Answers to Textbook Questions and Problems CHAPTER 8 Economic Growth I: Capital Accumulation and Population Growth Questions for Review 1. In the Solow growth model, a high saving rate leads to a large steady-state capital stock and a high level of steady-state output. A low saving rate leads to a small steady-state capital stock and a low level of steady-state output. Higher saving leads to faster economic growth only in the short run. An increase in the saving rate raises growth until the economy reaches the new steady state. That is, if the economy maintains a high saving rate, it will also maintain a large capital stock and a high level of output, but it will not maintain a high rate of growth forever. In the steady state, the growth rate of output (or income) is independent of the saving rate. 2. It is reasonable to assume that the objective of an economic policymaker is to maximize the economic well-being of the individual members of society. Since economic well-being depends on the amount of consumption, the policymaker should choose the steady state with the highest level of consumption. The Golden Rule level of capital represents the level that maximizes consumption in the steady state. Suppose, for example, that there is no population growth or technological change. If the steadystate capital stock increases by one unit, then output increases by the marginal product of capital MPK; depreciation, however, increases by an amount δ, so that the net amount of extra output available for consumption is MPK – δ. The Golden Rule capital stock is the level at which MPK = δ, so that the marginal product of capital equals the depreciation rate. 3. When the economy begins above the Golden Rule level of capital, reaching the Golden Rule level leads to higher consumption at all points in time. Therefore, the policymaker would always want to choose the Golden Rule level because consumption is increased for all periods of time. On the other hand, when the economy begins below the Golden Rule level of capital, reaching the Golden Rule level means reducing consumption today to increase consumption in the future. In this case, the policymaker’s decision is not as clear. If the policymaker cares more about current generations than about future generations, he or she may decide not to pursue policies to reach the Golden Rule steady state. If the policymaker cares equally about all generations, then he or she chooses to reach the Golden Rule. Even though the current generation will have to consume less, an infinite number of future generations will benefit from increased consumption by moving to the Golden Rule. 4. The higher the population growth rate is, the lower the steady-state level of capital per worker, and therefore there is a lower level of steady-state income per worker. For example, Figure 8-1 shows the steady state for two levels of population growth, a low level n1 and a higher level n2. The higher population growth n2 means that the line representing population growth and depreciation is higher, so the steady-state level of capital per worker is lower. In a model with no technological change, the steady-state growth rate of total income is n: the higher the population growth rate n is, the higher the growth rate of total income. Income per worker, however, grows at rate zero in steady state and, thus, is not affected by population growth. Problems and Applications 1. a. A production function has constant returns to scale if increasing all factors of production by an equal percentage causes output to increase by the same percentage. Mathematically, a production function has constant returns to scale if zY = F(zK, zL) for any positive number z. That is, if we multiply both the amount of capital and the amount of labor by some amount z, then the amount of output is multiplied by z. For example, if we double the amounts of capital and labor we use (setting z = 2), then output also doubles. To see if the production function Y = F(K, L) = K1/3L2/3 has constant returns to scale, we write: F(zK, zL) = (zK)1/3(zL)2/3 = zK1/3L2/3 = zY. Therefore, the production function Y = K1/3L2/3 has constant returns to scale. b. To find the per-worker production function, divide the production function Y = K1/3L2/3 by L: Y K1/3L2/3 = L L If we define y = Y/L, we can rewrite the above expression as: y = K1/3/L1/3. Defining k = K/L, we can rewrite the above expression as: y = k1/3 c. We know the following facts about countries A and B: δ = depreciation rate = 0.20, sa = saving rate of country A = 0.1, sb = saving rate of country B = 0.3, and y = k1/3 is the per-worker production function derived in part (b) for countries A and B. The growth of the capital stock Δk equals the amount of investment sf(k), minus the amount of depreciation δk. That is, Δk = sf(k) – δk. In steady state, the capital stock does not grow, so we can write this as sf(k) = δk. To find the steady-state level of capital per worker, plug the per-worker production function into the steady-state investment condition, and solve for k*: sk1/3 = δk. Rewriting this: k2/3 = s/δ k = (s/δ)3/2. To find the steady-state level of capital per worker k*, plug the saving rate for each country into the above formula: Country A: k = (sa/δ)3/2 = (0.1/0.2)3/2 = 0.35. Country B: k = (sb/δ)3/2 = (0.3/0.2)3/2 = 1.84. Now that we have found k* for each country, we can calculate the steady-state levels of income per worker for countries A and B because we know that y = k1/3: y*a = (0.35)1/3 = 0.71. y*b = (1.84)1/3 = 1.22. We know that out of each dollar of income, workers save a fraction s and consume a fraction (1 – s). That is, the consumption function is c = (1 – s)y. Since we know the steady-state levels of income in the two countries, we find Country A: c = (1 – sa)y = (1 – 0.1)(0.71) = 0.64. Country B: c = (1 – sb)y = (1 – 0.3)(1.224) = 0.86. d. If capital per worker is equal to 1 in both countries, we find the following values for income per worker and consumption per worker in each country: Country A: y = 1 and c = 0.9 Country B: y = 1 and c = 0.7. e. Using the following facts and equations, we calculate income per worker y, consumption per worker c, and capital per worker k: sa = 0.1. sb = 0.3. δ = 0.2. ko = 1 for both countries. y = k1/3. c = (1 – s)y. Country A Year k y = k1/3 c = (1 – sa)y i = say δk Δk = i – δk 1 1.00 1.00 0.90 0.10 0.20 −0.10 2 0.90 0.97 0.87 0.10 0.18 −0.08 3 0.82 0.93 0.84 0.09 0.16 −0.07 4 0.75 0.91 0.82 0.09 0.15 −0.06 5 0.69 0.88 0.79 0.09 0.14 −0.05 6 0.64 0.86 0.78 0.09 0.13 −0.04 7 0.60 0.84 0.76 0.08 0.12 −0.04 Country B Year k y = k1/3 c = (1 – sa)y i = say δk Δk = i – δk 1 1.00 1.00 0.70 0.30 0.20 0.10 2 1.10 1.03 0.72 0.31 0.22 0.09 3 1.19 1.06 0.74 0.32 0.24 0.08 4 1.27 1.08 0.76 0.32 0.25 0.07 5 1.34 1.10 0.77 0.33 0.27 0.06 6 1.40 1.12 0.78 0.34 0.28 0.06 7 1.46 1.13 0.79 0.34 0.29 0.05 Note that it will take seven years before consumption in country B is higher than consumption in country A. 2. a. The production function in the Solow growth model is Y = F(K, L), or expressed in terms of output per worker, y = f(k). If a war reduces the labor force through casualties, then L falls but k = K/L rises. The production function tells us that total output falls because there are fewer workers. Output per worker increases, however, since each worker has more capital. b. The reduction in the labor force means that the capital stock per worker is higher after the war. Therefore, if the economy were in a steady state prior to the war, then after the war the economy has a capital stock that is higher than the steady-state level. This is shown in Figure 8-2 as an increase in capital per worker from k* to k1. As the economy returns to the steady state, the capital stock per worker falls from k1 back to k*, so output per worker also falls. Hence, in the transition to the new steady state, the growth of output per worker is slower than normal. In the steady state, we know that the growth rate of output per worker is equal to zero, given there is no technological change in this model. Therefore, in this case, the growth rate of output per worker must be less than zero until the new steady state is reached. 3. a. We follow Section 8-1, “Approaching the Steady State: A Numerical Example.” The production function is Y = K0.4L0.6. To derive the per-worker production function f(k), divide both sides of the production function by the labor force L: Y K 0.4L0.6 = L L Rearrange to obtain: 0.4 Y K = . L Ł L ł Because y = Y/L and k = K/L, this becomes: y = k0.4. b. Recall that Δk = sf(k) – δk. The steady-state value of capital per worker k* is defined as the value of k at which capital per worker is constant, so Δk = 0. It follows that in steady state 0 = sf(k) – δk, or, equivalently, k * s = . f (k*) d For the production function in this problem, it follows that: k * s = . (k*)0.4 d Rearranging: (k*)0.6 = s d or 1/0.6 s k*= . Łdł Substituting this equation for steady-state capital per worker into the per-worker production function from part (a) gives: 0.4/0.6 s y*= . Łdł Consumption is the amount of output that is not invested. Since investment in the steady state equals δk*, it follows that 0.4/0.6 1/0.6 s s c*= f (k*)-dk*= -d . Łdł Łdł (Note: An alternative approach to the problem is to note that consumption also equals the amount of output that is not saved: 0.4/0.6 c*=(1-s) f (k*)=(1-s)(k*)0.4 =(1-s) s . Łdł Some algebraic manipulation shows that this equation is equal to the equation above.) c. The table below shows k*, y*, and c* for the saving rate in the left column, using the equations from part (b). We assume a depreciation rate of 15 percent (i.e., 0.1). (The last column shows the marginal product of capital, derived in part (d) below). s k* y* c* MPK - k* 0 0 0 0 0 0.10 0.51 0.76 0.69 0.52 0.20 1.62 1.21 0.97 0.06 0.21 1.75 1.25 0.99 0.02 0.22 1.89 1.29 1.01 −0.01 0.25 2.34 1.41 1.05 −0.11 0.30 3.17 1.59 1.11 −0.28 0.40 5.13 1.92 1.15 −0.62 0.50 7.44 2.23 1.12 −1.00 0.60 10.08 2.52 1.01 −1.41 0.70 13.03 2.79 0.84 −1.87 0.80 16.28 3.05 0.61 −2.37 0.90 19.81 3.30 0.33 −2.91 1.00 23.61 3.54 0.00 −3.48 Note that a saving rate of 100 percent (s = 1.0) maximizes output per worker. In that case, of course, nothing is ever consumed, so c* = 0. Consumption per worker is maximized at a rate of saving of somewhere between 21 and 22 percent—that is, where s equals capital’s share in output. This is the Golden Rule level of s. d. The marginal product of capital (MPK) is the change in output per worker (y) for a given change in capital per worker (k). To find the marginal product of capital, differentiate the per-worker production function with respect to capital per worker (k): MPK = 0.4k-0.6 = k0.40.6 . To find the marginal product of capital net of depreciation, use the equation above to calculate the marginal product of capital and then subtract depreciation, which is 15 percent of the value of the steady-state level of capital per worker. These values appear in the table above. Note that when consumption per worker is maximized, the value of the marginal product of capital net of depreciation is zero. 4. Suppose the economy begins with an initial steady-state capital stock below the Golden Rule level. The immediate effect of devoting a larger share of national output to investment is that the economy devotes a smaller share to consumption; that is, “living standards” as measured by consumption fall. The higher investment rate means that the capital stock increases more quickly, so the growth rates of output and output per worker rise. The productivity of workers is the average amount produced by each worker—that is, output per worker. So productivity growth rises. Hence, the immediate effect is that living standards fall but productivity growth rises. In the new steady state, output grows at rate n, while output per worker grows at rate zero. This means that in the steady state, productivity growth is independent of the rate of investment. Since we begin with an initial steady-state capital stock below the Golden Rule level, the higher investment rate means that the new steady state has a higher level of consumption, so living standards are higher. Thus, an increase in the investment rate increases the productivity growth rate in the short run but has no effect in the long run. Living standards, on the other hand, fall immediately and only rise over time. That is, the quotation emphasizes growth, but not the sacrifice required to achieve it. 5. a. An increase in the saving rate will shift the saving curve upwards, as illustrated in Figure 8-3. Since actual investment is now greater than breakeven investment, the level of capital per worker will increase and the steady-state level of capital per worker will be higher. The increase in capital per worker will increase output per worker. b. An increase in the depreciation rate will shift the break-even investment line upwards to (δ2 + n) as illustrated in Figure 8-4. Since actual investment is now less than break-even investment, the level of capital per worker will decrease and the steady-state level of capital per worker will be lower. The decrease in capital per worker will decrease output per worker. c. A reduction in the rate of population growth will shift the break-even investment line down and to the right to (δ + n2) as illustrated in Figure 8-5. Since actual investment is now greater than breakeven investment, the level of capital per worker will increase and the steady-state level of capital per worker will be higher. The increase in capital per worker will increase output per worker. d. The technological improvement increases output f(k), and as a result the saving curve shifts upwards as illustrated in Figure 8-6. Since actual investment is now greater than break-even investment, the level of capital per worker will increase and the steady-state level of capital per worker will be higher. The increase in capital per worker will increase output per worker. 6. First, consider steady states. In Figure 8-7, the slower population growth rate shifts the line representing population growth and depreciation downward. The new steady state has a higher level of capital per worker, k*2, and hence a higher level of output per worker. What about steady-state growth rates? In steady state, total output grows at rate n, whereas output perworker grows at rate 0. Hence, slower population growth will lower total output growth, but perworker output growth will be the same. Now consider the transition. We know that the steady-state level of output per worker is higher with low population growth. Hence, during the transition to the new steady state, output per worker must grow at a rate faster than 0 for a while. In the decades after the fall in population growth, growth in total output will transition to its new lower level while growth in output per worker will jump up but then transition back to zero. 7. If there are decreasing returns to labor and capital, then increasing both capital and labor by the same proportion increases output by less than this proportion. For example, if we double the amounts of capital and labor, then output increases by less than double. This may happen if there is a fixed factor such as land in the production function, and it becomes scarce as the economy grows larger. Then population growth will increase total output but decrease output per worker, since each worker has less of the fixed factor to work with. If there are increasing returns to scale, then doubling inputs of capital and labor more than doubles output. This may happen if specialization of labor becomes greater as population grows. Then population growth increases total output and also increases output per worker, since the economy is able to take advantage of the scale economy more quickly. 8. a. To find output per worker y we divide total output by the number of workers: Y Kaغ (1-u)Løß1-a = L L K a 1-a y= Lł (1-u) Ł y=ka(1-u)1-a K where the final step uses the definition k = . Notice that unemployment reduces the amount of L output per worker for any given capital–labor ratio because some of the workers are not producing anything. The steady state is the level of capital per worker at which the increase in capital per worker from investment equals its decrease from depreciation and population growth: sy = ( + n)k sk (1 – u)1– = ( + n)k 1 s 1-a k*=(1-u) . Łd+nł Finally, to get steady-state output per worker, plug the steady-state level of capital per worker into the production function: 1 a y*= (1-u*) s 1-a (1-u*)1-a Łd+nł Ł ł =(1-u*) Łd+nł Unemployment lowers steady-state output for two reasons: for a given k, unemployment lowers y, and unemployment also lowers the steady-state value k*. b. The steady state can be graphically illustrated using the equations that describe the steady state from part (a) above. Unemployment lowers the marginal product of capital per worker and, hence, acts like a negative technological shock that reduces the amount of capital the economy can maintain in steady state. Figure 8-8 shows this graphically: an increase in unemployment lowers the sf(k) line and the steady-state level of capital per worker. c. Figure 8-9 shows the pattern of output over time. As soon as unemployment falls from u1 to u2, output jumps up from its initial steady-state value of y*(u1). The economy has the same amount of capital (since it takes time to adjust the capital stock), but this capital is combined with more workers. At that moment the economy is out of steady state: it has less capital than it wants to match the increased number of workers in the economy. The economy begins its transition by accumulating more capital, raising output even further than the original jump. Eventually the capital stock and output converge to their new, higher steady-state levels. IN THIS CHAPTER, YOU WILL LEARN: ▪ the closed economy Solow model ▪ how a country’s standard of living depends on its saving and population growth rates ▪ how to use the “Golden Rule” to find the optimal saving rate and capital stock 1 Why growth matters ▪ Data on infant mortality rates: ▪ 20% in the poorest 1/5 of all countries ▪ 0.4% in the richest 1/5 ▪ In Pakistan, 85% of people live on less than $2/day. ▪ One-fourth of the poorest countries have had famines during the past 3 decades. ▪ Poverty is associated with oppression of women and minorities. Economic growth raises living standards and reduces poverty…. links to prepared graphs @ Gapminder.org notes: circle size is proportional to population size, color of circle indicates continent, press “play” on bottom to see the cross section graph evolve over time Income per capita and ▪ Life expectancy ▪ Infant mortality ▪ Malaria deaths per 100,000 ▪ Cell phone users per 100 people Why growth matters ▪Anything that effects the long-run rate of economic growth – even by a tiny amount – will have huge effects on living standards in the long run. annual growth rate of income per capita increase in standard of living after… …25 years …50 years …100 years 2.0% 64.0% 169.2% 624.5% 2.5% 85.4% 243.7% 1,081.4% Why growth matters ▪If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher from 2000–2010, the average person would have earned $2,782 more during the decade. The lessons of growth theory …can make a positive difference in the lives of hundreds of millions of people. These lessons help us ▪ understand why poor countries are poor ▪ design policies that can help them grow ▪ learn how our own growth rate is affected by shocks and our government’s policies 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 How Solow model is different from Chapter 3’s model 1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink 2. L is no longer fixed: population growth causes it to grow 3. the consumption function is simpler How Solow model is different from Chapter 3’s model 4. no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. cosmetic differences The production function ▪ In aggregate terms: Y = F (K, L) ▪ Define: y = Y/L = output per worker k = K/L = capital per worker ▪ Assume constant returns to scale: zY = F (zK, zL ) for any z > 0 ▪ Pick z = 1/L. Then Y/L = F (K/L, 1) y = F (k, 1) y = f(k) where f(k) = F(k, 1) The production function Output per worker, y The national income identity ▪ Y = C + I (remember, no G ) ▪ In “per worker” terms: y = c + i where c = C/L and i = I /L The consumption function ▪ s = the saving rate, the fraction of income that is saved (s is an exogenous parameter) Note: s is the only lowercase variable that is not equal to its uppercase version divided by L ▪ Consumption function: c = (1–s)y (per worker) Saving and investment ▪ saving (per worker) = y – c = y – (1–s)y = sy ▪ National income identity is y = c + i Rearrange to get: i = y – c = sy (investment = saving, like in chap. 3!) ▪ Using the results above, i = sy = sf(k) Output, consumption, and investment Output per Capital per worker, k Depreciation Depreciation δ = the rate of depreciation per worker, δk = the fraction of the capital stock Capital per worker, k Capital accumulation The basic idea: Investment increases the capital stock, depreciation reduces it. Change in capital stock = investment – depreciation Δk = i – δk Since i = sf(k) , this becomes: Δk = s f(k) – δk The equation of motion for k Δk = s f(k) – δk ▪ The Solow model’s central equation ▪ Determines behavior of capital over time… ▪ …which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., income per person: y = f(k) consumption per person: c = (1 – s) f(k) The steady state Δk = s f(k) – δk If investment is just enough to cover depreciation [sf(k) = δk ], then capital per worker will remain constant: Δk = 0. This occurs at one value of k, denoted k*, called the steady state capital stock. The steady state Investment Δk = sf(k) − δk Δk = sf(k) − δk Investment Δk = sf(k) − δk Δk = sf(k) − δk Investment Δk = sf(k) − δk A numerical example Production function (aggregate): Y FKL K L K L= ( , ) = = 1/2 1/2 To derive the per-worker production function, divide through by L: Y K L K1/2 1/2 1/2 L L= =L Then substitute y = Y/L and k = K/L to get y fk k= ( )= 1/2 A numerical example, cont. Assume: ▪ s = 0.3 ▪ δ = 0.1 ▪ initial value of k = 4.0 Approaching the steady state: A numerical example Year k y c i δk Δk 1 4.000 2.000 1.400 0.600 0.400 0.200 2 4.200 2.049 1.435 0.615 0.420 0.195 3 4.395 2.096 1.467 0.629 0.440 0.189 4 4.584 2.141 1.499 0.642 0.458 0.184 … 10 5.602 2.367 1.657 0.710 0.560 0.150 … 25 7.351 2.706 1.894 0.812 0.732 0.080 … 100 8.962 2.994 2.096 0.898 0.896 0.002 … ∞ 9.000 3.000 2.100 0.900 0.900 0.000 An increase in the saving rate An increase in the saving rate raises investment… …causing k to grow toward a new steady state: Investment δk CHAPTER 8 Economic Growth I Prediction: ▪ The Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run. ▪ Are the data consistent with this prediction? The Golden Rule: Introduction ▪ Different values of s lead to different steady states. How do we know which is the “best” steady state? ▪ The “best” steady state has the highest possible consumption per person: c* = (1–s) f(k*). ▪ An increase in s ▪ leads to higher k* and y*, which raises c* ▪ reduces consumption’s share of income (1–s), which lowers c*. ▪ So, how do we find the s and k* that maximize c*? kgold* = the Golden Rule level of capital, the steady state value of k that maximizes consumption. To find it, first express c* in terms of k*: c* = y* − i* = f (k*) − i* = f (k*) − δk* In the steady state: i* = δk* because Δk = 0. worker, k* worker, k* The transition to the Golden Rule steady state ▪ The economy does NOT have a tendency to move toward the Golden Rule steady state. ▪ Achieving the Golden Rule requires that policymakers adjust s. ▪ This adjustment leads to a new steady state with higher consumption. ▪ But what happens to consumption during the transition to the Golden Rule? Starting with too much capital Starting with too little capital Population growth ▪ Assume the population and labor force grow at rate n (exogenous): L = n L ▪ EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02). ▪ Then ΔL = n L = 0.02 ×1,000 = 20, so L = 1,020 in year 2. Break-even investment ▪ (δ + n)k = break-even investment, the amount of investment necessary to keep k constant. ▪ Break-even investment includes: ▪δ 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 is spread more thinly over a larger population of workers.) The equation of motion for k ▪With population growth, the equation of motion for k is: Δk = s f(k) − (δ + n) k The Solow model diagram The impact of population growth 1 Prediction: ▪ 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. ▪ Are the data consistent with this prediction? The Golden Rule with population growth To find the Golden Rule capital stock, express c* in terms of k*: c* = y* − i* = f (k* ) − (δ + n) k* c* is maximized when MPK = δ + n or equivalently, MPK − δ = n In the Golden Rule steady state, the marginal product of capital net of depreciation equals the population growth rate. Alternative perspectives on population growth The Malthusian Model (1798) ▪ Predicts population growth will outstrip the Earth’s ability to produce food, leading to the impoverishment of humanity. ▪ Since Malthus, world population has increased sixfold, yet living standards are higher than ever. ▪ Malthus neglected the effects of technological progress. Alternative perspectives on population growth The Kremerian Model (1993) ▪ Posits that population growth contributes to economic growth. ▪ More people = more geniuses, scientists & engineers, so faster technological progress. ▪ Evidence, from very long historical periods: ▪ As world pop. growth rate increased, so did rate of growth in living standards ▪ Historically, regions with larger populations have enjoyed faster growth. C H A P T E R S U M M A R Y 1. The Solow growth model shows that, in the long run, a country’s standard of living depends: ▪ positively on its saving rate ▪ negatively on its population growth rate 2. An increase in the saving rate leads to: ▪ higher output in the long run ▪ faster growth temporarily ▪ but not faster steady-state growth C H A P T E R S U M M A R Y 3. If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off. If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation. Solution Manual for Macroeconomics Gregory N. Mankiw 9781464182891, 9781319106058
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