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Chapter 03 – Growth and Accumulation 3 GROWTH AND ACCUMULATION FOCUS OF THE CHAPTER • In this chapter we study how potential outputthe output that would be produced if all factors were fully employedgrows over time. • To better accomplish this, we learn growth accounting and the fundamentals of neoclassical growth theory. Together, they tell us that output growth results both from improvements in technology and from increases in one or more of the inputs to the production processcapital, labor, and natural resources. Neoclassical growth theory also tells us that in the long run, growth in potential output results entirely from technological improvement. * Note: The authors in this chapter and the next use the term “long run” in a way that is inconsistent with the rest of the textbook. They should be saying “very long run.” SECTION SUMMARIES 1. Growth Accounting Output grows because of increases in factors of production like capital and labor, and because of improvements in technology. The production function provides a link between the level of technology (A), the amount of capital (K), labor (N), and other inputs used, and the amount of output (Y) created. The generic formula for the production function is: Y = AF(K,N) 3-1 Chapter 03 – Growth and Accumulation The Cobb-Douglas production function, a more specific formula, is frequently used as well, as it provides a good approximation of production in the actual economy. The formula for the Cobb- Douglas production function is: Y = AKN 1 pronounced “theta”, represents capital’s share of incometotal payments to capital, as a fraction of output, or (iK)/Y. (1 ) is labor’s share of income, given by (wN)/Y. To derive these results algebraically, you need one more fact: When the markets for capital and labor are in equilibrium (i.e., when the supply of capital equals the demand for capital, and the supply of labor equals the demand for labor), capital and labor are each paid their marginal product. For the Cobb-Douglas function, the marginal product of capital (MPK) is AKN. The marginal product of labor (MPL) is (1 )AKN 1 We can express our production function in terms of growth rates rather than levels: Y/Y = [(1 )x N/N] + [x K/K] + A/A The symbol pronounced “delta” means “change in”. The term Y/Y, then, should be interpreted as the growth rate of output. The terms N/N and K/K should be interpreted as the growth rates of labor and capital, respectively. The last term, A/A is the rate of improvement of technology, often called the growth rate of total factor productivity (TFP). It is the amount that output increases as a result of technological progress alone (plug in N/N = K/K = 0, and you’ll see why). Because growth in GDP per capita (output per person) tells us more about increases in the standard of living, it is useful to subtract the rate of population growth (N/N) from both sides, and write the above equation in per capita terms: y/y = (x k/k) + A/A The terms y and k represent output and capital per person: y = Y/N, k = K/N. (There is an implicit assumption here that the fraction of the population in the labor force is constant. This is why we can get away with using the terms “population” and “labor supply” interchangeably.) The terms (y/y) and (k/k) are the growth rates of output and capital per person: y/y = Y/Y N/N, and k/k = K/K N/N. The term K/N is often called the capital-labor ratio. 2. Empirical Estimates of Growth Since 1929, U.S. economic growth has averaged about 2.9 percent a year. Of this, estimates suggest that about 1.09 percent has been due to increases in the labor supply, 0.32 percent has been due to capital accumulation, and 1.49 percent has been the result of technological progress. Physical capital and labor are not the only inputs to production. Two other important factors of production are natural resources and human capitalthe skills and talents of workers. The 3-2 Chapter 03 – Growth and Accumulation shares of income (also called factor shares) of physical capital, human capital, and raw labor are estimated to be roughly 1/3 each. 3. Growth Theory: The Neoclassical Model Neoclassical growth theory studies the way that growth in the capital stock per worker affects the long-run level of per-capita potential output. A key result is that, while the rate of saving has a significant impact on the level of per capita potential output in the long run, the rate of improvement in technology entirely determines its growth rate. y y = f(k) In the constant returns to scale production function, capital & labor have diminishing marginal returns. k Figure 31 THE PER-CAPITA PRODUCTION FUNCTION To build our model, we begin with a few simplifying assumptions: (1) the level of technology is fixed, so that there is no growth in total factor productivity; (2) the production function has constant returns to scale (see Review of Technique 4), so that increasing the amount every input used in production will increase output by the same amount. A consequence of this second assumption is that all factors of production must have diminishing marginal productsas more of one input is added, and the others are held constant, each unit contributes less to output than did the previous one. (Buying more tractors for your construction company without hiring any more workers to drive them will not help increase your output much.) We also need to write our variables in per capita form; as before, y = Y/N and k = K/N. We write the per capita production function: y = f(k) 3-3 Chapter 03 – Growth and Accumulation (n + d)k We then consider the flows into and out kthe (flows out of k) stock of capital per worker. Investment increases the total stock of capital (K), sf(k) (flow into k) which increases k. It can be thought of as a flow into each worker’s pool of capital. Both depreciation and population growth decrease kdepreciation because it decreases the stock of functional capital, and population growth because steady– state k k it increases the number of workers sharing this Figure 32 capital. Both depreciation and population growth can be visualized as flows out of each worker’s STEADY-STATE IN THE NEOCLASSICAL MODEL pool of capital. When the flow into this pool is greater than the flows out, k grows. When the flows out of this pool are greater than the flow in, k shrinks. And when the flows in and out exactly balance, the level of capital per worker will remain fixed. We call this last case the steady-state, because it is the point at which the level of capital in each worker’s pool remains steady, or stable. It is the point of equilibrium in our model; we will find that the capital stock per worker grows or shrinks toward this point, and that once it gets there it stays. At least until some shock forces it to move. It is not difficult to express the dynamics described above as an equation. We know that saving must equal investment. If we assume that people save a constant fraction (s) of their incomes, we can write the flow into k as (s x y), or, using our per capita production function, as (s x f(k)). The standard assumption about depreciation is that a constant fraction (d) of the capital stock becomes obsolete each period. Using this assumption, we can express the flow out of each worker’s pool of capital that result from depreciation as (d x k). Similarly, when population grows at a constant rate (n), we can express the flow out of this pool resulting from population growth as (n x k). Putting all of these terms together, we get the following equation: k = sf(k) (n + d)k We find an expression for the steady-state by simply plugging in the requirement k = 0: sf(k*) = (n + d)k* k* represents the steady-state value of k. The steady-state value of y is y* = f(k*). The growth process can be studied graphically as well. Figure 32 graphs the flows into and out of k against the level of k. The outflows are graphed as a straight line, with slope (n + d). This is often called the investment requirement line, as it shows the amount that must be invested, if the capital stock is to remain constant. The slope of the line that represents the flow into k shrinks as 3-4 Chapter 03 – Growth and Accumulation k increases, because we have assumed it to have diminishing marginal returns. Where these two lines intersect, the flows into and out of k balance, and k is at its steady-state. Whether the savings line lies above the investment requirement line, so that k is increasing, or the investment requirement line lies above the savings line, so that k is decreasing, k always moves toward the steady-state. Using this graph, we can examine the consequences of changes in s, n, or d. An increase in the savings rate (s) will shift the savings line, sf(k), upward, increasing the steady-state capital-labor ratio, and hence the steady-state level of per capita potential output. It will also, temporarily, increase the growth rate of both y and k (remember, the growth rate of k is zero at the steady- state, and without improvements in technology per capita, output has no other reason to grow). An increase in either the rate of depreciation or the rate of population growth will increase the slope of the investment requirement line, decreasing the steady-state levels of k and y, and causing both to “grow”, temporarily, at a negative rate. (n + d)k A reduction in the savings rate s0 f(k) (flow into k at higher s) will decrease steady-state levels of capital and output . s1 f(k) (flow into k at low er s) steady– state, steady-state, steady– state, steady-state, lower s k lower s higher higher s s Figure 33 A DECREASE IN THE SAVINGS RATE REDUCES THE STEADY-STATE CAPITAL-LABOR RATIO When the level of technology is permitted to change, we add in another term: “g”, or the growth rate of technology. Technological improvement is represented on our graph as an upward shift in the savings line, now written s x Af(k). Notice that it causes the steady-state levels of k and y to rise. Notice also, though, that with the addition of technological growth our production function no longer has constant returns to scaledoubling capital and labor will more than double output. In order to fix this problem, growth theorists often assume that technology has a very particular characteristic: It is assumed to be labor augmenting, so that technological improvements increase the productivity specifically of labor. 3-5 Chapter 03 – Growth and Accumulation The production function, under this assumption, is written as follows: Y = F(K, AN) or, in its Cobb-Douglas form, Y = K (AN) 1 Now when we divide through by N we get something a little different: y = f(k,A) (trust me on this one), or y = k A 1 Now, when technology increases at a constant rate (the savings line shifts continually upward), the steady-state levels of k and y will grow at that same rate: %k* = %y* = %A This is not true when technology takes a more general form (the one where it affects the productivity of both capital and labor identically, for example). (n + d)k A technological improvement SA 1 f(k) will increase steady-state (im p roved technology) levels of capital and output. SA 0 f(k) (original technology ) steady– state, steady– state, lower A k higher A Figure 34 A TECHNOLOGICAL IMPROVEMENT INCREASES THE STEADY-STATE CAPITAL-LABOR RATIO 3-6 Chapter 03 – Growth and Accumulation Appendix The appendix discusses, in some detail, how to derive the fundamental growth equation Y/Y = [(1 )x N/N] + [x K/K] + A/A. There is one assumption needed to do this that markets are perfectly competitive, so that capital and labor are each paid their marginal products. Note: The fundamental growth equation takes a slightly different form when labor-augmenting technology is assumed, Y/Y = [(1 )x N/N] + [x K/K] + [(1 )x A/A] or, in per-capita terms, y/y = [x k/k] + [(1 )x A/A]. Just in case you were curious. KEY TERMS growth accounting growth theory production function Cobb-Douglas production function marginal product of labor (MPN) marginal product of capital (MPK) total factor productivity GDP per capita capital-labor ratio diminishing marginal returns convergence Solow residual human capital neoclassical growth theory steady-state equilibrium GRAPH IT 3 When two economies converge, their growth rates, and, in some cases, the levels of their output eventually become equal. This graph asks you to plot some historical growth rates for the U.S. and Japan between 1950 and 1970, to see if convergence was occurring. Table 3–1 provides these growth rates. All that you need to do is plot each country’s growth rate every year, and connect the dots. You will get two lines. You will notice that the rate of growth in per capita output is noticeably higher for Japan than for the U.S. during much of this period. It is this high rate of growth that has brought the Japanese standard of living into line with the standards of living enjoyed by others in the industrialized world. Convergence, however, cannot be said to have 3-7 Chapter 03 – Growth and Accumulation occurred unless Japan’s growth eventually slows, approaching the average growth of other nations. Can you see this happening in Chart 31? Can these data help you to argue that Japan’s high growth during this period was a transitory phenomenon? TABLE 3–1 Year Percentage growth in GDP Percentage growth in GDP (Japan) (U.S.) 1951 23.6 13.3 1952 11.8 3.2 1953 6.8 3.6 1954 5.5 1.7 1955 7.9 7.6 1956 10.4 3.5 1957 11.0 3.4 1958 7.5 0.2 1959 9.8 6.3 1960 14.5 2.4 1961 14.3 1.6 1962 7.7 6.1 1963 10.9 3.9 1964 13.4 5.3 1965 5.1 6.9 1966 13.2 8.2 1967 14.0 4.5 1968 16.8 7.9 1969 15.1 6.7 1970 16.8 5.7 3-8 Chapter 03 – Growth and Accumulation 25 20 15 10 5 0 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 Chart 31 PERCENTAGE CHANGE IN GDP FOR JAPAN AND THE U.S.: 19511970 THE LANGUAGE OF ECONOMICS 3 Stocks and Flows, or “About Your Bathtub” It is always important to know whether a given variable is a stock or a flow. It is easiest to understand the difference between stocks and flows in the context of your bathtub. The level of the water in your bathtub is a stock variableit rises and falls depending on the amount of water entering through the faucet, and, if the drain is unplugged, leaving through the drain. The amounts of water flowing into and out of the tub are, not surprisingly, flow variables. Let’s think through some examples of stock and flow variables. We already know that capital is a stock variable. We also know that investment is a flow into the capital stock, and that depreciation is a flow out of it. Population is a stock variable. Its level is affected by the birth rate (births are a flow into the pool of living, breathing people) and the death rate (deaths are a flow out of that pool). The unemployment rate, despite its name, is yet another stock variable. In Chapter 7 we will see that those losing their jobs and those entering the labor force are a flow into this pool, and that those who find jobs or leave the labor force are a flow out of it. 3-9 Chapter 03 – Growth and Accumulation REVIEW OF TECHNIQUE 3 Balancing Flows into and out of the Tub: Finding a Steady-State When more water flows into a bathtub than flows out of it, the level of water in the bathtub will rise. When more water flows out than in, that level will fall. Most importantly, when the flow of water in and out of this bathtub is exactly the same, the level of water will remain constant. This is all that a steady-state isa point at which the flows into and out of some “pool” balance. In the neoclassical growth model, each worker has an identical pool of capital. Investment is a flow into these pools. Depreciation is a flow out. Population growth, as it increases the number of pools without adding any more “water”, decreases amount available to each individual and can also be viewed as a flow out. Thus, at the steady-state, per capita savings = the flow out of each worker’s pool of capital caused by depreciation + the flow out of that pool caused by population growth. If we wanted to find the steady-state level of a population, we would find the point where the number of people being born was exactly equal to the number of people dying. There can, in some cases, be more than one steady-state, but that’s an issue for another day. CROSSWORD ACROSS 1 MPK assumed in neoclassical model 6 Type of variable, the growth rate of technology DOWN in this chapter is an example 2 Type of capital, includes knowledge 9 Kind growth theory studied in this chapter 3 _____ returns to scale 10 Growth rate of steady-state, per-capita output 4 If the growth rates of two countries become when TFP is fixed equal over time, they ________ 11 Type of output, its growth is modeled in this 5 Type of variable, capital is an example chapter 7 Affects the steady-state level of output, but not 13 Type of function, provides link between inputs its growth rate and outputs 8 Type of variable, investment is an example 14 Type of growth, reduces the steady-state 12 At the steady-state, flows into and out of the capital labor ratio capital stock are _________ 3-10 Chapter 03 – Growth and Accumulation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 FILL-IN QUESTIONS 1. ______________ helps us to determine how much of the growth in total output is the result of growth in different factors of production. 2. The ______________ provides a quantitative link between the inputs to production, and the output produced. 3. We call the amount by which output increases when 1 more unit of labor is used the ____________________. It ______________ as more labor is used in production. 4. When more output can be generated without using more inputs, it must be the case that ____________________ has increased. 5. Output per person (total output divided by population) is called ____________________. 6. We call the stock of machines and buildings used in production ______________ capital. 7. The stock of knowledge and skills is called ______________ capital. 8. The capital/labor ratio and output per person are constant at the ___________________. 3-11 Chapter 03 – Growth and Accumulation 9. If two countries have the same rate of population growth, and access to the same technologies, the rate at which their potential output grows will ______________ over time, and be identical at the steady state. 10. When they also have the same rate of savings, the ______________ of their potential output will be identical at this steady state as well. TRUE-FALSE QUESTIONS T F 1. An increase in the rate of population growth will change the rate at which per capita potential output grows in the steady state. T F 2. An increase in the rate of population growth will change the level of per capita potential output at the steady state. T F 3. An increase in the rate of population growth will change the rate at which total potential output grows in the steady state. T F 4. An increase in the savings rate will change the rate at which total potential output grows in the steady state. T F 5. An increase in the savings rate will immediately change the growth rate of total potential output. T F 6. After this, the growth rate will remain constant. T F 7. An increase in the rate of depreciation will change the levels of the capital/labor ratio and of per capita potential output at the steady state. T F 8. When we allow productivity to grow over time, the rate at which per capita potential output grows in the steady state is both positive and exogenous. T F 9. Two countries with the same rate of population growth, and with access to the same technologies will have the same level of output (and therefore income) at the steady state. T F 10. The growth rate of their potential output, at the steady state, will (also) be the same. MULTIPLE-CHOICE QUESTIONS 1. The rate of growth of potential output is the same as the b. growth rate of the economy a. business cycle 3-12 Chapter 03 – Growth and Accumulation c. trend path of GDP d. output gap 2. Which of the following can affect the growth rate of per capita potential output in the steady- state? c. rate of depreciation a. savings rate d. rate of productivity growth b. rate of population growth 3. Which of the following cannot affect the growth rate of total potential output in the steady- state? a. savings rate c. rate of depreciation b. rate of population growth d. (a) & (c) 4. Which of the following need to be equal, if two countries are to achieve the same growth rate of per-capita potential output at the steady state? c. rate of depreciation a. savings rate d. access to technology b. rate of population growth 5. Which of the following is a realistic estimate of capital’s share of income (the fraction of total output that is used to pay capital’s “wage”) in the U.S.? a. 0.25 c. 0.75 b. 0.5 d. 1 6. Which of the following would not increase labor’s productivity (measured as Y/N)? a. technological progress c. more natural resources b. an increase in the capital/labor ratio d. an increase in the population growth rate 7. Which of the following would not increase capital’s productivity (measured as Y/K)? a. technological progress c. more natural resources b. an increase in the capital/labor ratio d. an increase in the population growth rate 8. With the production function y = k¼, a 1% increase in the capital/labor ratio should increase per capita potential output by a. 0.25% c. 0.75% b. 0.5% d. 1% 9. Which of the following would not increase the stock of human capital? a. education c. more natural resources b. on the job training d. vaccinations 10. Human capital’s share of income in industrialized countries is roughly a. 0.1 c. 0.5 b. 0.33 d. 0.75 3-13 Chapter 03 – Growth and Accumulation CONCEPTUAL PROBLEMS 1. Can human capital depreciate? Does it have diminishing marginal returns? 2. Do natural resources depreciate? Do they have diminishing marginal returns? 3. Name two important assumptions at the foundation of the neoclassical model of growth. 4. Which of the following are stock variables? Which are flow variables? a) capital, b) per-capita GDP, c) depreciation, d) investment TECHNICAL PROBLEMS 1. Consider the following production function: Y = K1/4N3/4 a) What is capital’s share of income? b) Find an equation for the productivity of capital (Y/K). c) What is labor’s share of income? d) Find an equation for the productivity of labor (Y/N). e) Does this production function have constant returns to scale? (Translation: Do the exponents add to 1?) f) Write this production function in per capita terms. (Translation: Divide both sides by N.) 2. If in a fixed population the number of people in the labor force doubles, what will happen to a) the steady-state level of per capita potential output? b) the steady-state growth rate of per capita potential output? c) labor’s share of income? d) labor productivity? 3. If, instead, the number of people in the population doubles (you may assume that the number of people in the labor force doubles as well) what will happen to a) the steady-state level of per capita potential output? b) the steady-state growth rate of per capita potential output? c) labor’s share of income? d) labor productivity? 3-14 Chapter 03 – Growth and Accumulation 4. Now suppose that, in an economy initially at steady-state, there is an exogenous increase in the savings rate. Show how per capita output changes over time. 5. Use the growth accounting equation to answer the following question: If capital’s share of income is 25% and labor’s share of income is 75%, the stocks of both capital and labor increase by 50% (K/K = N/N = 0.5), and there is no technology growth, at what rate will potential output grow? Will the capital-labor ratio increase at all? 3-15

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