Chapter 3 National Income: Where It Comes From and Where It Goes a. Private saving is the amount of disposable income y-T, that is not consumed: =5,000-1,000-(250+0755,000-1,000) 750. Public saving is the amount of taxes the government has left over after it kes its purchases Public=T-G 1,000-1,000 is the sum of private saving and public saving S=S 750+0 b. The equilibrium interest rate is the value of r that clears the market for loanable funds. We already know that national saving is 750, so we just need to set it equal to investment. S=l 750=1,000-50r Solving this equation for r, we find: r=5% When the government increases its spending, private saving remains the same as before(notice that g does not appear in the sprivate above) while government saving decreases. Putting the new G into the equations abo Sprivate=750 APublic= T-G =1.000-1.250 Thus S=SP 750+(-250 =500 d. Once again the equilibrium interest rate clears the market for loanable fund 500=1.000-50r Solving this equation for r, we find: 7. To determine the effect on investment of an equal increase in both taxes and govern ment spending, consider the national income accounts identity for national saving National Saving =[Private Saving]+[Public Saving =ⅣY-T-C(Y-)+[T-G] We know that y is fixed by the factors of production. We also know that the change in consumption equals the marginal propensity to consume (MPc)times the change in disposable income. This tells us that △ National Saving=[△T-(MPCx(-△T)+[△T-△G [△T+(MPC×△T)]+0 =(MPC-1)△T
6. a. Private saving is the amount of disposable income, Y – T, that is not consumed: Sprivate = Y – T – C = 5,000 – 1,000 – (250 + 0.75(5,000 – 1,000)) = 750. Public saving is the amount of taxes the government has left over after it makes its purchases: Spublic = T – G = 1,000 – 1,000 = 0. Total saving is the sum of private saving and public saving: S = Sprivate + Spublic = 750 + 0 = 750. b. The equilibrium interest rate is the value of r that clears the market for loanable funds. We already know that national saving is 750, so we just need to set it equal to investment: S = I 750 = 1,000 – 50r Solving this equation for r, we find: r = 5%. c. When the government increases its spending, private saving remains the same as before (notice that G does not appear in the Sprivate above) while government saving decreases. Putting the new G into the equations above: Sprivate= 750 Spublic = T – G = 1,000 – 1,250 = –250. Thus, S = Sprivate + Spublic = 750 + (–250) = 500. d. Once again the equilibrium interest rate clears the market for loanable funds: S = I 500 = 1,000 – 50r Solving this equation for r, we find: r = 10%. 7. To determine the effect on investment of an equal increase in both taxes and government spending, consider the national income accounts identity for national saving: National Saving = [Private Saving] + [Public Saving] = [Y – T – C(Y – T)] + [T – G]. We know that Y is fixed by the factors of production. We also know that the change in consumption equals the marginal propensity to consume (MPC) times the change in disposable income. This tells us that ∆National Saving = [– ∆T – (MPC × ( – ∆T))] + [∆T – ∆G] = [– ∆T + (MPC × ∆T)] + 0 = (MPC – 1) ∆T. Chapter 3 National Income: Where It Comes From and Where It Goes 15
16 Answers to Textbook questions and problems The above expression tells us that the impact on saving of an equal increase in T and G depends on the size of the marginal propensity to consume. The closer the MPC is to 1, the smaller is the fall in saving. For example, if the MPC equals 1, then the fall in consumption equals the rise in government purchases, so national saving [Y-C(Y T)-GI is unchanged. The closer the MPC is to 0(and therefore the larger is the amount saved rather than spent for a one-dollar change in disposable income, the greater is the impact Because we assume that the mpc is less than 1. we xpect that national saving falls in response to an equal increase in taxes and govern left in Figure 3-3. The real interest rate rises, and investment falls curve shifts to The reduction in saving means that the supply of loanable funds S Figure 3-3 Investment, Saving 8. a. The demand curve for business investment shifts out because the subsidy increas- es the number of profitable investment opportunities for any given interest rate The demand curve for residential investment remains unchanged b. The total demand curve for investment in the economy shifts out since it repre- sents the sum of business investment which shifts out. and residential invest ment, which is unchanged. As a result the real interest rate rises as in figure 3-4
The above expression tells us that the impact on saving of an equal increase in T and G depends on the size of the marginal propensity to consume. The closer the MPC is to 1, the smaller is the fall in saving. For example, if the MPC equals 1, then the fall in consumption equals the rise in government purchases, so national saving [Y – C(Y – T) – G] is unchanged. The closer the MPC is to 0 (and therefore the larger is the amount saved rather than spent for a one-dollar change in disposable income), the greater is the impact on saving. Because we assume that the MPC is less than 1, we expect that national saving falls in response to an equal increase in taxes and government spending. The reduction in saving means that the supply of loanable funds curve shifts to the left in Figure 3–3. The real interest rate rises, and investment falls. 8. a. The demand curve for business investment shifts out because the subsidy increases the number of profitable investment opportunities for any given interest rate. The demand curve for residential investment remains unchanged. b. The total demand curve for investment in the economy shifts out since it represents the sum of business investment, which shifts out, and residential investment, which is unchanged. As a result the real interest rate rises as in Figure 3–4. 16 Answers to Textbook Questions and Problems S2 S1 I (r) I, S Investment, Saving r1 r2 r Real interest rate FFigure 3–2 igure 3–3
Chapter 3 National Income: Where It Comes From and Where It Goes Figure 8-4 三 B 1. An increase n desired Investment 2. the interest rate l1 c. The total quantity of investment does not change because it is constrained by the elastic supply of savings. The investment tax credit leads to a rise in business investment, but an offsetting fall in residential investment. That is, the higher interest rate means that residential investment falls (a shift along the curve), whereas the outward shift of the business investment curve leads business invest. ment to rise by an equal amount. Figure 3-5 shows this change. Note that +1=+2=S Figure 8-5 Business esiden investment 9. In this chapter, we concluded that an increase in government expenditures reduces national saving and raises the interest rate; it therefore crowds out investment by the full amount of the increase in government expenditure. Similarly, a tax cut increases
c. The total quantity of investment does not change because it is constrained by the inelastic supply of savings. The investment tax credit leads to a rise in business investment, but an offsetting fall in residential investment. That is, the higher interest rate means that residential investment falls (a shift along the curve), whereas the outward shift of the business investment curve leads business investment to rise by an equal amount. Figure 3–5 shows this change. Note that 9. In this chapter, we concluded that an increase in government expenditures reduces national saving and raises the interest rate; it therefore crowds out investment by the full amount of the increase in government expenditure. Similarly, a tax cut increases disposable income and hence consumption; this increase in consumption translates into a fall in national saving—again, it crowds out investment. Chapter 3 National Income: Where It Comes From and Where It Goes 17 S I2 I1 I, S A B 1. An increase in desired investment . . . 2. . . . raises the interest rate. Investment, Saving Real interest rate r Figure 3–4 Business Investment I2 B I1 B Residential Investment I2 R I1 R r r1 r2 r r1 r2 Figure 3–5 I11 22 IIIS BR BR +=+= Business investment Residential investment Figure 3–4
18 Answers to Textbook questions and Problems If consumption depends on the interest rate, then these conclusions about fiscal policy are modified somewhat. If consumption depends on the interest rate, then does saving. The higher the interest rate, the greater the return to saving. Hence, it seems reasonable to think that an increase in the interest rate might increase saving and reduce consumption. Figure 3-6 shows saving as an increasing function of the interest rate ure 8-6 S(r) Saving Consider what happens when government purchases increase. At any given level of the interest rate, national saving falls by the change in government purchases, as shown in Figure 3-7. The figure shows that if the saving function slopes upward, investment falls by less than the amount that government purchases rise; this happens because consumption falls and saving increases in response to the higher interest rate lence, the more responsive consumption is to the interest rate, the less government purchases crowd out investment 1, I L. S
If consumption depends on the interest rate, then these conclusions about fiscal policy are modified somewhat. If consumption depends on the interest rate, then so does saving. The higher the interest rate, the greater the return to saving. Hence, it seems reasonable to think that an increase in the interest rate might increase saving and reduce consumption. Figure 3–6 shows saving as an increasing function of the interest rate. Consider what happens when government purchases increase. At any given level of the interest rate, national saving falls by the change in government purchases, as shown in Figure 3–7. The figure shows that if the saving function slopes upward, investment falls by less than the amount that government purchases rise; this happens because consumption falls and saving increases in response to the higher interest rate. Hence, the more responsive consumption is to the interest rate, the less government purchases crowd out investment. 18 Answers to Textbook Questions and Problems Figure 3–7 S2(r) S1(r) I(r) I1 I I, S Investment, Saving ∆G r1 r r Real interest rate S(r) S Saving Real interest rate r Figure 3–6 Figure 3–6 Figure 3–7
Chapter 3 National Income: Where It Comes From and Where It Goes More Problems and applications to Chapter 3 1. a. A Cobb-Douglas production function has the form Y= AK L. In the appendix we showed that the marginal products for the Cobb-Douglas production function are MPL =(1-aY/L MPK =aY/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-O)Y/L R/P=MPK= oY/K Rewriting this (W/P=MPL×L=(1-0)Y. (R/PK=MPK×K=aY. Note that the terms(W/P)L and (r/p)k are the wage bill and total return to capi tal, respectively. Given that the value of a=0.3, then the above formulas indicat that labor receives 70 percent of total output, which is(1-0.3), and capital receives 30 percent of total output 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=AKL Let Y1 equal the initial value of output and Y2 equal final output. We know that a=0.3. We also know that labor L increases by 10 percent Y1=AK L Y Note that we multiplied L by 1.1 to reflect the 10-percent increase in the labor To calculate the percentage change in output, divide y2 by y: Y2- AK(1. AK 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=OAk-L1-a We know that a=0.3. We also know that labor(L)increases by 10 percent. Let (R/P), equal the initial value of the rental price of capital, and (R/P)2 equal the final rental price of capital after the labor force increases by 10 percent. To find (R/P)a, multiply l by 1.1 to reflect the 10-percent increase in the labor force (R/P)=0.3AK.L (R/P2=0.3AK07(11)07
More Problems and Applications to Chapter 3 1. a. A Cobb–Douglas production function has the form Y = AKα L1 – α . In the appendix we 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, which is (1 – 0.3), and capital receives 30 percent of total output. 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 = AKα L1 – 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: = = (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 (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. Chapter 3 National Income: Where It Comes From and Where It Goes 19 AK0.3(1.1L) 0.7 AK0.3L0.7 Y2 Y1