4
G
P
• In the last chapter, we learned that the rate of growth of potential output is determined by the rate of population growth, by changes in the level of natural resources, and by improvements in technology. In this chapter, we recognize that society’s choices affect these parameters.
• The first part of this chapter investigates the forces that bring about technological change, focusing in particular on the new ideas generated by the process of capital investment, and the fact that it is difficult, if not impossible, for any individual firm to capture all the benefits of these ideas. It also looks more carefully at convergence
 the question of whether economies with different levels of income will eventually achieve the same standard of living.
• The second part of this chapter looks at problems of population growth and development.
Using examples from high growth and low growth countries, this section examines the role of prudent policy in influencing the growth rate of a country.
Endogenous growth theory does something fairly radical
 it uses a production function with
increasing returns to scale. It is able to do this by assuming that there are external benefits associated with private investment, so that an individual firm does not reap all of the benefits from its own investment. New knowledge, inventions, and discoveries are a by-product of some kinds of investment, and all of these benefit society as well. Thus it turns out that capital can have a diminishing marginal product in the eyes of individual producers, but because of these external benefits, a constant marginal product in the eyes of society.
37

38 C HAPTER 4
(n + d)k
(flows out of k) sf(k )
(flow into k)
This tiny modification makes a world of difference. It was capital’s diminishing marginal product that gave us a savings curve with a declining slope, and it was this savings curve with the declining slope that guaranteed that the investment requirement line and the savings line would eventually intersect, so that there would be a steady-state.
When we allow capital to have a
Figure 4

1 steady– state k
THE NEOCLASSICAL GROWTH MODEL k constant marginal product, the savings curve becomes a straight line.
There is no steady-state.
When the savings curve is everywhere above the investment requirement line, as in Figure 4

2, we get ongoing growth. When it is everywhere below the investment requirement line (not shown), output and the capital stock eventually fall to zero, as depreciation and population growth erode each worker’s stock of capital more quickly than it can be replaced.
Interestingly, if the savings and investment requirement lines happen to coincide (if one is right on top of the other), the existing level of capital per worker will be in a steady-state. Therefore, the existing level of output per capita will also be in the steady-state. The growth rate of per capita output will be zero.
The simplest production function that will produce endogenous growth is:
Y = aK
Dividing both sides by N, we can express it in per capita terms: y = ak
The marginal product of capital in this function is constant and equal to a. (A 1 unit increase in the capital stock will increase output by an amount a.) Per capita savings curve is just equal to
sak, giving the savings line a constant slope equal to sa. The growth rate of per capita output,

k/k, is determined by the following equation:
 k = sf(k)

(n + d)k = sak

(n + d)k,
so that
 k /k = sa

(n + d).

G ROWTH AND P OLICY
39
The difference between the flows into and the flows out of the capital stock determine its growth rate, and, as a result, the growth rates of technology and of per-capita output.
}
 k sf (k ) = sa k
(flow into k)
(n + d)k
(flows out of k)
k
Figure 4

2
THE ENDOGENOUS GROWTH MODEL
Neoclassical and endogenous growth theories have different predictions regarding convergence
 the theory that growth rates and, under certain conditions, standards of living
(measured as levels of per capita output) should become equal across countries.
Neoclassical theory predicts that countries will absolutely converge
 will reach the same level of steady-state income, and that their steady-state income will grow at the same rate
 when they have the same rate of saving, the same rate of population growth, and have access to the same technologies. When population growth or savings rates differ, it predicts that they will
conditionally converge, or that, while their steady-state incomes will differ, the growth rate of these incomes will be the same. Endogenous growth theory does not predict any kind of convergence.
N OTE : One optional component of this section has not been covered. It combines elements of neoclassical and endogenous growth models to produce a model with two equilibria - one with low income and no growth, and another with high income and positive growth.
The first part of this section recognizes that the rate of population growth may itself depend on income. An optional section develops a model with endogenous population growth, assuming this to be true. The second part looks at the “miracle” of East Asian growth, and finds that it is due to high rates of saving, stable governments, and outward-looking economic policies. It does

40 C HAPTER 4 not discuss the currency crises that hit many of these economies in late 1997, or ask why none of these factors helped to prevent it. If you have to write a paper for your class, you might research this question further. It is interesting to wonder why the “tigers” of the 1980s became the
“dominos” of the late 1990s, and at this point many economists consider it an open question.
The following part of the section examines issues related to low growth countries like Ghana.
Almost of all the poor countries exhibit a low savings rate along with high population growth rate, resulting in a low standard of living. The last part of this section looks at the role of limited natural resources in constraining growth. However, technological progress enabling us to use less natural resource for production is the solution to this problem.
endogenous growth increasing returns to scale absolute convergence conditional convergence stable equilibrium unstable equilibrium golden-rule capital stock
China is often cited as one of the recent examples of the growth miracle in Asia. In the following graph, using data from Table 4–1, plot the growth rate of per capita GDP from 1994 to 2010. As a reference, the average per capita GDP growth for the U.S. during this period is also given in the graph.
Figure 4  3 GROWTH RATE OF PER CAPITA GDP , C HINA

G ROWTH AND P OLICY
41
TABLE 4–1
Year
1999
2000
2001
2002
2003
1993
1994
1995
1996
1997
1998
2004
2005
2006
2007
2008
2009
2010
Per capita Real GDP
2616
2822
3043
3294
3590
1590
1774
1931
2102
2276
2440
3916
4335
4855
5512
6014
6520
7130
Percent change from previous year
—
11.6
8.9
Endogenous growth theory gets its name from the fact that the growth rate of output is a function of the rate of saving. As the rate of saving is a variable we can affect (after all, we could save more), we can essentially choose how fast we want output to grow, weighing the costs of saving more against the benefits of faster growth. The rate of output growth is determined within our model

it is endogenous.

42 C HAPTER 4
A production function is said to exhibit constant returns to scale (CRTS) if whenever we double every input, output doubles. All inputs must be increased in a balanced way for this to be the case
 when we increase capital, labor must be increased by the same amount. The Cobb-
Douglas production function has constant returns to scale
 we can tell because its exponents add up to 1 (this is always a good, quick way to check).
We double K and L below to show, rigorously, that they cause output to double as well.
A(2K)

(2L) 1  
= A(2)

(2) 1  
(K)

(L) 1  
= A(2) [

+ (1
 
)]
(K) 
(L) 1  
= A(2) 1 (K)

(L) 1
 
= (2) (AK

L 1
 
) = 2Y
A mathematical note may help you through the algebra above:
(x) a
(x) b
can also be written as (x) a + b
A production function has increasing returns to scale if doubling all inputs more than doubles output; if it is in a form similar to that of the Cobb-Douglas function (factors of production raised to some power and then multiplied together), its exponents will add up to a number greater than one.
ACROSS
1 Has a constant marginal product for society but a diminishing marginal product for individuals
3 A rate, drives endogenous output growth
5 When countries "catch up" with others
6 In a CRTS production function, whenever we double inputs, this doubles
9 MPK in the eyes of individual firms
10 Technology growth in the neoclassical model is determined

G ROWTH AND P OLICY
43
1 2
6
9
5
10
7
8
3 4
DOWN
2 Type of convergence, requires same rate of saving
4 Returns to scale, produce continuing growth
7 When countries move from being centrally planned to free market they are in
______
8 There need not be a steady state when the MPK is ______
1. When an individual firm captures all the benefits of its investment, as is assumed in the neoclassical model of growth, capital will have a ______________ marginal product.
2. When it does not capture all these benefits, so that some of the ideas generated as a byproduct of their capital accumulation are beneficial to other firms, capital can have a
______________ marginal product.
3. A major difference between the neoclassical model of growth and endogenous models of growth is whether the production function has constant or increasing ______________.
4. Another difference is that endogenous growth models have no (non-zero) _____________.
5. In both neoclassical and endogenous growth models, long-term growth in per capita output results only from growth in ______________.
6. The rate of technology growth, in the neoclassical model, is an ______________ variable.
7. The rate of technology growth in endogenous growth models depends on the rate of capital accumulation, and therefore on the rate of ______________.
8. This is why, in the neoclassical model, ______________ cannot affect the long-term growth rate of per capita output, while in endogenous growth models it can.

44 C HAPTER 4
9. Neoclassical growth theory predicts ______________ convergence for countries with equal rates of saving, equal rates of population growth, and equal access to technology; it predicts
______________ convergence for those with different rates of saving or population growth.
10. Empirical evidence suggests that, over long periods of time, countries converge
_________________.
T F 1. The presence of increasing returns to scale in the production function will create a tendency toward monopolization (will make one large firm more efficient than many smaller ones).
T F 2. There is no steady-state in endogenous growth models.
T F 3. There can be no long-term growth in per capita output in the neoclassical growth model.
T F 4. There is long-term growth in per capita output in the neoclassical growth model, but it is independent of the rate of saving.
T F 5. There is no steady-state in the neoclassical growth model.
T F 6. There is a steady-state in the neoclassical model, but it is unaffected by changes in the rate of saving.
T F 7. Models of endogenous growth suggest that when countries have the same rates of depreciation and population growth, their potential output will grow at the same rate.
T F 8. The impact of higher saving on growth is transitory in the neoclassical model.
T F 9. The impact of higher saving on growth is transitory in models of endogenous growth.
1. An increase in the rate of saving in neoclassical growth theory increases the steady-state a. level of per capita output c. growth rate of per capita output b. growth rate of technology d. rate of capital accumulation

G ROWTH AND P OLICY
45
2. An increase in the rate of saving in models of endogenous growth increases everything except the a. rate of population growth c. growth rate of technology b. growth rate of per capita output d. rate of capital accumulation
3. The assumption that there are substantial external returns to capital is most reasonable when capital is in the form of a. machinery b. tools c. ideas d. all of the above
4. In endogenous growth theory, the marginal product of capital is a. constant c. decreasing in k b. increasing in k d. variable
5. In neoclassical growth theory, the marginal product of capital is a. constant c. decreasing in k b. increasing in k d. variable
6. The production function used in endogenous growth theory has _________ returns to scale
(for society). a. constant c. decreasing b. increasing d. variable
7. Some of these returns, however, are not captured by individual firms. We call these benefits
“___________.” a. internal c. sideways b. external d. fractured
8. As a result, the production function used in endogenous growth theory has _________ private returns to scale. a. constant b. increasing c. decreasing d. variable
9. When two countries move toward steady-states that have the same level and growth rate of potential output, they are said to converge a. absolutely c. relatively b. partially d. conditionally

46 C HAPTER 4
10. When two countries move toward steady-states with the same growth rate, but a different level of potential output, they are said to converge only a. absolutely c. relatively b. partially d. conditionally
1. What assumption changes when we move from neoclassical to endogenous growth theory?
2. Does this production function have constant returns to scale? Y = K

N 1  
Does this one? Y = K

N 2
 
3. How will an increase in the savings rate affect the growth rate of per capita output in an endogenous growth model?
4. How will an increase in the savings rate affect the growth rate of per capita output in the neoclassical growth model?
5. What can endogenous growth theory explain that neoclassical growth theory cannot?
What can’t endogenous growth theory explain?
(all from optional sections)
1. Consider the following neoclassical production function: Y = K 1/2 N 1/2 a) Write this production function in per capita form b) Find the golden rule level of the (steady-state) capital-labor ratio, when the rate of depreciation is 0.05, and the (exogenous) rate of population growth is 0.20.
(Translation: find the point at which the marginal product of capital


2
1 k


 n
 d .)
2. Find the rate of growth of the capital-labor ratio (

k/k) for the endogenous growth production function: f(k) = k. Assume that the rate of saving is 0.3, that the rate of depreciation is 0.05, and that the rate of population growth is 0.20.
3. How does the neoclassical model of growth change when we endogenize the rate of population growth (make n a function of y)? Are there any assumptions about the function
n(y) that are necessary to get the textbook’s result?
4. How can we generate a model in which some countries have low growth and low income and others have high growth, high income (a model with a growth trap)?