Finance expert
JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 23, NO. 1, MARCH 1988
Debt versus Equity under Asymmetric Information
M. P. Narayanan*
Abstract
In a world of asymmetric information in which only the insiders ktiow the quality of the firm, it is claimed that debt, even if it is risky, is more advantageous than outside equity because issuance of debt is less attractive to inferior firms. The advantage to debt arises from the fact that it can keep unprofitable firms out of the market, thus improving the average quality of firms in the market. This advantage exists even if the firms cannot be perfectly sorted in the signaling equilibrium.
I. Introduction
The traditional explanation of a firm's capital stmcture decision balances the benefits of interest tax shields against the expected bankmptcy costs. The depen- dence of this argument on the existence of corporate taxes has drawn cdticism (see Miller (1977)), since it does not explain the use of debt before the introduc- tion of corporate taxes. Several explanations that do not depend on corporate tax laws for the use of debt have appeared in the past decade. Jensen and Meckling (1976) argue that there are agency costs associated with both equity and debt financing and an optimum mixture minimizes the total agency costs. Ross (1977) suggests that the manager of a firm whose wages depend on current and future values of the firm will use debt to signal the quality of the firm (known only to him) to the market. The dependence of the wage on the current value of the firm gives him the incentive to signal, while a penalty in the case of bankmptcy dis- suades him from overstating the value. Leland and Pyle (1977) contend that the proportion of equity held by the owner-manager acts as a signal to the quality of the firm.
In this paper, a different explanation is provided for the use of debt. It is claimed that in a world of asymmetdc information in which the outsiders are less informed about the quality of the firms than the insiders, the use of debt by profit- able firms keeps the inferior firms out of the market even when the market is
' Graduate School of Business Administration, The University of Michigan, Ann Arbor, MI 48109-1234. An earlier version of this paper titled "Corporate Lemons and Capital Stmcture," was presented at finance workshops at Duke University and the Universities of Michigan and Southem Califomia. The author thanks the participants of these workshops and an anonymous JFQA referee for useful suggestions while retaining responsibility for any errors. Part of the research for this paper was done while the author was at the University of Florida whose financial support for the research the author gratefully acknowledges.
39
40 Journal of Financial and Quantitative Analysis
unable to perfectly distinguish between firms of different quality. The elimination of infedor firms from the market increases the average quality of the firms re- maining in the market. This benefits the firms that remain because if the market is unable to discriminate between them, it would value all of them at the average value.' Clearly, this explanation is independent of corporate tax laws.
The main result of this paper is that, when secudties of any given firm are being underpdced by the market, the firm will prefer debt to extemal equity. If we restdct ourselves to dsk-free debt, this result is tdvial. By definition, dsk-free debt cannot be mispdced and, hence, undervalued firms would prefer debt to underpdced equity. The issue gets complicated when debt is dsky since now if a firm is undervalued, both its debt and equity will be underpdced and it is not clear which mode of financing is beneficial to the firm. While this result is similar to that of Myers and Majluf (1984), there are very fundamental differences be- tween our models. Their model is based on the assumption that there is informa- tional asymmetry regarding not only the proposed project for which financing is required, but also the assets-in-place. Their results are ddven by the concem of the manager regarding the undervaluation of the assets-in-place when the firm goes to the capital market to finance the new project. In our model, we only need informational asymmetry regarding the new investment opportunity. Therefore, the results of our model will hold (unlike those of Myers and Majluf) even if (a) there is no asset-in-place, i.e., it is a newly fioated firm; (b) there is no informa- tional asymmetry regarding the assets-in-place, i.e., the firm is a mature, stable one; and (c) the new project is spun-off as a separate firm, or a separate class of stock and/or debt is issued to finance the new project. Moreover, in their model, insiders do not face any uncertainty at the time they need to make the investment decision; that is, they know the tme cash fiows that will occur in the future. Since all the proposed projects have positive net present values, the lack of uncertainty implies that debt financing is dsk free from the insiders' perspective. In equilib- dum, the outside investors also recognize this. Hence, debt is always dsk free in their model, and this, as stated earlier, makes the problem tdvial. In our model, investors face a dsky investment decision and, hence, debt is dsky. We shall discuss these and other differences between the models in greater detail in Sec- tion IV.
The paper is organized as follows. Section II sets up the basic model. Sec- tion III discusses the effect of informational asymmetry when the only source of capital is extemal equity. Section IV is the main section, in which there is a choice of both debt and equity financing, and where it is shown that dsky debt is preferred by good quality (and hence undervalued) firms. Section V concludes the paper.
II. The Model
Consider a world in which the output of firms is given by the equation,
(1) y = V(«,/) + e, >- ^ 0 ,
' In this paper, only semi-separating equilibria, that is, equilibria with different pools of firms, are considered. For the development of a fully separating equilibrium, see Narayanan (1985).
Narayanan 41
where y is the output, / is the investment, n is the production parameter or type of the firm, and e is a random shock with zero mean. V is the production function, and we assume that it is the same for all firms.^ Since the parameter n, which can vary across firms, represents firm-specific factors like the quality of manage- ment, philosophies, efficiency of the production facilities and processes, market- ing capabilities of the firm, etc., n can therefore be a vector. The investment / is an exogenous variable in our model and is assumed to be the same across firms. It may be viewed as the optimum investment that the firm wishes to make in a world of symmetdc information and is assumed to be independent of the financ- ing decision. G is the distribution function of e. Without loss of generality, we can assume that V is monotonically increasing in n, i.e., the better the type, the higher the expected output.3 Given this one-to-one relationship between V and n, V can be used as a proxy for n. For the sake of simplicity, we assume without loss of generality that the time value of money is zero.
Let Gy denote the distdbution function of the output of a firm of type V. From (1), it is seen that dGy/dV ^ 0 for all y, with stdct inequality for some y, i.e., the distdbution of the output of a supedor firm stochastically dominates (in the stdct first-order sense) that of an infedor firm. It must be noted that the ran- dom shock is the same for all firms in the group. This would be a reasonable assumption if e were to be considered as a shock due to economic factors affect- ing the group of firms as a whole, while inter-firm differences would be embod- ied in n.
Let F be the (objective) distribution function of the types V with support [V/,y^]. We assume that V is continuously distdbuted in this interval. For the market to show any interest in these groups of firms, Vf, must be strictly greater than /; that is, at least some firms must expect to make profits. Also, V^must be strictly less than /. In other words, firms with potentially unprofitable projects exist. This is an important condition in our model for it is the presence of such firms that makes dsky debt more attractive than outside equity. The assumption is entirely realistic since usually the difficulty lies in discovering profitable proj- ects, not unprofitable ones. In order to abstract from the effects of dsk aversion, we assume, as in Ross (1977), that the agents are dsk neutral. There are no taxes, bankmptcy costs, or other imperfections.
Next we specify the information stmcture of the model. The optimum in- vestment /, the production function, and the distribution functions G and F are assumed to be common knowledge. However, the type of the firm n is known only to the "insiders," i.e., the management of the firm. We ignore agency or moral hazard problems of the type discussed in Holmstrom (1979) or Jensen and
^ While the assumption that all firms have the same production function may seem to be too restrictive at the first glance, it is not necessarily so. We can rewrite (1) as
y = V(n,/,e) + e ,
where 9 is a vector of firm-specific factors like the industry classification, size, etc. The important difference between n and 9 is that, while n is known only to the insiders, 8 is common knowledge. In this case, all our results will hold conditional on 6. If, for example, 6 is the industry classification, we will obtain different capital stmcture strategies for different industry groups; but this is precisely what is observed in practice.
^ We are not concemed about the sign of dV/BI since we have assumed that / is exogeneously given. The usual assumption would be that dV/dl » 0.
42 Journal of Financial and Quanfifafive Analysis
Meckling (1976) between the management and current stockholders. In other words, the management makes decisions in the best interest of the current stock- holders.*• Thus, in our model, all information available to the management is shared with the current stockholders.
The owners' pdvate wealth w is insufficient to meet the investment needs of the firm,
(2) w < I for all firms.
w may be altematively interpreted as the intemal resources of the firm (retained eamings) since, in this model, the interests of owners and current shareholders are coincident, w is common knowledge and may differ from firm to firm. It is assumed that the distribution of w is independent of that of n and that w > 0 for all firms.
In this market, the value of any single firm in equilibrium will be the aver- age value of all the firms in the market. Thus, the uninformed investor (hereafter referred to as the market) will make an expected profit of zero. The insiders of any firm can possibly make positive expected profits from investments in their own firm. Their expected profits from any outside investments will be zero, since they have no information about the types of firms other than their own.
III. Only Equity Financing
The firms have to raise outside capital since their retained eamings fall short of the required investment. Let us first consider an economy in which the only source of capital is equity.^ Each firm needs to raise ( / - w ) . (To avoid unneces- sary notation, we omit subscdpts on w to indicate that intemal resources might vary from firm to firm.) It cannot raise more as this would signal the market that the firm is a "lemon" (V < I) and result in the undervaluation of the firm's stock. If the firm is profitable {V & / ) , it has no reason not to reinvest its intemal re- sources in its own projects as any outside investment would only result in zero expected profits.*
Given intemal resources w, a firm will enter the market only if its expected profits are nonnegative. Its expected profits depend on the market's valuation of the firm. The market value of the firm, in tum, depends on the market's per- ceived distribution of types V, given any level of intemal resources w. In equilib- dum, the perceived distribution of V given w would be the actual distdbution.
To formalize the above discussion, let F / V ) be the market's perceived dis-
•• Note that this does not preclude the management from acting against the interests of prospec- tive stockholders.
' If the firm could make a rights issue of stock, the problem of asymmetric information disap- pears since the manager is acting in the interests of the current shareholders. Therefore, when we state that w < /, we imply that current shareholders do not have the required funds. The current shareholders may be considered as entrepreneurs with limited funds whereas the outside investors may be considered as nonentrepreneurs looking for investments.
* This fact precludes the possibility of the "irrelevance" result, which can arise in the model of Myers and Majluf (1984). As the authors point out, when the current stockholders are allowed to actively rearrange their portfolios after the firm makes its investment and financing decisions, the financing decision is irrelevant in their model.
Narayanan 43
tdbution of the types of firms in the market, given that the only source of extemal capital is equity. Let
V =
The market assigns a value of Vj to all firms with intemal resources w. In order to raise (/ —w), the firm has to sell the fraction (/ —w)/v^ of its equity to outsiders. So, its expected profit/'^(V) is given by
(3) P ( V O = V[l - ( / - w ) / v j -w.
For any w, P^{V) is monotonically increasing in V at a constant rate. Also, P^iO) = - w < 0. Therefore, for any w, there exists V = V* such thatP,{V*) = 0.^ Only firms with V ^ V* will enter the market. In equilibdum, F^(V) = {condi- tional distdbution of V | V ^ V*}, i.e., market's perceived distdbution will be the same as the actual distribution of firms in the market.* Therefore, the per- ceived average value of the firms in the market, v ,̂ will be equal to the actual average value V .̂ Hence,
(4) P (V*) = V* [1 - ( / - w ) / V , ] - w = 0 .
The discussion above is meaningful only if Vf < V^, i.e., if at least some firms remain in the market. The proposition below shows that this condition will always hold. In fact, it proves a stronger result: that some lemons will always remain in the market in equilibrium.^ Proposition 1. There will always be some lemons in the market, i.e., for all w, V* < I. Proof. Suppose that V* s= /. By definition, P^iV*) = 0. Also V^ > V*, i.e., firm V* is overvalued. Since firm V* will make nonnegative profits even if it is correctly valued, it will make strictly positive profits if overvalued. Hence PgiV*) > 0, which is a contradiction. D Corollary 1.1.Vs> I. Proof. Follows from (4) and Proposition 1. The reason why lemons stay in the market is obvious. They are overvalued, and if the benefits dedved from overvaluation exceed the losses that are expected to occur, they would stay in the market.
The lowest type remaining in the market, V*, is a function of w, the intemal resources. The following corollary examines the relationship between V* and w. Corollary 1.2. V* is increasing in w. Proof. See Appendix.
' For notational simplicity, we suppress the dependence of V* on w. 8 F,(V) = [Fm-F{V*)]/[l - f ( V * ) ] . ' Note that, unlike in the Akerloff (1970) model, there is no market failure because of the pres-
ence of lemons. This is because the profitable firms are forced to remain in the market, this being the only way they can transfer their superior technologies into profits. Their profits are reduced by the presence of lemons, but by quitting the market they make no profits at all.
44 Journal of Financial and Quanfifafive Analysis
The result of Corollary 1.2 is intuitive. As w increases, the relative advantage from overvaluation is reduced since the proportion of outside equity goes down. Hence V* increases. Corollary 1.3. V*>0. Proof Since w > 0 and V5 > /, it follows from (4) that V* > 0. D
IV. Debt Financing
First consider a situation in which debt financing is the only way to raise extemal capital. Let D{V,B) be the expected value of the debt of a firm of type V when the face value of the debt is B. From (1), by stdct first-order stochastic dominance, it is clear thatD is nondecreasing in V, i.e.,Dy = dD/dV ̂ 0. D will be stdctly increasing in V if the debt is dsky. Whether the debt of any firm is dsky will depend, in addition to its type V, on the distdbution of e and the face value of debt B. It is further assumed that, for dsky debt, D is stdctly concave in V, i.e., Dyy < 0.1" This condition is a restdction on the distdbution function G. Many common distdbution functions satisfy the condition, i'
Let Pj(V) denote the expected profits of a firm of type V when its only source of extemal capital is debt
(5) — w
where B, the face value of the debt, is given by,
B
(6) / - w = '̂ B
is the market's perception of the distdbution of firms remaining in the market, given the condition that the only source of extemal capital is debt.'^ Equation (6) states that the expected value of the debt with face value B, as eval- uated by the market, must be equal to (/ - w), the required outside capital.
It follows from the assumption of stochastic dominance that /'̂ (VO is mono-
'" This assumption is only needed to prove Proposition 3 and Corollary 3.1. ' ' For example, if we use the option pricing model to evaluate the debt and equity of a firm.
= -d/dV
< 0 ,
where/is the density function of the standard normal and the rest of the notation is from Black and Scholes (1973).
'2 Fj is also a function of w and B but, for notational simplicity, we suppress these arguments.
Narayanan 45
tonically increasing in V. Also, P^iO) < 0. Therefore, there exists some type V^ such that
(7) p / y * ) = 0 .
In equilibdum, the perceived distribution of firms remaining in the market, Fj(V), must be the actual distdbution, i.e.,
(8) F^(V) = Conditional distdbution | v I V ^ V*\ .
Equations (6), (7), and (8) define the equilibdum and, by solving then simultane- ously, V^ and B can be obtained.
As the following proposition shows, when debt is dsky, there will be lem- ons in the market even if the only source of extemal capital is debt. Proposition 2. y | ^ /. If debt is dsky, V | < /, and if it is dsk free, V^ = I. That is, when the firm can issue only dsky debt, some lemons always remain in the market. Proof. By the Theorem of the Mean there exists some V̂ , > V^ such that
B
(9) I-w = jy dG^^iy) 0
From (5) and (7),
(10) w =
Adding (9) and (10), / = D{Vo,B) + V^ -D(V^,B) ^ V^, since Dy ^ 0, with the equality hold-
ing for dsk-free debt, n Corollary2.1. V[j>I. Proo/. From (5),/'rf(V) = V-D{V,B)-w
(11) = V -I -\- [ O ( V ^ , B ) - D(V,B)] , from (9) .
Therefore, PaiVo) = V Q - / . If V^ « / , ^^(Vo) ^ 0. This is impossible since PdiV) is monotonically increasing in V, and V̂ , > V$. Hence, V^>I. D Corollary 2.2. VJ is increasing in w. Proof. Similar to that of Corollary 1.2.
In order to obtain the intuition behind the fact that some lemons always remain in the market when debt is dsky, consider the profit of any firm as given in Equation (11). The term inside the braces represents the advantage (disadvan- tage) to overvaluation (undervaluation).!3 From Corollary 2.1, the term in the braces is positive for all lemons: lemons gain from overvaluation. If the extent of overvaluation is such that it overcomes the deficit ( V - / ) from being a lemon, the
" In the case in which firm types are common knowledge, this term would be zero for all types.
46 Journal of Financial and Quantitative Analysis
expected profits are positive and the lemon stays in the market. That this can occur can be verified by the fact that as V —» / the deficit (V-I)^ 0 while, by Corollary 2.1, the profit from overvaluation, [D(V£,,B) -D(V,B)], tends to some positive constant.
Since lemons stay in the market irrespective of the source of outside capital, the natural question that adses is whether there still exists any advantage for debt financing over equity financing. The answer is yes, as the next proposition shows. Proposition 3. Debt financing results in higher market value for the firms than equity financing. Proof. See Appendix. Corollary 3.1. There exists one and only one V > V | such that P^iV) = P,iV'). For all V < V',Pd(V) < P/V), and for all V > V',Pj{V) > P,(V). Proof. Obvious.
Proposition 3 states that the average quality of firms in the market increases if debt financing is used, resulting in higher market value for all of them. Corol- lary 3.1 shows that the firms in the upper end of the scale profit more from issu- ing debt than equity. So, given the choice between equity and debt financing, the best firms would prefer the latter. It is easily verified that an informationally con- sistent valuation function will be
Debt financing implies V ^ V* (12)
Equity financing implies V < V* .
This valuation would automatically eliminate all firms with types below V^ from the market. The following proposition shows that the above valuation function is unique. Proposition 4. The informationally consistent valuation function defined by (12) is unique. Proof. See Appendix.
The reason there are fewer lemons with debt is that it is a fixed claim. In states of bankmptcy, the fact that it is being overvalued is of no consequence to the firm because the equityholders get nothing. In most states in which the firm is solvent, the advantage of overvaluation does not increase with the firm's output. On the other hand, when equity is overvalued there is an advantage in every state. Since the probability of bankmptcy is relatively high for lemons, they pre- fer overvalued equity to overvalued debt.
The equilibrium is a pooling equilibdum with two separate pools, the divid- ing line being the type V^ firm. Firms with types V 5= yj= issue debt and enter the market. Firms with types V < Vjf do not enter the market. The market values all firms issuing debt at some "average" value V^, where Vp is given by Equation (9). By doing so, the market breaks even, i.e., makes zero expected profits, on the debt of all the firms entering the market.
The fact that lemons remain in the market in equilibdum results in a deadweight welfare loss compared to the full information case. When the types of firms are common knowledge, none of the lemons will enter the market since they do not expect to gain anything by issuing either debt or equity. So the aggre- gate output in the full information case would be higher than the aggregate output
Narayanan 47
in the signaling equilibrium. The most productive firms as a group will bear this loss as all firms of type lower than the average value V^ will benefit by the over- valuation of their debt. The benefit to overvalued firms does not fully offset the loss to undervalued firms. Since there is a deadweight loss, this signaling equilib- dum is dissipative in the sense of Rothschild and Stiglitz (1976), though the equilibrium is not perfectly discriminating as theirs is. It is this lack of perfect discrimination that enables lemons to enter the market and cause the deadweight loss.
The result that debt will be preferred by undervalued firms, though similar to that dedved by Myers and Majluf (1984), is based on a completely different line of reasoning. As stated in the Introduction, the informational asymmetry regarding the assets-in-place is cmcial to their model, while our results are driven by the existence of potential lemons. Moreover, in our model, investment is ri- sky, resulting in risky debt.
There are some interesting differences conceming the investment decision between our models. In their model, if the new project's net present value is not very large relative to the value of the asset-in-place, the firms will forego the new investment opportunity (even if it has positive net present value) rather than let the assets-in-place be devalued by the uninformed market, causing some loss of welfare. In our model, there are no such foregone profitable investment opportu- nities. The problem, on the other hand, is the possibility that even investments with negative net present values will be undertaken because of overvaluation. This will happen irrespective of the mode of financing since lemons are always present in the market. Thus, the loss of welfare in our model adses from a differ- ent source.
We considered in this section the two extreme modes of financing—the is- sue of equity or the issue of debt. We did not explicitly consider a mixture of debt and equity financing. There are two possible ways a mixture of debt and equity can produce greater market value for a supedor firm: (1) by increasing the quality of firms remaining in the market, or (2) if the signaling equilibdum is perfectly discdminating. But we know that the issue of equity would enable more lemons to enter the market thus lowedng the average quality of firms in the market. Also, perfect discrimination is not possible under the assumptions of our model since any debt-equity ratio that the best firm chooses can be imitated by the less profitable firms (that are also good, i.e., V > /) thus making a separate signaling equilibdum impossible (see Ross (1977) for a more detailed explanation). Thus, in our model, if firms can issue debt and equity, we will find a signaling equilib- dum in which only debt will be issued.
The model as presented above does not envisage a role for extemal equity in the firm's capital stmcture. All firms finance their investments by intemal capital (w) and debt. This was optimal in these models because debt kept more lemons out of the market than extemal equity without any concomitant "costs'' or disad- vantages. Therefore, for extemal equity to play a role in the firm's capital stmc- ture, some cost or disadvantage for debt should be introduced.'" As suggested by
I"* All previous signaling models of capital stmcture incorporate, either explicitly or implicitly, such a "cost." Ross (1977) assumes that the manager's wage schedule is a contingent contract with a penalty for bankmptcy. In Heinkel's (1982) model, the cost of debt is implicit as he assumes that the
48 Journal of Financial and Quantitative Analysis
Myers and Majluf (1984), agency costs as in Jensen and Meckling (1976) or Myers (1977), or bankmptcy costs as in Ross (1977), could be introduced into the model to provide a role for extemal equity, î
V. Conclusions
In this paper, we offer a new advantage to the use of debt. In a world in which only insiders know the quality of the firm, the use of debt acts as a barrier to entry of infedor firms. This improves the average quality of the firms in the market, thus benefiting everyone even when perfect discdmination is impossible. This advantage is unrelated to corporate tax laws and hence is equally applicable to situations or pedods when corporations do not pay taxes.
The model provides the normative result that financial managers should use debt financing (even if it is dsky) if they perceive that their firm is undervalued and use equity financing if they perceive it is overvalued.
There are several empidcal implications of this model.'* Bdefiy, we can conclude that: (1) Issuing safe securities is better than issuing risky ones. As shown in Proposi- tion 2, with risk-free debt, no lemon enters the market. This is consistent with the agency model of Jensen and Meckling (1976). The agency problem of excessive perquisite consumption by the owner-manager in their model can be costlessly resolved by issuing dsk-free debt. (2) Debt, even if it is risky, is always better than equity. This follows from Prop- osition 3. (3) It is better to build up financial reserves (by restdcting dividends, for exam- ple) so that higher proportions of capital needs can be supplied from intemal sources. This follows from Corollades 1.2 and 2.2, where it is shown that as the intemal funds increase, the average quality of the firms entering the market in- creases. This holds irrespective of the type of extemal financing. (4) When equity is issued, the stock pdce will fall. Though we did not prove this formally, this result is a direct implication of our model. Since any project fi- nanced with extemal equity is viewed as a lemon, the perceived present value of the firm and, hence, its stock pdce will fall. This loss of welfare to current stock- holders cannot be avoided even if the project is spun off as a separate firm (unlike in the Myers and Majluf (1984) model (in which the problem can be solved by such spin-offs).
higher the debt, the lower the value of the firm. Lee, Thakor, and Vora (1983) impose a cost on disclosure of firm type to bondholders (and bondholders only). By segmenting the investors into those who are willing to find out the firm's type and those who are not, and by constraining the information producers to be bondholders, they obtain an interior optimum. The only exception, to our knowledge, is the paper by Constantinides and Gmndy (1986), in which the insiders commit to using any excess cash generated through debt financing to repurchase stock. The insiders, in effect, agree to nullify any advantage gained through issuing overpriced debt by purchasing overpriced equity, mak- ing the existence of a nondissipative equilibrium (see Bhattacharya (1980)) feasible.
" A pooling equilibrium with bankmptcy costs that provide a role for extemal equity is dis- cussed in Narayanan (1986).
'* Some of these implications are similar to those derived by Myers and Majluf (1984); but, as stated earlier, the rationales are totally different.
Narayanan 49
Appendix
Proof of Corollary 1.2. Differentiating both sides of Equation (4) with respect to w.
+ ( / - H ' ) 5 V H - 1 = 0 .
From the above equation, dV*ldw ^ 0 implies dV^ldw > 0. If it were not so, the second term in the equation would be less than 1, since V* < V^. But dV*/dw « 0 and dV^/dw > 0 contradict the fact that any increase in V* increases V^. D Proof of Proposition 3. Let V* and V^ be the lowest types in the market under equity and debt financing, respectively. To prove the proposition, it needs to be shown that V} > V*. For dsk-free debt, this follows tdvially from Propositions 1 and 2. For dsky debt, suppose this is not tme, i.e., V^ « V*. PjiV) can be wdtten as
P/VO = V-D(V,B)-w.
For dsky debt, Dyy < 0. Therefore, Pj{V) is an increasing convex function of V with Pd(,O) = -w. From (3) we know that Pj(V) is an increasing linear function of V with P / 0 ) = - w. Therefore, if V^ « V*, Pj(V) > P,(V) for all V > V / .
When the only source of outside capital is equity, firms of type V such that V^ =s V < V* will not be in the market. If debt is the only source of capital, these firms will enter the market. P^(V) > P^(V) for all V > V | implies that the entry of these firms resulting from the use of debt will increase the expected profits of all the firms already in the market. But this is impossible since the entedng firms are all lemons (V* < I from Proposition 1). Since these lemons make positive expected profits, it must be at the expense of some of the firms already in the market and, hence, some of these firms must suffer a decrease in profits.
Thus, the supposition V^ ^ V* is incorrect. D Proof of Proposition 4. We can ignore all valuation functions in which debt im- plies a lower type than equity. It follows from Corollary 3.1 that, under such valuation functions, lemons with types V <V', where V" is given by P^iV') = P j ( ^ ' ) , will gladly issue equity. They profit from financing their projects with equity even if the market assigns only an average value to them. They would profit even more from overvaluation if the market thinks that they are the better types. This makes the market's valuation function informationally inconsistent.
Therefore, we consider only valuation functions in which debt implies a better type. Consider the following valuation function,
(Al) Debt implies V ^ V" and equity implies V < V,
where V ¥'V^. We show that (Al) cannot be informationally consistent. Casel.V >I.
Consider firms close to, but less than, V. These firms are profitable, but undervalued. If they issue debt, they will be overvalued and will make more profits since the average value of firms issuing debt, according to the market, is
50 Journal of Financial and Quantitative Analysis
greater than V. Therefore, firms in this range will shift to debt financing, making (Al) informationally inconsistent. Case2:V^<V' ^I.
No firms will issue equity since they will be branded as lemons. However, firms in the range [ V | , V ] will have an incentive to issue debt. For firms in this range, expected profits with debt financing under (Al) > expected profits with debt financing under (12) ^ 0.
The first inequality follows from the fact that the overvaluation of debt un- der (Al) is greater than that under (12), since the average perceived quality of firms issuing debt under (Al) is higher. The second inequality follows from Proposition 2 and the fact that all firms in the range [V/,V'] have values greater than V^. Therefore, firms in this range will resort to debt financing, making (Al) informationally inconsistent. Case3:V' <V^.
Again, no firm will issue equity. For firms in the range [V,V^), expected profit with debt financing under (Al) < expected profit with debt financing under (12) < 0 .
The first inequality follows from the fact that the overvaluation of debt un- der (Al) is less than that under (12), since the average perceived quality of firms issuing debt under (Al) is lower. The second inequality is the result of the fact that all firms in the range [V,V^) have values less than V^. Hence, no firm in this range would issue debt, making (Al) informationally inconsistent. D
Narayanan 51
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