Problem Set 1
FE411 Spring 2008
Rahman
Partial Answer Key
Comparing GDP Using Purchasing Power Parity
1) Consider two economies, Richania and Pooristan, which produce and consume only
bread and buildings. The table contains price and production data for each country:
Country
Bread
Produced per
Capita
Buildings
Produced per
Capita
Price of Bread
in Local
Currency
Richania
Pooristan
90
15
30
5
4
2
Price of
Buildings in
Local
Currency
100
10
a. Calculate the level of GDP per capita in each country, measured in its own
currency.
Richania: 90*4 + 30*100 = 3360 Richanian dollars (R$)
Pooristan: 15*2 + 5*10 = 80 Pooristanian dollars (P$)
b. Calculate the market exchange rate between the currencies of the two countries.
2 R$/1 P$ (look at traded good prices for this rate)
c. What is the ratio of GDP per capita in Richania to GDP per capita in Pooristan,
using the market exchange rate?
80 P$ * (2 R$/1 P$) = 160 R$. This calculation implies that Richania is
(3360 R$/160 P$) = 21 times wealthier than Pooristan.
d. Calculate the purchasing power parity (PPP) exchange rate between the two
currencies.
The basket of goods should be 3 units of bread and one building.
Richania: 3*4 R$ + 1*100 R$ = 112 R$
Pooristan: 3*2 P$ + 1*10 P$ = 16 P$
Thus the PPP exchange rate is 112 R$/16 P$ = 7 R$/1 P$
e. What is the ratio of GDP per capita in Richania to GDP in Pooristan, using the
PPP exchange rate?
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Problem Set 1
FE411 Spring 2008
Rahman
80 P$ * (7 R$/1 P$) = 560 R$. This calculation implies that Richania is
(3360 R$/560 P$) = 6 times wealthier than Pooristan.
Computing Growth Rates
2) Suppose x grows at 5% per time period (that is, xt = 1.05t), and y grows at 2% per
time period (that is, yt = 1.02t). Calculate the growth rate of z in each of the
following cases:
a.
b.
c.
d.
e.
f.
z = xy
z = x/y
z = y/x
z = x1/2y1/2
z = (x/y)2
z = x-1/3y2/3
0.05 + 0.02 = 0.07
0.05 – 0.02 = 0.03
0.02 – 0.05 = -0.03
(0.5*0.05) + (0.5*0.02) = 0.035
(2*0.05) - (2*0.02) = 0.06
(-0.33*0.05) + (0.67*0.02) = -0.33
How Do We Know Growth Over the Very Long Run?
3) The poorest countries in the world have a per capita income of about $600 today. We
can reasonably assume that it is nearly impossible to live on an income below half
this level (below $300). Per capita income in the United States in 2000 was about
$33,000. With this information in mind, consider the following questions.
a. For how long is it possible that per capita income in the United States has been
growing at an average annual rate of 2% per year?
300 * (1.02)t = 33,000. Solve for t to get t = ln(110)/ln(1.02) = 237.4 years
b. Some economists have argued that growth rates are mis-measured. For example,
it may be difficult to compare per capita income today with per capita income a
century ago when so many of the goods we can buy today were not available at
any price then. Suppose the true growth rate in the last century was 3% per year
rather than 2%. What would the level of per capita income in 1800 have been in
this case? Is this answer plausible?
For the twentieth century: x * (1.03)100 = 33,000
For the nineteenth century: y * (1.02)100 = x
So, first solve for x (the implied income for 1900), and then use this to solve for y
(the implied income for 1800). You should get y = 237 bucks. The answer is thus
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Problem Set 1
FE411 Spring 2008
Rahman
implausible if we believe mankind cannot survive on less than 300 dollars (valued
today). In other words, twentieth century growth could not have been that rapid.
Cobb-Douglas and Payments to the Factors of Production
4) Suppose that the production function is represented by the Cobb-Douglas function:
Y  AK 1 / 10 L9 / 10 .
a. What fraction of income do capita and labor receive?
Capitalists get one tenth of it, laborers get nine tenths of it. Yes, you can just look
at the exponents, but you should also be able to “prove” this.
b. Suppose immigration raises the labor force by 10 percent. What happens to total
output (in percentage terms)? The real rental price of capital (in percentage
terms)? The real wage (in percentage terms)?
First, note that MPK = (1 / 10) AK
9 / 10
9 / 10
L
K
MPL = (9 / 10) AK 1 / 10 L1 / 10  (9 / 10) A 
L
L
 (1 / 10) A 
K
9 / 10
, and that
1 / 10
So, when L rises by a factor of 1.1, Y rises by (1.1)9/10 = 8.96%.
Rental price of capital = MPK rises by (1.1)9/10 = 8.96%.
Real wage of labor = MPL falls by (1.1)-1/10 = 0.0095, or a 0.95% decrease.
c. Suppose the World Bank gives aid to a country that raises its capital stock by 10
percent. What happens to total output (in percentage terms)? The real rental
price of capital (in percentage terms)? The real wage (in percentage terms)?
Y rises by (1.1)1/10 = 0.96%.
MPK falls by (1.1)-9/10 = 0.918, or an 8.2% decrease.
MPL rises by (1.1)1/10 = 0.96%.
d. Suppose that a technological advance raises the value of the parameter A by 10
percent. What happens to total output (in percentage terms)? The real rental
price of capital (in percentage terms)? The real wage (in percentage terms)?
Note that the technological coefficient has no exponent on it (think about why that
is). So output, MPK and MPL all rise by 10%.
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