Intermediate Macroeconomics, Sciences Po, Answer Key to Problem Set 9

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1 Intermediate Macroeconomics, Sciences Po, 2014 Zsófia Bárány Answer Key to Problem Set 9 1. Ricardian Equivalence Consider a two-period economy in which the representative consumer maximizes the utility function U(c 1, c 2 ) = ln(c 1 ) + β ln(c 2 ) subject to the life-time budget constraint c 1 + c 2 /R = W, where 0 < β < 1, c i is consumption in period i = 1 or 2, W is the present value of after-tax life-time income and R = 1 + r, where r is the interest rate. (a) Derive the level of optimal consumption in the two periods. Provide economic intuition for your derivations. As in the lecture slides, we maximize a general utility function subject to an inter-temporal budget constraint. First, substitute the budget constraint into the objective function to eliminate c 2 and then maximize with respect to c 1 : max u(c 1 ) + β u [R (W c 1 )] Using the chain rule, the first-order condition for c 1 is u (c 1 ) β u [R (W c 1 )] R = 0 Note that the expression in the squared bracket is simply c 2, so we can rewrite it as: u (c 1 ) = β R u (c 2 ) Intuitively, we can express this condition in terms of the marginal rate of substitution: MRS = u (c 1 ) β u (c 2 ) = R which says that consumer maximize their utility when their desired allocation satisfies the condition that the internal relative value of c 1 in terms of c 2 equals to the market exchange value of these goods. 1

2 Using the utility function u(c) = ln(c), we have 1 c 1 = β R c 2 or c 2 = β R c 1. Finally, using this condition along with the budget constraint, we can derive c 1 = W and c 1+β 2 = β R W. 1+β (b) Suppose the consumer receives Y 1 and Y 2 and pays taxes T 1 and T 2 in periods 1 and 2. Use this model to explain the Ricardian equivalence of the timing of taxes. For simplicity assume that the number of consumers N = 1. Write down the government s budget constraint: T 1 + T 2 R = G 1 + G 2 R and the consumer s life-time income as W = Y 1 + Y 2 R ( T 1 + T 2 R If the present value of government spending, G 1 + G 2 /R, remains constant then any change of taxes in the current period must be eventually compensated by a proportional change of taxes in the future period to satisfy the government s budget constraint. Therefore consumer s wealth W is unaffected by the change in taxes and the optimal consumption levels remain the same because they are a function of the present discounted value of wealth. Therefore, the timing of taxes does not matter if the present value of government spending remains the same. Mathematically, this can be seen by combining the above two equations and showing that the budget constraint of consumers doesn t depend of taxes: W = Y 1 + Y ( 2 R G 1 + G ) 2 R Since the timing of taxes does not affect W, c 1 = W and c 1+β 2 = β R W 1+β remain unchanged. 2 )

3 When the government changes taxes in the first period by T 1, it will have to change taxes in the second period by T 2 = R T 1 to satisfy its inter-temporal budget constraint. For example, a cut in taxes in the first period by T 1 = 1, means an increase in government debt by 1, B = T 1 = 1 (a change in government savings by S g = B = 1) and requires higher taxation in the second period by T 2 = R T 1 = R. Households do not change their consumption, so the change in private savings is S p = Y 1 T 1 c 1 = 0 ( 1) 0 = 1. National saving S = S p + S g remain therefore unchanged. 3

4 (c) Suppose the consumer cannot borrow. Show that what you derived in part (b) might not hold. (c) Suppose the consumer cannot borrow. Show that what you derived in part (b) might not hold. We have to distinguish two cases: first, a consumer is a lender (i.e. consumes to the left of her endowment point). Then, the fact that there are limits on borrowing doesn t affect her consumption choices We have to distinguish two cases: first, a consumer is a lender (i.e. consumes at toall theand leftricardian of her endowment equivalence point). still Then, applies. the fact that there are limits on borrowing doesn t affect her consumption choices at all and Ricardian equivalence still assume applies. that a consumer actually wants to consume to the Next, right of her endowment point, so that she would be a borrower if market Next, assume conditions that aallowed. consumerwhen actually she wants cannot to consume borrow, toher theconsump- tion endowment will bepoint, c 1 = Yso 1 that T 1 and she would c 2 = Ybe 2 a Tborrower 2. A taxifincrease market conditions today will al- right of her reduce lowed. current When she consumption cannot borrow, even herifconsumption the consumer willknows be c 1 = that Y 1 future T 1 and c taxes 2 = Y will 2 T fall, 2. A tax increase today will reduce current consumption even if the so the timing of taxes matters and Ricardian Equivalence doesn t hold. Both cases are depicted in figure 1. consumer knows that future taxes will fall, so the timing of taxes matters and Ricardian Equivalence doesn t hold. Both cases are depicted in figure 1. c c IC 1 IC 2 IC 1 c c Figure 1: Ricardian equivalence and credit constraints Figure 1: Ricardian equivalence and credit constraints For For the the borrowing constraint constraint consumer consumer c 1 c = 1 Y = 1 Y T 1 1 T = 1 T = T 1. If 1 all. Ifconsumers all consumers are borrowing are borrowing constraint constraint and behave and like this, behave the change like this, in private the change savings is in S private p = Ysavings 1 T 1 c is S 1 = p 0 T = Y 11 ( T 1 ) = c 0. 1 So= if0current T 1 taxes ( T increase, T 1 > 0, S g = B = T 1 > 0, and national saving S = S p + S g 1 ) = 0. So if current taxes increase, T 1 > 0, S g = B = T increase. 1 > 0, and national saving S = S p + S g increase. (d) (d) Another proposed reason reason why why the Ricardian the Ricardian equivalence equivalence might not might hold isnot that hold consumers is that have consumers different life have horizon different than the life government. horizon than How the could government. How could you use the two-period model to show this? you use the two-period model to show this? This can be illustrated by having 4 two different consumers living in each period (consumer 1 is alive only in period 1 and consumer 2 only in period 2) and 3

5 This can be illustrated by having two different consumers living in each period (consumer 1 is alive only in period 1 and consumer 2 only in period 2) and a single government with a planning horizon that spans over the two periods. Assume that there is no bequest motive so that consumer 1 doesn t care about consumer 2 at all. In terms of the utility function set β = 0 so that consumers only care about current consumption and use two single-period budget constraints instead of one inter-temporal. In this case, consumer will consume everything in the period when they are alive. A tax cut in period 1 will increase consumption in period 1 even if there is no borrowing constraint. The consumer living in period 1 doesn t internalize the greater tax burden that falls on the consumer living in period 2, whose consumption will decrease. 2. The financial crisis: Consider a two-period consumption model with housing. Suppose there are no taxes and that housing is illiquid (i.e. it cannot be sold in the first period). The lifetime budget constraint can be written as: c + c 1 + r y + y + ph }{{ 1 + r } W Here total wealth (W ) consists of the present discounted value of income and the presented discounted value of housing wealth. (a) Credit market imperfections: Suppose that, due to asymmetric information, consumers face different interest rates for borrowing (r B ) and for lending (r L ). The difference between the borrowing and lending rates (r B r L )) is known as the interest rate spread. Assume that r B r L and suppose that an increase in credit risk causes a rise in the interest rate spread. What is the effect on the consumption of lending households? What about households that borrow? 5

6 Firstly, we denote the lifetime budget constraint of a lending household (s 0) by: c + c 1 + r L W L where W L = y + y + ph 1 + r L And the lifetime budget constraint of a borrowing household (s < 0) by: c + c 1 + r B W B where W B = y + y + ph 1 + r B In general, the effect of the increase in the interest rate spread on lending (unconstrained) and borrowing (constrained) households depends on the underlying cause of the change in spread. The spread can change because the borrowing rate (r B ) rises and/or the lending rate (r L ) falls. Suppose that financial institutions respond toborrowing the higher rate credit (r B ) rises risk and/or by increasing the lendingthe rate borrowing (r L ) falls. Suppose rate; this thatwill financial institutions the consumption respond to ofthe borrowing higher credit households risk by increasing to fall, but the the borrowing con- cause sumption rate; this will of lending cause thehouseholds consumptionwill of borrowing be unaffected. households to fall, but the consumption of lending households will be unaffected. C Slope = (1 + r L ) E Slope = (1 + r B ) IC B C Figure Figure 2: Increase 2: Increase in in the the borrowing rate Recall that the slope of the budget constraint is given by (1 + r). Given the Recall differences thatinthe lending slopeand ofborrowing the budget rates, constraint the budget isconstraint given by is(1 kinked + r). at Given the endowment the differences point (E), incorresponding lending andto borrowing the point on rates, the budget the line budget where consumers neither save nor borrow. As shown in figure 2, the rise in the borrowing rate causes the budget constraint to become steeper to the right of the endowment point (the borrowing 6 region ). In contrast, the change in the borrowing rate has no effect on the budget constraint to the left of the endowment point (the saving region ). Borrowing households therefore end up on a lower indifference curve after the change in the interest rate and so are worse off. The saving households remain on the same indifference curve and hence their welfare does not change.

7 (b) Limited commitment: Suppose now that there is no asymmetric information so that consumers face the same interest rate for borrowing and lending (r = r B = r L ). But now suppose that consumers are only allowed to borrow a certain fraction of the value of their housing wealth (θ) where 0 θ 1. constraint is kinked at the endowment point (E), corresponding to the point on the budget line where consumers neither save nor borrow. As shown in figure 2, the rise in the borrowing rate causes the budget constraint to become steeper to the right of the endowment point (the borrowing region ). In contrast, the change in the borrowing rate has no effect on the budget constraint to the left of the endowment point (the saving region ). Borrowing households therefore end up on a lower indifference curve after the change in the interest rate and so are worse off. The saving households remain on the same indifference curve and hence their welfare does not change. Alternatively, it is possible that financial institutions respond to the greater credit risk by lowering the interest rates they offer to saving households. This would correspond to the budget constraint becoming flatter in the saving region, but being unchanged in the borrowing region. In this case, the saving households are worse off while the borrowing households are unaffected. This case is represented in figure 3. C Slope = (1 + r L ) IC L E Slope = (1 + r B ) C Figure Figure 3: Decrease 3: in inthe thelending rate Of course, we may also have some combination of the above in which case both groups of households are made worse off. In either Note that there will also be substitution effects from the higher interest rate case spread; theifincrease the higher in spread the interest is duerate to higher spread borrowing has a negative rates, only income borrowers willonbe both affected types (less ofborrowing household. because current consumption is relatively effect more expensive); if the higher spread is due to lower lending rates, only lenders will be affected (less saving because 7 current consumption becomes relatively cheaper).

8 Note that there will also be substitution effects from the higher interest rate spread; if the higher spread is due to higher borrowing rates, only borrowers will be affected (less borrowing because current consumption is relatively more expensive); if the higher spread is due to lower lending rates, only lenders will be affected (less saving because current consumption becomes relatively cheaper). (b) Limited commitment: Suppose now that there is no asymmetric information so that consumers face the same interest rate for borrowing and lending (r = r B = r L ). But now suppose that consumers are only allowed to borrow a certain fraction of the value of their housing wealth (θ) where 0 θ 1. We can refer to θ as the loan-tovaluation ratio. The collateral constraint becomes: s(1 + r) θph Now suppose that the increase in credit risk causes a tightening in lending standards, which we can represent as a fall in the loan-tovaluation ratio (θ). What impact will this have on the consumption of households that lend? What about households that borrow? Note that because housing is illiquid the first period budget constraint is given by: But the collateral constraint says: c + s y s(1 + r) θph Combining these two equations we obtain: c y + θph 1 + r In figure 4, the endowment point (E) (where consumers neither save nor borrow) corresponds to the point on the x-axis where c = y. But, importantly, the kink in the budget constraint does not occur at this endowment point because of the collateral constraint. While consumers cannot consume the full value of their home in the first period, the collateral constraint implies that they can borrow against 8

9 Combining these two equations we obtain: c y + θph 1 + r In figure 4, the endowment point (E) (where consumers neither save nor borrow) corresponds to the point on the x-axis where c = y. But, importantly, the kink in the budget constraint does not occur at this endowment point because of the collateral constraint. While consumers cannot consume the full the value value of their of their home home in the first in the period, first the period collateral (i.e. constraint they are implies not completely they cancredit borrowrationed). against thethis valueimplies of their home that the in the kink firstinperiod the budget (i.e. they that constraint are not completely occurs at credit point rationed). A where This c = implies y + θph that. the kink in the budget constraint occurs at point A where c = y + θph 1+r 1+r. C E B A IC B C Figure 4: Tighter bank lending standards Figure 4: Tighter bank lending standards The tightening in lending standards, as represented by a lower θ, causes a leftwardtightening vertical shiftininlending this budget standards, constraint kink as represented from A to B. bythe a slope lower ofθ, the The causes budget aconstraint leftwardisvertical unaffected shift as the in interest this budget rate does constraint not change kink this from case. A to B. The slope of the budget constraint is unaffected as the interest In effect, rate does the tightening not change in lending thisstandards case. causes the consumption of (creditconstrained) borrowing households to fall. In contrast, it has no impact on the In effect, the tightening in lending standards causes the consumption of (credit-constrained) borrowing households to fall. In con- 7 trast, it has no impact on the consumption of lending households. Basically, for any given value of their housing wealth, borrowing households are forced to borrow less when lending standards become tougher. Note: Incidentally, there is significant evidence that changes in bank lending standards played an important role in the recent US subprime mortgage crisis. 9