The role of interest rate taxes in credit markets with divisible projects and asymmetric information

Journal of Public Economics 39 (1989) 33344. North-Holland

THE ROLE OF INTEREST RATE TAXES IN CREDIT MARKETS WITH DIVISIBLE PROJECTS AND ASYMMETRIC INFORMATION

David DE MEZA

University of Exeter, Exeter EX4 4RJ, UK

David C. WEBB*

London School of Economics, London WCZA ZAE, UK

Received September 1988, revised version received February 1989

This paper assumes that entrepreneurs have divisible projects with random payoffs. However, managerial skill is unobservable to outsiders. Projects are assumed to be financed through debt. A pooling equilibrium is shown to exist in which good and bad entrepreneurs sell debt at the same price. If both types of entrepreneur have a positive probability of bankruptcy, project scale will be too large. This provides a prima facie case for raising interest rates and hence the cost of capital through taxation. If in equilibrium only the poor entrepreneurs go bankrupt, there is an outlying possibility that interest rates are too low and a subsidy is called for.

1. Introduction

Banks can rarely identify the precise credit worthiness of potential borrowers. As a result, good risks typically end up paying higher interest rates than their characteristics merit, whilst poor risks obtain credit on terms that are better than actuarially fair. This raises questions of whether such an equilibrium is efficient and, if not, of what form of intervention is appropriate. In earlier papers [de Meza and Webb (1987, 1988)] we established that if projects are of fixed size and their quality is better known to entrepreneurs than to banks, then, under reasonable assumptions about the distribution of returns, an interest rate tax is efficient. This result contrasts with widely held intuitions concerning the performance of credit markets under asymmetric information [e.g. Akerlof (1970) Leland and Pyle (1977)] and may provoke suspicions that important elements of reality are missing from the model.

The reason the tax is beneficial in our 1987 paper is because it improves the average quality of projects financed. In its absence, low yielding projects

*We appreciate the constructive comments of two referees which have proved invaluable in revising this paper.

0047%2727/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

34 D. de Meza and D.C. Webb, Interest rate taxes

are undertaken only because of the implicit interest rate subsidy they receive from the good projects. If such projects had to pay the full opportunity cost of the funds involved they would not be profitable and so an interest rate tax efficiently counteracts the market generated subsidy. Good projects also have to pay the tax but being more profitable than the poorer projects are not eliminated and, by the indivisibility assumption, the tax does not affect the way they are implemented.

That projects are either accepted or rejected but never modified in response to changes in the cost of capital, appears as though it may bias results in favour of an interest rate tax, or at all events make the desirability of such a policy ambiguous. This paper shows that this is not normally the case.

In section 2 we set up a model with variable project scale in which the identity of the set of projects receiving finance is known but the productivity of individual entrepreneurs is unknown to banks. The structure of the model is similar to that of Bester (1985) and Milde and Riley (1988). However, both these papers are restricted to cases in which separation occurs under laissez- faire. Here we are primarily concerned with pooling equilibria. In section 3 the characteristics of an equilibrium are identified and then, in section 4, we examine the social efficiency of equilibrium. There we characterise the optimal linear interest income tax for a pooling equilibrium. It is found to be unambiguously positive if in equilibrium both types of entrepreneur have a positive probability of bankruptcy. The improvement in social efficiency arises through the constraining effect of the tax on the size of the projects that are undertaken. In conjunction with our earlier work this establishes that an interest rate tax, in eliminating the worst investments, not only leads to a better composition of projects undertaken but also, if project size is variable, to gains from each remaining project being operated at a more efficient scale. There is a qualification. If it is only poor quality entrepreneurs who go bankrupt it is possible that an interest rate subsidy is called for. A necessary condition is that the good state is an outlier which occurs with low probability.

2. The model

2.1. Entrepreneurs

Entrepreneurs indexed by i have projects which have a variable capital input K. Output is given by the random return stream:

e~if( K), f’ > 0, f” < 0,

where ;li> 0 is an index of project quality and 9 is common stochastic shock

D. de Meza and D.C. Webb, Interest rate taxes 35

to output or the price at which it sold. Introducing individual specific risk makes no essential difference to our results but is notationally messy. The higher the value of 1 the better the quality of the entrepreneur. The formulation of the payoff function (1) reflects the idea that given the state of the world, 6, the more able entrepreneurs always do better than the less able. To keep the subsequent analysis simple, without much loss we assume that the common stochastic shock 0 takes on only two values, 8 and 8, with a>S. The high value $ occurs with probability p and @ occurs with probability (1 -p). In this paper we assume two types of entrepreneur, high and low quality, h and 1, respectively, with A, >L,. Type h entrepreneurs form a proportion c( of the population.

Entrepreneurs are risk neutral, they have no initial wealth, and must obtain outside finance to undertake their projects. If entrepreneurs had financial resources of their own to invest or own assets that could serve as collateral, the analysis would be unchanged if a pooling equilibrium exists. However, as Bester (1985) shows, an entrepreneur’s willingness to accept collateral requirements or partial self-finance plays a signalling role, making a separating equilibrium more likely. Here we assume that outside finance takes the form of risky debt with the entrepreneur having limited liability, so that if he cannot repay the debt he defaults and forfeits the value of his project. The entrepreneur borrows K at a rate of interest r and in the relevant range (1 + r)K < enif but (1 + r)K 2 e~eif(K). Since the entrepreneur is risk neutral his objective function is to maximise

~i=p[83~if(K)-(l+r)K]+(l-p)max~3.if(K)-(l+r)K,O]. (2)

Taking r as given, the value of K which maximises (2) satisfies one of the following first-order conditions:

pe;lif’(K)+(l-p)eAif’(K)-(l+r)=O for@A,f(K)z(l+r)K, (3a)

P[eAif’(K)-(1 +r)]=O, for 0A,f(K)<(l +r)K. (W

These equations can be solved to yield the demand for K as a function of r.

2.2. Banks

Finance is supplied by risk-neutral banks. We assume two banks engaged in Bertrand competition. Banks cannot observe the characteristics of individual borrowers but do know the properties of the population of borrowers and the distribution of project parameters. The banks obtain finance at the


fixed deposit rate p and supply finance to maximise expected profit.’ Given that a particular contract involves a loan of size K and interest rate I, if it is the sole contract offered and attracts both types its expected profitability is:

n,=p(l +r)K+(l-p){ a min [e&f(K), (1 + r)K]

+(l-a)min[0LJ(K),(l+r)K]}-(l+p)K. (4)

If 0 co! c 1 the contract attracts a pool of applicants with average quality il=ai,+(l-LY)l,.

2.3. The game

We consider a three-stage game: Stage 1. Each bank announces a set of contracts of the form (( 1 + r), K). Stage 2. Each borrower applies for one contract at one bank. If both

banks offer the same contract, each applicant tosses a fair coin to determine the bank to which he applies (if he applies at all).

Stage 3. The bank decides whether to accept or reject applications. A bank will withdraw contracts which will be unprofitable given the contract offers of other banks. Borrowers who find that the contract they apply for is not funded select one of the contracts still on offer or else do not invest.

The equilibrium concept used corresponds to a Wilson equilibrium (1977). For a more detailed discussion of the relevant issues see Hellwig (1987).

3. Analysis of equilibrium

First we analyse the behaviour of entrepreneurs. From (2) the entrepreneurs’ iso-profit curves satisfy:

paAif’(K)dK +(l -p)eiJ’(K) dK -Kd(l +I) -(l +I) dK ~0,

for e&f(K)~(l+r)K, i=h, I, (5a)

p[??&f’(K) dK -K d( 1 + r) -( 1 + r) dK] = 0,

for CjA,f(K)<(l+r)K, i=h, I,

implying:

(5b)

d(1 +r) LiE(e)f’(K)-(1 +r) ----= dK K ’

for @J(K) 2 (1 + r)K,

‘We assume that p is fixed purely for simplicity. All of the results of the paper hold true if p is increasing in the volume of finance.

D. de Meza and D.C. Webb. Interest rate taxes 31

where E( 0) = @ + (1 - p)j, and

d(l+r) = A.iOf’(K) - (1 + r) dK K ’

for @f(K) <( 1 + r)K. (W

It can easily be checked from (6a) and (6b) that the iso-profit contours have a positive slope for K <K*, where K* satisfies (3a) and (3b), respectively, and a negative slope for K > K *. The iso-profit contours are drawn as concave in each region in (K,(l +r)) space. If one draws the demand curve and associated iso-profit curves in (K,(l + r)) space, the demand curve is discontinuous at the bankruptcy point, Ri, satisfying nif(K,e) =( 1 +p)K. TO the left of the bankruptcy point it satisfies (3a) and to the right (3b). The iso- profit curves in (5a) and (5b) are continuous between the two regions but will be decreasing immediately before the bankruptcy point and increasing immediately after, giving a wing shape.

From (3a) and (3b) we can see that if e&j(K) 2 (1 + r)K, then neither type of entrepreneur defaults so that the demand curve for both types satisfies (3a) set equal to zero. Since 2, <II,, and f” ~0, the demand curve in (( 1 + r), K) space for a type h lies to the right of that for a type 1. Now suppose that $A,f(K) < (1 + r)K but @&f(K) z(l + r)K so that the demand curve for a type 1 satisfies (3b) whilst that for a type h satisfies (3a). In this region the relative location of the demand curves depends upon whether 1,E(B)~;1,8. If &E(B) >A$ the demand curve for type h lies to the right of that for type 1. However, if &E(B) < A$, which is possible, the demand curve for type 1 lies to the right of type h. Finally, suppose that eA,f(K) <(I +r)K and @&f(K) < (1 + r)K, then the demand curves of both types satisfy (3b). Then since A,, > 1, the demand curve for type h lies to the right of that for type 1.

Now we consider the banks’ financial offers. By a standard implication of Bertrand competition, in equilibrium all contract offers earn banks zero expected profit. If banks identify the applicants’ types, then the loan supply function satisfies (4) set equal to zero with LX= 1 if the applicants are high- quality and with a=0 if the applicants are low-quality. For high-quality applicants the supply function in ((1 i-r), K) space is horizontal with (1 + r) = (1 + p) for e&f(K) 2 (1 + P)K becoming upward sloping with (1 + r) > (1 +p) when K is high enough to reverse the inequality. For the latter, the default threshold at which the supply function becomes upward sloping satisfies eA,f(k) =(l +p)K and is located to the left of that for the high- quality entrepreneurs. In the event that applicants are not identified and both types receive the same contract the loan supply function is given by (4) set equal to zero with O<u< 1. This supply function lies between those for types h and 1 with its location determined by the value of ~1. We now illustrate two possible equilibria.


6, Rh K

Fig. 1

Case 1 . In fig. 1 we illustrate the case where the parameters are such that equilibrium occurs in the region where both types of entrepreneur default in the bad state. Here Z denotes a contract offer, (( 1 +r), K), S denotes a loan supply curve, rc,, and x1 are the respective entrepreneurs’ iso-profit curves, and D, and D, their respective demand curves. There are two types of equilibria to consider: separating and pooling. It is useful to begin by considering the characteristics of a separating equilibrium.

A separating equilibrium where each bank offers two contracts, Z,, and Zr, must satisfy the following conditions:

(a) incentive compatibility: rci,(Zh) 2 rc,(Z,), rc,(Z,) 2 n,(Z,); (b) acceptability: rr,(Z,), ~~(2~) 2 0; (c) zero profit for banks: zB(Zh) = n,(Z,) = 0. As long as a is sufficiently small the pair of contracts (Z,,Z,) satisfies the

above requirements and constitutes a separating equilibrium of the game specified in the previous section. The reason is that the bank’s offer curve, curve s, lies above the high-quality entrepreneur’s iso-profit curve rr,,(Z3) passing though Z,. However, in fig. 1 this last condition is not satisfied since a is too high. Contract Z, is strictly preferred to the contracts (Z,,Z,) by both types, and as it lies on the pooling offer curve a separating equilibrium cannot exist.

For the contract Z, to constitute a pooling equilibrium of the postulated game it must be stable with respect to possible perturbations. The testing procedure is to suppose that at the first stage contracts such as Z, are


introduced alongside Z,. At the second stage, Z, would be demanded by high-quality borrowers leaving only low-quality borrowers demanding Z,. But then Z, would be unprofitable and would be withdrawn at the third stage. As a result low-quality borrowers must demand Z, which is unprofitable if taken by both groups. Hence, neither bank has an incentive to introduce Z,. If LY is small it can easily be checked that the pooling contract cannot be an equilibrium since it can be broken by offering an appropriate pair of contracts. In that situation (Z,, Z,) is the equilibrium.

The equilibrium contract, Z,, has the property that the slope of the type h entrepreneur’s iso-profit curve rr,, (Z,) equals the slope of the offer curve S. The slope of rc,, (Z,) is given by (6b) with ),,=A,. To calculate the slope of s take the total differential of (4) set equal to zero with O<cc< 1 and e&j’(K) <@&f(K) <( 1+ P)K <( 1 + r)K, and rearrange to get:

d(1 +r) -p(l+r)-_(l-p)Q;lf'(K)+(l+p)

dK PK (7)

where 1, = cr& + (1 - cr)ll,. Equating (6b) with ii = 1, to (7) we obtain:

p~%,f’( K) + ( 1 - p)tj(ctl, + ( 1 - a)Lr)f’( K) - ( 1 + p) = 0, (8)

which solves for the competitive equilibrium capital stock K,. For the equilibrium to entail both types going bankrupt in the bad state (K,>R,) it is easily seen that 8 should be high relative to fj, CI should not be too low, and 2, and R1 reasonably close.

An alternative method of locating the equilibrium is to equate the marginal product curve of the high-quality entrepreneurs D, in the region where e&f(K) ~(1 +p)K to the marginal cost curve kC derived from s to get:

??&f’(K)dK=Kd(l+r)+(l+r)dK (9)

and use the total differential of (4) set equal to zero with O<cr < 1 and ~L,f(K)<8&f(K)<(l+p)K<(l+r)K, to substitute for the right-hand side of (9) and again obtain (8). Note that in fig. 1 the MC curve is discontinuous at the points where each type goes bankrupt in the bad state, I?I and r?, respectively.

Case 2. In fig. 2 we illustrate an equilibrium in the region where only type 1 entrepreneurs default in the bad state. Again we assume that tc, the proportion of type h entrepreneurs, is sufliciently high to yield a pooling equilibrium.

It can be proved that in fig. 2 the contract at Z,, the preferred contract on s for type h, is a Wilson pooling equilibrium. The case illustrated is for

40

(1 +r:

D. de Meza and D.C. Webb, Interest rate taxes

Sh

* K

Fig. 2

m-(K) ’ ~(mJ-(K) so that II, lies to the right of D,. The equilibrium lies at the tangency point of ~~(2~) and the pooled offer curve S. Alternatively, the equilibrium value of K can be found by equating the marginal product curve of type h with the marginal cost curve, MC:

@&f’(K) dK + (1 - p)tj&J’( K) dK = K d( 1 + I) + (1 + r) dK.

Then take the total differential of (4) set equal to zero with O< c( < 1 @A,f(K) <( 1 + p)K <( 1 + r)K and fj&f(K) > (1 + r)K. On substituting resulting expression in the right-hand side of (lo), we find that competitive equilibrium investment level must satisfy:

4. Social efficiency

(10)

and the the

(11)

Here we examine the social efficiency of the pooling equilibria illustrated in cases 1 and 2. We consider these cases in turn and explore the role of simple linear taxes and subsidies on interest rates. The government cannot


(1 +r ‘)

(1 +P+T) (1 + P)

/

/

-

-

:

i

: ; : :

: i : : : :

. i : .

lx,’ R, lx,’ c(h Ke Kc K

Fig. 3

enforce a first-best separating equilibrium so that our argument is strictly second best2 Here we pay no attention to equity considerations.

Case 1. In fig. 1 the pooling equilibrium .Z4 Pareto-dominates the pair of contracts (Z,,Z,). However, the competitive equilibrium contract Z, is inefficient. The interesting feature is that the volume of investment, K,, is socially excessive. Interest rates are too low and with a single tax/subsidy instrument an interest rate tax achieves a rise in social efficiency.

Consider fig. 3 which illustrates a pooling equilibrium at Z,. The competitive equilibrium investment level K, satisfies (8) where D, intersects the private marginal cost of capital schedule AC. The (second-best) socially efficient pooling equilibrium occurs at the intersection of the social marginal product of capital given by the pooled demand curve D and the social marginal cost curve, (1 + p). That is, for social efficiency,

pei*f’(K) + (1 - p)@f’( K) - ( 1+ p) = 0, (12)

where i=al,+(l --a)& Comparing (12) and (8), given f”<O, the level of investment K, which satisfies (12) is less than the competitive equilibrium level determined by (8). In the figure, D, lies to the right of D and intersects the curve A?C to the right of where D intersects (1 +p).

An efficient pooling equilibrium is obtained by imposing an interest rate

‘The government is faced with exactly the same informational constraints as the banks. Hence, we restrict attention to the use of linear taxes to enforce the best pooling equilibrium.


tax, t =z/p, thereby raising the marginal cost of capitaL3 The competitive equilibrium now satisfies the following condition derived from (8):

P&f’(K) + (1 - p)&f’(K) - (1 + p + 7) = 0, (13)

where E,=cr&+(l -I+,. In fig. 3 the MC curve moves to the left, as does the bankruptcy point for type h. The competitive equilibrium remains in the region where type h goes bankrupt in the bad state.

Comparing (13) and (12), and evaluating both at K = K,, it is seen that the efficient tax satisfies:

~=~g(l-a)(&-&)f’(K,). (14)

According to (14) the efficient tax is zero if either c( = 1 or E,, = L1 and that tax is increasing in (1 -a), (A,,-&) and p. It should of course be remembered that this tax only applies in the region where we have a pooling equilibrium with both types of entrepreneur defaulting in the bad state.

The tax identified in (14) is that which is optimal for the pool of projects with the specified distribution of returns. In practice, bankers may be able to discriminate between different pools. The tax appropriate for each pool would differ. But even if the only feasible tax is uniform across pools, granted that there is a chance of default on all loans, it is certain that a positive tax, if sufficiently low, yields a potential Pareto improvement.

Case 2. Consider fig. 4 which illustrates a pooling equilibrium at 2, where in the bad state only the poor entrepreneurs go bankrupt. The optimal policy depends on whether E(O)%,f(K)~~AJ(K). Fig. 4 illustrates the case where the volume of investment is socially too small. The competitive equilibrium investment level satisfies (II), where D, intersects MC. Use of the social efliciency condition (12) reveals that at K = K, the optimal tax/subsidy is:

z=p(l -~)[A,E(fil-A@]f’(K,). (15)

From this we can see that if &E(8) <A,& as in fig. 4, the intersection of D, and MC lies to the left of the intersection of D and (1 +p) and a subsidy (z < 0) is called for. Expression (15) shows that if E(8) is close to 0 and I!? is an outlier which occurs with low probability, the’& provided 1, s not too much greater than I,, this result obtains. It might be thought that K, could be so far in excess of K, that optimality involves both types of entrepreneur going bankrupt in the bad state. This is impossible. The intuition is that in

3The interest rate tax, t =r/p, raises the cost to banks of obtaining deposits to (1 +p+ tp)=( 1 +p+~). In the present context p is fixed so that the interest rate tax is equivalent to a tax on deposits.

(1 +r ‘)

(1 +I4 4 (1 + P + 7)

L

D. de Meza and D.C. Webb, Interest rate taxes

Dh

. : : : : : : : : : . : : ; i : : : : : : : . : : : : : : : : : : : : : : : .

1

R,’ K,K, Kh K; K

Fig. 4

43

the region of K, there is a positive externality associated with type h’s increasing their investment. In selecting K, type h entrepreneurs ignore the expected benefit to type I’s whose marginal product in the bad state, the only state that matters to them, is in excess of (1 +r). But once investment exceeds f?, the externality becomes negative and so it can never be socially efficient for K to be at such a high level. Moreover, if, as a result of a subsidy, the equilibrium investment were K,, it is immediate that the good entrepreneurs would not go bankrupt. Hence, if in equilibrium good entrepreneurs are solvent in both states and A,&(0) <A,& it is optimal to introduce an interest rate subsidy, and this will not involve a switch to the regime in which the good entrepreneurs go bankrupt in the bad state.

Of course in the region where only type 1 entrepreneurs default we may have that &E(8) > ;C,& then the intersection of D, and MC lies to the right of the intersection of D and (1 + p) and a tax (T > 0) is required.

If a tax is required to yield a potential Pareto gain, then as illustrated in fig. 3, both groups are worse off in the post-tax equilibrium, Z6, than they were at Z,. By definition of a Pareto improvement the tax revenue could be distributed so as to make both groups better off. However, it is of interest to note that if the revenue from the tax, TK,, is returned in an equal lump-sum way to all borrowers, we move to Z6 directly below Z, at K,. It is easy to show that nh(Z,)<z,,(Z4) but rr,(Z,)>z1(Z4); low-quality borrowers gain at the expense of the high-quality borrowers.

44 D. de Meza and D.C. Webb. Interest rate taxes

5. Conclusions

It is hard to imagine that pooling equilibria are not the rule in credit markets. Rare is the banker who can be confident of an applicant’s innate chance of success simply by observing the size of his/her loan request. Taking other easily observed signals into account will seldom completely resolve the uncertainty either. The principal result of this paper suggests that giving a pooling equilibrium an interest rate tax will normally be beneficial even when its only effect is to influence project size and there is no improvement in the composition of projects undertaken. The conditions for a subsidy to be appropriate seem unlikely to apply. It is quite striking that when there is a chance that any project, however good, will default, an interest rate tax is appropriate. After all, in a pooling equilibrium the good entrepreneurs in effect pay an interest rate tax whilst the low-quality entrepreneurs receive a subsidy. Some projects are thus too small and others too large. Whether reducing the size of all projects is beneficial might therefore be expected to be sensitive to parameters such as the value of the elasticity of the investment function, but this is not so.

References

Akerlof, G.A., 1970, The market for lemons: Qualitative uncertainty and the market mechanism, Quarterly Journal of Economics 84, 488-500.

Bester, H., 1985, The level of investment in credit markets with imperfect information, Zeitschrift fur die gestame Staatwissenscaft, 503-515.

De Meza. D. and D.C. Webb. 1987. Too much investment: A oroblem of asvmmetric information, Quarterly Journal of Economics 102, 281-292. L

De Meza, D. and D.C. Webb, 1988, Credit market efficiency and tax policy in the presence of screening costs, Journal of Public Economics 36, l-22.

Hellwig, M., 1987, Some recent developments in the theory of competition in markets with adverse selection, European Economic Review Papers and-Proceedings 31, 319-325.

Leland. H. and D. Pvle. 1977. Informational asymmetries. financial structure and financial intermediation, The-Journal of Finance 32, 371-388.

Milde, H. and J.G. Riley, 1988, Signalling in credit markets, Quarterly Journal of Economics 103, 101-129.

Wilson, C., 1977, A model of insurance markets with incomplete information, Journal of Economic Theory 16, 167-207.

The role of interest rate taxes in credit markets with divisible projects and asymmetric information

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