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Carlin & Soskice: Macroeconomics 
 
 
 
 
 13 Exogenous Growth Theory 
 
 
Solutions to questions set in the textbook 
 
 
Please email w.carlin@ucl.ac.uk with any comments about the questions and 
answers. We would also be pleased to receive suggestions for additional questions 
(along with outline solutions), which can be added to the website resources. 
 
OXFORD H i g h e r E d u c a t i o n 
© Oxford University Press, 2006. All rights reserved. 
 
1 Chapter 13. Exogenous Growth Theory
1. In the Solow-Swan model explain what is meant by a steady state when population is constant
and there is no technological progress. Extend the concept to the case in which the population is
growing at a constant rate and in which there is a constant rate of technical progress.
ANSWER: The steady state is reached when investment is exactly enough to keep the stock of
capital unchanged (sY = δK). In this particular case Y,K are constant in absolute terms in the
steady state. With technological progress and population growth the equilibrium condition above
becomes sY = (n + x + δ)K, the condition implies that output per efficiency unit and capital
per efficiency unit are constant in steady state. As a consequence output per worker (capital per
worker) grows at a rate x in steady state.
2. What do empirical measurements of Solow’s residual tell us about the role of technological
progress in economic growth? What are their shortcomings in capturing the role of technology?
ANSWER: Measurements of Solow’s residual indicate that technology is a very important factor
in growth. For example, by looking at Table 3 in Chapter 13, it is immediately clear how important
technology is when its contribution is calculated by the SR. It has accounted for, in various periods
in the US, more than a half of GDP growth on average since 1948 to 2001, with peaks of almost
70%. The shortcomings of the Solow residual are linked with the identification of the residual
itself: the accounting identity only works in it simple formulation if the production function
exhibits constant returns to scale and markets are perfectly competitive. It is assumed that there
are only 3 inputs (K, L, A). However human capital and institutions including laws are important
growth factors, which are included with ‘technological progress’ in the Solow residual. Chapter
14 explores how some of these critiques have resulted in alternative models of economic growth.
3. Information technology is an important contributor to economic growth in modern economies.
Discuss how you would measure the impact of information technology on economic growth.
ANSWER: The impact of technology on growth is measured according to the growth accounting
identity given in Section 4.3. See Table 17.5 (Chapter 17) for an example for the US. See also
http://post.economics.harvard.edu/faculty/jorgenson/papers/IT_G7_economies_05012005.pdf for
a recent set of estimates for the advanced economies.
4. Assume that at time zero there is a rich country and a poor country. You are in a Solow world
without technical progress. When the countries are observed at a later date, they are characterized
by the same standard of living. Is this consistent with (a) absolute convergence; (b) conditional
convergence? Specify your assumptions about technology, demography, tastes and policy.
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ANSWER: Assume that there is no technological progress in the model and that the two economies
have the same institutions, demography, tastes and policy. If further we assume that they have the
same rate of population growth, we know that at time 0 the difference between the two countries
is given by the capital stock. However, in the long run they will reach the same steady state and
therefore this is consistent with absolute convergence. However, if we change our assumption to
include a fixed technology (or different institutions) more efficient in the initially poor country
we know that in the steady state the initially poor country will reach a higher standard of liv-
ing and therefore the situation described in the question may also be consistent with conditional
convergence.
5. Can shocks to technology explain macroeconomic volatility?
ANSWER: It is possible for shocks to technology to explain macroeconomic volatility. Shocks
to technology lead to changes in the growth rate of output per worker (see Fig. 13.10 and Fig.
13.11). If there is a higher rate of technology growth this suddenly raises the rate of growth
of output and output per worker. In general, however, we have to be cautious about explaining
all volatility through long-run economic processes such as technological growth. As argued in
Chapter 15, fluctuations in aggregate demand are likely to explain much of the short-run volatility
in the economy.
6. “Technology remains the dominant engine of growth, with human capital investment in second
place.” (Robert Solow, Nobel Prize Lecture, 1987) Do you agree? Explain your reasoning.
ANSWER: This question can be used to revise the role of technology as the only source of long-
run productivity growth in the Solow–Swan model and to discuss the role of physical and human
capital in accounting for the level of productivity and hence in growth rates along the convergence
path (Mankiw, Romer and Weil as discussed in Section 5.3). If further we consider that human
capital can play a substantial role in the diffusion of technology (Section 5.4) we cannot fully
agree with Solow. This is a good question to revisit after Chapter 14.
7. If the savings rate falls permanently, what happens to welfare in the Solow-Swan model?
ANSWER: We can approach this question looking at welfare as output per worker or as consump-
tion per worker. If welfare is measured as output per worker, a fall in the saving rate will certainly
lower the steady state welfare of an economy since the new steady state is at a lower capital labour
ratio and hence has lower output per worker. However, if welfare is interpreted as consumption
per worker the effect of a reduction of the saving rate depends on whether the initial saving rate
was above or below the Golden Rule saving ratio (saving ratio maximizing consumption in the
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steady state).
8. Use the analytical tools acquired in this chapter to analyze the introduction of a welfare system
on the long-run performance of the economy financed through taxation. Does it matter whether
lump-sum or proportional taxes are raised?
ANSWER: See Chapter 13, Section 7. Yes, there will be differences in the model’s behaviour. A
lump-sum tax will shift the savings function down by the same amount at all levels of capital per
capita, whilst a proportional tax will have a proportional impact on savings.
1.1 Problems and questions for discussion
QUESTION A: In the Solow model, assume the economy is initially in a steady state in which there is
no growth of output per capita. At time t there is a sudden fall in the growth rate of the population
and it remains at the new lower growth rate. Describe what happens to output growth and to the
growth of living standards in the economy and why. Now assume there is labour augmenting
technical progress: how does that affect the growth of the economy in response to a fall in the
population growth rate?
QUESTION A: ANSWER: To answer this question, it is best to use the ‘alternative Solow diagram’.
The question is asking for the reverse of the exercise in Fig. 7 and 8. If we assume that the rate
of population growth falls from n1 to n2 at time t, then using the equation gY = σKgK + σLn,
we know that output growth falls immediately by σL(n1 − n2). This is shown in the alternative
Solow diagram by the downward shift of the gY curve. The economy then adjusts along the
new gY curve to the new steady state at wheregY = n2. Since output growth falls by less than
population growth, there is an immediate rise in the growth of output per head (from its initial
growth rate of zero) at t followed by a gradual decline in the growth of output per head to zero
once the new steady state is reached. The economic effects of the fall in population growth are
best understood by using the standard Solow diagram. Lower population growth means there is
‘too much’ capital to equip new entrants to the labour force at the existing capital-labour ratio so
the growth of the capital-labour ratio jumps up. The capital labour ratio thereafter rises slowly to
its new steady state level, with its growth rate falling back to zero. If there is a constant rate of
labour-augmenting technical progress, this simply adjusts the growth rates of output and of output
per capita by the constant rate of technical progress, x.
QUESTION B: Consider an economy which experiences civil war. How will this affect the short-run
performance of the economy? Describe the steady state to which the economy returns after the
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end of the civil war. Are there any long-run effects of civil war that may affect the economy
many years after the civil war ended? (Hint: Discuss the effect of civil war on human capital.)
[Background reading: For the long-run effects of civil wars see Ghobarah, H, P. Huth and B.
Russett (2003). ‘Civil Wars Kill and Maim People–Long after the Shooting Stops’, American
Political Science Review 97/2: 189-202.]
QUESTION B: ANSWER: This question requires a number of comparative statics exercises to be
performed similar to those in Section 2.4. These should be illustrated using the set of diagrams
exemplified in Fig. 13.6. Civil war may affect a number of the variables in the Solow-Swan
modelling framework. Civil wars cost many lives (absolute fall in L) and destroy the productive
capacity of the economy (absolute fall in K). Barro (2005) estimates that the Spanish Civil War
1935-38 lead to a 31% loss of real per capita GDP for Spain. Uncertainty increases the risk of
default in the economy and thus leads to a lower savings rate s. Civil wars have long term effects
long after the actual fighting stopped. Disease may continue to kill numerous people and lead
to permanently lower fertility rates. Years of conflict will have reduced available human capital
and the economy will be lagging in terms of technological transfers. A thoughtful answer would
include a discussion of how all these effects play out (using Ghobarah et al.).
QUESTION C: In an economy characterized by a Cobb-Douglas production function and exogenous
labour-augmenting technical progress, labour’s share of income is 70% and the depreciation rate
is 3% per annum. The economy is in a steady state with GDP growth at 4% per year and with a
capital output ratio of 2. Find the saving rate and the marginal product of capital. At time t the
saving rate in this economy increases to a new constant level, with the outcome that the economy
converges to the Golden Rule steady state. What is the new savings rate, capital output ratio and
marginal product of capital? Use diagrams with time on the horizontal axis to sketch the path of
the capital-output ratio, the marginal product of labour and of consumption per effective unit of
labour.
QUESTION C: ANSWER: The economy is characterized by: v∗ = (K/Y )∗ = 2; δ = 0.03 and
gY = n + x = 0.04. Using Domar’s rule, we can calculate the initial steady state saving rate:
v∗ = s/(n+x+δ), which implies that s = 0.07×2 = 0.14. From the Cobb Douglas production
function, MPK = 0.3(K/Y ) = 0.15. The Golden Rule savings rate is equal to capital’s share so
sG = 0.3. Hence the economy in the initial equilibrium has a savings rate below the Golden Rule
rate. In the new steady state using Domar’s rule again, we have (K/Y )∗∗ = 0.3/0.07 = 4.28
and the MPK = 0.07 (since by the Golden Rule, MPK = n+x+ δ). The sketches will show:
the K/Y rising gradually at time t and converging to its higher long-run value; the MPK falling
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gradually to its new lower long-run value; and consumption per efficiency unit initially dropping
(since when the savings ratio goes up, consumption immediately falls: ((1− 0.3) / (1− 0.14) =
0.81) and then gradually rising to the new level, which is higher than it was initially.
QUESTION D: Use the models discussed in this chapter to explore the possible implications for growth
in output and in output per capita in Australia of the data presented in Fig1.1 Using the right-hand
scale, the chart shows the trend in the fertility rate (the average number of children born to a
woman in her life-time). Measured against the left scale are shown (a) the rate at which girls
stay on to the final year of secondary education and (b) the female participation rate in the labour
force.
QUESTION D: ANSWER: The implications are generally positive; a better educated labor force
will have a positive impact on growth, the fall in the fertility rate will make the economy richer while
the increase in female participation will make sure that the labor force does not fall in the future.
QUESTION E: The data in the table show the GDP per capita relative to the EU25 average adjusted
for purchasing power parity for the regions of the Czech Republic and Austria for 1995 and 2002.
The EU25 average is set equal to 100 in each year. Note that the same purchasing power parity
is used for all the regions within a country. Provide a concise description of the data presented in
the table. Show how the concepts of absolute and conditional convergence can be used to discuss
the evolution of living standards in these two countries.
Czech Republic 1995 2002 Austria 1995 2002
Prague 129 153 Vienna 184 173
Strední Cechy 54 55 Kärnten 107 100
Jihozápad 67 61 Steiermark 105 103
Severozápad 67 64 Oberösterreich 118 113
Severovýchod 60 57 Salzburg 144 134
Jihovýchod 63 60 Tirol 132 124
Strední Morava 59 52 Vorarlberg 131 126
Moravskoslezko 66 57
Source: Eurostat website: Regional breakdown of GDP per capita: Percentage of EU-25 average
1We are grateful to Steve Dowrick for generously providing this data. The sources for the data are: Participation: Reserve
Bank of Australia, ’Australian Economic Statistics 1964-96’; Australian Bureau of Statistics, ’Labour Force (prelim) 1997-
2001’ Fertility: Australian Bureau of Statistics, ’Births, Australia’ (Cat. no. 3301.0). School Retention: Apparent Retention
Rates to Year 12 - from Figure 33 in Collins, Cherry, Jane Kenway and Julie McLeod (2000), ’Factors Influencing Educational
Performance of Males and Females in School and their Initial Destinations after Leaving School’, Commonwealth of Australia,
Ausinfo, Canberra.
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Fertility, Education and Female Labour Force Participation: Australia
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Figure 1: Fertility, Education and Female Labour Force Participation: Australia
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QUESTION E: ANSWER: As we can see from the table there is a large variation in growth even
within countries. Comparing the two capital city regions, it is clear that the Austrian one had a much
higher GDP per capita in 1995, however it seems that the Czech Republic (i.e. region including Prague)
is closing the gap. This is consistent with absoluteconvergence. However this is only true of the
Prague and Vienna regions. When we look at the regional breakdown, we notice that there are still large
differences among regions. Therefore there is some support for conditional convergence as well. One
plausible hypothesis is that the Prague and Vienna regions may have the same steady state, given by the
same technology etc. (e.g. driven by foreign direct investment in the Prague region) but in 1995 Prague
was further away from that steady state. The large catch-up in such a short time period is consistent
with absolute convergence. The large disparity in GDP levels between regions within a country may
however suggest a process driven by the conditional convergence of each region to a separate steady
state identified by local variations in technological progress, human capital levels or growth promoting
institutions. This example suggests that country-level data may obscure deep divisions that exist among
regions in economic development.
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