Problem Set 6
FE411 Spring 2007
Rahman
Due Date: Thursday, April 26th (at the beginning of class)
INSTRUCTIONS: Use your own paper to answer the questions. Please turn in your
problem sets with your name clearly marked on the front page and all pages stapled
together. You are encouraged to work together, but you must hand in your own work.
You must show your work for credit and answer in complete sentences when appropriate
(such as when the question asks you to “describe” or “explain”).
1) Capital Flows Within Europe
Consider Europe as a whole. Europe’s entire capital stock in the steady state may be
accurately described by the following equation:
1
k ss Eur
  A  1
  Eur Eur 
 

where  Eur and AEur is the European-average investment rate and European-average
productivity, respectively.  is the depreciation rate of capital.
Now consider a small open country within Europe, like Greece. Greece’s capital stock in
the steady state may be accurately described by the following equation:
 A

k ss Greece   Greece 
 rEur 
1
1
where AGreece is the productivity of Greece, and rEur is the rental rate of capital for all of
Europe.
a) In a couple of sentences, can you explain why the expression for European capital
differs so much from Greek capital? In particular, why is it that the investment
rate in Greece does not affect the stock of capital in Greece?
If Greeks save more (that is, γGreece increases), Greeks will not get any more
capital than they already have. The reason is that the extra savings will likely
flow abroad to other European countries. Similarly, if Greeks save very little,
they will probably have capital flow into the country. On the other hand, Europe
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Problem Set 6
FE411 Spring 2007
Rahman
as a whole can be considered a closed economy – in this case savings in Europe
must equal investment in Europe.
b) Suppose that Greeks save a lot (that is, γGreece is very high). Would you expect
Greece to be a net borrower or a net lender in international capital markets. Why?
We would expect Greece to be a net lender. Those extra savings are allowed to
flow to the most productive places in Europe. When that happens, Greek savers
lend to other parts of Europe, and those other European countries get to have
more capital.
c) Suppose that AGreece / AEur rises (that is, relative productivity in Greece rises).
How would you expect this to change the flow of capital?
We should expect capital to flow into Greece. Why? Because capital flows to
where it is most productive (i.e., where its marginal product is highest).
d) Finally, suppose that the European rental price of capital, rEur , doubles. By what
factor will the level of GDP per worker change in Greece? For this problem,
assume that the value of α, capital’s share in the production function, is 0.5.
If the European rental price of capital doubles (from rEur with the old steady-state
to 2 rEur with the new steady state), we can write the new steady-state level of
output per worker as:
y new ss  A
1
1
 

 2reur




1
A
1
1
 

 reur




1


1
 1  1
old  1 
   y ss  
2
2
Assuming α = 0.5, you should find that
1
y new ss   
2
0.5
0.5
1
   y old ss
2
That is, the new steady-state income level in Greece falls by half when the
European rental rate of capital doubles.
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Problem Set 6
FE411 Spring 2007
Rahman
Chapter 12:
3) a) Denoting Qd as the quantity demanded and Qs as the quantity supplied, at the
market-clearing price, the two must equal each other. That is:
Qd  Qs , 100  P  P, P  50 .
b) At a tax rate of τ for each good, the quantity demanded for any given price does not
change because the price paid by the consumer remains unaffected. However, the
quantity supplied for any given price decreases to a factor of (1-τ), as government
collects from the supplier. Specifically,
Qd  100  P, Qs  (1   ) P .
Setting Qd  Qs and solving for P gives us our equilibrium price:
Peq 
100
(2   )
To find the equilibrium quantity for the price, we substitute and get:
Qeq  (1   ) Peq 
100(1   )
(2   )
c) In order to solve for the tax rate that will maximize government revenue, we must
write the revenue function of the tax rate. Government revenue for each good is τPeq.
Since the amount of the good sold is Qeq, we know that government revenue is equal to
τPeqQeq. Thus:
Max (Qeq Peq ) 

 (1   )100 2
.
(2   ) 2
Maximizing this expression means taking the derivative with respect to τ and setting it to
zero. You have to use the Quotient Rule, and it’s a bit messy…ultimately, you should get
τ = 2/3.
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Problem Set 6
FE411 Spring 2007
Rahman
4) Consider an extension of the Solow model to encompass a second type of capital,
called government capital, which consists of publicly funded infrastructure such as roads
and ports. Let x denote the quantity of government capital per worker, k the quantity of
physical capital per worker, and y the quantity of output per worker. The economy’s
production function (in per-worker terms) is
y  Ak 1/ 3 x1/ 3
We assume that the government collects a fraction τ of national income in taxes and
spends all of this revenue producing government capital. We also assume that a constant
fraction γ of after-tax income is invested in producing physical capital. Both government
capital and physical capital depreciate at rate δ. The equations describing how
government capital and physical capital change over time are thus
x  Ak 1 / 3 x1 / 3  x
k   (1   ) Ak 1 / 3 x1 / 3  k
a) Solve for the steady-state level of output per worker.
In steady-state level of output per worker, the quantities of government capital
per worker and physical capital per worker will not change over time. Therefore,
x  0  Ak 1 / 3 x1 / 3  x , and
k  0   (1   ) Ak 1 / 3 x1 / 3  k
We know have 2 equations with 2 unknowns. Working through the algebra and
solving for the steady-state values of xss and kss, we get:
x ss 
A3 (1   ) 2
3
k ss 
A3 2 (1   ) 2 
3
Plugging in these values into the production function will now yield the steadystate level of output per worker:
y  Ak
ss
1/ 3 1/ 3
x
 A3 2 (1   ) 2  
 A

3


1/ 3
 A3 (1   ) 2 


3


1/ 3

A3
2
( )(1   )
b) What value of τ will maximize output per worker in the steady state?
Page 4 of 5
Problem Set 6
FE411 Spring 2007
Rahman
The value of τ that will maximize output per worker is the same value that will
maximize (τ)(1-τ), as A3γ/δ2 is a constant. Therefore, τ = ½, a tax rate of 50%.
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