Liquidity constraints and the Ricardian equivalence theorem.

Liquidity Constraints and the Ricardian Equivalence Theorem

1. INTRODUCTION

Econometric tests have been commonly interpreted as showing that consumption is too sensitive to current income to be consistent with the permanent income hypothesis (PIH), for example, in Flavin (1981).

The excess sensitivity of consumption to current income is often thought to be the consequence of some proportion of consumers being liquidity constrained, due to credit market imperfections. Further, it is often claimed that if consumers are liquidity constrained, then the Ricardian equivalence theorem (RET) (Barro 1974) is false. Since more direct tests of the RET, which estimate the effects of changes in government debt on aggregate consumption (Seater 1982, Kormendi 1983, Aschauer 1985) and on interest rates (Plosser 1982 and Evans 1985), have been interpreted as failing to reject the RET, it seems that we are left with a mysterious set of conflicting econometric results (Seater 1985).

Hayashi (1987) makes the important point that the interpretation of the econometric tests of the PIH that indicate that some consumers are liquidity constrained does not necessarily imply that the RET is false empirically. Hayashi shows, in two different models in which the credit market is imperfect, that the RET holds even though current consumption is to sensitive to current income to be consistent with the PIH. In showing that the empirical failure of the PIH is not sufficient for empirical failure of the RET, Hayashi's results provide a reconciliation of the "conflicting" econometric evidence on the PIH and the RET within one model of consumption behavior.

This paper reveals that Hayashi's result depends upon the creditor's receiving a positive partial payment if the debtor chooses to default. The term default is used here to mean the failure of a borrower to completely fulfill the terms of a debt obligation. In this case it is shown that a bond-financed tax cut, followed by a tax increase of equal present value, induces lenders to reduce the amount they lend, at a given interest rate, by the same amount as the initial tax cut. Since this leaves the intertemporal budget constraints of borrowers unchanged, the RET holds.

The contribution of this paper is that it reveals that whether the RET holds depends upon the specification of the payment to the lender by the borrower when the borrower defaults. If the creditor expects to receive nothing from the debtor if the debtor defaults, which is perhaps the more empirically relevant case, the analysis shows that the RET does not hold.

The model developed below treats Hayashi's (1987) result that the RET holds even when the credit market, is imperfect, as a special case of the model. The model has the following properties: 1) individuals default if their consumption is higher after they default than when they honor their debt; 2) as a consequence of individuals' having the option of defaulting, after a nonnegative level of borrowing the rate at which individuals can borrow is higher than the interest rate at which they can lend; 3) consumption is sensitive to changes in current income that leave the present value of lifetime income unchanged; and 4) the RET may or may not hold, depending upon the exact specification of the debt contract. Property 2 is consistent with the observation that typically individuals face higher interest rates when they borrow than when they lend. Property 3 is consistent with the standard interpretation of econometric tests of the PIH. The specification of the model when the RET holds is consistent with the standard interpretations of econometric tests of the RET.

2. MODEL

Consider a two-period model of the loan market. In period one, borrowers know their period-one income, [y.sub.1], and know they will receive in period two income [y.sub.2.sup.h] with probability h and income [y.sub.2.sup.1] with probability 1-h, where [y.sub.1] [is less than] [y.sub.2.sup.1] [is less than] [y.sub.2.sup.h] and h [Epsilon] (0,1). This information is also known to lenders in period one, who also observe the amount of debt owed by a borrower to any given lender. In period two, lenders observe at zero cost the realized income of borrowers.

Each borrower solves the maximization problem, (1a) max U([c.sub.1]) + [Beta] EU([c.sub.2]), subject to the period budget constraints, (1b) [c.sub.1] = [y.sub.1] + B - [T.sub.1]; (1c) [c.sub.2s.sup.n] = [y.sub.2.sup.s] - (1+r) B - [T.sub.2]; (1d) [c.sub.2s.sup.d] = [y.sub.2.sup.s] - [z.sub.s] - [T.sub.2]; where s = l,h. U(.) is a utility function with the properties, [U.sup.'] [is greater than] 0, [U.sup."] [is less than] 0, [U.sup.'](0) = [affinity] and [U.sup.'] [affinity] = 0. A discount factor is indicated as [Beta]. The expectations operator E is conditioned on information known in period one. First period consumption is [c.sub.1], and B is the amount borrowed in the first period, at interest rate r. Quantities, [T.sub.1] and [T.sub.2] are lump sum taxes levied in period one and period two respectively. The expression [c.sub.2s.sup.n] is the level of second-period consumption when the borrower honors the debt contract made in the first period; [c.sub.2s.sup.d] is the level of second-period consumption when the borrower defaults on the debt contract made in the first period. Variable [z.sub.s] is the partial payment received in period two by the lender if the borrower defaults. Equation (1c) is the budget constraint faced by the borrower in period two if the borrower honors the debt contract made in the first period. Equation (1d) is the budget constraint faced by the borrower in period two if the borrower defaults.

Using equations (1), the borrower's maximization problem can be rewritten as (2) [Mathematical Expression Omitted]

From equation (2), it is readily apparent a borrower defaults in the second second whenever (3) (1+r)B [is greater than] [z.sub.s] for s = l, h since in this case, [c.sub.2s.sup.d] [is greater than] [c.sub.2s.sup.n]. Equation (3) indicates that borrowers default whenever the level of their consumption after they default is higher than if they pay off the principal and interest on the debt incurred in period one.

The specification of [z.sub.s] is given by (4) [z.sub.s] = (max [0, [y.sub.2.sup.s] - [T.sub.2] - [c.sup.*]], B (1 + r)).

Equation (4) is consistent with personal bankruptcy law in the United States, which places an upper bound on the amount of assets or income that can be seized by the lender and prevents individuals from avoiding taxes by declaring bankruptcy (see Epstein 1986). The maximum level of consumption a borrower may obtain if he defaults, [c.sup.*], is exogenous in this model. Following United States bankruptcy law it is assumed that [c.sup.*] [is greater than] 0. Since equation (2) implies that [z.sub.s] [is greater than or equal to] 0, if the borrower defaults, the lender loses at most (1 + r)B.

The sequence of events if the borrower defaults is as follows. Given second-period income, first taxes [T.sub.2] are paid. Second, the borrower may consume up to [c.sup.*]. Given equations (1d) and (4), the level of consumption under default is given by [c.sub.2s.sup.d] = min[[c.sup.*], [y.sub.2.sup.s] - [T.sub.2]]. Third, [z.sub.s] is paid to the lender in partial payment of the principal and interest owed.

It is in the specification of [z.sub.s], given by equation (4), that the model developed here is more general than. Hayashi's model. Using the notation of this paper, Hayashi assumes that [y.sub.2.sup.s], [c.sup.*] and [T.sub.2] always take on values such that [z.sub.s] [is greater than] 0 for s = l, h. Hence, Hayashi considers only the case where the lender receives a positive partial payment when the borrower defaults. If [z.sub.s] [is greater than] 0, the partial payment depends on the borrower's period-two tax liability. As a consequence, the amount the lender is willing to lend is a function of the borrower's future tax liability. This paper generalizes Hayashi's result, because [z.sub.s] is not restricted to be positive. In the case where [z.sub.s] = 0, for some s, the results of a bond-financed tax cut on consumption are exactly the opposite of Hayashi's. As is shown below, if [z.sub.s] = 0 for some s, the RET does not hold.

Given the probability distribution for period-two income and equation (4), the distribution for [z.sub.s] is given by (5) [Mathematical Expression Omitted]

The assumption that [y.sub.2.sup.l] [is less than] [y.sub.2.sup.h] implies that [z.sub.l] [is less than or equal to] [z.sub.h]. For levels of loans such that (1+r) B [is less than or equal to] [z.sub.l], borrowers do not default. For levels of interest rates and borrowing such that [z.sub.l] [is less than or equal to] (1+r)B [is less than or equal to] [z.sub.h], borrowers default when their low level of income [y.sub.2.sup.l] is realized. For (l + r)B [is greater than] [z.sub.h], borrowers always default.

The supply side of the loan market consists of a large number of lenders who are assumed to be risk neutral and face an exogenously given cost of loanable funds, i. The loan market is assumed to be competitive. Lenders know the borrowers' maximization problem given by equation (2), and hence know equation (3). In addition, lenders know the probability distribution for the period-two income of borrowers, and equation (5), the distribution of the payment they receive if the borrower defaults. Given this information, lenders know that the probability of default P on a loan contract with principal B and interest rate r is given by (6) [Mathematical Expression Omitted]

The expected profit from lending B at interest rate r is defined as [Mathematical Expression Omitted]. Since lenders are risk neutral, competition among lenders assures that in equilibrium all loans earn zero expected profits. Imposing the zero expected profit condition and using equation (6) yields the loan schedule, (7) [Mathematical Expression Omitted] where

[B.sub.1] = [z.sub.1]/(1+i); (8) [B.sub.2] = ([hz.sub.h] + (1-h) [z.sub.1])/(1+i). Quantities [B.sub.1] and [B.sub.2] are the levels of debt at which the loan schedule has kinks. Since lenders receive at least [z.sub.l] with probability one if a borrower defaults, lenders will to lend up to [B.sub.1] at interest rate i. The maximum amount that lenders will lend is given by [B.sub.2], the maximum lenders can lend and still expect to receive (1+i) [B.sub.2]. For B [is greater than] [B.sub.2], borrowers default for all realizations of period-two income; hence, for levels of debt greater than [B.sub.2], the loan schedule does not exist.

3. EFFECTS OF A BOND-FINANCED TAX CUT

Assume that the government borrows at the same rate of interest as lenders. Consider the following fiscal policy:

[dT.sub.1] = -D; (9) [dT.sub.2] = (1+i)D = -(1+i) [dT.sub.1]. In period one the tax cut of [dT.sub.1] is financed by borrowing, D. In the second period, taxes are increased by the amount of the principal and interest on the debt issued in period one. This policy can be thought of as a tax cut that leaves the present value of government spending unchanged. It is the effect of this type of fiscal policy that the RET addresses.

PROPOSITION I: If the fiscal policy given by equation (9) is enacted and if

[z.sub.s] [is greater than] 0 for s = l,h, then the RET holds.

PROOF: Define [c.sup.1] = [y.sub.1] + [B.sub.1] - [T.sub.1] and [c.sup.2] = [y.sub.1] + [B.sub.2] - [T.sub.1]. Then (0, [c.sup.1]] [union] ([c.sup.1], [c.sup.2]] gives the feasible set for period-one consumption, [c.sub.1]. First note that the feasible set for [c.sub.1] does not change in response to the fiscal policy given by equation (9), since using equation (4) for [z.sub.1] [is greater than] 0 and equations (8) and (9), [dc.sup.1] = [dB.sub.1] - [dT.sub.1] = [-dT.sub.2]/(1+i) - [dT.sub.1] = (1+i) [dT.sub.1]/(1+i) - [dT.sub.1] = 0, and [dc.sub.2] = [dB.sub.2] - [dT.sub.1] = 0 using equation (4) for [z.sub.h] [is greater than] 0 and equations (8) and (9). Next, suppose [c.sub.1] is constant such that [c.sub.1] [Epsilon] (0, [c.sup.1]). Then [c.sub.1] = [y.sub.1] + B - [T.sub.1] and dB = [dT.sub.1]. For [c.sub.1] [Epsilon] (0, [c.sup.1]), B lies in the range (0, [B.sup.1]) so the borrower never defaults. Hence [Mathematical Expression Omitted] for s = l,h. So using equation (9), [Mathematical Expression Omitted]. Now suppose [c.sub.1] is constant such that [c.sub.1] [Epsilon] ([c.sub.1], [c.sup.2]). Then [c.sub.1] = [y.sub.1] + B - [T.sub.1] and dB = [dT.sub.1]. In this case the borrower defaults if s = l, in which case [Mathematical Expression Omitted] = [c.sup.*]. Since [c.sup.*] is exogenous, if s = l in period two, then [c.sub.2] does not change. For s = h, [Mathematical

Expression Omitted]. Taking the total differential of this equation, using equation (4) for [z.sub.h] [is greater than] 0, and equations (7), (8), and (9), yields the result [Mathematical Expression Omitted]. This completes the proof of Proposition I.

The intuition behind Proposition I is as follows. The period-one tax cut equals the decline in the expected present value of the ability of borrowers to pay back debt in the second period. As a consequence, the levels of borrowing at which the loan schedule has kinks, [B.sub.1] and [B.sub.2], decline by the same amount as the tax cut. Hence feasible period-one consumption remains unchanged. Holding period-one consumption constant, for all feasible period-one consumption, the level of second-period consumption does not increase. Since [z.sub.s] [is greater than] 0, [z.sub.s] is a function of a borrower's future tax liability, for s = l,h. As a consequence, lenders take into account a borrower's future tax liability when determining the amount they will lend at a particular interest rate. When the government enacts the tax policy given by equation (9), lenders reduce the amount they will lend by the exact amount as the initial tax cut. This leaves the borrower's intertemporal budget constraint unchanged, which implies that the RET holds.

While the RET holds, current consumption may be responsive to changes in current income. Let current income increase by [dy.sub.1], in such a way that the expected present value of lifetime income discounted at interest rate i remains unchanged. This results in a change in the level of feasible period-one consumption, provided [Mathematical Expression Omitted]. This property of the model and Proposition I is interesting because it provides a reconciliation, within one model of consumption behavior, of the seemingly conflicting econometric evidence that rejects the PIH and accepts the RET. This is due to the model's property that the RET holds and that consumption is sensitive to changes in current income, due to the credit market's being imperfect.

PROPOSITION II: If the fiscal policy given by equation (9) is enacted and if

[z.sub.l] = 0 and [z.sub.h] [is greater than] 0, then the RET does not hold.

PROOF: Suppose that [z.sub.l] = 0 and [z.sub.h] = [Mathematical Expression Omitted]. Then, from equation (8), [B.sub.1] = 0 and [B.sub.2] = [hz.sub.h]/(1+i). Define [c.sup.1] = [y.sub.1] + [B.sub.1] - [T.sub.1] = [y.sub.1] - [T.sub.1] and [c.sup.2] = [y.sub.1] + [B.sub.2] - [T.sub.1]. Then (0, [c.sup.1]] [union] ([c.sup.1], [c.sup.2]] gives the feasible set for period-one consumption [c.sub.1]. Note that [dc.sup.1] = [-dT.sub.1], and using equation (9), [dc.sup.2] = [dB.sub.2] - [dT.sub.1] = [-hdT.sub.2]/(1+i) - [dT.sub.1] = -(1-h) [dT.sub.1]. Since [dT.sub.1] [is less than] 0, the fiscal policy given by equation (9) enlarges the feasible set of period-one consumption. This is sufficient to show that the RET does not hold if [z.sub.l] = 0 and [z.sub.h] [is greater than] 0. This completes the proof of Proposition II.

The intuition behind Proposition II is as follows. If the tax reduction is not enacted, and the borrower's period-two income equals the lower level, [Mathematical Expression Omitted], then the borrower defaults and the lender receives a zero partial payment, (i.e., [z.sub.l] = 0). When the tax policy given by equation (9) is enacted, the borrower still defaults if [Mathematical Expression Omitted] is realized. Since the tax cut does not affect the ability of the borrower to make a partial payment to the lender, if the borrower defaults, the lender still receives a zero partial payment. As a consequence, the debt level [B.sub.1] does not change due to the tax cut. For the borrower's higher level of period-two income, [Mathematical Expression Omitted], the ability to repay debt decreases by the increase in period-two taxes. As a consequence of lenders' being risk neutral, the maximum amount lenders will lend declines by less than the reduction in period-one taxes. Hence, the feasible set for period-one consumption increases.

One interpretation of Propositions I and II is that a bond-financed tax cut affects only the consumption of borrowers who, if they default, pay nothing in partial payment to their lender.

The above model can be reinterpreted to incorporate the case of two types of borrowers: those with high period-two income, [Mathematical Expression Omitted], who make up proportion h of the total population of borrowers, and those with low period-two income, [Mathematical Expression Omitted], who make up proportion (1-h) of the total population of borrowers. Borrowers with low period-two income always default. Lenders know that there are two types of borrowers, and know the proportions of each type in the total population of borrowers. Both types of borrowers are assumed to have the same period-one income [y.sub.1]. As a consequence, the lender cannot distinguish between the different types of borrowers by observing their period-one income. Under this specification, Propositions I and II have a slightly different interpretation. If both types of borrowers always pay a positive partial payment, then the RET holds. However, if borrowers with low period-two income pay the creditor a zero partial payment, then the RET does not hold.

4. CONCLUSIONS AND SOME EMPIRICAL EVIDENCE

The analysis indicates that the RET will hold, even when the loan market is imperfect, so long as the amount that lenders are willing to lend is a function of a borrower's future tax liability. This is the case when lenders are always partially compensated when a borrower defaults. This result suggests that the existence of imperfect loan markets does not necessarily imply that the RET is false.

However, there exists empirical evidence that lenders often receive nothing in partial payment when borrowers default which, given Proposition II, implies that the RET does not hold. Data collected on 1946 and 1969 by the Administrative Office of the U.S. Courts shows in 82 to 87 percent of bankruptcy cases, the lender received no partial payment from the borrower (Stanley and Girth 1971, p. 28). If the process that determines whether a borrower defaults has been stable over time, these data reveal that lenders may expect to receive a zero partial payment when at least some types of borrowers default.

MARC HAYFORD is assistant professor of economics, Loyola University of Chicago.