3. Long-Run Economic Growth

A. International Growth Comparisons
B. Growth Accounting

A. Simple Steady State Model

B. Solow Growth Model

C. Solow Model Implications

A. Constant Marginal Productivity of Capital
B. Savings and GDP Growth
C. The Rich and Poor and Convergence

A. Growth Accounting Equation
B. Elasticity and Income Share
C. Constant Returns and Elasticities

Comparing growth rates across countries as in Table 3-1 is not a problem despite differences in currencies because growth rates are independent of the units of measurement. But when we compare levels of output or income across countries as in Figure 3-1 we run into the problem of differences in currencies. U.S. output is measured in dollars and Japanese output is measured in yen. The first step in comparing levels is to convert currencies using foreign exchange rates. Output in yen can then be expressed as output in dollars by multiplying by the dollar-yen exchange rate.

We still have one significant problem - differences in the cost of living. For example, if we express welfare in terms of income per worker the income in dollars per worker in Japan may be identical to that of the United States. However, if the cost of living in Japan is higher because of housing, food and other costs, this direct comparison of incomes does not reveal actual differences in purchasing power and the standard of living.

Let's compare two U.S. States, which avoids the small complication of converting different currencies using exchange rates. The median annual household income in Virginia in 2000 was $46,677, while that of Mississippi was $31,330. Does this mean the average household in Virginia was better off? Not necessarily because the median value of owner-occupied housing in 2000 was $125,000 in Virginia versus $71,400 in Mississippi. Incomes may be lower on average in Mississippi but the cost of living, at least in terms of housing, was also lower. Output and income levels must be corrected for purchasing power parity in order to make valid comparisons.

Exchange Rate - The price of one currency in terms of another currency. For example, the exchange rate between the yen and the dollar may be 100 yen = $1.00. This means that you need to pay a price of 100 yen to get $1.00, or pay $1.00 in exchange for 100 yen. Exchange rates can be fixed or floating. Fixed means that they stay at the same value as set by the government. Floating means that they fluctuate day to day according to the market. Purchasing Power Parity (PPP) - The PPP exchange rate is represents the quantities of money in each currency that would buy exactly the same basket of goods in both countries. For example, say a certain basket of goods costs 1,000 Yen in Japan, and the same basket costs $10 in the U.S.. The PPP rate would be 100 Yen = $1. The PPP exchange rate will often differ from the actual exchange rate.

International comparisons are made easier using data provided by Alan Heston, Robert Summers and Bettina Aten with the Center for International Comparisons at the University of Pennsylvania. The Penn World Tables (http://pwt.econ.upenn.edu/) provide real national income accounts converted to U.S dollars based on purchasing power parity for 179 countries for some or all of the years 1950-2000. The Organization for Economic Cooperation and Development (OECD) also lists PPP exchange rates for its member countries (Annual National Accounts for OECD Member Countries, http://www.oecd.org/).

The Japan-U.S. exchange rate makes an interesting study of the role of purchasing power parity. In 1980 the market exchange rate was 227 yen to the dollar. The PPP exchange rate (based on a basket of goods and services represented by GDP) was close to the actual at 231 yen to the dollar. The exchange rates were very different in 2000: the actual rate was 108 yen per dollar and the PPP exchange rate was 155 yen per dollar. While the yen had appreciated in value over those 20 years (it now takes fewer yen to buy one dollar), the smaller decline in the PPP exchange rate indicates that it has become relatively more costly to live in Japan, which has offset some of their gain in the actual exchange rate.

The growth accounting equation (2) gives us a foundation for evaluating historical economic growth and its causes. Empirical studies of economic growth generally follow three steps:

1. Determine the growth rates of aggregate output (ΔY/Y), labor (ΔL/L), and capital (ΔK/K) over some period of time.

2. Estimate the elasticities of output with respect to capital, ε_K, and to labor, ε_L. This is made somewhat easier if we assume markets are competitive, which implies that the elasticities are equal to income shares that are observable (see Appendix B). For the United States the elasticity of output with respect to capital has been about 0.3, and the elasticity of output with respect to labor about 0.7.

3. Calculate the contribution of the growth in capital (ε_K ΔK/K) and the growth in labor (ε_L ΔL/L) to the growth in output (ΔY/Y). The difference between the growth in aggregate output and the contributions from capital and labor is attributed to multifactor productivity change. Productivity change is treated as an unexplained residual.

Two well known studies that follow this growth accounting procedure are those of Robert Solow and Edward Denison, which are summarized in Table 3-2. One interesting result of these studies is the smaller contribution made by the growth in capital. Increasing aggregate output depends more on increases in multifactor productivity than on real capital. Denison suggested that almost two-thirds of the increase in multifactor productivity (0.66% of the 1.02%) was due to advances in knowledge. Improved resource allocation (e.g., movement of workers from farms to the cities) and economies of scale added 0.23% and 0.26% to output growth, respectively.

Table 3-2. Solow and Denison Studies of U.S. Growth
percent change per year
	Solow	Denison
Period Covered	1909 - 1949	1929 - 1982
Total Output, ΔY/Y	2.9	2.92
Capital Inputs, ΔK/K	0.32	0.56
Labor Inputs, ΔL/L	1.09	1.34
multifactor Productivity, ΔA/A	1.49	1.02
Sources: Robert Solow, "Technical Change and the Aggregate Production Function", Review of Economics and Statistics, August 1957; Edward F. Denison, Trends in American Economic Growth, The Brookings Institution, 1985.

We can take our own stab at estimating the growth in multifactor productivity using published data on real GDP, hours worked and the private capital stock. For example, Table 3-3 presents these data published by the Bureau of Economic Analysis (Dept. of Commerce) and Bureau of Labor Statistics (Dept. of Labor).

Table 3-3. U.S. Growth Accounting
	1987	2003	Annual Average Percent Change
Output: Real GDP Quantity Index (2000=100)	65.958	105.749	2.99 %
Capital: Private Nonresidential Fixed Assets Index (2000=100)	69.663	105.714	2.64 %
Labor: Total Private Aggregate Weekly Hours Index (2002=100)	79.77	98.62	1.33 %
Sources: Real GDP from Bureau of Economic Analysis (BEA), National Income and Product Accounts, Table 1.1.3 (http://www.bea.gov/bea/dn/nipaweb/index.asp); Nonresidential fixed assets from BEA, Fixed Assets, Table 4.2 (http://www.bea.gov/bea/dn/FA2004/SelectTable.asp); and Total private aggregate weekly hours from Bureau of Labor Statistics, Current Employment Statistics Survey, Series CES0500000040 (http://www.bls.gov/ces/home.htm)

Given the estimates for the elasticities of output with respect to capital (ε_K=0.3) and labor (ε_L=0.7) published by others (or we could calculate income shares as explained in the Appendix) we can easily calculate the growth in multifactor productivity:

	Real GDP growth rate = Productivity growth rate + εK • fixed assets growth rate + εL • labor growth rate
	2.99 = Productivity growth rate + 0.3 • 2.64 + 0.7 • 1.33
	Productivity growth rate = 2.99 - 0.79 - 0.93
	Productivity growth rate = 1.27 % per year average

We calculate an average growth rate of multifactor productivity of 1.27 percent per year. The Bureau of Labor Statistics (BLS) calculated average growth of multifactor productivity of 0.8 percent per year over this same period. Calculating multifactor productivity is not as simple as the calculation we provided above. Some corrections the BLS includes in their calculation are (Bureau of Labor Statistics, BLS Handbook of Methods, Chapter 10, http://www.bls.gov/opub/hom/homtoc.htm):

• BLS excludes the following outputs from real GDP growth: general government, nonprofit institutions, paid employees of private households, and the rental value of owner-occupied dwellings.

• Real stocks are constructed as vintage aggregates of historical investments (in real terms) in accordance with an "efficiency" or service flow concept (as distinct from a price or value concept). The efficiency of each asset is assumed to deteriorate only gradually during the early years of an asset's service life and then more quickly later in its life.

• The hours of employees of government enterprises are excluded. Also, the hours at work for each of 1,008 types of workers classified by their educational attainment, work experience and gender are aggregated using an annually chained (Tornqvist) index. The growth rate of total labor is therefore a weighted average of the growth rates of each type of worker where the weight assigned to a type of worker is its share of total labor compensation. The resulting aggregate measure of labor input accounts for both the increase in raw hours at work and changes in the skill composition (as measured by education and work experience) of the work force.

This last correction, weighting labor hours by skill, is probably the most serious. Earlier we said that multifactor productivity includes improvements in labor such as from better education or training. However, productivity increases arising from improving work skills through training or education may not show up in the multifactor productivity statistic reported by BLS. Consequently the BLS estimate of multifactor productivity growth should be smaller than our quick calculation.

Neoclassical growth models identify a steady state where the rates of change in output, capital and labor are constant. A steady state is something like an equilibrium condition. If there is no shock to the economy then markets will maintain a constant rate of growth. The question we ask is what determines the rate of growth.

Steady State - a condition of constant rates of growth in economic measures. With no technological change, a steady state is represented by identical constant growth rates in the labor force, total output, and the level of capital.

First we start with the growth accounting equation and assume that technology is constant, i.e., there is no productivity growth. In other words, in equation (2) ΔA/A = 0. Thus, we can simplify equation (2) as follows:

ΔY = εK ΔK + εL ΔL  Y            K            L

(3)

where, ΔA/A = 0.

A steady state with no technological change implies that output per worker and the capital-labor ratio are constant. Capital must grow at the same rate as the labor force. If capital grows faster or slower than the quantity of labor then the economy is not in a steady state rate of growth. Another way of looking at it is if there is no technological change there should be no reason to increase or decrease the amount of capital every worker is given. New entrants to the labor force will be given a level of capital identical to all other workers.

If labor supply grows at some constant rate, n, then ΔL/L = n. Since the level of capital per worker is constant in this model of a steady state then capital must grow at the same rate as the labor force, or ΔK/K = n. Equation (3) can be rewritten:

ΔY = εK n + εL n  Y

(4)

where,

ΔL/L = ΔK/K = n
n = labor force growth rate

Equation (3) doesn't tell us much because the elasticities of output with respect to capital, ε_K, and labor, ε_L, are unknown. We can give it some meaning by making a key assumption about the form of the production function, which determines the elasticities.

We assume the production function has constant returns to scale. Constant returns to scale simply means that if the labor force grows at 2% per year and capital grows at 2% per year (the capital-labor ratio in this steady state model is constant) then output also grows at 2% per year. Declining returns to sale in macroeconomic growth models implies that as the labor force grows (and the level of capital grows with it) a country would get poorer in terms of output (and real income) per worker. Total output would not increase as fast as the labor force. Thomas Malthus applied the concept of declining returns when he conjectured in 1798 (An Essay on the Principle of Population) that the world population would eventually outgrow the capability to produce food. Increasing returns to scale implies that output grows faster than the labor force. The only thing a country would need to do for its residents to become wealthier is to increase its labor force while maintaining the capital-labor ratio.

Returns to Scale - the change in production that occurs when all resources are proportionately increased (increased by the same percentage). If labor, capital, and all other inputs to the production process increase by 1%, does output increase by more than 1% (increasing returns to scale), less than 1% (decreasing returns to scale), or exactly 1% (constant returns to scale)?

When we assume constant returns to scale we can show mathematically (see Appendix C) that ε_K + ε_L = 1. Thus, equation (4) simplifies to reveal that the growth rate of output equals the growth rate of the labor force in steady state as shown in equation (5).

ΔY = n  Y

(5)

The surprising implication of this model is that the level of capital, K, has no affect on the long-term growth rate of an economy. The workers in one country can be loaded with the best tools and factories while another country's workers may be barely equipped, yet the steady-state aggregate economic growth rates of both countries can be identical if their populations are growing at the same rates. While output per worker will be greater for the country with the higher capital-labor (K/L) ratio, output per worker will not change in either country unless there is a change in total factor productivity. In other words, in steady state the economies of countries will grow at the same rate as their populations but the output of each individual worker remains unchanged. Economies can grow faster than their populations only if productivity improves.

In the opening of this section we made the simplifying assumption that technological change is assumed to be zero and total factor productivity remains unchanged. This is a very strong assumption. If instead technological change, A, in the production fucntion equation (1) and growth accounting equation (2) is some positive number then it should be obvious that economic growth is a combination of technological change and the growth rate of the labor force. This is more realistic but it doesn't change our primary observation that the level of capital does not affect the long-run growth rate of the economy in steady state.

The representation for technological change, A, in equation (1) is itself a strong assumption. This implies that technology advances at a constant rate over time and is neutral in that it relates to capital and labor in the same way. In other words, we get the same rate of productivity improvement regardless of the relative levels of capital and labor. The model can be modified to account for differences in technological change between the factors of production. For example, we could introduce a labor productivity factor, which would result in maintaining a steady state capital-output rather than capital-labor ratio (referred to as Harrod neutrality). We still get a similar result. Output grows at the same rate as the labor force plus the rate of technological change. The assumptions and result are perhaps more consistent with what we actually observe in most economies, but do not change the implications of the model.

So far we have established that the steady-state growth rate of aggregate output depends only on the rate of growth of the labor force (and technological change). The capital-labor ratio influences only the level and not the steady-state growth rate of an economy. For example, let's consider a rich and a poor country. Assume the per worker output of the rich country is 50% greater than that of the poor country. If the labor force growth rate of both countries is 2% per year then we can expect the aggregate output of both countries to double in 35 years. After 35 years the per worker output of both countries will remain unchanged and the rich country will still produce 50% more output per worker than the poor country.

The 2% growth and doubling in 35 years is a simple rule-of-thumb that we get from the "Rule of 70." The rule of 70 states that the approximate number of years it takes a variable to double is 70 divided by the annual growth rate.

Increasing the level of capital and the capital-labor ratio may not contribute to a permanently higher economic growth rate but it does give a short-term boost to output per worker and the standard of living. If countries can choose the amount of capital employed and raise or lower the capital-labor ratio is more capital always better? The next surprising result of our analysis is that a higher output per worker is not necessarily better. To see this result we need to expand the neoclassical model to include savings and investment in the Solow growth model.

A constant returns to scale production function was a key assumption that allowed us to simplify the growth accounting equation. In the Solow growth model we again put the constant returns assumption to use in a slightly different way.

A production function that represents constant returns to scale is represented in equation (6). We still assume that technological change, A, is zero. The constant returns production function, as noted earlier, implies that if we multiply both the quantity of labor and the quantity of capital by some positive number, z, then we also multiply output by z.

z Y = f (z K, z L)

(6)

where,

z = some positive number

Y = total real output
K = economy's use of capital
L = economy's use of labor
f (z K, z L) = aggregate production given inputs of capital and labor

If we assume the value of z is equal to 1/L, then we can convert the aggregate production function in equation (6) to a per worker production function in equations (7) and (8)

Y = f (K/L, 1) L

(7)

or,

Y = f (K/L) L

(8)

So far nothing has changed from what was presented in the previous section. We have simply transformed the aggregate production function to a per worker basis, which is illustrated in Figure 3-3. Per worker output, Y/L, in Figure 3-3 is determined by the capital-labor ratio, K/L.

Figure 3-3. The Production Function.

The production function has a positive slope because an increase in capital per worker, K/L, results in an increase in output per worker, Y/L. The slope of the production function represents the marginal product of capital. Each one unit increase in the capital-labor ratio increases output per worker by an amount equal to the marginal product of capital. The bowed shape of the production function implies diminishing marginal productivity of capital. Each incremental increase in capital with the amount of labor held constant (i.e., the capital-labor ratio increases) produces progressively smaller increases in output.

Marginal Product - the change in the quantity of total output resulting from a unit change in a variable input, keeping all other inputs unchanged. The marginal product of capital is the change in output resulting from a 1 unit change in capital with the quantity of labor and other inputs held constant.

Second, we assume the national savings rate is some fixed fraction of total output as shown in equation (9):

S = s • Y

(9)

where,

S = total savings per period
s = national savings rate

Savings on a per worker basis is shown in Equation (10). Output per capita is again plotted as a function of the capital-labor ratio in Figure 3-4. Output per capita is multiplied by the fixed savings rate, s, to obtain the lower dashed savings per capita line.

S = s • Y L          L

(10)

Figure 3-4. The Production and Savings Functions.

Third, investment is by definition equal to the change in the capital stock plus depreciation. We must not only equip new workers with capital but must also replace existing equipment that wears out or becomes obsolete. The change in the capital stock required to equip new workers is the labor force growth rate, n, times the current level of capital, or n • K. Let d represent the capital depreciation rate, or the fraction of capital that wears out each year. The amount of capital that must be replaced every year because of depreciation is the depreciation rate, d, times the level of capital, or d • K. Thus, total investment in steady state is represented by equation (11).

I = n • K + d • K    = (n + d) • K

(11)

where,

I = total investment per period
d = the depreciation rate on existing capital stock.

Investment on a per worker basis is presented in equation (12) and illustrated in Figure 3-5.

I  = (n + d) • K  L                  L

(12)

Figure 3-5. Investment.

Fourth, we assume a steady state condition that savings equals investment. Steady state in Figure 3-6 is represented by the point the savings line crosses the investment line. If the capital-labor ratio is less than the steady-state level savings exceeds the level of investment needed to maintain this low capital-labor ratio. The extra savings is invested in new capital, which makes the capital-labor ratio increase until the steady-state point is reached. Similarly, if the capital-labor ratio is to the right of the steady state point we have the opposite situation. Savings is less than the amount needed to finance the investment required to replace depreciated capital and to equip new workers. The capital-labor ratio declines until the steady state level is reached.

Figure 3-6. Steady State.

One of the common empirical observations made when comparing economic growth rates across countries is the inverse relationship between wealth and population growth. The poorest countries generally appear to have the highest rate of population growth. The sociological reasons for high population growth rates among the poor are beyond the scope of this course. But we can use the Solow growth model to see why there is a relationship between population growth rates and wealth.

Consider two countries with identical production functions, savings and capital depreciation rates but different population growth rates. The identical production function assumption simply means that the same technology is available to both countries. Any differences in wealth are not due to political barriers that prevent the export of new technology from one country to another.

We can illustrate the effect of different population growth rates on wealth in Figure 3-7. Country B has a higher population and labor force growth rate than country A. This difference is reflected in the different investment lines in Figure 3-7 labelled I_A for country A and I_B for country B. The I_B investment line is higher than the I_A investment line because the slope, n + d, is higher, which reflects to the higher population growth rate, n. The consequence of the higher population growth rate is the output per worker is lower.

Figure 3-7. Population and Wealth.

The logic is pretty simple. A given rate of savings will support a some amount of investment in capital goods. With more people continually entering the labor force in a country with the higher population growth rate the available capital goods must be spread more thinly. Each of the new and existing workers get fewer tools to work with and consequently produce less output. While the aggregate economic growth rate of a poor country may be greater than a wealthy country (because of the higher population and labor force growth rates), the poor country will remain poor relative to the wealthy country.

We can look at the effect of the savings rate on wealth in the same way we considered population growth except that we assume now the population growth rates of two countries are identical while the savings rates differ. A change in the savings rate, s, is required for an economy to move to a different capital-labor ratio. We show this in Figure 3-8 with a low savings rate country A and high savings rate country B. A poor country with low savings must operate at a low capital-labor ratio. A wealthy country with a higher savings rate may operate at a higher capital-labor ratio. A poor country can increase its wealth by increasing its savings rate thereby moving from a low K/L steady state to a higher K/L steady state over time.

Figure 3-8. Savings and Wealth.

Why is one country poor and another wealthy? One answer lies in the capital-labor ratio. The wealthy country must have a higher capital-labor ratio than the poor country. But differences in capital-labor ratios and wealth do not affect aggregate economic growth rates in the neoclassical model. The steady state rate of growth of aggregate output of an economy is equal to the labor force growth rate. While one country may be poorer because of a high population and labor force growth rate, its aggregate economic growth rate may still be higher. But it won't catch up unless it can raise its capital-labor ratio.

For the poor country to catch up it must increase the level of capital employed per worker. However, the cruel dilemma of many poor countries is that increased investment requires savings to finance that investment. A higher savings rate means less current consumption. A higher savings rate may be impossible when only the basic necessities of living can be afforded in the poor country.

There is an incentive for capital to flow from the rich country to the poor country because of the declining marginal productivity of capital. If we were to move one unit of capital from the rich to poor country we would give up some small amount of output in the rich country but gain a much larger amount of output in the poor country. In the neoclassical model there is an economic incentive for convergence of the capital-labor ratios across countries.

We have seen an example of convergence with the impressive growth if the Asian "tigers" (such as Japan, Hong Kong, and Korea) over the last 50 years. The infrastructure of the east Asian nations had been decimated by World War II. Capital-labor ratios were comparatively very low. Through very high savings and investment rates their capital-labor ratios progressively grew. Economic growth came not just from labor force and technological change like other industrialized countries, but also because of an increase in their capital-labor ratios, which moved them up the production curve. This rapid capital intensification really can't go on forever. The declining marginal productivity of capital makes it uneconomical to sustain growth over the long term by increasing capital inputs alone. In fact the 1990s suggest that the envious growth rates of the Asian tigers has come to an end. The Asian tigers spent 40 years catching up and now they are just like us.

So why do some countries remain terribly poor? The answers are varied but often relate to barriers to the international flow of investment. For example:

• corruption diverts money from investment to consumption by the corrupt,
• barriers to investment by non-citizens,
• lack of a legal system to protect private ownership of land, resources, or businesses,
• political instability increases uncertainty about the security of investment, and
• a labor force that does not have the necessary education or training.

While these explanations have appeal they still seem inadequate to explain differences in wealth across countries that last decades and even centuries. In the next section we will briefly describe a recent development in economic growth theory, the endogenous growth model, that attempts to explain why differences may persist and convergence may not occur.

If wealth increases with savings and investment is a higher savings rate always better? In our final application of the Solow growth model the surprising answer is no. It is not output per person that indicates wealth or welfare but consumption per person. We are supposedly better off when we can consume more.

There are only two things we can do with the output we produce: we either consume it or use it as investment to produce other goods and services. Consumption is the difference between total output and investment (or savings) as represented by equation (13).

C = Y - I     = Y - (n + d) • K

(13)

And consumption in equation (13) can be represented on a per worker basis by dividing through by the size of the labor force, L, in equation (14):

C = Y - (n + d) • K L     L                  L

(14)

Consumption per worker is shown in Figure 3-9 as the distance between output per worker and investment per worker.

Figure 3-9. Consumption.

Notice in Figure 3-9 that as we increase the capital-labor ratio from a zero starting point the distance between output and investment gets progressively larger. Output rises faster than required investment and consumption per worker rises as the capital-labor ratio increases. At some point a maximum distance is reached and any further increases in the capital-labor ratio reduces the distance between output and investment per worker. Consumption begins to decline because of the curvature of the production function and the declining marginal productivity of capital. Maximum consumption occurs at the point where the slope of the per worker production function equals the slope of the investment line.

An increase in the capital-labor ratio has two opposing effects on consumption. First, a higher capital-labor ratio enables each worker to produce more output, which allows for an increase in consumption. Second, a higher capital-labor ratio requires more ongoing investment to replace worn out capital and equip new workers, which reduces consumption.

The implied level of consumption in Figure 3-9 is plotted in Figure 3-10, which reveals the effect of the trade-off between more output and more investment on consumption. Starting with low levels of capital, increases in the capital-labor ratio allow workers to consume progressively more. At some point a maximum is reached where the declining marginal productivity of capital is no longer large enough to support the level of investment required and consumption begins to decline. The point of maximum consumption is known as the Golden Rule capital-labor ratio (E.S. Phelps, "The Golden Rule of Accumulation: A Fable for Grown men," American Economic Review, 55, September 1965, 638-643). The Golden Rule suggests that it is possible to save and invest too much.

Figure 3-10. Optimal Consumption.

We can identify the implications of the endogenous growth model by starting with a simple aggregate production function with the assumption of a constant rather than declining marginal productivity of capital. To simplify our model we also assume the number of workers does not change and accept the output and labor force growth rate relationship implied by the neoclassical model. This simplification means that the growth rate of aggregate output will be identical to the growth rate of output per worker in this model. We have in equation (15) a direct relationship between aggregate output and the level of physical capital.

Y = α • K

(15)

where,

Y = total real output
K = economy's use of physical capital
α = positive constant

In equation (15) each additional single unit of physical capital, K, increases aggregate output, Y, by α units regardless of the level of capital employed. Because α does not depend on the level of capital we have constant marginal productivity of capital. The growth rate of output is equal to the growth rate of capital as shown in equation (16).

ΔY / Y = ΔK / K

(16)

We add to this model a representation for savings in equation (17). Aggregate savings is assumed to be constant at a fixed proportion, s, of total output.

S = s • Y    = s • α • K

(17)

Investment equals additions to the level of capital plus depreciation, equation (18), just as it did in the neoclassical model:

I = ΔK + d • K

(18)

In steady state aggregate savings must equal aggregate investment:

s • α • K = ΔK + d • K

(19)

Rearrange equation (19) to solve for the change in the physical capital stock:

ΔK = s • α • K - d • K

(20)

Divide both sides of equation (20) by K to get the growth rate of the physical capital stock:

ΔK / K = s • α - d

(21)

Substitute the solution for the growth rate of capital stock in equation (21) back into the output growth equation (15) and we get the result in equation (22) that the growth rate of output is a function of the savings rate:

ΔY / Y = s • α - d

(22)

The result that the growth rate of both total output and output per worker are a function of the savings rate is a significant departure from the neoclassical model. Savings affects long-run growth in the endogenous growth model because higher rates of saving lead to greater investment in human capital, which contributes to greater labor productivity.

Figure 3-11 presents a scatter plot of annual average savings rates (as a percentage of real GDP per worker) and real GDP per worker growth rates across countries over the period 1950 through 2000. The dashed linear regression line in Figure 3-11 indicates that each 10 percent increase in the average savings rate increases the growth rate of real GDP per worker by 0.45 percent. The United States falls right in the middle of the graph with an average savings rate of 19.2 percent and a real GDP per worker growth rate of 1.95 percent.

Figure 3-11. Relationship between savings and growth rate of real GDP per worker.
Source: Penn World Tables (http://datacentre.chass.utoronto.ca/pwt/alphacountries.html).

The neoclassical growth model suggested that standards of living of the poor and wealthy countries should converge on each other. The assumed declining marginal productivity of capital suggests that investment should migrate from the wealthy countries to the poorer countries, which have lower capital-labor ratios and higher marginal productivity from new capital investment. While there is some capital migration and some poor countries have converged on the wealthy in recent decades (in particular, the Asian tigers such as Hong Kong and japan), this usually seems the exception rather than the case.

The endogenous growth model provides an explanation for the absence of convergence. When there is a constant or even increasing marginal productivity of capital we should not expect the migration of investment from wealthy to poor countries as implied by the neoclassical model. Moreover. while there may be no barriers to the movement of physical capital from wealthy to poor countries, human capital cannot be transferred so easily. Differences in standards of living and economic growth rates may be sustained over time and we may not see convergence.

Production is a function of the economy's use of capital, K, labor, L, and a productivity index, A.

Y = A • f(K, L, t )

(A1)

where,

Y = total real output
A = productivity index
K = economy's use of capital
L = economy's use of labor
t = time
f(K, L, t ) = aggregate production given inputs of capital and labor at a point in time

Take the total derivative of equation (A1) with respect to time, with df(.) used as a abbreviation for df(K,L,t):

dY = dA f(K, L, t ) + A • df(.) dK + A • df(.) dL + A • df(.) dt      dt                          dK   dt            dL   dt            dt

(A2)

We can make equation (A2) easier to read by abbreviating the time derivations:

ΔY = ΔA f(K, L, t ) + dY ΔK + dY ΔL                                  dK          dL

(A3)

where,

dY = A • df(.)
ΔY = dY/dt
ΔK = dK/dt
ΔL = dL/dt
ΔA = dA/dt

Divide equation (A3) through by Y, or the equivalent A • f(K, L, t):

ΔY = ΔA + dY ΔK + dY ΔL  Y       A     dK   Y     dL   Y

(A4)

Multiply the second term on the right-hand side of equation (A4) by K/K and the third term by L/L.

ΔY = ΔA + dY K ΔK + dY L ΔL  Y       A     dK Y  K      dL Y  L

(A5)

Finally, we can simplify equation (A5) by representing the term (dY•K)/(Y•dK) as the elasticity output with respect to capital and the term (dY•L)/(Y•dL) as the elasticity of output with respect to labor. For example the elasticity of output with respect to capital is the percent change in output, dY/Y, divided by the percent change in labor, dK/K.

ΔY = ΔA + εK ΔK + εL ΔL  Y       A            K            L

(A6)

where,

ΔY = output growth rate
 Y
ΔA = productivity growth rate
 A
ε_K = elasticity of output with respect to capital
ΔK = capital growth rate
 K
ε_L = elasticity of output with respect to labor
ΔL = labor growth rate
 L

Equation (A6) is the basic growth accounting equation that decomposes growth in output into three parts:

1. from productivity growth, ΔA/A
2. from increased capital inputs, ΔK/K
3. from increased labor inputs, ΔL/L

Because elasticities are not observable (without knowing form of production function), they cannot be used in empirical analysis. But we can transform elasticities into income shares that can be calculated using government GDP/income survey data.

Assume a perfectly competitive market. The first-order condition of the microeconomic profit maximization problem is that inputs are paid the value of their marginal products, as represented in equation (B1).

dY = r dK    p

(B1)

where,

dY = change in output arising from a 1 unit change in the level of capital
dK
r = rental price of capital
p = price of output

The definition of the elasticity of output with respect to capital is presented in equation (B2):

εK = dY K         dK   Y

(B2)

where,

ε_K = elasticity of output with respect to capital
Y = total real output
K = economy's use of capital

Substituting the profit maximization first-order condition that dY/dK = r/p from equation (B1) into equation (B2), we get the result in equation (B3) that the elasticity of output with respect to capital is equal to the income share to capital. The income share of capital equals the rents paid to the owners of capital for the use of their equipment, r K, divided by the value of total sales, p Y.

εK = r K         p Y

(B3)

The same procedure can be used for labor where the profit maximization first-order condition is dY/dL = w/p, where w is the wage rate.

If we assume constant returns to scale the elasticities of output with respect to capital and labor sum to 1. We start with the standard profit equation (C1).

Profits = p • Y - r • K - w • L

(C1)

where,

p = price of output
Y = total real output
r = rental price of capital
K = economy's use of capital
w = real wage rate
L = economy's use of labor

Some basic microeconomics (that we will not show here) reveals that constant returns to scale implies total profits equal zero as in equation (C2).

0 = p • Y - r • K - w • L

(C2)

or,

p • Y = r • K + w • L

(C3)

Divide both sides of equation (C3) by p • Y:

p • Y = r • K + w • L p • Y    p • Y    p • Y

(C4)

Equation (C4) simplifies to:

1 = r • K + w • L       p • Y    p • Y

(C5)

Equation (C5) shows that the income shares of capital (rK/pY) and labor (wL/pY) sum to 1. In the previous section we showed that elasticities of output are equivalent to income shares. Thus the elasticities of output with respect to capital and labor sum to 1.

1 = εK + εL

(C5)