Yes
Use the social security model developed in chapter 10 to answer this question. Each period there are N old people and N' young people, with N' = (1 + n). Suppose that the government establishes a social security program in period T, which provies a social security benefit of b ( in terms of consumption goods) for each old person forever. In period T the government finances the benefits to current old by issuing debt(borrowing from young persons in period T). This debt is then paid off in period T +1 through lump-sum taxes on the young. In periods T +1 and later, lump-sum taxes on the young finance social security payment to the old. (a) Write down the period budget constraint and present value (life time) budget constraint for the old alive at time T with and without the social security program. Show, using diagrams, that the old alive at time T benefits from the social security program. (6 points) (b) Write down the period budget constraint and present value (life time) budget constraint for the young alive at time T with and without the social security program. Show, using diagrams, that the young alive at time I benefits from the social security program. (6 points) (d) Write down the present value budget constraint of the cohort born in peiord T +1. What is the effect of social security program on that cohort? How does this depend on the real interest rate and the population growth rate?(8 points) (c) Write down the present value budget constraint of people born in period T+2 and later with and without the social security program. Are they benefit from the program. How does this depend on the real interest rate and the population growth rate?(6 points) Chapter 10 Credit Market Imperfections: Credit Frictions, Financial Crises, and Social Security Learning Objectives After studying Chapter 10, students will be able to: 10.1 Construct the basic credit market imperfections problem for the consumer, with a kinked budget constraint. 10.2 Adapt the credit markets model to deal with asymmetric information. O 10.3 Show how limited commitment makes collateral important in the credit markets model. O 10.4 Show how pay-as-you-go social security works, and demonstrate what conditions are required so that it increases economic welfare. 10.5 Show how fully-funded social security programs function, and explain their economic role. O In Chapter 9, we explored the basic elements of consumer behavior in credit markets-how consumers act to smooth consumption over time in response to changes in their incomes and in market interest rates. As well, we studied the aggregate effects of changes in government tax policy. A key theoretical result from Chapter 9 is the Ricardian equivalence theorem, which states that a change in the timing of taxes can have no effects on consumer behavior or interest rates, provided that some special conditions hold. The Ricardian equivalence theorem provides us with a firm foundation for understanding the circumstances under which government tax policy will matter. In particular, as discussed in Chapter 90, the Ricardian equivalence theorem will not hold if the tax burden is not shared equally among consumers, if there is intergenerational redistribution resulting from a change in taxes, if there are tax distortions, or if there are credit market imperfections. The cases under which Ricardian equivalence does not hold have practical importance in at least two respects. First, credit market imperfections, or "frictions, o fail, are key to understanding some Social Security Programs Social security programs are government-provided means for saving for retirement. As such they are programs that help individuals smooth their consumption over their lifetimes. But if credit markets work well, then why do we need the government to provide us with consumption- smoothing services? As macroeconomists, if we want to provide a rationale for social security, we must be able to find some type of credit market failure that the government can correct. One purpose of this section is to explore this idea, and to examine how social security systems work in practice. THEORY CONFRONTS THE DATA The Housing Market, Collateral, and Consumption Figure 10.6 shows the relative price of housing in the United States, as measured by the Case-Shiller 20-city home price index, divided by the consumer price index, normalized to equal 100 in January 2000. A remarkable feature of the figure is the large increase in the relative price of housing to the peak in 2006. In particular, the purchasing power of the average house in the United States increased by almost 80% between 2000 and 2006. The U.S. housing stock then lost most of this accumulated value from 2006 to 20 180 120 150 Relative Price of Housing: January 2000 - 100 160 130 120 110 100 2004 200 WW RUT In Figure 10.7. we show the percentage deviations from trend in aggregate consumption. Note in Figure 10.6 that the relative price of housing continues to increase through the 2001 recession, when consumption declines below trend and does not begin recovering until 2003. The 2001 recession was relatively mild, as was the decline in consumption, in part because the value of housing as collateral continued to increase through the recession. Consumers were then able to continue to finance their consumption by borrowing against the value accumulated in their houses. Once the relative price of housing starts to decrease in 2006, this coincides with a subsequent decrease in consumption relative to trend, and the rapid decrease in consumption below trend in 2008-2009. What we see in Figures 10.6 and 10.7 is consistent with the idea that the value of collateral in credit markets contributes in an important way to the behavior of aggregate consumption expenditures. 2 1.5 0.5 Percentage Deviation From Trend -0.5 -1 -1.5 2002 2004 2006 2010 2012 2014 2016 2008 Year Figure 10.7 Percentage Deviations from Trend in Aggregate Consumption In comparing this figure to Figure 10.6 note the contraction in consumption folowing the 2001 recession, and the decrease in consumption relative to trend beginning in 2007, all consistent with the idea that the value of collateral is important for aggregate consumption There are essentially two types of programs: pay-as-you-go and fully funded social security, though in practice social security could be it. Pay-As-You-Go Social Security LO 10.4 Show how pay-as-you-go social security works, and demonstrate what conditions are required so that it increases economic welfare. In the United States, social security operates as a pay-as-you-go system, in that taxes on the young are used to finance social security transfers to the old. While public discussion may make it appear that the system is in fact fully funded, as the difference between social security tax revenue and social security benefits is used to purchase interest-bearing federal government securities, this is merely an accounting convention and is unimportant for the economic consequences of U.S. social security. To see the implications of pay-as-you-go social security for the distribution of wealth over time and across consumers, we use the basic credit market model of Chapter 9 but modify it to accommodate intergenerational redistribution of income by the government. Assume for simplicity that social security has no effect on the market real interest rater, which we suppose is constant for all time. Each consumer lives for two periods, youth and old age, and so in any period there is a young generation and an old generation alive. Let N denote the number of old consumers currently alive, and the number of young consumers currently alive. Assume that N'= (1 + r) N (10-9) so that the population is growing at the rate n, just as in the Solow growth model used in Chapters 7 and 8 though here people are finite lived. A given consumer receives income y when young and income y' when old, and we allow (as in Chapter 9O) for the fact that incomes can differ across consumers. For simplicity, assume that government spending is zero in all periods. Now, suppose that no social security program exists before some date T, and that before date the taxes on the young and old are zero in each period. Then, pay-as-you-go social security is established at date and continues forever after. Here, for simplicity we suppose that the social security program guarantees each old-age consumer in periods T and later a benefit of b units of consumption goods. Then, the tax for each old consumer in periods Tand after ist'= -6. The benefits for old consumers must be financed by taxes on the young, and we assume that each young consumer is taxed an equal amount, t. Then, because total social security benefits equal total taxes on the young, we have Nb = N2, (10-10) and so, using Equation (10-9) to substitute for N'in Equation (10-11), we can solve fort, obtaining b (10-11) 1+ How do consumers benefit from social security Cream Yararam is introduced in perind pain as these How do consumers benefit from social security? Clearly, the consumers who are old when the program is introduced in period T gain, as these consumers receive the social security benefit but do not have to suffer any increase in taxes when they are young, In Figure 10.8 the lifetime budget constraint of a consumer who is old in period Tis AB if there is no social security program, where the slope of AB is - (1+r) and the endowment point with no social security is E, determined by disposable income of y when young and y' when old. With the social security program, this consumer receives disposable income y when young and y'+b when old and has an endowment point given by E, on the budget constraint DF (with slope - (1 + r)) in the figure. The optimal consumption bundle shifts from H to J, and the consumer is clearly better off because his or her budget constraint has shifted out and he or she is able to choose a consumption bundle on a higher indifference curve. - Future Consumption - Current Consumption Figure 10.8 Pay-As-You-Go Social Security for Consumers Who Are Old in Period T in the period when social security is introduced, the old receive a social security berett. The budget constraint of an old consumer shifts from A to DF, and he or she is clearly better of What happens to consumers born in periods T and later? For these consumers, in Figure 10.9. the budget constraint would be AB without social security, with an endowment point at E, and the budget constraint having slope - (1 + r). With social security, disposable income when young is y-t=y-from Equation (10-11) and disposable income when old is y'+b, and the endowment point shifts to E, in the figure on the budget constraint DF. Because the market real interest rate has not changed, the slope of DF is -(1+r). The slope of EE is - (1 + n), so in the figure we have shown the case where n > . In this case, the budget constraint shifts out for this consumer, with the . (ith) Clth) Social Security (Pay-as-you-gol T THI T-2 H T+2- Endurment Economy N old Yold Nyoo young " young / Infinite Periods In each penod: two generation Everyone lives for a periods I old n cexo) N old population growth rate N young Y ) G: Is this In each period : b cexo) PAYGSS ss transfer to each old perod BC of the government No G No B population size young pare to optime da Case I NOSS for population size old end every period 1 EN CPAYG) b Case2. Start to implement Is at period T EN N. c ta in 01:51 Step 2 of 13 Done To develop a model with pay-as-you-go social security program we assume the following: The social security has no effect on market interest rater, and this is constant all the time. Each consumer lives for only two periods, young age and old age. If N denote the number of old consumer currently alive, and N' young consumer presently alive and if the population grows the rate n; then N'=(1+n)N .... (1) A given consumer receives income y when young and y' when old. The government pending is zero at all time. 01:51 Step 3 of 13 Done Suppose hat he government establishes a social security program at time T that provide a social security benefit of b for each old person forever. In period T the government finances the benefits of the current old by issuing a debt. The debt is then paid in period T+1 through lump-sum taxes on the young. In periods T+1 and later, lump-sum taxes on the old finances social security payments to the old. 01:51 Step 4 of 13 Done y+ b The consumers' lifetime wealth before the program is given by wey+ After the benefit program, which levied no tax liability on either young and old, the life time wealth of the young is given by we = y + The old receives the benefit without suffering the increase in taxes, their current income increases by the amount of benefit. The budget line of the old is given in the panel A in the figure below. The line AB is the budget line initially, the old chooses the point Hover the endowment point E. After the introduction of the benefit program the current income of the old increases by b, this shifts he budget line to DF, given his preferences the old chooses the point J and his endowment point is E2. Here we see that the old moves to a higher indifference curve and thus the old benefits from the program. 01:52 Step 5 of 13 Done future consumption I, F B y-b E. H E current consumption Figure Use the social security model developed in chapter 10 to answer this question. Each period there are N old people and N' young people, with N' = (1 + n). Suppose that the government establishes a social security program in period T, which provies a social security benefit of b ( in terms of consumption goods) for each old person forever. In period T the government finances the benefits to current old by issuing debt(borrowing from young persons in period T). This debt is then paid off in period T +1 through lump-sum taxes on the young. In periods T +1 and later, lump-sum taxes on the young finance social security payment to the old. (a) Write down the period budget constraint and present value (life time) budget constraint for the old alive at time T with and without the social security program. Show, using diagrams, that the old alive at time T benefits from the social security program. (6 points) (b) Write down the period budget constraint and present value (life time) budget constraint for the young alive at time T with and without the social security program. Show, using diagrams, that the young alive at time I benefits from the social security program. (6 points) (d) Write down the present value budget constraint of the cohort born in peiord T +1. What is the effect of social security program on that cohort? How does this depend on the real interest rate and the population growth rate?(8 points) (c) Write down the present value budget constraint of people born in period T+2 and later with and without the social security program. Are they benefit from the program. How does this depend on the real interest rate and the population growth rate?(6 points) Chapter 10 Credit Market Imperfections: Credit Frictions, Financial Crises, and Social Security Learning Objectives After studying Chapter 10, students will be able to: 10.1 Construct the basic credit market imperfections problem for the consumer, with a kinked budget constraint. 10.2 Adapt the credit markets model to deal with asymmetric information. O 10.3 Show how limited commitment makes collateral important in the credit markets model. O 10.4 Show how pay-as-you-go social security works, and demonstrate what conditions are required so that it increases economic welfare. 10.5 Show how fully-funded social security programs function, and explain their economic role. O In Chapter 9, we explored the basic elements of consumer behavior in credit markets-how consumers act to smooth consumption over time in response to changes in their incomes and in market interest rates. As well, we studied the aggregate effects of changes in government tax policy. A key theoretical result from Chapter 9 is the Ricardian equivalence theorem, which states that a change in the timing of taxes can have no effects on consumer behavior or interest rates, provided that some special conditions hold. The Ricardian equivalence theorem provides us with a firm foundation for understanding the circumstances under which government tax policy will matter. In particular, as discussed in Chapter 90, the Ricardian equivalence theorem will not hold if the tax burden is not shared equally among consumers, if there is intergenerational redistribution resulting from a change in taxes, if there are tax distortions, or if there are credit market imperfections. The cases under which Ricardian equivalence does not hold have practical importance in at least two respects. First, credit market imperfections, or "frictions, o fail, are key to understanding some Social Security Programs Social security programs are government-provided means for saving for retirement. As such they are programs that help individuals smooth their consumption over their lifetimes. But if credit markets work well, then why do we need the government to provide us with consumption- smoothing services? As macroeconomists, if we want to provide a rationale for social security, we must be able to find some type of credit market failure that the government can correct. One purpose of this section is to explore this idea, and to examine how social security systems work in practice. THEORY CONFRONTS THE DATA The Housing Market, Collateral, and Consumption Figure 10.6 shows the relative price of housing in the United States, as measured by the Case-Shiller 20-city home price index, divided by the consumer price index, normalized to equal 100 in January 2000. A remarkable feature of the figure is the large increase in the relative price of housing to the peak in 2006. In particular, the purchasing power of the average house in the United States increased by almost 80% between 2000 and 2006. The U.S. housing stock then lost most of this accumulated value from 2006 to 20 180 120 150 Relative Price of Housing: January 2000 - 100 160 130 120 110 100 2004 200 WW RUT In Figure 10.7. we show the percentage deviations from trend in aggregate consumption. Note in Figure 10.6 that the relative price of housing continues to increase through the 2001 recession, when consumption declines below trend and does not begin recovering until 2003. The 2001 recession was relatively mild, as was the decline in consumption, in part because the value of housing as collateral continued to increase through the recession. Consumers were then able to continue to finance their consumption by borrowing against the value accumulated in their houses. Once the relative price of housing starts to decrease in 2006, this coincides with a subsequent decrease in consumption relative to trend, and the rapid decrease in consumption below trend in 2008-2009. What we see in Figures 10.6 and 10.7 is consistent with the idea that the value of collateral in credit markets contributes in an important way to the behavior of aggregate consumption expenditures. 2 1.5 0.5 Percentage Deviation From Trend -0.5 -1 -1.5 2002 2004 2006 2010 2012 2014 2016 2008 Year Figure 10.7 Percentage Deviations from Trend in Aggregate Consumption In comparing this figure to Figure 10.6 note the contraction in consumption folowing the 2001 recession, and the decrease in consumption relative to trend beginning in 2007, all consistent with the idea that the value of collateral is important for aggregate consumption There are essentially two types of programs: pay-as-you-go and fully funded social security, though in practice social security could be it. Pay-As-You-Go Social Security LO 10.4 Show how pay-as-you-go social security works, and demonstrate what conditions are required so that it increases economic welfare. In the United States, social security operates as a pay-as-you-go system, in that taxes on the young are used to finance social security transfers to the old. While public discussion may make it appear that the system is in fact fully funded, as the difference between social security tax revenue and social security benefits is used to purchase interest-bearing federal government securities, this is merely an accounting convention and is unimportant for the economic consequences of U.S. social security. To see the implications of pay-as-you-go social security for the distribution of wealth over time and across consumers, we use the basic credit market model of Chapter 9 but modify it to accommodate intergenerational redistribution of income by the government. Assume for simplicity that social security has no effect on the market real interest rater, which we suppose is constant for all time. Each consumer lives for two periods, youth and old age, and so in any period there is a young generation and an old generation alive. Let N denote the number of old consumers currently alive, and the number of young consumers currently alive. Assume that N'= (1 + r) N (10-9) so that the population is growing at the rate n, just as in the Solow growth model used in Chapters 7 and 8 though here people are finite lived. A given consumer receives income y when young and income y' when old, and we allow (as in Chapter 9O) for the fact that incomes can differ across consumers. For simplicity, assume that government spending is zero in all periods. Now, suppose that no social security program exists before some date T, and that before date the taxes on the young and old are zero in each period. Then, pay-as-you-go social security is established at date and continues forever after. Here, for simplicity we suppose that the social security program guarantees each old-age consumer in periods T and later a benefit of b units of consumption goods. Then, the tax for each old consumer in periods Tand after ist'= -6. The benefits for old consumers must be financed by taxes on the young, and we assume that each young consumer is taxed an equal amount, t. Then, because total social security benefits equal total taxes on the young, we have Nb = N2, (10-10) and so, using Equation (10-9) to substitute for N'in Equation (10-11), we can solve fort, obtaining b (10-11) 1+ How do consumers benefit from social security Cream Yararam is introduced in perind pain as these How do consumers benefit from social security? Clearly, the consumers who are old when the program is introduced in period T gain, as these consumers receive the social security benefit but do not have to suffer any increase in taxes when they are young, In Figure 10.8 the lifetime budget constraint of a consumer who is old in period Tis AB if there is no social security program, where the slope of AB is - (1+r) and the endowment point with no social security is E, determined by disposable income of y when young and y' when old. With the social security program, this consumer receives disposable income y when young and y'+b when old and has an endowment point given by E, on the budget constraint DF (with slope - (1 + r)) in the figure. The optimal consumption bundle shifts from H to J, and the consumer is clearly better off because his or her budget constraint has shifted out and he or she is able to choose a consumption bundle on a higher indifference curve. - Future Consumption - Current Consumption Figure 10.8 Pay-As-You-Go Social Security for Consumers Who Are Old in Period T in the period when social security is introduced, the old receive a social security berett. The budget constraint of an old consumer shifts from A to DF, and he or she is clearly better of What happens to consumers born in periods T and later? For these consumers, in Figure 10.9. the budget constraint would be AB without social security, with an endowment point at E, and the budget constraint having slope - (1 + r). With social security, disposable income when young is y-t=y-from Equation (10-11) and disposable income when old is y'+b, and the endowment point shifts to E, in the figure on the budget constraint DF. Because the market real interest rate has not changed, the slope of DF is -(1+r). The slope of EE is - (1 + n), so in the figure we have shown the case where n > . In this case, the budget constraint shifts out for this consumer, with the . (ith) Clth) Social Security (Pay-as-you-gol T THI T-2 H T+2- Endurment Economy N old Yold Nyoo young " young / Infinite Periods In each penod: two generation Everyone lives for a periods I old n cexo) N old population growth rate N young Y ) G: Is this In each period : b cexo) PAYGSS ss transfer to each old perod BC of the government No G No B population size young pare to optime da Case I NOSS for population size old end every period 1 EN CPAYG) b Case2. Start to implement Is at period T EN N. c ta in 01:51 Step 2 of 13 Done To develop a model with pay-as-you-go social security program we assume the following: The social security has no effect on market interest rater, and this is constant all the time. Each consumer lives for only two periods, young age and old age. If N denote the number of old consumer currently alive, and N' young consumer presently alive and if the population grows the rate n; then N'=(1+n)N .... (1) A given consumer receives income y when young and y' when old. The government pending is zero at all time. 01:51 Step 3 of 13 Done Suppose hat he government establishes a social security program at time T that provide a social security benefit of b for each old person forever. In period T the government finances the benefits of the current old by issuing a debt. The debt is then paid in period T+1 through lump-sum taxes on the young. In periods T+1 and later, lump-sum taxes on the old finances social security payments to the old. 01:51 Step 4 of 13 Done y+ b The consumers' lifetime wealth before the program is given by wey+ After the benefit program, which levied no tax liability on either young and old, the life time wealth of the young is given by we = y + The old receives the benefit without suffering the increase in taxes, their current income increases by the amount of benefit. The budget line of the old is given in the panel A in the figure below. The line AB is the budget line initially, the old chooses the point Hover the endowment point E. After the introduction of the benefit program the current income of the old increases by b, this shifts he budget line to DF, given his preferences the old chooses the point J and his endowment point is E2. Here we see that the old moves to a higher indifference curve and thus the old benefits from the program. 01:52 Step 5 of 13 Done future consumption I, F B y-b E. H E current consumption Figure
