0. Lecture 12 Economic Growth

1. Economic Growth
Explain improvements in standards of living (GDP per capital) along time
Explain differences across countries
learn how our own growth rate is affected by shocks and our government’s policies
Solow Growth Model

2. some statistics
In Uganda, 96% of people live on less than $2/day. (data link)
2.8 billion people live on less than $2/day (1.1 billion under $1/day)
GDP per capita
Chad in 1960: $1212, in 2000: $908
Venezuela in 1960: $7840, in 2000: $6420
Korea in 1960: $1495, in 2000: $15875
H.K. in 1960: $3090, in 2000: $26698
source: The Elusive Quest for Growth, by William Easterly. (MIT Press, 2001)

3. Huge effects from tiny differences
percentage increase in standard of living after…
annual growth rate of income per capita
…25 years
…50 years
…100 years
2.0%
64.0%
169.2%
624.5%
2.5%
85.4%
243.7%
1,081.4%
These calculations show that a one-half point increase in the growth rate has, in the long run, a HUGE impact on the standard of living.

4. Long term growth effect
Rule of 72: 1% growth rate, approximately takes 72 years to double GDP
What will happen if China keeps 10% growth rate and US keeps 3% growth rate (US per capita GDP $42,000 China $6800)
The $449 billion is in “today’s dollars” (i.e. measured in the prices that prevailed in the first quarter of 2002).
In case you’re wondering how I did this calculation:
Computed actual quarterly growth rate of real income per capita from 1989:4 through 1999:4.
Added one-fourth of one-tenth of one percent to each quarter’s actual growth rate.
Computed what real income per capita would have been with the new growth rates.
Multiplied this hypothetical real income per capita by the population to get hypothetical real GDP.
Computed the difference between hypothetical and actual real GDP.
Cumulated these differences over the period 1990:1-1999:4.
Like the original real GDP data, the cumulative difference was in 1996 dollars. I multiplied this amount by 10%, the amount by which the GDP deflator rose between 1996 and 2002:1, so the final result ($449 billion) is expressed in 2002:1 dollars, which I simply call “today’s dollars.”
DATA SOURCE:
Real GDP, GDP deflator - Dept of Commerce, Bureau of Economic Analysis.
Population - Dept of Commerce, Census Bureau.
All obtained from “FRED” - the St. Louis Fed’s database, on the web at

5. World Distribution of Income

6. World Income Map
Income varies greatly among nations. More than half the world’s population lives in countries that are poorer than the U.S. was in Source: World Bank.

7. South vs. North

9. Real GDP per capita, 1975–2003
Some countries, such as China and Ireland, have been very successful at achieving a rapid rise in real GDP per capita. Others, such as Argentina, have been much less successful. Still others, such as the countries of the former U.S.S.R., have slid backward.
(Data for the former U.S.S.R. are from the period 1975–2001.) Source: World Bank.

10. Life Expectancy and Income (Preston, 1976)

12. Heights of Males and Females in China

13. Happiness and Income

14. The Solow Model
due to Robert Solow, won Nobel Prize for contributions to the study of economic growth
a major paradigm:
widely used in policy making
benchmark against which most recent growth theories are compared
looks at the determinants of economic growth and the standard of living in the long run

15. How Solow model is different from Chapter 3’s model
K is no longer fixed: investment causes it to grow, depreciation causes it to shrink.
L is no longer fixed: population growth causes it to grow.
The consumption function is simpler.
No G and T
It’s easier for students to learn the Solow model if they see that it’s just an extension of something they already know, the classical model from Chapter 3. So, this slide and the next point out the differences.

16. Production
Initially assume constant population (L) and no technology change
Production of goods and services:
Constant Returns to Scale:

17. Production
Letting z = 1/L, we get the production function in per capita terms:
y = Y/L = output per worker
k = K/L = capital per worker
Constant Returns to Scale  size of the economy does not affect the relationship between capital per worker and output per worker

18. Production Decreasing MPK:
This implies the following shape for the production function:

19. Production y
MPK is the slope of this curve.
f(k)
Low MPK
High MPK k

20. Production
Cobb-Douglas case:

21. Demand Assume a closed economy with no government: NX = G = 0
Assume that people save a fraction s of their income (and therefore consume 1 – s),

22. Demand
Substituting:
In equilibrium:

23. Capital Accumulation
Two elements determine how the capital stock changes over time:
Investment: addition of new plants and equipment (makes capital stock rise)
Depreciation: wearing out of existing capital stock (makes capital stock fall)

24. Capital Accumulation
In other words:

25. Capital Accumulation
f(k) c
sf(k) y i k

26. Capital Accumulation
Investment higher than depreciation  capital stock increases
Depreciation higher than investment  capital stock increases

27. Capital Accumulation Steady-state capital stock (k*):
Steady state output, consumption, investment:

28. Determining the capital–labor ratio in the steady state

29. Capital Accumulation
Low k  high MPK  high returns from investment  capital stock grows
High k  low MPK  low returns from investment  capital stock decreases
In both cases, the economy converges to the steady state (long-run equilibrium)

30. Capital Accumulation
Cobb-Douglas:
In steady state:

31. Increase in Savings Ratek
s2f(k)
s1f(k) k

32. Increase in Savings Rate
Higher s means that more resources will be dedicated to investment  higher capital stock in steady state
Therefore, output per capita will be also higher

33. Golden Rule
What is the relationship between steady-state consumption and savings rate?
Two conflicting forces:
Higher s  higher output  higher the amount of resources available for consumption  c* 
Higher s  lower the proportion of income allocated to consumption  c* 

34. Golden Rule For low values of s, c* increases with s
For high values of s, c* decreases with s
Golden Rule: capital stock implied by the savings rate such that c* is maximized

35. Golden Rule
More formally:
But in steady state:

36. Golden Rule
Golden Rule: find k* such that c* is maximized

37. Golden Rule k
f(k)
MPK =  k

38. Golden Rule k f(k) sgf(k) k sg is the savings rate that implies kg*:
MPK = 
sgf(k) k

39. The relationship of consumption per worker to the capital–labor ratio in the steady state

40. Golden Rule
Cobb-Douglas case:
Golden Rule: MPK = 

41. Golden Rule
In steady state:

42. Transition to Golden Rule
Case 1: s > sg, i.e., steady-state capital too high. Decrease s in order to reach sg k
s2f(k)
sgf(k) k

43. Transition to Golden Rulek y k* y*
kg*
yg* t t c i i*
cg* c*
ig* t t

44. Transition to Golden Rule
Case 2: s < sg, i.e., steady-state capital too low. Increase s in order to reach sg k
sgf(k)
sf(k) k

45. Transition to Golden Rulek y
kg* y*
yg* k* t t c i
cg*
ig* c* i* t t

46. Transition to Golden Rule
If the economy begins above the golden rule (s too high), consumption increases in all future periods  decrease in s leads to welfare improvement
If the economy begins below the golden rule (s too low), consumption falls during transition  there is a tradeoff between consuming today or in the future

47. International Evidence on Investment Rates and Income per Person
Graph to be updated for final version of this PowerPoint presentation.

48. Population Growth
Assume that the population--and labor force-- grow at rate n. (n is exogenous)
EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02).
Then L = n L = 0.02  1000 = 20, so L = 1020 in year 2.

49. Break-even investment
( + n)k = break-even investment, the amount of investment necessary to keep k constant.
Break-even investment includes:
 k to replace capital as it wears out
n k to equip new workers with capital (otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers)

50. The equation of motion for k
With population growth, the equation of motion for k is
k = s f(k)  ( + n) k
actual investment
break-even investment
Of course, “actual investment” and “break-even investment” here are in “per worker” magnitudes.

51. The Solow Model diagram
Investment, break-even investment
Capital per worker, k
k = s f(k)  ( +n)k
( + n ) k
sf(k) k*

52. The impact of population growth
Investment, break-even investment
( +n2) k
( +n1) k
An increase in n causes an increase in break-even investment,
sf(k)
k2*
leading to a lower steady-state level of k.
k1*
Capital per worker, k

53. Prediction: Higher n  lower k*.
And since y = f(k) , lower k*  lower y* .
Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.
Of course, the converse is true, as well: a fall in s (caused, for example, by tax cuts or government spending increases) leads ultimately to a lower standard of living.
In the static model of Chapter 3, we learned that a fiscal expansion crowds out investment. The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment. If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labor productivity, and income per capita to fall toward a new, lower steady state. (If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)

54. International Evidence on Population Growth and Income per Person

56. Clark 2005, p Fig 1

57. The Golden Rule with Population Growth
To find the Golden Rule capital stock, we again express c* in terms of k*:
c* = y*  i*
= f (k* )  ( + n) k*
c* is maximized when MPK =  + n
or equivalently, MPK   = n

58. Technology Progress
Rewrite the production function to incorporate technology change:
E = efficiency of labor
E  L = effective workers
Assume: Technological progress is labor-augmenting: it increases labor efficiency at the exogenous rate g:

59. Technology Progress Assume that E grows at rate g
Therefore E  L grows at rate n + g
Redefine all variables in terms of effective workers:
k = K/EL = capital per effective worker

60. Technology Progress
Then y = Y/EL (= output per effective worker) is given by:
Similarly for consumption and investment:

61. Technology Progress Therefore, the equations are the same as before
The only change is in the law of motion for k. Capital per effective worker:
Increases with investment
Decreases with physical depreciation
Also decreases because there are more effective workers to share the existing capital (higher L and E)

62. Technology Progress Then:
In steady-state, capital per effective worker is fixed:

63. Technology Progress
( +n+g)k
sf(k) k k1 k* k2

64. Technology Progress
In steady state, income, consumption and investment per effective worker are also constant over time:

65. Technology Progress
Therefore capital, income, consumption and investment per worker grow at the rate g in steady-state:

66. Steady-State Growth Rates in the Solow Model with Tech. Progress
Symbol
Variable
Capital per effective worker
k = K/ (L E )
Output per effective worker
y = Y/ (L E )
Output per worker
Explanations:
k is constant (has zero growth rate) by definition of the steady state
y is constant because y = f(k) and k is constant
To see why Y/L grows at rate g, note that the definition of y implies (Y/L) = yE. The growth rate of (Y/L) equals the growth rate of y plus that of E. In the steady state, y is constant while E grows at rate g.
Y grows at rate g + n. To see this, note that Y = yEL = (yE)L. The growth rate of Y equals the growth rate of (yE) plus that of L. We just saw that, in the steady state, the growth rate of (yE) equals g. And we assume that L grows at rate n.
(Y/ L ) = y E g
Total output
Y = y E L
n + g

67. Technology Progress
This follows since steady-state variables are constant and E is growing at the rate g
Therefore, the inclusion of technology progress in the Solow model can generate sustained long-run growth

68. Technology Progress
Moreover, total capital, output, consumption and investment grow at the rate n+g in steady state:
Given that steady-state variables are constant and EL is growing at the rate n+g

69. Golden Rule Consumption per effective worker in steady state:
Golden Rule: find k* s.t. c* is maximized:

70. Government Policies to raise the rate of productivity growth
Improving infrastructure
Would increased infrastructure spending increase productivity?
There might be reverse causation: Richer countries with higher productivity spend more on infrastructure, rather than vice versa
Infrastructure investments by government may be inefficient, since politics, not economic efficiency, is often the main determinant

71. Government Policies to raise the rate of productivity growth
Building human capital
There’s a strong connection between productivity and human capital
Government can encourage human capital formation through educational policies, worker training and relocation programs, and health programs
Another form of human capital is entrepreneurial skill
Government could help by removing barriers like red tape
Encouraging research and development
Government can encourage R and D through direct aid to research

72. What determined whether/when new technology adopted?
Why is technological breakthroughs progress so unequal across countries?
What determined whether/when new technology adopted?
Geography view: importance of ecology, climate, disease environment, geography, in short, factors outside human control.
Institutions view: importance of man-made factors; especially organization of society that provide incentives to individuals and firms.
History’s accidents: some countries are unlucky and trapped in underdevelopment.
Possible fundamental answers to these questions:
Geography: Jared Diamond (1997), Ken Pomeranz (2000).
Genetic/social improvement: Gregory Clark (2007).
Culture: Landes (1999).
Relative prices: Bob Allen’s work (in press).
Institutions: North and Weingast (1989), AJR Atlantic Traders, La Porta et al. Legal Origins.
Interaction of resources, institutions, and technology: Engerman and Sokoloff (1997, 2005), AJR Reversal of Fortune, AJR Colonial Origins.
􀁺 Jared Diamond:
– Importance of geographic and ecological differences in
agricultural technology and availability of crops and animals.
􀁺 Jeff Sachs:
– "Economies in tropical ecozones are nearly everywhere poor, while
those in temperate ecozones are generally rich" because "Certain
parts of the world are geographically favored. Geographical
advantages might include access to key natural resources, access to
the coastline and sea…, advantageous conditions for agriculture,
advantageous conditions for human health."
– "Tropical agriculture faces several problems that lead to reduced
productivity of perennial crops in general and of staple food crops in
particular" …
– "The burden of infectious disease

73. The Geography Factor Possible fundamental answers to these questions:
Geography: Jared Diamond (1997), Ken Pomeranz (2000).
Genetic/social improvement: Gregory Clark (2007).
Culture: Landes (1999).
Relative prices: Bob Allen’s work (in press).
Institutions: North and Weingast (1989), AJR Atlantic Traders, La Porta et al. Legal Origins.
Interaction of resources, institutions, and technology: Engerman and Sokoloff (1997, 2005), AJR Reversal of Fortune, AJR Colonial Origins.

74. The Institutions Factor
Possible fundamental answers to these questions:
Geography: Jared Diamond (1997), Ken Pomeranz (2000).
Genetic/social improvement: Gregory Clark (2007).
Culture: Landes (1999).
Relative prices: Bob Allen’s work (in press).
Institutions: North and Weingast (1989), AJR Atlantic Traders, La Porta et al. Legal Origins.
Interaction of resources, institutions, and technology: Engerman and Sokoloff (1997, 2005), AJR Reversal of Fortune, AJR Colonial Origins.

75. Institutions and Economic Performances
Possible fundamental answers to these questions:
Geography: Jared Diamond (1997), Ken Pomeranz (2000).
Genetic/social improvement: Gregory Clark (2007).
Culture: Landes (1999).
Relative prices: Bob Allen’s work (in press).
Institutions: North and Weingast (1989), AJR Atlantic Traders, La Porta et al. Legal Origins.
Interaction of resources, institutions, and technology: Engerman and Sokoloff (1997, 2005), AJR Reversal of Fortune, AJR Colonial Origins.

76. Institutions and Economic Performances
Possible fundamental answers to these questions:
Geography: Jared Diamond (1997), Ken Pomeranz (2000).
Genetic/social improvement: Gregory Clark (2007).
Culture: Landes (1999).
Relative prices: Bob Allen’s work (in press).
Institutions: North and Weingast (1989), AJR Atlantic Traders, La Porta et al. Legal Origins.
Interaction of resources, institutions, and technology: Engerman and Sokoloff (1997, 2005), AJR Reversal of Fortune, AJR Colonial Origins.

77. But institutions are complicated: identification problem
Good institutions are correlated with many other good things. Theories about institutions are thus very difficult to test.
The study of the causal role of institutions on economic growth is therefore complicated by concerns about endogeneity.
For example, the United States is rich; it has good institutions; it has high levels of education; it has a common law heritage; it has a temperate climate.
Good institutions are difficult to pin down precisely. We want to be very careful to disentangle different causal effects and isolate the effect of interest.

78. But institutions are also endogenous
Institutions could vary because underlying factors differ across countries: Geography, ecology, climate
Montesquieu’s story:
– Geography determines “human attitudes”
– Human attitudes determine both economic performance and political system.
– Institutions potentially influenced by the determinants of income
􀁺 Montesquieu:
– “The heat of the climate can be so excessive that the body
there will be absolutely without strength. So, prostration will
pass even to the spirit; no curiosity, no noble enterprise, no
generous sentiment; inclinations will all be passive there;
laziness there will be happiness,”
– "People are ... more vigorous in cold climates. The inhabitants
of warm countries are, like old men, timorous; the people in
cold countries are, like young men, brave".
􀁺 Moreover, Montesquieu argues that lazy people tend to
be governed by despots, while vigorous people could
be governed in democracies; thus hot climates are
conducive to authoritarianism and despotism.

79. Factor Prices
So far, we solved the model without any reference to wages and rental rates (factor prices)
We just focused on how income is generated, but not on how it is distributed
Assume that a competitive firm hires capital and labor to generate output

80. Factor Prices Assuming Cobb-Douglas technology:
Then the problem for this firm is given by:

81. Factor Prices First-order condition for K implies that:
In steady-state, the real rental rate is fixed (since k is fixed)

82. Factor Prices First-order condition for L implies that:
In steady-state, the real wages increase at the rate g (since k is fixed and E grows at the rate g)

83. Factor Prices
Assume that capital is initially below the steady-state. Then k will evolve according to the following path: k k* t

84. Factor Prices Rental rate: initially high (low k implies high MPK)
decreases over time as capital accumulates and MPK decreases
R/P t

85. Factor Prices Define wage in terms of efficiency units as: Then :
initially low (low k implies low MPL  labor abundant relative to capital)
increases over time as capital accumulates and MPL increases
constant in steady state

86. Factor Prices This means that real wages (w/P):
Grow faster than g during the transition
Grow at the rate g in steady-state t

87. Growth Accounting
Want to be able to explain why and how countries grow
There are many sources of growth
First step is to decompose aggregate growth into its components:
Growth in the labor force
Growth in capital
Growth in productivity

88. Sources of Economic Growth
Assume Cobb-Douglas Production Function
Take log and differentiating

89. Computing TFP z: Total Factor Productivity (TFP) or “Solow Residual”
Y is GDP, K is aggregate capital, N is number of workers
Need to know α

90. What is “z” Human Capital (Education) Technological Progress
Externality: environmental Issues
Institutional Effect
Firm Organization
Patent Protection
Corruptions

91. Labor Share in the Cobb-Douglas Production Function
Firm optimization:
First-order condition with respect to N:
Labor share is wN/Y. Here:

92. Result Can use average labor share as measure of
Labor share (total wages divided by GDP) in the U.S. is about 64%
Estimate α to be 0.36
Can now compute TFP as:

93. Total Factor Productivity in the U.S.

94. Decomposing Growth Rates
Taking logs of production function:
The same applies to log differences:
Log differences are approximately equal to percentage changes:

95. Growth Decomposition for the U.S.

96. Growth Decomposition for the Asian Tigers

97. Growth Accounting for China

98. Human Capital in China

99. Human Capital in China 99

100. Technology in China
Innovation
New Goods
Patents

101. Management and Productivity
Patents
Management Score
101
Note: European firms only as uses the European Patent Office database

102. Policies to promote growth
Saving Rate
Human capital investment
Encouraging technological progress
Right Institutions

103. Growth empirics: Confronting the Solow model with the facts
Solow model’s steady state exhibits balanced growth - many variables grow at the same rate.
Solow model predicts Y/L and K/L grow at same rate (g), so that K/Y should be constant.
This is true in the real world.
Solow model predicts real wage grows at same rate as Y/L, while real rental price is constant.
Also true in the real world.
Check out the second paragraph on p.220: It gives a nice contrast of the Solow model and Marxist predictions for the behavior of factor prices, comparing both predictions with the data.

104. Convergence
Solow model predicts that, other things equal, “poor” countries (with lower Y/L and K/L ) should grow faster than “rich” ones.
If true, then the income gap between rich & poor countries would shrink over time, and living standards “converge.”
In real world, many poor countries do NOT grow faster than rich ones. Does this mean the Solow model fails?

105. Convergence No, because “other things” aren’t equal.
In samples of countries with similar savings & pop. growth rates, income gaps shrink about 2%/year.
In larger samples, if one controls for differences in saving, population growth, and human capital, incomes converge by about 2%/year.

106. Convergence
What the Solow model really predicts is conditional convergence - countries converge to their own steady states, which are determined by saving, population growth, and education. And this prediction comes true in the real world.