.50 N 0.50
Where, Y is the real GDP and K and N are the amounts of capital and labor. We have the
following information about this economy:
Labor grows at an annual rate of 2% per year ? n = 0.02
Annual depreciation rate of capital is 3% ? d = 0.03
Saving rate is 25% ? s = 0.25
a. Find the steady state levels of k and y. Show this case in a nice and readable Solow diagram
(the graph must be fully labeled with the values of k*, y*, (d + n)×k* and s×y*.
b. If “this year” capital per worker was k = 9, at what rate would it change from “this year” to
“the next year” (in other words, calculate ?k/k)? What if k was equal to 16? Show these
cases in a Solow diagram. From this exercise, what do you conclude about the relationship
between k and ?k/k? What is the economics behind this relationship?
c. Suppose that “this year” we are in steady state with K = 2,500 and Y = 500. Assuming steady
state, if you look at this economy “next year”, what will you find for the values of K, N, Y, k,
and y?
d. Find the Golden-Rule level of k. What are the Golden Rule levels of consumption and GDP
per worker? What value of the saving rate would achieve the Golden-Rule level of k? Show
the whole thing graphically.
Question 2:
An economy’s technology is described by the following Cobb-Douglas production function.
Y = AK 0.25 N0.75
Where, Y is the real GDP and K and N are the amounts of capital and labor. Currently the
technology parameter equals A = 16. We have the following additional information about this
economy:
Labor grows at an annual rate of 1% per year ? n = 0.01

Annual depreciation rate of capital is 4% ? d = 0.04
Saving rate is 20% ? s = 0.20
a. Find the steady state levels of k and y. For the sake of argument let’s suppose that this
solution pertains to the year 2016. Show this case in a nice and readable Solow diagram (the
?
?
?
?
graph must be fully labeled with the values of 2016
, 2016
, ( + )2016
, and × 2016
.
?
?
b. Technology improves by 5% from 2016 to 2017. Find the values of 2017 , 2017 . Show all
the shifts with the 2016 and 2017 equilibrium values in a Solow diagram.
c. Now here is the cool part. Consider the graph below that you have seen several times before.
It shows that, unlike what Cobb-Douglas predicts, the relationship between k and y for the
U.S. was more or less linear (no diminishing returns to capital). One way to resolve this
conflict was to argue that improvements in technology (know-how) had caused the function
to shift up over years so that the actual (blue) points observed in the graph are the steadystate k-y pairs lying on different production functions. Assume that technology improves one
?
?
more time in 2018 by the same 5%. Find the values of 2018
and 2018
. Do the three k*-y*
pairs for 2016, 2017, and 2018 lie on a straight line? If yes, what is the equation of this line?

120,000

100,000

GDP per Worker (y)

Y/N
? = 0.25

80,000

2011

60,000

40,000

20,000

1950

0
0

50

100

150

200

Capital Per Worker (k)

250

300

350

Question 3:
Generally, the third-world countries that are at the beginning of their economic development are
hungry for capital. They either like to consume less and save more to accumulate capital, or they
like to receive lots of capital from other, more advanced countries through foreign investments.
On the other hand, advanced capitalist countries, such as the U.S., are crazy about research and
development activities. They try to encourage such activities through tax breaks and other types
of subsidies to research labs or universities. Why is that? Can our model explain this?
Suppose there are two countries, Poor and Rich. They have the same Cobb-Douglas production
functions (just to make our calculations less cumbersome):
= 100 × 0.20
In both countries technology improves at the rate of 5% per year, but Poor has a much smaller
capital per worker than Rich. The following are the numbers for two years:
Year Capital per Worker in Poor Capital per Worker in Rich
Year 1
100
2,000
Year 2
200
2,100
Calculate the percentage contributions of improvements in technology and increases in capital
per worker to the growth rates of GDP per worker in these two countries between these two
years. Fill in the following table. What accounts for the difference in the results?
Country
Poor
Rich

Contribution of
Contribution of
Capital per Worker Technological Progress
%
%
%
%