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Answers to Textbook Questions and Problems CHAPTER 3 National Income: Where It Comes From and Where It Goes Questions for Review 1. The factors of production and the production technology determine the amount of output an economy can produce. The factors of production are the inputs used to produce goods and services: the most important factors are capital and labor. The production technology determines how much output can be produced from any given amounts of these inputs. An increase in one of the factors of production or an improvement in technology leads to an increase in the economy’s output. 2. When a firm decides how much of a factor of production to hire or demand, it considers how this decision affects profits. For example, hiring an extra unit of labor increases output and therefore increases revenue; the firm compares this additional revenue to the additional cost from the higher wage bill. The additional revenue the firm receives depends on the marginal product of labor (MPL) and the price of the good produced (P). An additional unit of labor produces MPL units of additional output, which sells for P dollars per unit. Therefore, the additional revenue to the firm is P  MPL. The cost of hiring the additional unit of labor is the wage W. Thus, this hiring decision has the following effect on profits: ΔProfit = ΔRevenue – ΔCost = (P  MPL) – W. If the additional revenue, P  MPL, exceeds the cost (W) of hiring the additional unit of labor, then profit increases. The firm will hire labor until it is no longer profitable to do so—that is, until the MPL falls to the point where the change in profit is zero. In the equation above, the firm hires labor until ΔProfit = 0, which is when (P  MPL) = W. This condition can be rewritten as: MPL = W/P. Therefore, a competitive profit-maximizing firm hires labor until the marginal product of labor equals the real wage. The same logic applies to the firm’s decision regarding how much capital to hire: the firm will hire capital until the marginal product of capital equals the real rental price. 3. A production function has constant returns to scale if an equal percentage increase in all factors of production causes an increase in output of the same percentage. For example, if a firm increases its use of capital and labor by 50 percent, and output increases by 50 percent, then the production function has constant returns to scale. If the production function has constant returns to scale, then total income (or equivalently, total output) in an economy of competitive profit-maximizing firms is divided between the return to labor, MPL  L, and the return to capital, MPK  K. That is, under constant returns to scale, economic profit is zero. 4. A Cobb–Douglas production function has the form F(K,L) = AKαL1–α. The text showed that the parameter α gives capital’s share of income. So if capital earns one-fourth of total income, then = 0.25. Hence, F(K,L) = AK0.25L0.75. 5. Consumption depends positively on disposable income—i.e. the amount of income after all taxes have been paid. Higher disposable income means higher consumption. The quantity of investment goods demanded depends negatively on the real interest rate. For an investment to be profitable, its return must be greater than its cost. Because the real interest rate measures the cost of funds, a higher real interest rate makes it more costly to invest, so the demand for investment goods falls. 6. Government purchases are a measure of the value of goods and services purchased directly by the government. For example, the government buys missiles and tanks, builds roads, and provides services such as air traffic control. All of these activities are part of GDP. Transfer payments are government payments to individuals that are not in exchange for goods or services. They are the opposite of taxes: taxes reduce household disposable income, whereas transfer payments increase it. Examples of transfer payments include Social Security payments to the elderly, unemployment insurance, and veterans’ benefits. 7. Consumption, investment, and government purchases determine demand for the economy’s output, whereas the factors of production and the production function determine the supply of output. The real interest rate adjusts to ensure that the demand for the economy’s goods equals the supply. At the equilibrium interest rate, the demand for goods and services equals the supply. 8. When the government increases taxes, disposable income falls, and therefore consumption falls as well. The decrease in consumption equals the amount that taxes increase multiplied by the marginal propensity to consume (MPC). The higher the MPC is, the greater is the negative effect of the tax increase on consumption. Because output is fixed by the factors of production and the production technology, and government purchases have not changed, the decrease in consumption must be offset by an increase in investment. For investment to rise, the real interest rate must fall. Therefore, a tax increase leads to a decrease in consumption, an increase in investment, and a fall in the real interest rate. Problems and Applications 1. a. According to the neoclassical theory of distribution, the real wage equals the marginal product of labor. Because of diminishing returns to labor, an increase in the labor force causes the marginal product of labor to fall. Hence, the real wage falls. Given a Cobb–Douglas production function, the increase in the labor force will increase the marginal product of capital and will increase the real rental price of capital. With more workers, the capital will be used more intensively and will be more productive. b. The real rental price equals the marginal product of capital. If an earthquake destroys some of the capital stock (yet miraculously does not kill anyone and lower the labor force), the marginal product of capital rises and, hence, the real rental price rises. Given a Cobb–Douglas production function, the decrease in the capital stock will decrease the marginal product of labor and will decrease the real wage. With less capital, each worker becomes less productive. c. If a technological advance improves the production function, this is likely to increase the marginal products of both capital and labor. Hence, the real wage and the real rental price both increase. d. High inflation that doubles the nominal wage and the price level will have no impact on the real wage. Similarly, high inflation that doubles the nominal rental price of capital and the price level will have no impact on the real rental price of capital. 2. a. To find the amount of output produced, substitute the given values for labor and land into the production function: Y = 1000.51000.5 = 100. b. According to the text, the formulas for the marginal product of labor and the marginal product of capital (land) are: MPL = (1 – α)AKαL–α. MPK = αAKα–1L1–α. In this problem, α is 0.5 and A is 1. Substitute in the given values for labor and land to find the marginal product of labor is 0.5 and marginal product of capital (land) is 0.5. We know that the real wage equals the marginal product of labor and the real rental price of land equals the marginal product of capital (land). c. Labor’s share of the output is given by the marginal product of labor times the quantity of labor, or 50. d. The new level of output is 70.71. e. The new wage is 0.71. The new rental price of land is 0.35. f. Labor now receives 35.36. 3. A production function has decreasing returns to scale if an equal percentage increase in all factors of production leads to a smaller percentage increase in output. For example, if we double the amounts of capital and labor output increases by less than double, then the production function has decreasing returns to scale. This may happen if there is a fixed factor such as land in the production function, and this fixed factor becomes scarce as the economy grows larger. A production function has increasing returns to scale if an equal percentage increase in all factors of production leads to a larger percentage increase in output. For example, if doubling the amount of capital and labor increases the output by more than double, then the production function has increasing returns to scale. This may happen if specialization of labor becomes greater as the population grows. For example, if only one worker builds a car, then it takes him a long time because he has to learn many different skills, and he must constantly change tasks and tools. But if many workers build a car, then each one can specialize in a particular task and become more productive. 4. a. A Cobb–Douglas production function has the form Y = AKαL1–α. The text showed that the marginal products for the Cobb–Douglas production function are: MPL = (1 – α)Y/L. MPK = αY/K. Competitive profit-maximizing firms hire labor until its marginal product equals the real wage, and hire capital until its marginal product equals the real rental rate. Using these facts and the above marginal products for the Cobb–Douglas production function, we find: W/P = MPL = (1 – α)Y/L. R/P = MPK = αY/K. Rewriting this: (W/P)L = MPL  L = (1 – α)Y. (R/P)K = MPK  K = αY. Note that the terms (W/P)L and (R/P)K are the wage bill and total return to capital, respectively. Given that the value of α = 0.3, then the above formulas indicate that labor receives 70 percent of total output (or income) and capital receives 30 percent of total output (or income). b. To determine what happens to total output when the labor force increases by 10 percent, consider the formula for the Cobb–Douglas production function: Y = AKαL1–α. Let Y1 equal the initial value of output and Y2 equal final output. We know that α = 0.3. We also know that labor L increases by 10 percent: Y1 = AK0.3L0.7. Y2 = AK0.3(1.1L)0.7. Note that we multiplied L by 1.1 to reflect the 10-percent increase in the labor force. To calculate the percentage change in output, divide Y2 by Y1: YY2 = AKAK0.3 (0.31.1L0.7L)0.7 1 =(1.1)0.7 =1.069. That is, output increases by 6.9 percent. To determine how the increase in the labor force affects the rental price of capital, consider the formula for the real rental price of capital R/P: R/P = MPK = αAKα–1L1–α. We know that α = 0.3. We also know that labor (L) increases by 10 percent. Let (R/P)1 equal the initial value of the rental price of capital, and let (R/P)2 equal the final rental price of capital after the labor force increases by 10 percent. To find (R/P)2, multiply L by 1.1 to reflect the 10-percent increase in the labor force: (R/P)1 = 0.3AK–0.7L0.7. (R/P)2 = 0.3AK–0.7(1.1L)0.7. The rental price increases by the ratio ((RR// PP))21 = 0.30.3AKAK-0.7-(0.71.1L0.7L)0.7 0.7 =(1.1) =1.069 So the rental price increases by 6.9 percent. To determine how the increase in the labor force affects the real wage, consider the formula for the real wage W/P: W/P = MPL = (1 – α)AKαL–α. We know that α = 0.3. We also know that labor (L) increases by 10 percent. Let (W/P)1 equal the initial value of the real wage, and let (W/P)2 equal the final value of the real wage. To find (W/P)2, multiply L by 1.1 to reflect the 10-percent increase in the labor force: (W/P)1 = (1 – 0.3)AK0.3L–0.3. (W/P)2 = (1 – 0.3)AK0.3(1.1L)–0.3. To calculate the percentage change in the real wage, divide (W/P)2 by (W/P)1: ((WW // PP))21 = (1-(10.3-0.3) AK) AK0.3 (0.31.1L-L0.3)-0.3 -0.3 =(1.1) = 0.972 That is, the real wage falls by 2.8 percent. c. We can use the same logic as in part (b) to set Y1 = AK0.3L0.7. Y2 = A(1.1K)0.3L0.7. Therefore, we have: Y A(1.1K)0.3 L0.7 Y2 = AK 0.3L0.7 1 0.3 =(1.1) =1.029 This equation shows that output increases by about 3 percent. Notice that α zY1 ▪ If decreasing returns to scale, Y2 0 Returns to scale: Example 2 FKL( , ) = K L2 + 2 F zK zL( , ) = (zK)2 +(zL)2 = z K2( 2 +L2) 2 increasing returns = z F K L( , ) to scale for any z > 1 Assumptions 1. Technology is fixed. 2. The economy’s supplies of capital and labor are fixed at: K K= and L L= Determining GDP Output is determined by the fixed factor supplies and the fixed state of technology: Y FK L= ( , ) The distribution of national income ▪ determined by factor prices, the prices per unit firms pay for the factors of production ▪ wage = price of L ▪ rental rate = price of K Notation How factor prices are determined ▪ Factor prices are determined by supply and demand in factor markets. ▪ Recall: Supply of each factor is fixed. ▪ What about demand? Demand for labor ▪ Assume markets are competitive: each firm takes W, R, and P as given. ▪ Basic idea: A firm hires each unit of labor if the cost does not exceed the benefit. ▪ cost = real wage ▪ benefit = marginal product of labor Marginal product of labor (MPL ) ▪Definition: The extra output the firm can produce using an additional unit of labor (holding other inputs fixed): MPL = F (K, L +1) – F (K, L) MPL and the production function L labor CHAPTER 3 National Income Diminishing marginal returns ▪ As one input is increased (holding other inputs constant), its marginal product falls. ▪ Intuition: If L increases while holding K fixed machines per worker falls, worker productivity falls. CHAPTER 3 National Income ANSWERS MPL and labor demand L Y MPL 0 0 n.a. If L = 3, should firm hire more or less 1 10 10 labor? 2 19 9 Answer: MORE, because the benefit 3 27 8 of the 4th worker (MPL = 7) exceeds 4 34 7 its cost (W/P = 6) 5 40 6 If L = 7, should firm hire more or less 6 45 5 labor? 7 49 4 Answer: LESS, because the 7th 8 52 3 9 54 2 worker adds MPL = 4 units of output but costs the firm W/P = 6. 10 55 1 MPL and the demand for labor The equilibrium real wage Determining the rental rate ▪ We have just seen that MPL = W/P. ▪ The same logic shows that MPK = R/P: ▪ Diminishing returns to capital: MPK falls as K rises ▪ The MPK curve is the firm’s demand curve for renting capital. ▪ Firms maximize profits by choosing K such that MPK = R/P. The equilibrium real rental rate The neoclassical theory of distribution ▪ States that each factor input is paid its marginal product ▪ A good starting point for thinking about income distributio How income is distributed to L and K W Total labor income =L =MPL L P R Total capital income = K =MPK K P If production function has constant returns to scale, then Y = MPL L MPK K +  labor income nationalcapital incomeincome The Cobb-Douglas production function ▪ The Cobb-Douglas production function has constant factor shares: α = capital’s share of total income: capital income = MPK × K = αY labor income = MPL × L = (1 – α )Y ▪ The Cobb-Douglas production function is: Y AK L=  1− where A represents the level of technology. The Cobb-Douglas production function ▪Each factor’s marginal product is proportional to its average product: MPK=AK L−1 1−=  Y K MPL= −(1 )AK L − = (1−)Y L CHAPTER 3 National Income 34 Labor productivity and wages ▪Theory: wages depend on labor productivity ▪U.S. data: period productivity growth real wage growth 1960-2013 2.1% 1.8% 1960-1973 2.9% 2.7% 1973-1995 1.5% 1.2% 1995-2013 2.3% 2.0% Explanations for rising inequality 1. Rise in capital’s share of income, since capital income is more concentrated than labor income 2. From The Race Between Education and Technology by Goldin & Katz ▪ Technological progress has increased the demand for skilled relative to unskilled workers. ▪ Due to a slowdown in expansion of education, the supply of skilled workers has not kept up. ▪ Result: Rising gap between wages of skilled and unskilled workers. Outline of model A closed economy, market-clearing model Supply side DONE ✓❑factor markets (supply, demand, price) DONE ✓❑determination of output/income Demand side Next ➔❑determinants of C, I, and G Equilibrium ❑goods market ❑loanable funds market Demand for goods and services Components of aggregate demand: C = consumer demand for g&s I = demand for investment goods G = government demand for g&s (closed economy: no NX ) Consumption, C ▪ Disposable income is total income minus total taxes: Y – T. ▪ Consumption function: C = C (Y – T ) ▪ Definition: Marginal propensity to consume (MPC) is the change in C when disposable income increases by one dollar. The consumption function Investment, I ▪ The investment function is I = I (r ) where r denotes the real interest rate, the nominal interest rate corrected for inflation. ▪ The real interest rate is: ▪ the cost of borrowing ▪ the opportunity cost of using one’s own funds to finance investment spending So, I depends negatively on r The investment function Government spending, G ▪ G = govt spending on goods and services ▪ G excludes transfer payments (e.g., Social Security benefits, unemployment insurance benefits) ▪ Assume government spending and total taxes are exogenous: G G= and T T= The market for goods & services ▪ Aggregate demand: CY T Ir G( − +) ( )+ ▪ Aggregate supply: Y FKL= ( , ) ▪ Equilibrium: Y CY T Ir G = ( − +) ( )+ The real interest rate adjusts to equate demand with supply. The loanable funds market ▪ A simple supply–demand model of the financial system. ▪ One asset: “loanable funds” ▪ demand for funds: investment ▪ supply of funds: saving ▪ “price” of funds: real interest rate Demand for funds: investment The demand for loanable funds . . . ▪ comes from investment: Firms borrow to finance spending on plant & equipment, new office buildings, etc. Consumers borrow to buy new houses. ▪ depends negatively on r, the “price” of loanable funds (cost of borrowing). Loanable funds demand curve Supply of funds: saving ▪ The supply of loanable funds comes from saving: ▪ Households use their saving to make bank deposits, purchase bonds and other assets. These funds become available to firms to borrow and finance investment spending. ▪ The government may also contribute to saving if it does not spend all the tax revenue it receives. Types of saving Private saving = (Y – T ) – C Public saving = T – G National saving, S = private saving + public saving = (Y –T ) – C + T – G = Y – C – G Notation: Δ = change in a variable ▪For any variable X, ΔX = “change in X ” Δ is the Greek (uppercase) letter Delta Examples: ▪If ΔL = 1 and ΔK = 0, then ΔY = MPL. More generally, if ΔK = 0, then MPL= YL . ▪Δ(Y − T ) = ΔY − ΔT , so ΔC = MPC × (ΔY − ΔT ) = MPC ΔY − MPC ΔT Budget surpluses and deficits ▪ If T > G, budget surplus = (T – G ) = public saving. ▪ If T 0 ▪ big tax cuts: ΔT < 0 ▪ Both policies reduce national saving: S Y CY T G= − ( − −)  G S   T C S CASE STUDY: The Reagan Deficits Are the data consistent with these results? 1970s 1980s T – G –2.2 –3.9 S 19.6 17.4 r 1.1 6.3 I 19.9 19.4 T–G, S, and I are expressed as a percent of GDP All figures are averages over the decade shown. Mastering the loanable funds model (continued) Things that shift the investment curve: ▪some technological innovations ▪to take advantage of some innovations, firms must buy new investment goods ▪tax laws that affect investment ▪e.g., investment tax credit An increase in investment demand Saving and the interest rate ▪ Why might saving depend on r ? ▪ How would the results of an increase in investment demand be different? ▪ Would r rise as much? ▪ Would the equilibrium value of I change? An increase in investment demand when saving depends on r An increase in investment demand raises r, which induces an increase in the quantity of saving, which allows I to increase. ▪Total output is determined by: ▪the economy’s quantities of capital and labor ▪the level of technology ▪ Competitive firms hire each factor until its marginal product equals its price. ▪ If the production function has constant returns to scale, then labor income plus capital income equals total income (output). ▪ A closed economy’s output is used for consumption, investment, and government spending. ▪ The real interest rate adjusts to equate the demand for and supply of: ▪ goods and services. ▪ loanable funds. ▪ A decrease in national saving causes the interest rate to rise and investment to fall. ▪ An increase in investment demand causes the interest rate to rise but does not affect the equilibrium level of investment if the supply of loanable funds is fixed. Solution Manual for Macroeconomics Gregory N. Mankiw 9781464182891, 9781319106058

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