To consider the effect of maturity on loan demand, it is necessary to construct a model with at least three periods, so that the effect of extending the maturity from one to two periods can be analyzed. To simplify the algebra and obtain closed form solutions, we introduce some strong assumptions, none of which, however, is essential to our results. The list of assumptions is the following:
(i) there are three periods;
(ii) there is no uncertainty;
(iii) the utility function is intertemporally separable and defined over non-durables and cars; utility is separable between cars and non-durables;
(iv) the relative price of cars and non-durables is fixed and equal to 1; the depreciation rate is 0;
(v) the income process is exogenous; income is substantially higher in the third period.
(vi) cars can only be bought in the first period, and cannot be sold in subsequent periods;
(vii) consumers can finance a fraction 3 of their value, where 3 is between 0 and 1;
(viii) there is a single asset and a single liability; the interest rate on the former is lower than that on the latter; the asset cannot be held in negative quantities; the liability can be used only to finance car purchases;
(ix) loans can have maturity (n) equal to 1 or to 2; if the maturity is 2, consumers can choose how much to repay each period. Each payment, however, has to be non-negative, that is consumers cannot borrow more money in subsequent periods.
Assumptions (i) to (iv) are not particularly important and are made only for the purpose of obtaining a closed form solution. The assumption concerning the income process (assumption (v)) serves two purposes. First, by making the income process exogenous we simplify the algebra. Second, the fact that income is much higher in the third period compared to the first two periods makes the problem interesting in that it gives the consumer an incentive to move resources from the future to the present, that is, to borrow. To borrow money you may with the help of speedy-payday-loans.com and Speedy Payday Loans online.
Assumption (vi) is stronger. By letting consumers buy a car only in the first period, we avoid having to model car purchases in every period, and we can abstract from transaction costs. In addition, assumption (vi) allows us to abstract from the issue of timing in car purchases, that is conceivably affected by credit conditions. As mentioned above, the primary motivation for this rather restrictive assumption is analytical tractability; the fact that our focus in this paper is on the interest rate and maturity sensitivity of loan demand, conditional on the decision to buy a car, provides an additional justification.
Assumptions (vii) to (ix) specify the nature of the loans available to the individual consumer. They incorporate our definition of liquidity constraints in the dimensions we mentioned above: the difference between lending and borrowing rates, and the requirement to collateralize the loan with the value of the car. Note also that consumers in this model can only borrow in the first period. If one wants to compare the results of our model to an ideal benchmark in which there are no liquidity constraints, one can relax these constraints; that is one can make the interest rates on loans and assets the same, allow consumers to borrow more than the value of their cars, and give them the possibility of borrowing in the second period to transfer resources from the third to the second period. The assumption that in the two-period maturity case the repayment in each period has to be greater than or equal to zero (assumption (ix)) is consistent with the basic institutional setting in the car loan market.
Given these assumptions, the consumer solves the following maximization problem:
and the terminal condition, h = 0- Here y and p are income and non-durable consumption in period i, and h is the asset at the end of period i, which pays interest rate rl. K is the value of the car purchased by the individual in the first period, of which a proportion 3 is financed, while rb is the borrowing rate. Note that by assumption rl < rb. P is the first payment on the loan in the second period. P is the second payment on the loan in the third period, which may be non-zero when the maturity equals to two periods. The first three constraints say that net assets cannot be negative. The next two constraints dictate that the financing share has to lie between 0 and 1. The sixth constraint says that the first payment cannot be negative, i.e. the consumer cannot finance more than the value of the car. The last constraint says that the consumer can at most pay back the entire value of the loan. With each one of the constraints above we associate a Kuhn-Tucker multiplier denoted by (k = 1,…, 7). Below we discuss certain aspects of the possible equilibria when the maturity of the loan is one or two periods.
2.1 Characterizing the solution
When the maximum maturity is one, P is constrained to be equal to 3X (1 + r) . In this case some consumers, depending on the pattern of their income and preferences, will be at corners; that is they will set either 3 =1 or 3 = 0. Such points correspond to kinks in the intertemporal budget constraint (IBC). Others, instead, will be on flat parts of the IBC and the equilibrium will be described by a tangency condition relating the ratio of marginal utilities to the relevant intertemporal price. In addition, the presence of liquidity constraints in this model can also distort the allocation between durables and non-durables: for consumers who want to transfer resources from the future to the present, cars will become relatively more attractive as they constitute the only way consumers can borrow. Speedy Payday Loans is the best way to borrow cash and spend it for some purchase you have being looking forward to to buy.
Note that given the difference between lending and borrowing rates, no consumer will simultaneously choose h > 0 and 3 > 0.7 Hence, if h > 0, the optimal finance share is zero, and the Euler equation links the consumption in periods 1 and 2 to the interest rate rl:
A second case to consider is one in which the finance share is zero but the first period assets h are also zero. The Euler equation for these consumers is characterized by a slack term:
In other words, these consumers face a shadow interest rate that lies between the borrowing and the lending rates.
In either case, since the maximum maturity is one period, consumers with 3 = 0 will not have a chance to borrow in period 2 and move resources from period 3 to period 2. The equilibrium condition will therefore be either an Euler equation involving rl (if h > 0), or an Euler equation involving a notional (but unobservable) interest rate that is higher than the lending rate (if h = 0).
For consumers who borrow a positive amount (but less than K), so that 0 < 3 < 1, the interest rate that enters the Euler equation is rb. Their Euler equation is:
Consumers who want to borrow more than K, will not be able to do so. These consumers will set 3 =1, and will have a Kuhn-Tucker multiplier entering the intertemporal first order condition:
Since these consumers will not be able to increase their finance share even if maturity is extended to two periods, we focus on consumers with interior values of the finance share.
Consumers with 0 < 3 < 1 will set h =0 and may have h equal or greater than zero. Although we cannot solve for the optimal finance shares for these consumers without further assumptions on the form of the utility function, we can characterize the optimal car value. If the optimal 3 is less that one we can show that in equilibrium Uc1 = Vf – When the optimal 3 is one, however, the equilibrium condition for K involves a Kuhn-Tucker multiplier:
If V” < 0, this implies that the optimal K increases when the liquidity constraint is binding.
The case of the two-period maturity is more complex, as we have to consider both the allocation between periods 1 and 2, and the allocation between periods 2 and 3. Given our assumptions, consumers can now choose how much to repay in each period, as long as the repayment amounts are non-negative. This structure assumes more flexibility in the repayment schedule than what is actually observed in practice, but by focusing on the least constraining case, we avoid making specific assumptions about the repayment schedules and obtain more general results.
In characterizing the equilibrium when maturity is equal to two periods, it is useful to make the following observations. First, 3hh = 0. Thus, if h > 0 and h > 0, then 3 = 0. This means that, to the extent that we are interested in interior values of the finance share (i.e. in 0 < 3 < 1), we can concentrate on cases where the assets are zero in at least one period. We next summarize our findings for this case under different repayment schedules.
If the repayment amount in period 2 is zero (i.e. P = 0), then it is possible that the consumer finances a fraction of the car purchase even if the first period or second period assets are positive. Hence, there are three conceivable subcases: (a) h = h = 0, (b) h > 0,h = 0, and (c) h = 0, h > 0. However, given the difference in the borrowing and lending rates, it is not possible to have simultaneously P = 0 and h > 0.
If the repayment in period 2 is positive but less than the entire value of the loan (0 < P < 3 K (1 + rb)), then it is not possible to have 0 <3< 1, h = 0 and h > 0 at the equilibrium, nor can we have 0 <3< 1, h > 0 and h = 0. For a positive repayment amount in period 2 and interior values of the finance share, the only relevant case is thus the one in which both h = 0 and h = 0. The consumer is here trying to move resources from the last period of his/her life to the earlier ones.
Finally, we allow consumers to repay the entire loan in the first period (P = 3K(1 + rb)). In this case the optimal finance shares when h = h = 0 or h = 0,h > 0 coincide with those obtained when the maximum maturity is equal to 1, while, as in the one-period maturity case, it is not possible to simultaneously have 0 < 3 < 1 and h > 0.
Similar to the one-period maturity case, we can show that for consumers with 0 < 3 < 1 and P > 0, the Euler equation is:
while if P = 0, it is Ucx = (1 + r6)2ft2Uc3. And as before, we can characterize the optimal K for consumers with 0 < 3 < 1 through the relationship, Vk = Uc±-
In short, our discussion of the solution of the model for the cases in which the financing share is an interior point may be summarized in the following Table:
NA means that the optimal share is not a feasible solution. To derive an explicit expression for the optimal 3’s, and compare the effects that the interest rate has in the various cases, we have to be specific about the form of the utility functions for non-durable consumption U (C), and durable consumption V(K). In the Appendix we give the expressions for the optimal 3’s in the table assuming isoelastic functions,9 that is, U(C) = C1_7/1 — 7, and V(K) = K1_7/1 — 7. Which of the entries is relevant, depends on the particular assumptions one makes about the income process, the interest rates and the parameters of the utility function. We should point out that these assumptions are not used in the empirical part. They are only made in this section to help us study the effects of a maturity increase on the finance share.
2.2 Liquidity Constraints and the Effects of a Maturity Increase
To study the effects that an increase in the maximum maturity has on the loan demand of liquidity constrained consumers, one has to compare the optimal finance share in the one-period maturity case to the optimal finance share in the two-period case. The discussion in the previous subsection should immediately make clear that this is a very difficult task. Even when one assumes specific functional forms, the expressions for the optimal finance shares depend on which specific constraint is binding in each case; depending on the particular values of the multipliers and the values of each period’s assets (which of course are endogenously determined in the model) one can obtain a variety of optima, and an evaluation of the effects of a maturity increase requires a large set of bilateral comparisons.
In analyzing the effects of maturity, it is useful to distinguish between two cases: the case where the optimal finance share takes interior values, and the case where the consumer is at a corner (this corresponds to 3 = 0, or 3 =1). As mentioned above, for consumers who are at an interior and decide to pay off the entire amount of the loan in the second period, maturity has no effect on loan demand. These consumers are obviously not constrained in the sense considered in this paper. However, in many cases consumers at an interior solution will have different optimal shares depending on whether the maximum maturity is one or two periods. To show how the optimal finance shares change as a function of maturity, we look at some specific numerical examples below.
For consumers at a corner, the argument is less clear cut. Suppose that the consumer’s preferences and income path are such that he/she sets 3 = 0 in both the case where the maximum maturity is one, and the case where the maximum maturity is two. Then a maturity extension obviously has no effects on the fraction financed, even though the consumer may very well be liquidity constrained. The same applies to the case 3 =1- Hence the experiment of a maturity increase will not be very useful in identifying liquidity constrained consumers who are at corners. To the extent of course that a consumer switches from 3 = 0 in the one-period maturity case to 3 > 0 in the two period case, the maturity increase experiment can be informative. The former cases show that our test could have, under certain circumstances, limited power.
Now consider the case 0 < 3 < 1. To get an idea of how loan demand responds to interest rate and maturity changes, we use the formulas given in the Appendix to compute the optimal finance shares for alternative income paths and interest rates. The results from such a numerical example are depicted in Figure 1. We assume that a = 0.9, and 7 = 0.8. To make the problem interesting, we assume a rising income path with a relatively high income in the last period; in particular, we set yi = 1.5, y2 = 2, and y$ = 4. We then compute the optimal finance shares, as well as each period’s consumption and assets, for each possible equilibrium scenario and for 80 different values of the borrowing interest rate r,. As required by the theoretical model, rb is specified to be greater than r, the latter being set equal to 10 percent. To characterize the optimal financing shares, we compute all the quantities in the Table and check which, among those that do not violate any of the constraints imposed by the model, correspond to the highest value of utility.
Comparing the utility levels at each borrowing rate, we find that the consumer would choose h = h = 0, 0 < P < 3X (1 + r) , 0 < 3 < 1, if a two period loan is available, and h = h = 0, 0 < 3 < 1 if maximum maturity is equal to one period. In Figure 1 we plot the optimal finance shares for each one of these two cases as a function of the borrowing rate. The graph exhibits three interesting features. First, in both cases the lines are downward sloping indicating that loan demand is negatively related to the interest rate. Second, the line corresponding to a longer maturity (m = 2) lies above the line for the one-period maturity case. Hence, the graph suggests that for the consumers with the assume features, longer maturities are associated with higher finance shares. Third, the slopes of the lines depicting the finance share as a function of interest rate depend on whether the maximum maturity is one or two periods. This indicates that maturities affect loan demand interactively with the interest rate (this is also evident in the formulas for the optimal finance share in the Appendix).
Finally, the implications of the model described by assumptions (i)-(ix) can be compared to an ideal benchmark in which there are no liquidity constraints. Suppose that the borrowing rate rb is equal to the lending rate r*, that borrowing is not restricted to the first period only (or, equivalently, that buyers can roll over car loans), and that consumers can borrow an amount that exceeds the value of their car (that is 3 is unconstrained). Then it is easy to see that the length of the maturity is irrelevant for consumers’ decisions. Given that there are no constraints on 3, consumers can borrow as much as they want in the first period; and given that they can also borrow in the second period, longer maturities are no longer the only way by which resources can be transferred from the third to the second period.
In summary, the simple model considered in this section has the following empirical implications:
(1) In the absence of credit rationing, loan demand is independent of maturity. When liquidity constraints are binding, however, loan demand will $ in most cases $ be an increasing function of maturity.
(2) In the presence of binding liquidity constraints, maturity affects loan demand interactively with the interest rate.
It is important to note that, while these general implications form the basis of our empirical tests, we do not take the exact results or functional forms at face value. The empirical investigation of these implications will have to deal with several challenges, ranging from the endogeneity of interest rate and maturity, to issues associated with sample selection bias and corner solutions. We discuss these issues extensively in the next section.
Figure 1: Optimal Finance Shares: