Solutions to Exercises in
Introduction to Economic Growth
Charles I. Jones
(with Chao Wei and Jesse Czelusta)
Department of Economics
Berkeley, CA 94720-3880
September 18, 2001 1
2 The Solow Model
Exercise 1. A decrease in the investment rate.
A decrease in the investment rate causes the~ curve to shift down: at any
given level of k, the investment-technology ratio is lower at the new rate of sav-
Assuming the economy began in steady state, the capital-technology ratio is
now higher than is consistent with the reduced saving rate, so it declines gradually,
as shown in Figure 1.
Figure 1: A Decrease in the Investment Rate
The log of output per worker y evolves as in Figure 2, and the dynamics of
~ ~ ~
t00▯▯1owth rate are shown in Figure 3. Recall that ~ = ▯log k and k=k =
s k ▯ (n + g + d).
The policy permanently reduces the level of output per worker, but the growth
rate per worker is only temporarily reduced and will return to g in the long run. 2
Figure 2: y(t)
Figure 3: Growth Rate of Output per Worker
Exercise 2. An increase in the labor force.
The key to this question is to recognize that the initial effect of a sudden in-
crease in the labor force is to reduce the capital-labor ratio since k ▯ K=L and K
is ﬁxed at a moment in time. Assuming the economy was in steady state prior to
the increase in labor force, k falls from k to some new level k 1 Notice that this is
a movement along the sy and (n+d)k curves rather than a shift of either schedule:
both curves are plotted as functions of k, so that a change in k is a movement along
the curves. (For this reason, it is somewhat tricky to put this question ﬁrst!)
At k 1 sy > (n + d)k , 1o that k > 0, and the economy evolves according to
the usual Solow dynamics, as shown in Figure 4.
Figure 4: An Increase in the Labor Force
(n + d) k
k1 k* k
In the short run, per capita output and capital drop in response to a inlarge ﬂow
of workers. Then these two variables start to grow (at a decreasing rate), until in
the long run per capita capital returns to the original level, k . In the long run,
nothing has changed!
Exercise 3. An income tax.
Assume that the government throws away the resources it receives in taxes.
Then an income tax reduces the total amount available for investing and shifts the
investment curve down as shown in Figure 5. 4
Figure 5: An Income Tax
~ ** ~* k
The tax policy permanently reduces the level of output per worker, but the
growth rate per worker is only temporarily lowered. Notice that this experiment
has basically the same results as that in Exercise 2.
For further thought: what happens if instead of throwing away the resources it
collects the government uses all of its tax revenue to undertake investment?
Exercise 4. Manna falls faster.
Figure 6 shows the Solow diagram for this question. It turns out, however,
that it’s easier to answer this question using the transition dynamics version of the
diagram, as shown in Figure 7. When g rises to g ,=k turns negative, as shown in
Figure 7 and A=A = g , the new steady-state growth rate.
To see what this implies about the growth rate of y, recall that
_ ~ A_ k 0
y = ~ + A = ▯ ~ + g :
So to determine what happens to the growth rate of y at the moment of the change
in g, we have to determine what happens tok=k at that moment. As can be seen in
Figure 7, or by algebra, this growth rate falls to g ▯ g < 0 — it is the negative of
the difference between the two horizontal lines.
Substituting into the equation above, we see that _=y immediately after the 5
Figure 6: An Increase in g
~ ~ ~
k** k* k
Figure 7: An Increase in g: Transition Dynamics
k** k* 6
increase in g (suppose this occurs at time t = 0) is given by
_ j = ▯(g ▯ g ) + g = (1 ▯ ▯)g + ▯g > 0:
Notice that this value, which is a weighted average of g and g, is strictly less than
After time t = 0, y _=y rises up to g (which can be seen by looking at the
dynamics implied by Figure 6). Therefore, we know that the dynamics of the
growth rate of output per worker look like those shown in Figure 8.
Figure 8: Growth Rate of Output per Worker
Exercise 5. Can we save too much?
From the standard Solow model, we know that steady-state output per capita is
▯ s 1▯▯ ▯
given by y = ( n+d ) . Steady-state consumption per worker is (1 ▯ s)y , or
▯ ▯ ▯
▯ s 1▯▯
c = (1 ▯ s) n + d :
From this expression, we see that an increase in the saving rate has two effects.
First, it increases steady-state output per worker and therefore tends to increase
consumption. Second, it reduces the amount of output that gets consumed.
To maximize c , we take the derivative of this expression with respect to s and
set it equal to zero:
▯ ▯ ▯ ▯ ▯1
@c ▯ s 1▯▯ ▯ s1▯▯
= ▯ + (1 ▯ s) ▯ = 0:
@s n + d 1 ▯ ▯ (n + d)1▯▯
Rearrange the equation, we have
1 ▯ s▯ ▯
1 = ;
s▯ 1 ▯ ▯
s = ▯:
The saving rate which maximizes the steady-state consumption equals ▯.
Now turn to the marginal product of capital, MPK. Given the production
function y = k , the marginal product of capital is ▯. Evaluated at the steady
state value k , ▯ ▯
▯ (▯▯1) n + d
MPK = ▯(k ) = ▯ :
When the saving rate is set to maximize consumption per person, s = ▯, so
that the marginal product of capital is
MPK = n + d:
That is, the steady-state marginal product of capital equals n + d when consump-
tion per person is maximized. Alternatively, this expression suggests that the net
marginal product of capital — i.e. the marginal product of capital net of depre-
ciation — is equal to the population growth rate. This relationship is graphed in
Figure 9. 8
Figure 9: Can We Save Too Much?
If s > ▯, then steady-state consumption could be increased by reducing the
saving rate. This result is related to the diminishing returns associated with capital
accumulation. The higher is the saving rate, the lower is the marginal product of
capital. The marginal product of capital is the return to investing — if you invest
one unit of output, how much do you get back? The intuition is clearest if we
set n = 0 for the moment. Then, the condition says that the marginal product of
capital should equal the rate of depreciation, or the net return to capital should be
zero. If the marginal product of capital falls below the rate of depreciation, then
you are getting back less than you put in, and therefore you are investing too much.
Exercise 6. Solow (1956) versus Solow (1957).
a) This is an easy one. Growth in output per worker in the inital steady state is
2 percent and in the new steady state is 3 percent.
b) Recall equation (2.15)
_ k B_
y = ▯ k + B
y_ _ _
y ▯ k B
Initial S.S. .02 1/3*(.02) 2/3*(.02)=.0133
New S.S. .03 1/3*(.03) 2/3*(.03)=.0200
Change .01 1/3*(.01) 2/3*(.01) = .0067 9
In other words, Solow (1957) would say that 1/3 of the faster growth in output
per worker is due to capital and 2/3 is due to technology.
c) The growth accounting above suggests attributing some of the faster growth
to capital and some to technology. Of course this is true in an accounting sense.
However, we know from Solow (1956) that faster growth in technology is itself
the cause of the faster growth in capital per worker. It is in this sense that the
accounting picture can sometimes be misleading. 10
3 Empirical Applications of Neoclassical Growth Models
Exercise 1. Where are these economies headed?
From equation (3.9), we get
▯ ▯ 1▯▯ ! 1▯▯
^ = ^K hA = ^K e (u▯uU:S:A;
^ (n + 0:075)
where the (^) is used to denote a variable relative to its U.S. value and x = n+g+d.
The calculations below assume ▯ = 1=3 and = :10, as in the chapter.
Applying this equation using the data provided in the exercise leads to the
following results for the two cases: Case (a) maintains the 1990 TFP ratios, while
case (b) has TFP levels equalized across countries. The Ratio column reports the
ratio of these steady-state levels to the values in 1997.
^97 (a)^y Ratio (b) ^ Ratio
U.S.A. 1.000 1.000 1.000 1.000 1.000
Canada 0.864 1.030 1.193 1.001 1.159
Argentina 0.453 0.581 1.283 0.300 0.663
Thailand 0.233 0.554 2.378 0.259 1.112
Cameroon 0.048 0.273 5.696 0.064 1.334
The country furthest from its steady state will grow fastest. (Notice that by
furthest we mean in percentage terms). So in case (a), the countries are ranked by
their rates of growth, with Cameroon predicted to grow the fastest and the United
States predicted to grow the slowest. In case (b), Cameroon is still predicted to
grow the fastest while Argentina is predicted to grow the slowest.
Exercise 2. Policy reforms and growth.
The ﬁrst thing to compute in this problem is the approximate slope of the re-
lationship in Figure 3.8. Eyeballing it, it appears that cutting output per worker in
half relative to its steady-state value raises growth over a 37-year period by about 2
percentage points. (Korea is about 6 percent growth, countries at the 1/2 level are
about 4 percent, and countries in their steady state are about 2 percent). 11
a) Doubling A will cut the current value of y=A in half, pushing the economy
that begins in steady state to 1/2 its steady state value. According to the calculation
above, this should raise growth by something like 2 percentage points over the next
b) Doubling the investment rateKswill raise the steady state level of output
per worker by a factor of 21▯▯) according to equation (3.8). If ▯ = 1=3, then
this is equal to2 ▯ 1:4. Therefore the ratio of current output per worker to
steady-state output per worker falls to 1=1:4 ▯ :70, i.e. to seventy percent of its
steady-state level. Dividing the gap between 1/2 and 1.0 into tenths, we are 3/5ths
of the way towards 1/2, so growth should rise by 3=5 ▯ (:02) = 1:2 percentage
points during the next 37 years.
c) Increasing u by 5 years of schooling will raise the steady state level of output
per worker by a factor of exp ▯ 5 according to equation (3.8). = :10, then
this is equal to 1.65, and the ratio of current output per worker to steady-state
output per worker falls to 1=1:65 ▯ :60, i.e. to sixty percent of its steady-state
level. Dividing the gap between 1/2 and 1.0 into tenths, we are 4/5ths of the way
towards 1/2, so growth should rise by 4=5 ▯ (:02) = 1:6 percentage points during
the next 37 years.
Exercise 3. What are state variables?
Consider the production function
Y = K (AH) :
Dividing both sides by AL yields
y = k h1▯▯ :
Use the (~) to denote the ratio of a variable to A and rewrite this equation as
~ = k h :
Now turn to the standard capital accumulation equation:
K = s K ▯ dK:
Using the standardtechniques, this equationcan be rewrittenin terms of the capital-
technology ratio as
k = sKy~▯ (n + g + d)k: 12
In steady state, k = 0 so that
sK sK ▯ 1▯▯
k = ~ = k h ;
n + g + d n + g + d
▯ ▯ 1
k = sK 1▯▯ h:
n + g + d
Substituting this into the production func~ = k h we get
▯ s ▯ 1▯▯ ▯ s ▯ 1▯▯
~ = K h h 1▯▯ = K h:
n + g + d n + g + d
Finally, note th~ = y=A; hence
▯ ▯ ▯
y (t) = sK 1▯▯ hA(t);
n + g + d
which is the same as the equation (3.8).
Exercise 4. Galton’s fallacy.
In the example of the heights of mother and daughter, it is true that tall mothers
tend to have shorter daughters and vice versa. Under the assumption of indepen-
dent, identical (uniform) distributions of the heights of mothers and daughters, we
have the following chart:
5’1 5’2 5’3 5’4 5’5 5’6 5’7 5’8 5’9 5’10
0 10 10 10 10 10 10 10 10 10
Mothers with height 5’1” have zero chance of having shorter daughters because no
one can be shorter than 5’1”. Mothers with height 5’2” h10chance of having
daughters with height 5’1”. Other cases can be reasoned in the same way.
In the above example, there is clearly no convergence or narrowing of the dis-
tribution of heights: there is always one very tall person and one very short person,
etc., in each generation. However, we just showed that in spite of the fact that the
heights of mothers and daughters have the same distribution (non-converging), we
still can observe the phenomenon that tall mothers tend to have shorter daughters,
and vice versa. Let the heights correspond to income levels, and consider observ-
ing income levels at two points in time. Galton’s fallacy implies that even though 13
we observe that countries with lower initial income grow faster, this does not nec-
essarily mean that the world income distribution is narrowing or converging.
The ﬁgures in this chapter are not useless, but Galton’s fallacy suggests that
care must be taken in interpreting them. In particular, if one is curious about
whether or not countries are converging, then simply plotting growth rates against
initial income is clearly not enough. The ﬁgures in the chapter provide other types
of evidence. Figure 3.3, for example, plots per capita GDP for several different
industrialized economies from 1870 to 1994. The narrowing of the gaps between
advanced countries is evident in this ﬁgure. Similarly, the ratios in Figure 3.9 sug-
gest a lack of any narrowing in the distribution of income levels for the world as a
Exercise 5. Reconsidering the Baumol results.
As in Figure 3.3, William Baumol (1986) presented evidence of the narrow-
ing of the gaps between several industrialized economies from 1870. But De Long
(1988) argues that this effect is largely due to “selection bias”. First, only countries
that were rich at the end of the sample (i.e., in the 1980s) were included. To see
the problem with this selection, suppose that countries’ income levels were like
women’s heights in the previous exercise. That is, they are random numbers in
each period, say drawn with equal probability from 1,2,3,...,10. Suppose we look
only at countries with income levels greater than or equal to 6 in the second period.
Because of this randomness, knowing that a country is rich in the second period
implies nothing about its income in the ﬁrst period — hence the distribution will
likely be “wider” in the ﬁrst period than in the second, and we will see the appear-
ance of convergence even though in this simple experiment we know there is no
convergence. The omission of Argentina from Baumol’s data is a good example of
the problem. Argentina was rich in 1870 (say a relative income level of 8) but less
rich in 1987 (say a relative income level of 4). Because of its low income in the
last period, it is not part of the sample and this “divergent” observation is missing.
This criticism applies whenever countries are selected on the basis of the last
observation. What happens if countries are selected on the basis of being rich for
the ﬁrst observation? The same argument suggests that there should be a bias to-
ward divergence. Therefore, to the extent that the OECD countries were already
rich in 1960, the OECD convergence result is even stronger evidence of conver-
For the evidence related to the world as a whole, there is clearly no selection
bias — all countries are included.
Exercise 6. The Mankiw-Romer-Weil (1992) model. 14
From the Mankiw-Romer-Weil (1992) model, we have the production function:
▯ ▯ 1▯▯▯▯
Y = K H (AL) :
Divide both sides by AL to get
▯ ▯▯▯ ▯▯
y k h
A A A
Using the (~) to denote the ratio of a variable to A, this equation can be rewritten as
~ = k h :
Now turn to the capital accumulation equation:
K = sKY ▯ dK:
As usual, this equation can be written to describe the evolution of k as
k = s y~▯ (n + g + d)k:
Similarly, we can obtain an equation describing the evolution of h as
h = sHy~▯ (n + g + d)h:
In steady state, 0 andh = 0. Therefore,
k = ~;
n + g + d
h = n + g + d~:
Substituting this relationship back into the production function,
▯ ▯▯▯ ▯▯
~ ~▯ sK sH
~ = k h = n + g + d~ n + g + d~ :
Solving this equation~ yields the steady-state level
( ▯ ▯▯▯ ▯ ▯)1▯▯▯▯
▯ sK sH
~ = n + g + d n + g + d : 15
Finally, we can write the equation in terms of output per worker as
( ) 1
▯ s ▯ ▯ s ▯ ▯ 1▯▯▯▯
y (t) = K H A(t):
n + g + d n + g + d
Compare this expression with equation (3.8),
▯ ▯ ▯
▯ sK 1▯▯
y (t) = hA(t):
n + g + d
In the special case ▯ = 0, the solution of the Mankiw-Weil-Romer model is
different from equation (3.8) only by a constant h. Notice the symmetry in the
model between human capital and physical capital. In this model, human capital
is accumulated by foregoing consumption, just like physical capital. In the model
in the chapter, human capital is accumulated in a different fashion — by spending
time instead of output. 16
4 The Economics of Ideas
Exercise 1. Classifying goods.
Rival Goods Nonrival Goods
Trade secret for
Degree of Music from a
Tropical rainforest compact disc
A chicken and a rainforest are clearly rivalrous — consumption of either by one
person reduces the amount available to another. Private goods like a chicken have
well-deﬁned property rights which make them excludable to a very high degree.
For some rainforests, property rights appear to be less well-deﬁned.
The trade secret for Coca-Cola is a nonrivalrous idea. Although not protected
by a patent, the good is protected by trade secrecy (although Pepsi and other soft
drinks do imitate the formula). Music from a compact disc is fundamentally a
collection of 0’s and 1’s and so is also nonrivalrous. The degree of excludability is
a function of the property rights system. Within the U.S. the enforcement appears
to be fairly strong, but this is less true in some other countries, where pirating of
compact discs is an issue.
The lighthouse (a tower ﬂashing lights to provide guidance to ships at night)
is sometimes thought of as a public good. Notice that it is not truly nonrivalrous
— if a million ships wanted to use one lighthouse, there would be some crowding
effects. Excludability is, as always, a function of the markets in place. See the 17
next exercise for a discussion of this point by Ronald Coase (the 1991 Nobel Prize
winner in economics).
Similarly, clean air is not truly nonrivalrous. If I breathe a molecule of air,
then you cannot breathe the same molecule (at least at the same moment in time).
In terms of excludability, however, it is very difﬁcult to monitor an individual’s
consumption of air and charge her for it.
Exercise 2. Provision of goods.
Chicken are rivalrous and highly excludable. The market does a good job of
providing chicken on a supply and demand basis.
The trade secret for Coca-Cola is nonrivalrous and partially excludable. One
might think that the disclosure of the trade secret would seriously weaken the com-
petitive position of Coca-Cola, but in practice, this doesn’t seem to be the case.
Other soft drinks, while not produced with exactly the same ingredients, are close
substitutes, yet Coca-Cola is a large and prosperous company. Similarly, the lack
of use of any ofﬁcial mechanism like patents to protect intellectual property rights
does not appear to be a serious problem in this industry — we see innovations like
diet soft drinks and New Coke.
Music from a compact disc is nonrivalrous and partially excludable (however,
the speciﬁc compact disc is rivalrous). Markets should provide music because there
is an incentive for proﬁts in the music production business. However, once a com-
pact disc is produced, it is easy to replicate. Governments typically intervene to
protect intellectual property rights so that individuals can beneﬁt from their mu-
sic talents and will have the incentives to produce better music. An interesting
issue is whether or not China should protect the intellectual property rights of U.S.
A tropical rainforest is rivalrous but only partially excludable. For example,
the pollution that occurs when rainforests are burned is a serious externality on
neighbors (not always within the same country). Does the owner of the land have
the right to burn it and pollute the neighbor’s air, or does the neighbor have the right
to clean air on his property? If these rights are well-deﬁned, then the Coase theorem
suggests that negotiation might achieve the efﬁcient outcome. In the absence of
well-deﬁned property rights on this issue, some government involvement may be
necessary. Other issues related to tropical rainforests also arise, such as biodiversity
and global warming.
Similar issues apply to clean air more generally.
The lighthouse has been used (by Mill, Pigou, Samuelson, and others) as a
classic example of a public good that should be provided by the government. The 18
claim is that it is impossible to charge passing ships for their use of the lighthouse
and that the marginal cost of allowing one more ship to pass is zero.
Ronald Coase in “The Lighthouse in Economics” (Journal of Law and Eco-
nomics, October 1974:357-376) provides an excellent discussion of the history of
lighthouses as an example of a public good in economics. Coase shows that in fact
lighthouses in Britain in the 17th and 18th centuries were often very successfully
provided by the private market system in conjunction with patents granted by the
government. Individuals would apply to the government for authorization to build
and maintain a lighthouse in exchange for the right to charge any ship that docked
in nearby ports a speciﬁed fee (based on the size of the ship, etc.).
Exercise 3. Pricing with increasing returns to scale.
a) C = wL = w( 100 + F)
b) C(Y ) = w( 100+ F)
c) dC = w
d) C = w + Fw; d(C=Y )< 0;
Y 100 Y dY
e) ▯ = PY ▯ C(Y ) = w Y ▯ w( Y + F) = ▯wF < 0.
100 100 19
5 The Engine of Growth
Exercise 1. An increase in the productivity of research.
Figure 11: An Increase in ▯
gA/ δ’ gA/ δ LA/A
Figure 12: The Growth Rate of Technology
As shown in Figures 11 and 12, an increase in ▯ causes a temporary increase in
the growth rate of technology: at the initial level of L , research is more productive
and the economy produces more ideas. Over time, the growth rate falls as AA 20
Figure 13: The Level of Technology
decreases whenA > n. In the long run, the growth rate of technology returns to
n. The long-run level effect of an increase in ▯ is shown in Figure 13.
Exercise 2. Too much of a good thing?
From equation (5.11), we have
▯ ▯ ▯
▯ sK 1▯▯ ▯ R
y (t) = (1 ▯ sR) L(t):
n + gA+ d gA
To maximize output per worker along a balanced growth path, take the derivative
with respect tR s : ▯
@ y (t) @ (1 ▯ R )R
= B ;
@ sR @ R
▯ ▯ ▯
B = n + g + d g :
@ y (t)
The maximum occurs when this derivative is equal to zero, @ s = 0
implies that R
1 ▯ 2sR= 0;
Notice that the ﬁrst timeRsappears, it enters negatively to reﬂect the fact
that more researchers mean fewer workers producing output. The second time, it
enters positively to reﬂect the fact that more researchers mean more ideas, which
increases the productivity of the economy= s1 achieves the balance between
▯ 1 R 2
these two effects. Rf > 2, the negative effect of more researchers overpowers
the positive effect. As a result, we can have too much of a good thing.
Exercise 3. The future of economic growth.
a) From equation (5.6), we have
0 = ▯ ▯ (1 ▯ ▯) ;
which implies that
▯ A L _ 2
= = A = :
1 ▯ ▯ A L A 3
b) From equation (5.7), we have
1 ▯ ▯
Assuming 1▯▯ in the world economy is the same as that calculated from the ad-
vanced countries, we get
g = 2▯ :01 = 2= :0067:
A 3 3
The long-run steady-state growth rate of the per capita output in the world economy
is also equal to this value.
c) gyis different fromA given in the question becausLA is substantially
higher than population growth. This means that the advanced countries are in tran-
sition stage wherA L =L is rising in theydata. g is the long-run steady-state growth
rate, while the current leveA is reﬂects transition dynamics. The implication
is that, holding everything else constant, growth rates may decline substantially in
d) The fact that many developing countries are starting to engage in R&D sug-
gests that the decline in the growth Anmay not occur for a long time. The
decline in growth suggested by part (c) may therefore be postponed for some time.
Exercise 4. The share of the surplus appropriated by inventors. 22
Denote ▯ as the proﬁt captured by the monopolist. The monopolist chooses
the price P to maximize ▯, where
▯ = (P ▯ MC)Q(P) = (P ▯ c)(a ▯ bP):
Setting @▯[email protected]
= 0 yields the ﬁrst-order condition:
a ▯ bP ▯ b(P ▯ c) = 0:
Solving for P, we ﬁnd the monopolist chooses a price of
a + bc
and earns proﬁts
▯ ▯▯ ▯
a + bc a + bc
▯ ▯ = ▯ c a ▯
▯ 2b ▯▯ ▯ 2
a ▯ bc a ▯ bc
= (a ▯ bc) :
The potential consumer surplus, CS, is the area of the shaded triangle in Figure
1 a 1 2
CS = ( ▯ MC)(a ▯ bMC) = (a ▯ bc) :
2 b 2b
Therefore, the ratio of proﬁt to consumer surplus is 1/2. 23
6 A Simple Model of Growth and Development
Exercise 1. The importance of A versus h in producing human capital.
Equation (6.8) can be rewritten to isolate the effect of schooling as
y (t) = Z(t)e
where Z is the collection of all of the other terms (e.g. the one involving the
physical investment rate and the A (t) term).
This can be compared to the model in Chapter 3, where equation (3.8) and the
fact that h = euin that chapter imply
y (t) = Z 3t)e ;
where Z 3epresents some other terms from Chapter 3, similar to those that make
up Z above.
In Chapter 3, we used a fact documented by Mincer (1974) and many other
labor economists that an additional year of schooling — u — tends to generate
proportional effects on the wage, and therefore on output per worker. In particu-
lar, the labor market evidence from a wide range of countries suggests that a one
year difference in schooling translates into about a ten percent difference in wages.
Since the wage is proportional to output per worker (recall that the marginal prod-
uct of labor in this model is (1▯ ▯)Y=L), this leads us to choose a value of .10 for
in Chapter 3:
= = :10:
Recall that the derivative of a log is like a percentage change, so this expression
can be written as
▯u = = :10;
i.e. a one unit change in u leads to