Chapter 1
Introduction to Macroeconomics
Answers to Textbook Exercises
Analytical Exercises
1. What are the key differences between microeconomics and macroeconomics?
Macroeconomists examine the economy in the large, focusing on feedback from one
component of the economy to another and studying the total level of production and
employment. In contrast, microeconomics deals with the economy in the small.
Microeconomists study the markets for single commodities, examining the behavior of
individual households and businesses. They focus on how competitive markets allocate
resources to create producer and consumer surplus, as well as on how markets can go wrong.
These two groups of economists also differ in their view of how markets work.
Microeconomists assume that imbalances between demand and supply are resolved by
changes in prices. Rises in prices bring forth additional supply, and falls in prices bring forth
additional demand, until supply and demand are once again in balance. Macroeconomists
consider the possibility that imbalances between supply and demand can be resolved by
changes in quantities rather than in prices. That is, businesses may be slow to change the
prices they charge, preferring instead to expand or contract production until supply balances
demand.
2. How would a sharp increase in oil prices reduce productivity growth?
In response to a sharp increase in oil prices, firms redirect their capital expenditures from
capital that uses more energy to capital that uses less energy; firms retire a large share of
their most energy-intensive capital and begin to substitute workers for energy use wherever
possible. Both of these tend to reduce output per worker.
3. How would the entry of a large number of young people into the labour force reduce
productivity growth?
Young workers are less experienced and do not have the same skill level as older workers.
Hence, with the entry of a large number of young people into the labour force there will be a
fall in the average level of labour-force experience, which will reduce productivity growth.
4. How can changes in expectations lead to a stock market crash and a recession?
Suppose there is a sudden change in the mood of the investors, who start to believe that there
will be a reduction in the profitability of a certain major sector of the economy. Investors will
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then try to sell a large part of their stocks in that sector, precipitating a stock market crash. The
stock market crash will result in a loss of wealth, reducing consumption expenditures and
creating a recession.
5. Roughly, how much higher is measured real GDP per worker in 2005 than it was in
1973?
Based on Figure 1.8, it has grown from roughly $13,000 in 1973 to roughly $17,000 in 2005,
both measured in 1971 dollars—an increase of about 31%. (However, this increase has not
been evenly distributed: some of the jobs in the economy pay little more than they did in 1973;
and some pay much more).
Policy Exercises
1. What have been the main factors that have contributed to Canadian economic growth
since Confederation? Briefly outline Canada´s economic development since
Confederation.
Canada’s economic development since Confederation can be explained in terms of growth in
its labour and capital endowments and improvements in technology over time that have
contributed to increases in multi-factor productivity. The staples theory is consistent with this
explanation, in that it emphasizes the role of land, natural resources, and primary products in
Canada’s early economic development. The early staples (fish, fur, timber, wheat) provided
the basis for Canada’s economic growth and the main impetus for capital accumulation and
the development of the country’s industrial base.
From Confederation to World War I (1867-1913) the national economy followed two distinct
periods: low economic growth until the beginning of the wheat boom in 1986 and
accelerating growth thereafter. The years 1914-1945 were turbulent with high volatility in
economic activity, due to the Great Depression and the two World Wars. The modern postWWII is characterized by significant economic growth that divides into two sub-periods: the
sub-period 1946-1973 was one of rapid economic growth and prosperity; and the sub-period
1973-present that has been characterized by a slowdown of economic growth and economic
challenge.
2. Why do you think Canada did better than Argentina in the twentieth century?
Even though the two countries started from the same position at the start of the twentieth
century, Canada grew much faster than Argentina during the century. As a result, Canadian
living standards today are much higher than Argentina’s. The main reason is the difference
in economic policies followed by the two countries. Argentina followed bad polices for
growth, such as policies resulting in high inflation, high budget deficits, low expenditures on
education and on law and order, and no sustained support for research and development. On
the other hand, Canada followed policies that were conducive to long-run growth, such as
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maintaining low inflation and budget deficits, high expenditures on education and
infrastructure and more spending on research and development.
3. What are some possible reasons for the increase in the value of the Canadian dollar in
recent years?
There are two major reasons for the decline in the value of the Canadian dollar. First, because
Canada has a relatively large primary goods sector, the rise in the world prices of
commodities, which are quoted in US dollars, has meant that Canada will receive a larger
number of Canadian dollars for its exports of primary commoditiesenergy, forestry, mining
and agricultural products. This, in effect, has increased the demand for the Canadian dollar in
the foreign exchange market. Second, Canada in recent years has adopted low inflation
polices and prudent fiscal policies that have resulted in sustained government budget
surpluses. These two trends have led to the increase in the value of the Canadian dollar by
increasing the demand for it in the foreign exchange markets.
4. What are the prominent features of monetary and fiscal policies in Canada in recent
years?
The Bank of Canada continues its policy of maintaining the inflation rate between 1% and
3%. The inflation rate has been within this range since 1992. The target has been easier to
maintain with the government’s fiscal success in recent years of turning its budget deficits of
the 1980s and early 1990s into surpluses. The government budget has been in a surplus since
1996. With these surpluses, as well as the good performance of the economy, the government
has been able to reduce the federal debt to GDP ratio from its peak of 71% in 1995-1996 to
34.9% at the end of 2003. The federal budget has continued to be in surplus to date and is
expected to be in a surplus in the subsequent years.
5. Discuss some of the important macroeconomic events in the U.S. in the recent years.
The 1990s was a remarkable decade of economic boom. The Clinton administration's
economic programs of deficit reduction and the lowering of trade barriers, together with the
wave of rapid productivity growth driven by the information technology, contributed to the
longest period of economic expansion in U.S. history. This decade long growth ended in the
slowdown of 2001-2002, which was due to the bursting of the bubble in IT industry, the
terrorist attacks of September 11, 2001, and the subsequent decline in the travel and tourism
industry. The Bush administration reacted by introducing tax cuts to boost the US economy.
However, these tax cuts combined with the large military expenditures following September
11, 2001, have created large budget deficits and low public savings that may reduce
economic growth. A weakening US dollar is also worrisome, as a potential sharp decline of it
may contribute to a global recession.
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Chapter 2
Measuring the Macroeconomy
Answers to Textbook Exercises
Analytical Exercises
1. Are capital goods—large turbine generators, jet airliners, bay-spanning bridges—
intermediate goods or final goods? How are they included in GDP?
Capital goods are included in GDP. The convention is not that goods like large turbine
generators, jet airliners, and bay-spanning bridges are intermediate goods useful in the
production of items that consumers, the government, or foreigners want to use to boost
their utility. The convention is that these capital goods are part of final output, and so
they are included in GDP under the category “investment spending.”
This convention is somewhat arbitrary. A consistent case could be made that this
classification is in error.
2. How do the labour and other factors of production that go into producing
intermediate goods get ultimately counted in GDP?
Through the final goods that the intermediate goods are used to produce. The wheat that a
farmer grows is an intermediate good, and is not included in GDP. But the loaf of bread
made from the wheat and sold in the supermarket is included in GDP. Hence ultimately
the farmer’s labour, the capital he or she uses, and the land he or she uses is included in
real GDP in this indirect way.
3. Explain whether or not and why the following items are included in the calculation
of GDP:
a. Increases in business inventories
b. Sales of existing homes
c. The fees earned by real estate agents on selling existing homes
d. Income earned by Canadians living and working abroad
e. Purchases of IBM stock by your brother
f. Purchase of a new tank by the Defense Ministry
g. Rent that you pay to your landlord
a. Yes; an increase in business inventories is included in investment because it increases
the economy’s capital stock.
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b. No; the sale of an existing home is simply a transfer of assets within the household
sector: one household sells and another household buys, but no goods or services are
produced by the exchange.
c. Yes; real estate agent fees are included in real GDP because they are payment for
providing a final service to households—the service of matching homeowners who wish
to sell with those who wish to buy.
d. No; income earned by Canadians living and working abroad is included in GNP—
Gross National Product—but is not included in GDP—Gross Domestic Product.
e. No; one again this is a transfer of assets: one person sells, another person buys. The
amount of investment that IBM undertakes to increase its capital stock is unaffected by
this purchase.
f. Yes; the purchase of a new tank is definitely a government purchase, and is included in
GDP.
g. Yes; rent is payment for a final service—“housing services.”
4. Which interest rate concept--the nominal interest rate or the real interest rate--do
lenders and borrowers care more about? Why?
The real interest rate is the more important concept. The nominal interest rate does not
tell you whether it is cheap or expensive to borrow because it does not correct for
inflation. A nominal interest rate of 20 percent per year seems high. But if inflation is
proceeding at 30 percent per year, the real interest rate is an extraordinarily low –10% per
year: the amount of money a borrower repays—principal plus interest—after a year buys
10% fewer goods then than the original amount borrowed did at the moment that the loan
is made. Hence look at the real, not the nominal, interest rate to determine how cheap or
expensive it is to borrow in terms of its effect on borrowers’ and lenders’ purchasing
power, which is what they ultimately care about.
5. Which is the more important measure for assessing an economy's performance, real
GDP or nominal GDP?
Real GDP. Once again, nominal GDP does not correct for changes in the price level. A
rise in nominal GDP might be due to inflation, or might be due to an increase in
production. You need to look at real GDP to ascertain which.
6. Consider the two definitions of the nominal exchange rate, e and e f in Section 2.2.
(a) Write down the real exchange rates  and  f corresponding to e and e f and
interpret them.
(b)Using the definitions  and  f in (a) show that:
 /   e / e  P f / P f  P / P
and
 f / 
f
 e f / e f  P / P  P f / P f .
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Interpret these relations.
(c) As we will discuss in Chapter 15, the theory of relative purchasing power asserts
that the real exchange rate is constant over time. If this is true, how would you
simplify the two expressions in part (b) of the question? Interpret your results.
a)
  e
(1)
Pf
P
and
(2) 
f
 ef 
P
.
Pf
 is the domestic real exchange rate; that is the number of baskets of domestic goods and
services that are required in order to buy one basket of foreign goods and services.
 f is the foreign real exchange rate; that is the number of baskets of foreign goods and services
that are required in order to buy one basket of domestic goods and services.
Notice that  
1

f
, so that the two exchange rates are the inverse to each other.
b) Taking the natural log on both sides of equation (1) in (a) and totally differentiating the
resulting expression we get:
 log    log( e 
Pf
)   log e   log P f   log P
P
or equivalently,
(3)
 /   e / e  P f / P f  P / P .
Similarly, taking the natural log on both sides of equation (2) in (a) and totally differentiating the
resulting expression we get:
 log 
f
  log( e f 
P
)   log e f   log P   log P f
f
P
or equivalently,
(4)
 f / 
f
 e f / e f  P / P  P f / P f .
Equation (3) states that the proportional rate of change of the real domestic exchange rate is
equal to the proportional rate of change of the domestic nominal exchange rate plus the inflation
differential between the foreign and the home country.
Equation (4) states that the proportional rate of change of the real foreign exchange rate is equal
to the proportional rate of change of the foreign nominal exchange rate plus the inflation
differential between the home and the foreign country.
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c) Under relative purchasing power parity (PPP), the real exchange rate is constant over time, so
that  /    f /  f  0 .Using these relations in equations (3) and (4), respectively, we get:
e / e  P f / P f  P / P
and
e f / e f  P / P  P f / P f .
These equations state that, under relative PPP, the proportional rate of change in the nominal
exchange rate is exactly equal to the inflation differential between the two countries.
7. Suppose the nominal GDP grows at the rate of 2 percent per year, while at the same time
prices increase at the rate of 1 percent per year. What is the growth rate of real GDP?
Real GDP, Y, is the ratio of nominal GDP, X say, divided by the price level P: Y=X/P. Taking
the proportional rates of change on both sides of this equation, we see that the growth rate of Y
is the difference between the growth rate of X and the growth rate of P. Thus, in this case, the
growth rate of real GDP will be 1% (2%-1%).
8. Suppose the price level in the US increases at the rate of 1 percent per year, the price
level in Canada increases at the rate of 2 percent per year, and the value of the US dollar
relative to the Canadian dollar increases at the rate of 1 percent per year. What is the rate
of change in the real exchange rate?
Using equation (3) from 6(a) above, we find that the rate of change in the real exchange rate is
2% (1%+2%-1%).
Policy Exercises
1. In a particular year the (short-term) nominal interest rate on three-month Treasury bills
averaged 10.0%, and the GDP deflator rose from 50.88 to 55.22. What was the annual
rate of inflation? What was the real interest rate?
a. Were real interest rates higher in year A describe above, or in year B, when the
(short-term) nominal interest rate on three-month Treasury bills was 4.8%, and the
inflation rate was 2.6%?
b. Which interest rate concept--the nominal interest rate or the real interest rate--do
lenders and borrowers care more about? Why?
In year A the inflation rate—the proportional rate of increase in the GDP deflator—
was 55.22/50.88 – 1 = 8.53% per year. Thus the real interest rate in that year was 10.0%
–8.53% = 1.467% per year: lower than the real interest rate in year B of 4.8% – 2.6% =
2.2% per year.
The real interest rate is the more important concept. The nominal interest rate does not
tell you whether it is cheap or expensive to borrow because it does not correct for
inflation. A nominal interest rate of 20 percent per year seems high. But if inflation is
proceeding at 30 percent per year, the real interest rate is an extraordinarily low –10% per
year: the amount of money a borrower repays—principal plus interest—after a year buys
7
10% fewer goods then than the original amount borrowed did at the moment that the loan
is made. Hence look at the real, not the nominal, interest rate to determine how cheap or
expensive it is to borrow in terms of its effect on borrowers’ and lenders’ purchasing
power, which is what they ultimately care about.
2. Suppose in a particular year the GDP deflator rose at an annual rate of 2.6%, and the
short-term interest rate on three-month Treasury bills averaged 4.8% What was the
(short-term) nominal interest rate? What was the (short-term) real interest rate?
The short-term nominal (safe) interest rate was 4.8% per year. The short-term real
(safe) interest rate was the difference between the nominal interest rate and the
inflation rate: 4.8% - 2.6% = 2.2% per year.
3. In 1992 the implicit GDP deflator (in 1992 dollars) was equal to 100; in 1993 it was equal
to 101.54. What was the annual rate of inflation between 1992 and 1993? In 1994 the
implicit GDP deflator (in 1992 dollars) was equal to 102.70. What was the annual rate of
inflation between 1993 and 1994?
The annual rate of inflation between 1992 and 1993 was the proportional increase in the
101.54  100
 100%  1.54% . The annual rate of
GDP deflator between 1992 and 1993:
100
inflation between 1993 and 1994 was the proportional increase in the GDP deflator
102.70  101.54
 100%  1.14%
between 1993 and 1994:
101.54
4. In 1992 both nominal GDP and real GDP (measured in 1992 dollars) were equal to
$702,393 billion. By 1997 nominal GDP had risen to $885,022 billion, and the implicit
GDP deflator had risen to 108.59. What was real GDP in 1997? What was the average
rate of real GDP growth between 1992 and 1997?
Dividing nominal GDP in 1997 by the implicit GDP deflator (and remembering that a
value of 100 for the implicit GDP deflator tells us that nominal GDP is equal to real
GDP), we calculate that real GDP in 1997 was $815,012 billion. Let us denote the
average rate of growth of real GDP during the five years from 1992-1997 as x.
Then we will have Real GDP in 1997=(1+x)5  Real GDP in 1992. Hence,
x=(Real GDP in 1997  Real GDP in 1992)1/5 –1 = 0.03=3%.
5. Use the data in the following table to answer the questions below:
Real GDP
Year
1981
1991
2001
(billions)
$551.30
$692.25
$1,011.09
Labour Force
(millions)
12.22
14.33
16.25
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What was real GDP per worker in 1981, 1991, and 2001? How fast did real GDP per
worker grow between 1981 and 1991? Between 1991 and 2001?
Levels of GDP per worker
1981 $45,115
1991 $48,307
2001 $62,221
Annual rates of growth are calculated as follows. Let us denote the average rate of growth
of real GDP per worker during the ten years from 1981-1991 as x. Then we will have
Real GDP per worker in 1991=(1+x)5  Real GDP per worker in 1981. Hence, x=(Real
GDP per worker in 1991  Real GDP per worker in 1981)1/10 –1 = 0.007=0.7%.
Similarly, can can calculate the average rate of growth of real GDP per worker during the
ten years from 1991-2001 as =(Real GDP per worker in 2001  Real GDP per worker in
1991)1/10 –1 = 0.026=2.6%.
6. Consider the following data for Canada for 2000 in millions of dollars.
Consumption
spending
Investment spending
Government
purchases
Net exports
Capital consumption
allowances
Indirect taxes less
subsidies
594,089
194,177
219,816
55,397
135,781
127,745
a. Using this data calculate Gross Domestic Product (GDP) at market prices.
b. Calculate GDP at factor cost.
c. Calculate Net Domestic Product (NDP)at factor cost.
a. GDP at market prices = C+I+G+NX=594,089+194,177+219,816+55,397=1,063,479
b. GDP at factor cost = GDP at market prices–Indirect taxes less subsidies
=1,063,479–127,745 = 935,734
c. NDP at factor cost = GDP at factor cost – Capital consumption allowances
= 935,734 – 135,781 = 799,953
7. Consider the following data for Canada for 2000 in millions of dollars.
Wages, salaries & supplementary labour income
Corporation profits before taxes
Government business enterprise profits before taxes
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545,110
129,821
11,832
Interest & miscellaneous investment income
Accrued net income of farm operators from production
Net income of non-farm unincorporated businesses, including rents
Inventory valuation adjustment
Capital consumption allowances
Indirect taxes less subsidies
Net income from abroad
53,933
1,758
63,962
-3,431
135,781
127,745
-22,368
a. Using this data calculate Net Domestic Product (NDP) at factor cost.
b. Calculate Gross Domestic Product (GDP) at market prices.
c. Calculate Net National Product (NNP) at factor cost?
a. NDP at factor cost = Wages, salaries & supplementary labour income
+ Corporation profits before taxes
+Government business enterprise profits before taxes
+ Interest & miscellaneous investment income
+ Accrued net income of farm operators from production
+ Net income of non-farm unincorporated businesses, including
rents
+ Inventory valuation adjustment
=545110+129821+11832+53933+1758+63962–3431
=802,985
b. GDP at market prices = NDP at factor cost + capital consumption allowances
+ Indirect taxes less subsidies
= 802,985+135,781+127,745
=1,066,511
c. NNP at factor cost = NDP at factor cost + Net income from abroad
= 802,985–22,368
= 780,617
8. How does the level of the stock market today compare with what it was in 1999?
What were the reasons for the decline in stock prices in 2001?
The stock prices in 2007 are still slightly lower than in the late 1990s. The reason for the
dramatic decline in stock prices in 2001 was the disillusionment of the investors about the
Information Technology (IT) sector in the early 2000s. In the 1990s investors were overoptimistic about the future of the IT sector. When these hopes were dashed, stock prices fell
dramatically in 2001. This was compounded by the September 11 terrorist attacks and the
corporate corruption scandals in the U.S., headed by Enron and World Com. Since then stock
prices have recovered to a large extent, but even in 2007 they have not reached their levels in the
late1990s.
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Chapter 3
The Theory of Economic Growth
Answers to Textbook Exercises
Analytical Exercises
1. Explain the effects of an increase in the rate of depreciation of capital in the neoclassical growth model and in Romer´s endogenous growth model.
First consider the neoclassical model. In Figure (i) below, the initial steady state
equilibrium is at point A, where the original saving per worker curve,

s  Kt / Lt   E1 , intersects the “required investment per worker” curve
( 0  n)  K t / Lt . An increase in the depreciation rate from  0 to 1 will shift the
required investment schedule up to (1  n)  ( K t / Lt ) . With higher depreciation of
capital, at the initial capita-labour ratio ( K / L) 0 the total amount of saving will not be
sufficient to maintain K/L at a constant level. Hence, K/L will start to fall. The new steady
state equilibrium will be at point B, with a lower steady state capital-labour ratio ( K / L)1 .
In Romer´s endogenous growth model, the saving per worker function,
s  Yt / L  s  A  K t / L , will be a straight line through the origin, as shown in Figure (ii)
below. An increase in the depreciation rate from  0 to 1 will shift the required
investment schedule up from ( 0  n)  K t / Lt to (1  n)  ( K t / Lt ) . Then corresponding
to any capital-labour ratio, (Kt/L)0 say, the difference between savings per worker and the
required investment per worker will be smaller, which will reduce the growth rate.
Recall, in the Romer model growth never comes to a halt; it is the growth rate that will be
affected.
Figure (i): The Neoclassical Model
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Figure (ii): The Romer Model
2. Suppose the economy’s aggregate production function is given by
12
Yt  Kt   E  Lt 
0.5
0.5
where Yt is aggregate output, K t is capital, Lt is labour and E is the efficiency of labour.
Assume that the capital’s depreciation rate is 3 percent per year, the rate of population
growth is 1 percent per year, and the efficiency of labour is E=1.
a) Suppose that the saving rate is 10 percent of GDP. What is the steady-state
capital-labour ratio? What is the value of output per worker in the steady-state?
b) Suppose that the saving rate is 15 percent of GDP. What is the steady-state
capital-output ratio? What is the value of output per worker along the economy’s
steady-state growth path?
a. First derive the steady state capital-labour ratio, which is given by setting equation
(3.10) equal to zero; that is,
Yt
K
 (  n)  t .
Lt
Lt
Next, consider the Cobb-Douglas production function with E=1:
0  s
Yt  K t 
 
Lt  Lt 
(i)

(ii)
Substituting for Y/L from equation (ii) into (i), we obtain
K
0  s   t
 Lt


K
  (  n)  t .
Lt

(iii)
Now divide both sides of equation (iii) by Kt/Lt to obtain
 1
K 
0  s   t   (  n) .
(iv)
L
 t 
Equation (iv) can be solved for Kt/Lt, which gives us its steady state level as
1
K t    n   1

 .
Lt  s 
(v)
Finally, substituting for Kt/Lt from equation (v) into (ii), we obtain the steady state level
of output per worker

Yt    n   1

(vi)
 .
Lt  s 
If δ=0.03, n=0.01, s=0.10, and α=0.5. This gives us the steady state output per worker as
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0.5
Yt  0.03  0.01  0.51

 2.5 units of output per worker.

Lt  0.10 
b. If δ=0.03, n=0.01, s=0.15, and α=0.5. This gives us the steady state output per worker
as
0.5
Yt  0.03  0.01  0.51

 3.75 units of output per worker.

Lt  0.15 
The increase in the saving rate from 0.10 to 0.15 increases the output per worker from 2.5
to 3.74.
3. What happens to the steady-state capital-output ratio if the rate of technological
progress increases? Would the steady-state growth rate of output per worker
increase, decrease or remain in the same position?
If the rate of technological progress (i.e., the growth rate of labour efficiency) increases,
the steady-state capital-output ratio falls but the steady-state growth rate of output per
worker increases. Faster technological progress leads to a less capital-intensive
economy, but to an economy with a higher and more rapidly growing efficiency of
labour.
4. According to the marginal productivity theory of distribution, in a competitive economy
the rate of return on a dollar's worth of capital--its profits or interest--is equal to
capital's marginal productivity. If this theory holds and the marginal productivity of
capital is indeed:
dY/dK =   (Y/K)
How large are the total earnings received by capital? What share of total output will be
received by the owners of capital as their income?
Since the marginal product of capital is the amount earned by a unit of capital, the total
income arising out of the economy’s entire capital stock is:
K  (dY/dK) =   (Y/K)  K =   Y
So a share  of total output is received by the owners of capital as their income. This
provides a way to estimate the diminishing-returns parameter  in the production
function: simply look at all the income earned by capital owners and see how large a
share it is of total output. (Of course, this works only if the marginal productivity theory
of distribution is in fact true.)
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5. What is meant by convergence in growth theory? What is the evidence regarding
convergence?
Two types of convergence have been analyzed in the literature, absolute convergence and
conditional convergence. The hypothesis of absolute convergence states that poor
economies tend to grow faster per capita than rich ones. This concept assumes that
countries are similar in other characteristics regarding saving rates, population growth
rates, production technologies, education and human capital, government policies and the
like. On the other hand, conditional convergence takes into account the fact that countries
may differ in these important characteristics. Hence, conditional convergence states that
poor economies tend to grow faster in terms of per capita income than rich ones, provided
that we take into account differences in country characteristics.
A large number of recent studies have used actual economic data to analyze the
hypothesis of economic convergence across different countries, and also different
provinces/states within countries. The empirical findings have been mixed with some
studies supporting convergence and other studies rejecting convergence in favour of
divergence.
6. What are some of the possible shortcomings in the predictions of the neoclassical growth
model? How do endogenous growth models attempt to overcome these possible
shortcomings?
Paul Romer, of Stanford University, and Robert Lucas, of the University of Chicago,
pointed out in the 1980’s that there were some empirical regularities that contradict the
predictions of the neoclassical growth model. First, the model suggests that countries
with similar rates of growth of population should eventually converge to the same steady
state point. This cannot account for the disparities we observe in standards of living
around the world, where poor countries seem to have standards of living that are
permanently below those of the rich countries.
Another empirical regularity that contradicts the predictions of the neoclassical growth
model is with regards to the growth rate of output per worker. Recall in the neoclassical
model after a technological improvement there will be an increase in the growth rate of
output per worker, but this growth rate will gradually decline to its original level in the
new steady state, which is zero. Romer, considering growth rates over centuries,
discovered that over such long periods growth seems to have accelerated over time, and
in recent decades it has been relatively stable.
Another shortcoming of the neoclassical growth model is that it relies on exogenous
factors to generate ongoing growth in the economy. Ideally, economists would like to
explain the effects of policy changes on growth, and recommend policies that would
foster growth. We would like to explain, for example, whether free trade and economic
integration would have any effect on growth.
15
Policy Exercises
1. Suppose because of the elimination of the federal budget deficit the national saving rate
is boosted from 16% to 20% of real GDP. Suppose further that the rate of labour force
growth is 1% per year, the depreciation rate is 3% per year, the level of the efficiency of
labour E is 30,000, and that the diminishing-returns-to-capital parameter  is 1/3.
Suppose that these rates continue into the indefinite future. What would be the steady
state level of output per worker before the budget deficit was eliminated? What would be
the steady state level of output per worker after the budget deficit is eliminated?
First derive the steady state capital-labour ratio, which is given by setting equation (3.10)
equal to zero; that is,
Yt
K
 (  n)  t .
Lt
Lt
Next, consider the Cobb-Douglas production function with E=1:
0  s
(i)

Yt  K t 
    E 1
Lt  Lt 
(ii)
Substituting for Y/L from equation (ii) into (i), we obtain
K
0  s   t
 Lt


K
  E 1  (  n)  t .
Lt

(iii)
Now divide both sides of equation (iii) by Kt/Lt to obtain
 1
K 
0  s   t   E 1  (  n) .
(iv)
 Lt 
Equation (iv) can be solved for Kt/Lt, which gives us its steady state level as
1
K t    n   1

 .
Lt  s  E 1 
(v)
Finally, substituting for Kt/Lt from equation (v) into (ii), we obtain the steady state level
of output per worker

Yt    n   1

 E 1 .
(vi)
1 
Lt  s  E 
Before the elimination of the budget deficit we had δ=0.03, n=0.01, s=0.16, E=30,000,
and α=0.33. This gives us the steady state output per worker as
16
Yt  0.03  0.01 


Lt  0.16  30,000 0.67 
0.33
0.331
 30,000 0.67  59,385 units of output.
After the elimination of the budget deficit, s jumps to 0.20. The new steady state output
per worker then becomes
0.33
Yt  0.03  0.01  0.331

 30,000 0.67  66,280 units of output.

0.67
Lt  0.20  30,000 
2. What are the long-run costs as far as economic growth is concerned of a policy of taking
money that would reduce the national debt—and thus add to national savings—and
distributing it as tax cuts instead? What would be the long-run benefits of such a policy? How
could we decide whether such a policy was a good thing or not?
The short-run benefits of this policy would be to boost consumption now and in the near
future, as less of today’s output is used for investment and more is available for
consumption. The long-run costs are likely to be a lower national savings rate, a lower
steady-state capital-output ratio, and a lower level of output-per-worker along the
economy’s long-run steady-state growth path. Whether the short-run benefits outweigh
the long-run costs depends on how we balance off improvements in consumption now
versus probable costs in terms of reduced consumption in the future. We would need to
decide how much we value the present compared to the future to be able to decide on
whether this policy was a good thing.
3. Suppose because of the computer revolution the efficiency of labour E increases from
30,000 to 40,000. Suppose the rate of labour force growth were to remain constant at 1
percent per year, the depreciation rate were to remain constant at 3 percent per year,
and the saving rate were to remain constant at 20 percent per year. Assume that the
diminishing-returns-to-capital parameter  is 1/3. What is the change in the steady-state
output per worker?
Following the same steps as in the answer to policy exercise 1, we obtain the steady state
level of output per worker

Yt    n   1

 E 1 .
(vi)
1 
Lt  s  E 
Before the computer revolution we had δ=0.03, n=0.01, s=0.20, E=30,000, and α=0.33.
This gives us the steady state output per worker as
0.33
Yt  0.03  0.01  0.331

 30,000 0.67  66,280 units of output per

Lt  0.20  30,000 0.67 
worker.
After the computer revolution E jumps to 40,000. The new steady state output per worker
then becomes
17
Yt  0.03  0.01 


Lt  0.20  40,000 0.67 
0.33
0.331
 40,000 0.67  88,375 units of output per
worker.
4. Suppose that a sudden disaster–an epidemic, say– reduces a country's population and
labour force, but does not affect its capital stock. What is the immediate effect of the
epidemic on output per worker? On the total economy-wide level of output? What
happens subsequently?
In the figure below, start at the steady state equilibrium point A, with the capital labour
ratio (K/L)0. As the epidemic decreases the number of workers without affecting the
economy’s capital stock, it increases the capital-labour ratio to (K/L)1. Total output per
worker jumps upward because the capital-labour ratio has increased. However, total
output falls: there are fewer workers in the economy. Subsequently the economy’s level
of output per worker falls (or grows more slowly) as the economy re-converges to its
steady state.
5. Suppose that environmental regulations lead to a slowdown in the rate of growth of the
efficiency of labour in the production function, but also lead to better environmental quality.
Should we think of this as a “slowdown” in economic growth or not?
How much is environmental quality improved? It is entirely possible that these
regulations lead to a speedup in the “true” rate of economic growth if the better
environmental quality is worth more than the output foregone. It is also possible that
these regulations lead to a slowdown in the “true” rate of economic growth if the
18
regulations are inefficient—if they reduce production by a lot, and improve the
environment by only a little. We would need to know a lot more to assess the net impact
on “true” economic growth.

Y K 
6. Consider the Barro growth model, where the production function is t   t   Gt1 ,
Lt  Lt 
and Gt    Kt . Assume there is no depreciation or population growth.
a. For this model, derive the steady state growth rate.
b. Derive the growth maximizing level of taxes.
a. First let us simplify the equations that are given. Substituting for Gt

Y K 
from Gt    Kt into the production function t   t   Gt1 , we obtain
Lt  Lt 

Yt  1 
     1  K t
Lt  Lt 
(ii)
With no population growth Lt is constant. For simplicity set Lt=1.
Next, recall equation (3.10), which gives us the evolution of Kt/Lt:
K 
Y
K
 t   s  t  (  n)  t .
Lt
Lt
 Lt 
(i)
Substituting for Yt/Lt from equation (ii) into (i), and setting Lt=1, we obtain the
expression for the evolution of of Kt/Lt as
K 
 t   s  1  K t  (  n)  K t
 L 
Dividing both sides of this equation by Kt, we obtain the growth rate of K as
K t
 s  1  (  n) .
Kt
(iii)
With Lt constant, the growth rate of Y will be the same as the growth rate of K. This is
clear from equation (i).
b. To find the growth maximizing taxes τ*, differentiate equation (iii) and set it equal to
zero to obtain
1
  n 

 *  
 s  (1   ) 
19
Chapter 4
The Reality of Economic Growth:
History and Prospect
Answers to Textbook Exercises
Analytical Exercises
1. Would an increase in the saving and investment share of Canada’s total output raise
growth in productivity and living standards?
The answer would be “no” if the saving-investment rate was greater than the diminishing
returns to scale parameter, but this appears not to be the case. An increase in the savinginvestment rate would raise growth in output per worker. It would increase living
standards in the future after the economy has substantially converged to its steady-state
growth path. However, it would reduce living standards in the present and near future as
resources that would otherwise have been devoted to consumption are diverted to
investment.
2. Many observers project that by the end of the twenty-first century the population of the
industrialized countries will be stable. Using the Solow growth model, what would such a
downward shift in the growth rate of the labour force do to the growth of output per
worker and to the growth of total output (consider both the effect on the steady-state
growth path, and the transition from the "old" positive population growth to the "new"
zero population growth steady-state growth path)?
A move to zero population growth would raise the steady-state capital output ratio and
raise the level of output per worker along the steady-state growth path. As long as the
economy was in the transition to the new steady-state growth path, output per worker
would be growing more rapidly than before. Once the steady-state growth path is
reached, output per worker growth would be the same: along the steady-state growth
path, output per worker growth is determined by technological progress.
The rate of growth of total output would, of course, slow down: no growth in the number
of workers means slower growth in total output.
3. What are the arguments for having a strong patent system to boost economic growth?
What are the arguments for having a weak system of protections of "intellectual
property"? Under what systems do you think that the first will outweigh the second?
Under what circumstances do you think that the second will outweigh the first?
20
Governments face a difficult dilemma. If their patent laws are strong, then much of the
modern technology in the economy will be restricted in use: either restricted to being
used only by the inventor, or restricted because the inventor is charging other firms high
licensing fees to use the technology (or not letting them use it at all). Thus a government
that enacts strict patent laws is pushing the average level of technology used in its
factories and businesses at some particular moment far below the level that could be
achieved at that particular moment.
On the other hand, if the patent laws are weak--so that they provide little protection to
inventors and innovators--then the profits that inventors and innovators earn will be low.
Why then should businesses devote money and resources to research and development?
They will not. And the pace of innovation, and thus of technological improvement, will
slow to a crawl.
This dilemma cannot be evaded. Governments must determine the circumstances under
which the benefits from diffusing knowledge widely outweigh the reduced pace of
domestic innovation, and the circumstances under which providing more incentives for
invention and innovation outweigh the restricted distribution of knowledge and the
resulting gap between typical and best practice technology.
4. What steps do you think that international organizations--the UN, the World Bank, or the
IMF--could take to improve political leaders' incentives to follow growth-promoting
policies?
A very hard and open-ended question. The answer that the great and good of the world
have decided upon is “neoliberalism”: offer countries incentives to reduce their barriers
to trade, and advise them that more aid will be forthcoming if they privatize industries
and shrink the size of the government.
However, as it says in the text, whether this is the right answer is unclear. Whether such
policies will in fact lead to convergence rather than continued divergence is still an open
question.
5. Suppose somebody who hasn't taken any economics courses were to ask you why
humanity escaped from the Malthusian trap--of very low standards of living and slow
population growth rates that nevertheless put pressure on available natural resources and
kept output per worker from rising--in which humanity found itself between the year
8000 B.C.E. and 1800. What answer would you give?
First, the demographic transition: once incomes rose high enough, population growth
rates slowed down—and slower population growth means higher steady-state capital
output ratios and a richer world. What made incomes rise high enough for the
demographic transition to begin? An acceleration of technological progress caused by the
long slow accumulation of technologies for growing food, manipulating matter, and
transmitting information that together raised the number of brains thinking of inventions
21
and the ability of those brains to communicate with each other. And once you have
passed critical technological mass, technological progress continues rapidly.
6. Suppose somebody who hasn't taken any economics courses were to ask you why it is
that some countries are so very, very much poorer than others in the world today. What
answer would you give?
Rich countries are rich today because they (a) are culturally close enough to the heartland
of the industrial revolution to take advantage of productive industrial technologies, (b)
have relatively uncorrupt governments that encourage savings and investment, and (c)
have passed through the demographic transition and thus have low rates of labour force
growth.
Poor countries are poor today because they have not been culturally close to the heartland
of the industrial revolution, hence technology transfer has been slow and uneven, and
have not had uncorrupt governments that encourage savings and investment. Because
they are poor vicious circles have set in: the capital goods needed to boost investment and
capital intensity are very expensive; the demographic transition has not set in because in
places of high mortality and low productivity your ability to avoid starving to death in
your old age depends on having enough children that some will survive to support you.
7. The endogenous growth theorists, led by Stanford’s Paul Romer, argue that it is a
mistake to separate the determinants of the efficiency of labour from investment—
that investments both raise the capital-worker ratio and increase the efficiency of
labour as workers learn about the new technology installed with the purchase of
new, modern capital goods. If the endogenous growth theorists are correct, is the
case for government policies to boost national savings and investment rates
strengthened or weakened? Why?
Strengthened. The benefits to higher savings and investment are increased because
they are bringers of faster productivity growth as well as higher capital-output ratios.
8. Suppose that population growth depends on the level of output per worker, so that:
(1) n = (.0001)  [(Y/L) - $200]
the population growth rate n is zero if output per worker equals $200, and that each $100
increase in output per worker raises the population growth rate by 1% per year.
Suppose also that the economy is in its Malthusian regime, so that the rate of increase of the
efficiency of labour E is zero and output per worker is given by:
  


(2)
Yt  s   1 

 E0

Lt  n   
22
with the diminishing-returns-to-investment parameter  = .5, with the depreciation rate  =
.04, and with the efficiency of labour E0 = $100.
a. Suppose that the savings rate s is equal to .08, 8% per year. Graph (on the same set of axes)
steady-state output-per-worker (Y/L) as a function of the population growth rate n from
equation (2) and the population growth rate n as a function of output-per-worker (Y/L)
from equation (1).
b. Where do the curves cross? For what levels of output per worker Y/L and population
growth n is the economy (i) on its steady-state path, and (ii) at its Malthusian rate of
population growth?
c. Suppose that the savings rate were to rise by an infinitesimal amount--say by onehundredth of one percentage point, from .08 to .0801. Calculate approximately how the
equilibrium position of the economy would change. By how much--and in which direction-would steady-state output per worker change? By how much--and in which direction-would the population growth rate change?
a. The graph (on the same set of axes) of steady-state output-per-worker (Y/L) as a
function of the population growth rate n from equation (2), and the population growth
rate n as a function of output-per-worker (Y/L) from equation (1).
23
b. At an output per worker level of $200, population growth is zero. At a population
growth rate of zero, steady-state output per worker is $200.
c. At an output per worker level of $200.167 the economy is in Malthusian equilibrium
with a population growth rate of .00167% per year
9. Suppose we have our standard growth model with s = 20%, n = 1%, g = 1%, and  = 3%.
Suppose that the current level of the efficiency of labour E is $10,000 per year, and that
the current level of capital per worker is $50,000.
Suppose further that the parameter  in the production function:

Yt  K t 
1
    Et 
Lt  Lt 
is equal to one:  = 1.
a. What can you say about the future growth of output per worker in this economy? Can
you write down an equation for what output per worker will be at any date in the
future?
24
b. Suppose that the savings rate s were not 20% but 15%. How would the future growth of output
per worker be different?
c. Why aren’t the normal tools of analysis and rules of thumb of the growth model much
use when  = 1? (Consider the shape of the production function, and what that says
about diminishing returns to investment.)
a. If  = 1, then Yt/Lt = Kt/Lt. The proportional growth rate of output per worker is then
equal to s –  – n. So if we write y0 for the level of output per worker at time zero, then at
any time t in the future output per worker will be:
Yt
t
 y0  1  s    n 
Lt
In this case, for these parameter values, the solution is:
Yt
t
 $50,000  1.16
Lt
b. For a savings rate of 15%:
Yt
t
 $50,000  1.11
Lt
c. Well, consider our expression for the speed of convergence to the steady state: that the
economy closes a fraction (1–)(n+g+) of the gap to the steady state each year. This
means that when  =1, there is no convergence to the steady-state growth path at all—so
there seems to be no point in analyzing it: it no longer meets the definition of an
equilibrium.
Policy Exercises
1. Take a look in the back of this book at the rate of growth of real GDP per worker in
Canada over the period 1991 to 2001. How large was the "trend" component of growth
per year? How large was the "cycle" component of growth per year?
First computed the average growth rate of real GDP per worker for the period 19912001. Real GDP per worker in 1991 was $54,137 (in 1992 prices), while it was $62,236
(in 1992 prices) in 2001. If x is the average annual rate of growth of real GDP per
worker in this decade, then we will have x=(Real GDP per worker in 2001  Real GDP
per worker in 1991)1/10 –1 = 0.014=1.4%. This is the trend annual rate of growth of real
GDP per worker
Any deviation of the growth rate in a year from the trend level of the growth rate is due
to cyclical factors. For example, the growth rate of real GDP per worker between 1991
and 1992 was 1.4 per cent, which the same as the trend rate of growth of output per
worker.
25
Between 1992 and 1993 output per worker grew at the rate of 1.12 per cent, which was
0.28 percentage points below trend.
Between 1993 and 1994 output per worker grew at the rate of 3.81 per cent, which was
2.41 percentage points above trend.
Between 1994 and 1995 output per worker grew at the rate of 1 per cent, which was 0.4
percentage points below trend.
Between 1995 and 1996 output per worker grew at the rate of 0.86 per cent, which was
0.54 percentage points below trend.
Between 1996 and 1997 output per worker grew at the rate of 1.47 per cent, which was
0.07 percentage points above trend.
Between 1997 and 1998 output per worker grew at the rate of –0.17 per cent, which was
1.57 percentage points below trend.
Between 1998 and 1999 output per worker grew at the rate of 2.48 per cent, which was
1.08 percentage points above trend.
Between 1999 and 2000 output per worker grew at the rate of 3.06 per cent, which was
1.66 percentage points above trend.
Between 2000 and 2001 output per worker grew at the rate of –1.07 per cent, which was
2.47 percentage points below trend.
2. Take a look at the first three columns of the "Canadian Macroeconomic Data" table
at the back of the book. Identify the years that are business cycle peaks-years that
are followed by a decline in real GDP or real GDP per worker. Are the two possible
sets of business cycle peaks the same? Calculate economic growth rates between
business cycle peaks, and make a table of these annual peak-to-peak growth rates. Is
this a good way of looking for and estimating changes in long-run economic growth
trends in Canada? Why or why not?
The data is from 1961 until 2001. If real GDP is used then 1981 and 1990 are the
only two years in which the business cycle peaked. If the real GDP per worker is
used then the business cycle peaked in 1974, 1977, 1979, 1985, 1989, 1997 and
2000. These two sets of business cycle peaks are not the same.
To calculate the average economic growth rate x between years t and t+n. use the
formula Yt(1+x)n=Yt+n, or x=[Yt+n/Yt]1/n–1, where Y is either GDP or GDP per
worker. Using this formula, the average rate of growth of real GDP between 1981
and 1990 was 2.78%.
26
Similarly, the average rate of growth of real GDP per worker
between 1974 and 1977 was 0.94%,
between 1977 and 1979 was –0.47%
between 1979 and 1985 was 0.15%
between 1985 and 1989 was 0.50%
between 1989 and 1997 was 0.58%
between 1997 and 2000 was 1.83%
This is not a good way of looking for and estimating long run economic growth
trends. To look at long-run growth trends one needs to eliminate all business cycle
elements (i.e., estimating potential outputs), and calculate the growth rates
between potential outputs. The procedure we have used above does not do that.
3. Take a look at the relative purchasing-power-parity compared levels of GDP per
worker for the G-7 economies--Germany, France, Britain, Italy, Canada, Japan,
and the U.S. Have they drawn closer together in levels of GDP per worker in the
past fifty years?
Yes. They have grown closer together since 1950. However, the U.S. has grown
faster since the early to mid-1990s than the other major economies in the world; see
Figure 4.10.
4. What items of news have you read in the past week that you would classify as shifts
in economic policies that encourage growth?
Anything that leads one to be more optimistic about the savings-investment rate expected
in the future, or that decreases the expected labour force growth rate over the past
century. Anything that increases expected economic efficiency or technological
development.
5. What items of news have you read in the past week that you would classify as shifts
in macro policies that discourage growth?
Anything that leads one to be less optimistic about the saving-investment rate expected in
the future, or that increases the expected labour force growth rate over the past century.
6. What items of news have you read in the past week that you would classify as shifts in
micro policies that encourage growth?
Anything that increases expected economic efficiency or technological development.
7. What items of news have you read in the past week that you would classify as shifts
in micro policies that discourage growth?
27
Anything that decreases expected economic efficiency or technological development.
8. Do you believe that over the next three decades the lower income countries of the
world will catch up to--or at least draw nearer in relative terms to--the high income
countries? Why or why not?
It’s a judgment call. There are lots of economic forces that should more convergence. But
in the past there were the same forces, and convergence has not (or has not yet)
happened.
9. Discuss the possible explanations for the productivity gap between Canada and the
US. What kind of government policies would help in reducing the productivity gap?
See Box 4.1 for details. There are several main causes for the slowdown: a) The most
prominent explanation is the tripling of world oil prices by the OPEC cartel in 1973, in
the wake of the third Arab-Israeli war. Productivity growth slowed at almost exactly the
same time that oil prices skyrocketed; b) Industrialized economies have spent a fortune
on environmental protection in the past generation, and have received big benefits from
this investment, but the gains aren't included in measured GDP; c) In the 1970s the
baby-boom generations began to enter the labour force. This generation is very large. The
relatively young labour force had many more workers with little experience than did the
labour force of the 1960s and 1950s. There must be other explanations as well.
28