Problem Set 4
FE411 Spring 2009
Rahman
Some Answers
1) A Variation on the Romer Model
Consider the following variation to the model discussed in chapter 8.2 of the book:
Y  A1 / 2 (1   A ) L
^
1
A    A L

There is just one difference – we’ve changed the exponent on A in the production of the
output good so that there is now a diminishing marginal product to ideas in that sector.
a) Provide an economic interpretation for each equation (one or two sentences for each)
Output increases with technological progress, but at a DECREASING rate. Output
also increases with the number of people working in production.
Technological progress rises with the number of people working in research.
b) What is the growth rate of knowledge in this economy? What is the growth rate of
output per person in this economy?
The growth rate of knowledge is simply the second equation above. As long as some
people are working in research, knowledge grows at a positive rate forever.
However, because this knowledge exhibits diminishing returns in PRODUCTION, the
long run growth rate of output per person in this economy will be zero.
c) Graph output per person over time (use the ratio-scale for the y-axis).
y
time
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Problem Set 4
FE411 Spring 2009
Rahman
2) Some Numbers for the Romer Model
Consider again the one-country model of section 8.2. Suppose the parameters of the
model have the following values: At0 = 100 (this is the level of technology at time t0,
which is right now), γA = 0.06, μ = 3000, and L = 1000.
a) What is the growth rate of output per person in this economy?
The growth rate of y in the long run will just be the growth rate of A. This of course
^
1
is given by the equation A    A L . Plugging in our values, we have

(1/3000)*(0.06)*(1000) = 0.02. The economy grows 2% every year.
b) What is the initial level of output per person now (at t0)? What will the level of
output per person be after 100 years?
GDP per person is given by y  A (1   A ) , which at time t0 is 100*0.94 = 94.
This value grows at 2% every year, so in 100 years it will be 94*(1.02)100 = 681.
c) Now consider the following changes, one at a time: a doubling of the initial stock of
knowledge, At0, a doubling of the research share γA, a doubling of research productivity
(1/μ), and a doubling of the population L. How would your answer to parts a) and b)
change in each case?
Doubling initial stock of knowledge increases initial GDP per person to 188, but does
not change the growth rate (in 100 years GDP per person will be 1362).
Doubling the research share (from 6% to 12%) actually lowers initial GDP per
person (to 88), but doubles growth (to 4%). In this case in 100 years GDP per
person will be 88*(1.04)100 = 4444.4, which is a lot more!
Doubling research productivity does nothing to initial per person GDP, but doubles
the growth rate to 4%.
Doubling the population does not change initial per person GDP, but doubles the
growth rate to 4%.
d) If you could advocate just one of the changes considered in part c), which would you
choose? Write a couple of sentences arguing for your choice.
The last two seem to be the best. There is no short term cost (unlike increasing the
share of researchers), and you get to grow much faster.
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Problem Set 4
FE411 Spring 2009
Rahman
3) Nonrivalry
Explain whether the following goods are rivalrous or nonrivalrous:
a) Beethoven’s Fifth Symphony – non-rivalrous, it is at heart an “idea”
b) a portable music player - rivalrous
c) Monet’s painting Water Lilies – rivalrous, there are only a few of these in the world.
d) the method of public key cryptography – non-rivalrous
e) fish in the ocean – rivalrous, even though there are like a gazillion of them, they are
depleting even as we speak. The number of the goods that currently exist is not the issue,
it’s the nature of how you consume it that is important.
4) The Two-Country Model
Consider the two-country model of section 8.2. Suppose that the cost-of-copying
function is
A
 c   i  1
 A2




where 0 < β < 1. Assume that the two countries have labor forces of equal size.
a) Using this function, solve for the steady-state ratio of technology in the leading
country to technology in the follower country (that is, A1/ A2) as a function of the values
of γA in the two countries. Show how this depends on the value of β, and explain what is
going on.
In the steady state, the growth of A1 equals the growth of A2. Therefore:
 A,1 L  A, 2 L

i
c
Rearranging and solving for μc, we get,
 
 c   A, 2   i
  A,1 
Setting the above steady state condition equation to the specified cost-of-copying
function, we get
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Problem Set 4
FE411 Spring 2009
Rahman
 
A 
 c   A, 2  i   1 
 A2 
  A,1 

i
Rearranging, we find the solution to be:
1
 A1    A,1  

   
 
A
2
A
,
2
  

Without the exponent, the ratio of technology in Country 1 to Country 2 would be
determined proportionally by the ratio of the fraction of the labor force employed
in R&D. However because we assume 0<β<1, the ratios will not be
proportional. As the value of β falls to zero, the proportional difference in the
level of technology between the two countries grows extremely large, and as the
value of β rises to one, the proportional difference in the level of technology
between the two countries matches the proportional difference in the fraction of
worker devoted to R&D.
b) Assume now that β = ½, μi = 10, γA,1 = 0.2, γA,2 = 0.1. Calculate the steady-state ratio
of technology in Country 1 to technology in Country 2.
If we assume β = ½, μi = 10, γA,1 = 0.2, and γA,2 = 0.1, we can solve the previous
equation to get:
1
2
 A1    A,1    0.2 


  

   0.1   4
 A2    A, 2 
That is, the steady-state ratio of technology in Country 1 to technology in Country
2 is 4.
5) Technological Change in Two Sectors
Suppose that people consume only two goods, cheese and bread. They consume these
two goods in a fixed ratio: One slice of bread is always eaten with one slice of cheese.
Both cheese and bread are produced using only labor as an input. Their production
functions are
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Problem Set 4
FE411 Spring 2009
Rahman
Yb  Ab Lb
Yc  Ac Lc
where Yb is the quantity of bread, Yc is is the quantity of cheese, Lb is the amount of
labor devoted to producing bread, and Lc is the amount of labor devoted to producing
cheese. The total quantity of labor in the economy, L, is constant, and Lb + Lc = L.
In the year 2000, Ab = Ac = 1. But technological progress takes place at different speeds
in the two industries. Specifically, annual grow of Ab is 2%, and annual grow of Ac is
1%, where these growth rates of technology are exogenous.
a) What quantities of labor will be devoted to producing bread and cheese in 2000?
In any given year, the production of bread must equal the production of cheese in this
economy. That is, Yb  Yc , always. Knowing that the productivity of each good is
equal at this point in time, we can solve for the quantity of labor devoted to each
sector as follows:
Yb  Yc
Ab Lb  Ac Lc
Lb  Lc
Since Lb  Lc  L, Lb  Lc  L / 2 . The labor force will be equally split between the
two sectors.
b) What will the growth rate of total output be in 2000?
To calculate the growth rate of total output, we first calculate the growth rates of
each sector by taking the natural log of both sides and differentiating with respect to
time. For the bread sector:
ln( Yb )  ln( Ab )  ln( Lb ),
d
d
ln( Yb )  ln( Ab )  ln( Lb ),
dt
dt
^
^
^
Yb  Ab  L b
Similarly, for the cheese sector, we get:
^
^
^
Yc  Ac  L c
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Problem Set 4
FE411 Spring 2009
Rahman
We know the value for the growth rate of productivity in both sectors. Furthermore,
in our answer to part a, we found that labor is currently equally divided among the
two sectors. Thus, the growth of labor in one sector must be offset by the growth of
^
^
labor in the other - L b  ( L c ) . We substitute in these values and get:
^
^
Yb  2%  L b
and
^
^
Yc  1%  L c
Setting the growth rate of output in the bread sector to the growth rate of output in
^
^
the cheese sector, we find that L b  0.5% and L c  0.5% . So, the growth in GDP
is equal to the growth of each sector, which is 1.5%.
c) Draw a graph showing the growth rate of output from the year 2000 onward. Show
whether growth rises, falls, or stays constant, and explain why it does so. What will the
growth rate of output be in the long run?
We know that the growth rate of output right now is equal to 1.5%. But, productivity
in the bread sector rises by a greater percent than in the cheese sector. Bread
production would rise faster than cheese production and because one piece of bread is
consumed with one piece of cheese, labor resources are continually shifted into the
cheese sector. Over time, nearly the whole of the economy’s resources will be shifted to
the cheese sector with a minimal amount devoted to the high productivity bread sector.
The economy’s growth rate asymptotically approaches 1% , the growth rate of
productivity in the cheese sector.
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