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Some recent developments in the theory of competition in markets with adverse selection ∗
European
Economic
Review 3 1 (1987) 3 19-325.
North-Holland
SOME RECENT DEVELOPMENTS COMPETITION IN MARKETS WITH Martin
IN THE THEORY OF ADVERSE SELECTION*
HELLWIG
University of Bonn, D-5300 Bonn I, Federal Republic of Germany
1. Introduction Over the past 20 years, economists have become more and more interested in the functioning of markets with incomplete information. The simplest case is that of one-sided incomplete information, where one side of the market has private information about some relevant characteristic of the good that is being traded. Thus in the used-car market the seller is likely to have private information about the quality of the car that he is offering. In the insurance market, somebody who buys insurance may have private information about his own personal accident risk. In the loan market, an entrepreneur who wants to finance a project is likely to have some private information about his prospects and hence about the probability that the loan will actually be repaid. In each of these cases, the informed party’s behaviour will depend on its information. Thus in the insurance market, a consumer who believes that his accident probability is high will buy more insurance than a consumer who believes that his accident probability is low. If the other side of the market is aware of this ‘adverse selection’ phenomenon, then it will try to infer the underlying private information from the observed behaviour of the informed party. The question is how this inference process affects the functioning of the market. In response to this question, one strand of the literature, beginning with Spence (1973) Rothschild and Stiglitz (1976) and Wilson (1977), stresses the role of sorting devices that facilitate the transmission of information from informed to uninformed agents in the market. As an example, consider the role of deductibles in the insurance market. Given a set of available insurance contracts, a high-risk agent will presumably choose a contract with a lower deductible than a low-risk agent. Conversely an agent who chooses a contract with a low or zero deductible is more likely to have a high accident *Research support from the Deutsche bereich 303 is gratefully acknowledged. 0014-2921/87/$3.50
0
Forschungsgemeinschaft
1987, Elsevier Science Publishers
through
B.V. (North-Holland)
the Sonderforschungs-
320
M. Hellwig, Theory
of competition in markets with adoerse selection
probability than one who chooses a contract with a high deductible. In this way, the size of the deductible that an agent accepts may serve as a sorting device. For markets that have such sorting devices, the main conclusions of the literature are: (i) If the market has an equilibrium at all, then in equilibrium, the sorting device is actually used to transmit information from one side of the market to the other. (ii) However, such a market may not have an equilibrium at all. Most of this literature is not very explicit about what exactly is meant by ‘competition’ in a market with adverse selection, i.e., what is the game-theoretic structure of the strategic interaction between informed and uninformed agents in the market. To some extent, this neglect is due to a reluctance to narrow the intuitive concept of ‘competition’ down to one specific game-theoretic model. However, two new contributions by Cho and Kreps (1986) and by myself (forthcoming) suggest that the conclusions which have been reached are very sensitive to the details of the game-theoretic specification that one looks at. Seemingly minor modifications of the formal model have a major impact on the predicted outcome of ‘competition’ in a market with adverse selection. In the remainder of this paper, I shall compare the different specifications and try to give some insight into why the details of the game-theoretic modelling of competition matter as much as they do. 2. The strategic modelling of competition In a market with adverse selection, the assumption that prices are set by a Walrasian auctioneer is even more unpalatable than in an ordinary market. Therefore, most of the literature on competition with adverse selection is based on the Bertrand paradigm of price-setting firms with constant marginal and average costs which offer to serve an arbitrary number of customers at the prices they set. I compare three distinct versions of this approach. The first version corresponds to the work of Rothschild and Stiglitz (1976) or Wilson (1977). This version involves a two-stage game such that at stage 1, the uninformed agents in the market offer some contracts, and at stage 2, the informed agents choose among these contract offers. In my own work, I look at a three-stage game in which at stage 3, the uninformed agents may reject whatever contract applications they have received at date 2.’ Finally, Cho and Kreps (1986) analyse a three-stage game in which the informed agents moue first by announcing a signal. Then at stage 2, the uninformed agents make contract offers contingent on the signals that have been sent, and at stage 3, the informed agents choose among the contract offers that they have received. Given the sequential character of these games, I shall be concerned with ‘In a nonsequential
setting,
this specification
was first discussed
by Grossman
(1979).
M. Hellwig, Theory of competition in markets with adverse selection
321
sequentid equilibria in the sense of Kreps and Wilson (1982). The usual Nash condition that each agent’s strategy be a best response to the other agents’ strategies is applied not only to the overall game, but to any decision node in the game tree regardless of whether in equilibrium this node is actually reached or not. In the two-stage game of Rothschild and Stiglitz, this sequential rationality condition requires that an informed agent’s strategy at stage 2 prescribes an optimal choice among the available contracts regardless of what contracts have been made available at date 1. To be specific, I consider Bester’s (1985b) model of a credit market with E entrepreneurs and B banks. Each entrepreneur belongs to one of two types. For i= 1,2, an entrepreneur of type i has a project which requires an investment I and bears a return Ri, where Bi= R,>O with probability pi and ai=0 with probability 1 -pi. The parameters pi, R,, i= 1,2 satisfy the inequalities
so that a project of type 1 has both, a lower expected return and a higher risk than a project of type 2. However, a project of type 1 does have a higher return realization in the case of success. Each entrepreneur has an initial wealth w >O, which is insullicient to finance the investment I. Therefore, he needs a loan from a bank. Banks have no direct knowledge of an entrepreneur’s type. Even ex post they can only observe whether his project has succeeded or not; they do not observe the actual return realization. Given this information structure, a loan contract is a triple (L, r, C) such that L is the size of the loan, r (the interest) is the debtor’s payment when his project succeeds, and C (the collateral) is the debtor’s payment when his project fails. Entrepreneurs have a type-independent, strictly increasing, strictly concave von Neumann-Morgenstern utility function u( .), which satisfies u(O)= - CO. , Therefore, type i’s payoff from a contract (L, r, C) is equal to p,u(w+R,-r-(I-L))+(l
-p,)u(w-C-(Z-L)).
(1)
Banks are risk neutral and have a constant marginal cost of funds equal to one. A bank’s payoff from a contract (L, r, C) with an entrepreneur of type i is equal to pir+(l -pJC-L.
(2)
From (1) and (2), one sees that any contract (L, r, C) is trivially equivalent to the contract (I, r+l- L, C+I - L) which involves full debt finance of the entrepreneur’s project. Therefore, there is no loss of generality in restricting
322
M. Hellwig, Theory of competition
in markets with adverse selection
the analysis to contracts (I, r, C) with full debt finance, which for short are written as (I, C). Expressions (1) and (2) suggest that under complete information, all loans should be made under contracts with zero collateral requirements. If C were positive, then a reduction in C together with an appropriate increase in r would shift some of the risk of project failure from the entrepreneur to the bank, which is advantageous because the bank is risk neutral, and the entrepreneur is not. However, expressions (1) and (2) also suggest that under incomplete information, the collateral C may serve as a signal of the entrepreneur’s type, i.e., as a sorting device. Because an entrepreneur of type 2 has a lower failure probability, he is more willing to accept an increase in C in return for a reduction in r than an entrepreneur of type 1. Now consider the two-stage game of Rothschild, Stiglitz and Wilson. At stage 1, the banks make contract offers (r, C). At stage 2, the entrepreneurs choose among the available offers. According to Rothschild, Stiglitz and Wilson, the only candidate for a pure-strategy sequential-equilibrium outcome is the pair of contracts (r:, Crf), (rz, CT) such that (i) r; = I/p,, CT =O, (ii) rt = [I -( 1 -pz)Cz]/p,, and CT > 0 is just high enough so that type 1 entrepreneurs do not prefer (r:, CT) to (rz, Cy). If only the contracts (r:, CT) and (rz, C:) are offered at stage 1, then at stage 2, type 1 entrepreneurs choose (rl, C:), and type 2 entrepreneurs choose (rT, Cq). For the type 2 entrepreneurs, this perfect sorting has the advantage that the interest rz corresponds to their actual success probability pz so that they do not subsidize the type 1 entrepreneurs. However, the disadvantage is that the collateral Cz > 0 imposes a risk on them, which under complete information the banks would be willing to bear. This cost of sorting will exceed the benefit if the population share 2 of type 1 entrepreneurs is small so that the population average success probability p= Lpi + (1 - A)p2 is close to pz and the costs of subsidizing type 1 entrepreneurs are small. In this case, the contract constellation (r:, C:), (rr, CT) is strictly Pareto dominated by some pooling contract (f, c). Given the available offers (rf,CT), (rf,Cf), a deviation to (Y,c) by some bank will be profitable so that (r:, C:), (r:, C:) ca nnot correspond to a sequential equilibrium. In this situation, consider the contract (r**, C**) which the type 2 entrepreneurs prefer most among those contracts (r, C) that break even on the population average so that pr + (1 -p)C 2 I. If p is close to p2, then an offer of (r**, C**) by all banks yields the contract constellation that type 2 entrepreneurs prefer most among all offer constellations (pooling or sorting) that entail non-negative profits for the banks. Trivially then, the two-stage game cannot have a sequential-equilibrium outcome other than (r**, C**). However, (r**, C**) itself can not be a sequential-equilibrium outcome because it can always be upset by a contract (3, C) with i C** such that (i) type 2 entrepreneurs prefer (P, C) to (r**, C**), (ii) type 1
M. Hellwig, Theory
ofcompetitionin markets
with adverse selection
323
entrepreneurs prefer (r**, C**) to (?, C), and (iii) pZ(?-C)>I -C>p(?-C) so that (?, C) makes a profit on type 2 entrepreneurs, but a loss on the population average. In summary, if R is close to zero, and p is close to p2, the two-stage game of Rothschild, Stiglitz and Wilson fails to have an equilibrium in pure strategies. As an alternative, consider the three-stage game in which a bank may reject an entrepreneur’s loan application at the third stage. I claim that if ,? is small so that the two-stage game has no sequential equilibrium in pure strategies, then the optimal pooling contract (r**, C**) does correspond to a sequential equilibrium of the three-stage game. In the three-stage game, a deviation to the separating contract (i, C) is unprofitable because under the equilibrium strategies no entrepreneur applies for (f, C). Entrepreneurs do not apply for (i, C) because under the equilibrium strategies, such applications are always rejected at the third stage. To see this, suppose that such an application was accepted. Then at stage 3, the bank that has offered (?, C) must believe that its applications form an above-average sample of the population. Consequently the banks that have offered (r**, C**) must have a below-average sample of the population. Since the contract (r**, C**) only breaks even at the population average, applications for this contract must then be rejected at stage 3. However, given these different rejection strategies for stage 3, at stage 2 all entrepreneurs must apply for (?, C) rather than ** p* ), contrary to the assertion that (?, C) receives an above-average (r 3 sample of applications. The assumption that applications at (?. C) may not be rejected must therefore be false. In the three-stage game, the optimal pooling contract (r**, C**) can never be upset by the separating contract (;. C). One can show that not only the optimal pooling contract, but also the separating contract pair (rT, CT), (rz, Ct) as well as any pooling contract that Pareto dominates the pair (rf, Cf), (rr, C:) corresponds to a sequential equilibrium of the three-stage game. However, an application of the Kohlberg and Mertens (1986) stability criterion shows that except for the optimal pooling contract, all these sequential-equilibrium outcomes lack certain desirable robustness properties. The optimal pooling contract (r**, C**) seems to be the most plausible outcome of the three-stage game. The addition of another stage has thus led to a dramatic reversal of the Rothschild, Stiglitz and Wilson results. The three-stage game always has a sequential equilibrium in pure strategies. Moreover the most plausible sequential equilibrium involves pooling rather than sorting whenever pooling is Pareto preferred. The latter conclusion is again reversed if one looks at the model of Cho and Kreps (1986) in which the informed agents move first. Suppose that at stage 1, each entrepreneur announces a collateral C that he wants to put up. At stage 2, each bank announces to each entrepreneur an interest payment r that it requires for a loan with the announced collateral C. Finally, at stage 3, each entrepreneur chooses among the offers that he has received.
324
M. HeLlwig, Theory
ofcompetitionin markets
with adverse selection
In this game again, the separating pair (r:, CT), (rr, Ct) always corresponds to a sequential equilibrium. In this equilibrium, at stage 1, an entrepreneur of type i announces the collateral CT. At stage 2, all banks offer him the interest rate r:, and at stage 3, he chooses one of the banks. If one of the entrepreneurs were to deviate say to the collateral announcement C**, then the banks would believe that this announcement was more likely to come from a type 1 than a type 2 entrepreneur. Given this belief they would ask for interest rates close to [I -(l -pJC**]/p, rather than r**. Therefore such a deviation from the equilibrium strategies is undesirable both for type 1 and type 2 entrepreneurs; the separating equilibrium cannot be upset. One can show that any pooling contract that Pareto dominates the separating contract pair (r:, CT), (r$, CT) is also a sequential-equilibrium outcome of the Cho-Kreps game. However, one now finds that the Kohlberg-Mertens stability criterion singles out the separating outcome (rT,C:), (r:, Cy). If the informed rather than the uninformed agents move first in the game, the pooling rather than the separating equilibria lack robustness. 3. Concluding remarks The preceding discussion shows that three different game theoretic formalizations lead to three different predictions of how a competitive market with adverse selection will work. In the two-stage game of Rothschild, Stiglitz and Wilson, a pure-strategy sequential equilibrium fails to exist if the proportion of ‘good’ types in the population is large. In the same situation, an analysis based on the three-stage game in which the uninformed agents move first predicts the emergence of the optimal pooling contract, whereas the optimal separating contract constellation should emerge if the informed agents move first. The discrepancy between these results corresponds to a clash between two underlying intuitive principles. On the one hand, pooling contracts are vulnerable because at least locally, they provide ‘good’ types with an incentive to distinguish themselves from ‘bad’ types in order to get better conditions. On the other hand, Bertrand competition should lead to (constrained) Pareto optimal outcomes. If the proportion of ‘good’ types in the population is large, the first of these principles points to the separating contract constellation (r:, CT), (rf, Cly) whereas the second points to the pooling contract (r**, C** ). In the three-stage games, the first principle prevails if the informed agents move first, and the second principle prevails if the uninformed agents move first. The discrepancy between the predictions of different game-theoretic models extends to other issues such as the question whether in a market with adverse selection, rationing can be an equilibrium phenomenon. If the model
M. Hellwig,
Theory
ofcompetitionin
markets
with adverse selection
325
of the credit market is modified to allow for an inelastic supply of funds to the banks, then the three-stage game in which the uninformed agents, i.e., the banks, move first supports the contention of Stiglitz and Weiss (1981) that equilibrium credit rationing is possible. Given that the equilibrium involves pooling, the banks may not wish to respond to an excess demand for loans by raising interest rates because this would have an adverse effect on the population mix in the market. On the other hand, the Cho-Kreps specitication in which the informed agents move first will support Bester’s (1985a, 1985b) contention that equilibrium credit rationing cannot occur in a market with an effective sorting device. Bester’s argument rests on the Rothschild-Stiglitz-Wilson assertion that if a sorting device is available, then it will be used. As we have seen, this assertion is true when the informed agents move first, but not necessarily when the uninformed agents move first. For the applied economist who studies, e.g., the regulation of credit or insurance markets, the discrepancy between the predictions of the different game-theoretic models presents a fundamental dilemma. On the one hand, it is important that he determine which of the different models is appropriate for the market at hand. On the other hand, this may be impossible to determine because matters like the order in which people make moves, which make all the difference for the game-theoretic specification, may not be observable and may not even be fixed in a given market.
References Bester, H., 1985a, Screening vs. rationing in credit markets with imperfect information, American Economic Review 85G855. Bester, H., 1985b, The role of collateral in credit markets with imperfect information, Discussion paper no. A-30 (Sonderforschungsbereich 303, University of Bonn, Bonn). Cho, I.K. and D. Kreps, 1986, Signalling games and stable equilibria, Discussion paper (Graduate School of Business, Stanford University, Stanford, CA). Grossman, H., 1979, Adverse selection, dissembling, and competitive equilibrium, Bell Journal of Economics 10, 336343. Hellwig, M.F., 1986, A sequential approach to modelling competition in markets with adverse selection, Discussion paper (Sonderforschungsbereich 303, University of Bonn, Bonn) forthcoming. Kohlberg, E. and J.F. Mertens, 1986, On the strategic stability of equilibria, Econometrica 54, 1003-1038. Kreps, D. and R. Wilson, 1982, Sequential equilibria, Econometrica 50, 863-894. Riley, J., 1979, Informational equilibrium, Econometrica 47, 331-359. Rothschild, M. and J.E. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics 90,629-650. Spence, M., 1973, Job market signalling, Quarterly Journal of Economics 87, 355-374. Stiglitz, J.E. and A. Weiss, 1981, Credit rationing in markets with imperfect information, American Economic Review 71, 393410. Wilson, C., 1977, A model of insurance markets with incomplete information, Journal of Economic Theory 16, 167-207.
Economic
Review 3 1 (1987) 3 19-325.
North-Holland
SOME RECENT DEVELOPMENTS COMPETITION IN MARKETS WITH Martin
IN THE THEORY OF ADVERSE SELECTION*
HELLWIG
University of Bonn, D-5300 Bonn I, Federal Republic of Germany
1. Introduction Over the past 20 years, economists have become more and more interested in the functioning of markets with incomplete information. The simplest case is that of one-sided incomplete information, where one side of the market has private information about some relevant characteristic of the good that is being traded. Thus in the used-car market the seller is likely to have private information about the quality of the car that he is offering. In the insurance market, somebody who buys insurance may have private information about his own personal accident risk. In the loan market, an entrepreneur who wants to finance a project is likely to have some private information about his prospects and hence about the probability that the loan will actually be repaid. In each of these cases, the informed party’s behaviour will depend on its information. Thus in the insurance market, a consumer who believes that his accident probability is high will buy more insurance than a consumer who believes that his accident probability is low. If the other side of the market is aware of this ‘adverse selection’ phenomenon, then it will try to infer the underlying private information from the observed behaviour of the informed party. The question is how this inference process affects the functioning of the market. In response to this question, one strand of the literature, beginning with Spence (1973) Rothschild and Stiglitz (1976) and Wilson (1977), stresses the role of sorting devices that facilitate the transmission of information from informed to uninformed agents in the market. As an example, consider the role of deductibles in the insurance market. Given a set of available insurance contracts, a high-risk agent will presumably choose a contract with a lower deductible than a low-risk agent. Conversely an agent who chooses a contract with a low or zero deductible is more likely to have a high accident *Research support from the Deutsche bereich 303 is gratefully acknowledged. 0014-2921/87/$3.50
0
Forschungsgemeinschaft
1987, Elsevier Science Publishers
through
B.V. (North-Holland)
the Sonderforschungs-
320
M. Hellwig, Theory
of competition in markets with adoerse selection
probability than one who chooses a contract with a high deductible. In this way, the size of the deductible that an agent accepts may serve as a sorting device. For markets that have such sorting devices, the main conclusions of the literature are: (i) If the market has an equilibrium at all, then in equilibrium, the sorting device is actually used to transmit information from one side of the market to the other. (ii) However, such a market may not have an equilibrium at all. Most of this literature is not very explicit about what exactly is meant by ‘competition’ in a market with adverse selection, i.e., what is the game-theoretic structure of the strategic interaction between informed and uninformed agents in the market. To some extent, this neglect is due to a reluctance to narrow the intuitive concept of ‘competition’ down to one specific game-theoretic model. However, two new contributions by Cho and Kreps (1986) and by myself (forthcoming) suggest that the conclusions which have been reached are very sensitive to the details of the game-theoretic specification that one looks at. Seemingly minor modifications of the formal model have a major impact on the predicted outcome of ‘competition’ in a market with adverse selection. In the remainder of this paper, I shall compare the different specifications and try to give some insight into why the details of the game-theoretic modelling of competition matter as much as they do. 2. The strategic modelling of competition In a market with adverse selection, the assumption that prices are set by a Walrasian auctioneer is even more unpalatable than in an ordinary market. Therefore, most of the literature on competition with adverse selection is based on the Bertrand paradigm of price-setting firms with constant marginal and average costs which offer to serve an arbitrary number of customers at the prices they set. I compare three distinct versions of this approach. The first version corresponds to the work of Rothschild and Stiglitz (1976) or Wilson (1977). This version involves a two-stage game such that at stage 1, the uninformed agents in the market offer some contracts, and at stage 2, the informed agents choose among these contract offers. In my own work, I look at a three-stage game in which at stage 3, the uninformed agents may reject whatever contract applications they have received at date 2.’ Finally, Cho and Kreps (1986) analyse a three-stage game in which the informed agents moue first by announcing a signal. Then at stage 2, the uninformed agents make contract offers contingent on the signals that have been sent, and at stage 3, the informed agents choose among the contract offers that they have received. Given the sequential character of these games, I shall be concerned with ‘In a nonsequential
setting,
this specification
was first discussed
by Grossman
(1979).
M. Hellwig, Theory of competition in markets with adverse selection
321
sequentid equilibria in the sense of Kreps and Wilson (1982). The usual Nash condition that each agent’s strategy be a best response to the other agents’ strategies is applied not only to the overall game, but to any decision node in the game tree regardless of whether in equilibrium this node is actually reached or not. In the two-stage game of Rothschild and Stiglitz, this sequential rationality condition requires that an informed agent’s strategy at stage 2 prescribes an optimal choice among the available contracts regardless of what contracts have been made available at date 1. To be specific, I consider Bester’s (1985b) model of a credit market with E entrepreneurs and B banks. Each entrepreneur belongs to one of two types. For i= 1,2, an entrepreneur of type i has a project which requires an investment I and bears a return Ri, where Bi= R,>O with probability pi and ai=0 with probability 1 -pi. The parameters pi, R,, i= 1,2 satisfy the inequalities
so that a project of type 1 has both, a lower expected return and a higher risk than a project of type 2. However, a project of type 1 does have a higher return realization in the case of success. Each entrepreneur has an initial wealth w >O, which is insullicient to finance the investment I. Therefore, he needs a loan from a bank. Banks have no direct knowledge of an entrepreneur’s type. Even ex post they can only observe whether his project has succeeded or not; they do not observe the actual return realization. Given this information structure, a loan contract is a triple (L, r, C) such that L is the size of the loan, r (the interest) is the debtor’s payment when his project succeeds, and C (the collateral) is the debtor’s payment when his project fails. Entrepreneurs have a type-independent, strictly increasing, strictly concave von Neumann-Morgenstern utility function u( .), which satisfies u(O)= - CO. , Therefore, type i’s payoff from a contract (L, r, C) is equal to p,u(w+R,-r-(I-L))+(l
-p,)u(w-C-(Z-L)).
(1)
Banks are risk neutral and have a constant marginal cost of funds equal to one. A bank’s payoff from a contract (L, r, C) with an entrepreneur of type i is equal to pir+(l -pJC-L.
(2)
From (1) and (2), one sees that any contract (L, r, C) is trivially equivalent to the contract (I, r+l- L, C+I - L) which involves full debt finance of the entrepreneur’s project. Therefore, there is no loss of generality in restricting
322
M. Hellwig, Theory of competition
in markets with adverse selection
the analysis to contracts (I, r, C) with full debt finance, which for short are written as (I, C). Expressions (1) and (2) suggest that under complete information, all loans should be made under contracts with zero collateral requirements. If C were positive, then a reduction in C together with an appropriate increase in r would shift some of the risk of project failure from the entrepreneur to the bank, which is advantageous because the bank is risk neutral, and the entrepreneur is not. However, expressions (1) and (2) also suggest that under incomplete information, the collateral C may serve as a signal of the entrepreneur’s type, i.e., as a sorting device. Because an entrepreneur of type 2 has a lower failure probability, he is more willing to accept an increase in C in return for a reduction in r than an entrepreneur of type 1. Now consider the two-stage game of Rothschild, Stiglitz and Wilson. At stage 1, the banks make contract offers (r, C). At stage 2, the entrepreneurs choose among the available offers. According to Rothschild, Stiglitz and Wilson, the only candidate for a pure-strategy sequential-equilibrium outcome is the pair of contracts (r:, Crf), (rz, CT) such that (i) r; = I/p,, CT =O, (ii) rt = [I -( 1 -pz)Cz]/p,, and CT > 0 is just high enough so that type 1 entrepreneurs do not prefer (r:, CT) to (rz, Cy). If only the contracts (r:, CT) and (rz, C:) are offered at stage 1, then at stage 2, type 1 entrepreneurs choose (rl, C:), and type 2 entrepreneurs choose (rT, Cq). For the type 2 entrepreneurs, this perfect sorting has the advantage that the interest rz corresponds to their actual success probability pz so that they do not subsidize the type 1 entrepreneurs. However, the disadvantage is that the collateral Cz > 0 imposes a risk on them, which under complete information the banks would be willing to bear. This cost of sorting will exceed the benefit if the population share 2 of type 1 entrepreneurs is small so that the population average success probability p= Lpi + (1 - A)p2 is close to pz and the costs of subsidizing type 1 entrepreneurs are small. In this case, the contract constellation (r:, C:), (rr, CT) is strictly Pareto dominated by some pooling contract (f, c). Given the available offers (rf,CT), (rf,Cf), a deviation to (Y,c) by some bank will be profitable so that (r:, C:), (r:, C:) ca nnot correspond to a sequential equilibrium. In this situation, consider the contract (r**, C**) which the type 2 entrepreneurs prefer most among those contracts (r, C) that break even on the population average so that pr + (1 -p)C 2 I. If p is close to p2, then an offer of (r**, C**) by all banks yields the contract constellation that type 2 entrepreneurs prefer most among all offer constellations (pooling or sorting) that entail non-negative profits for the banks. Trivially then, the two-stage game cannot have a sequential-equilibrium outcome other than (r**, C**). However, (r**, C**) itself can not be a sequential-equilibrium outcome because it can always be upset by a contract (3, C) with i
M. Hellwig, Theory
ofcompetitionin markets
with adverse selection
323
entrepreneurs prefer (r**, C**) to (?, C), and (iii) pZ(?-C)>I -C>p(?-C) so that (?, C) makes a profit on type 2 entrepreneurs, but a loss on the population average. In summary, if R is close to zero, and p is close to p2, the two-stage game of Rothschild, Stiglitz and Wilson fails to have an equilibrium in pure strategies. As an alternative, consider the three-stage game in which a bank may reject an entrepreneur’s loan application at the third stage. I claim that if ,? is small so that the two-stage game has no sequential equilibrium in pure strategies, then the optimal pooling contract (r**, C**) does correspond to a sequential equilibrium of the three-stage game. In the three-stage game, a deviation to the separating contract (i, C) is unprofitable because under the equilibrium strategies no entrepreneur applies for (f, C). Entrepreneurs do not apply for (i, C) because under the equilibrium strategies, such applications are always rejected at the third stage. To see this, suppose that such an application was accepted. Then at stage 3, the bank that has offered (?, C) must believe that its applications form an above-average sample of the population. Consequently the banks that have offered (r**, C**) must have a below-average sample of the population. Since the contract (r**, C**) only breaks even at the population average, applications for this contract must then be rejected at stage 3. However, given these different rejection strategies for stage 3, at stage 2 all entrepreneurs must apply for (?, C) rather than ** p* ), contrary to the assertion that (?, C) receives an above-average (r 3 sample of applications. The assumption that applications at (?. C) may not be rejected must therefore be false. In the three-stage game, the optimal pooling contract (r**, C**) can never be upset by the separating contract (;. C). One can show that not only the optimal pooling contract, but also the separating contract pair (rT, CT), (rz, Ct) as well as any pooling contract that Pareto dominates the pair (rf, Cf), (rr, C:) corresponds to a sequential equilibrium of the three-stage game. However, an application of the Kohlberg and Mertens (1986) stability criterion shows that except for the optimal pooling contract, all these sequential-equilibrium outcomes lack certain desirable robustness properties. The optimal pooling contract (r**, C**) seems to be the most plausible outcome of the three-stage game. The addition of another stage has thus led to a dramatic reversal of the Rothschild, Stiglitz and Wilson results. The three-stage game always has a sequential equilibrium in pure strategies. Moreover the most plausible sequential equilibrium involves pooling rather than sorting whenever pooling is Pareto preferred. The latter conclusion is again reversed if one looks at the model of Cho and Kreps (1986) in which the informed agents move first. Suppose that at stage 1, each entrepreneur announces a collateral C that he wants to put up. At stage 2, each bank announces to each entrepreneur an interest payment r that it requires for a loan with the announced collateral C. Finally, at stage 3, each entrepreneur chooses among the offers that he has received.
324
M. HeLlwig, Theory
ofcompetitionin markets
with adverse selection
In this game again, the separating pair (r:, CT), (rr, Ct) always corresponds to a sequential equilibrium. In this equilibrium, at stage 1, an entrepreneur of type i announces the collateral CT. At stage 2, all banks offer him the interest rate r:, and at stage 3, he chooses one of the banks. If one of the entrepreneurs were to deviate say to the collateral announcement C**, then the banks would believe that this announcement was more likely to come from a type 1 than a type 2 entrepreneur. Given this belief they would ask for interest rates close to [I -(l -pJC**]/p, rather than r**. Therefore such a deviation from the equilibrium strategies is undesirable both for type 1 and type 2 entrepreneurs; the separating equilibrium cannot be upset. One can show that any pooling contract that Pareto dominates the separating contract pair (r:, CT), (r$, CT) is also a sequential-equilibrium outcome of the Cho-Kreps game. However, one now finds that the Kohlberg-Mertens stability criterion singles out the separating outcome (rT,C:), (r:, Cy). If the informed rather than the uninformed agents move first in the game, the pooling rather than the separating equilibria lack robustness. 3. Concluding remarks The preceding discussion shows that three different game theoretic formalizations lead to three different predictions of how a competitive market with adverse selection will work. In the two-stage game of Rothschild, Stiglitz and Wilson, a pure-strategy sequential equilibrium fails to exist if the proportion of ‘good’ types in the population is large. In the same situation, an analysis based on the three-stage game in which the uninformed agents move first predicts the emergence of the optimal pooling contract, whereas the optimal separating contract constellation should emerge if the informed agents move first. The discrepancy between these results corresponds to a clash between two underlying intuitive principles. On the one hand, pooling contracts are vulnerable because at least locally, they provide ‘good’ types with an incentive to distinguish themselves from ‘bad’ types in order to get better conditions. On the other hand, Bertrand competition should lead to (constrained) Pareto optimal outcomes. If the proportion of ‘good’ types in the population is large, the first of these principles points to the separating contract constellation (r:, CT), (rf, Cly) whereas the second points to the pooling contract (r**, C** ). In the three-stage games, the first principle prevails if the informed agents move first, and the second principle prevails if the uninformed agents move first. The discrepancy between the predictions of different game-theoretic models extends to other issues such as the question whether in a market with adverse selection, rationing can be an equilibrium phenomenon. If the model
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of the credit market is modified to allow for an inelastic supply of funds to the banks, then the three-stage game in which the uninformed agents, i.e., the banks, move first supports the contention of Stiglitz and Weiss (1981) that equilibrium credit rationing is possible. Given that the equilibrium involves pooling, the banks may not wish to respond to an excess demand for loans by raising interest rates because this would have an adverse effect on the population mix in the market. On the other hand, the Cho-Kreps specitication in which the informed agents move first will support Bester’s (1985a, 1985b) contention that equilibrium credit rationing cannot occur in a market with an effective sorting device. Bester’s argument rests on the Rothschild-Stiglitz-Wilson assertion that if a sorting device is available, then it will be used. As we have seen, this assertion is true when the informed agents move first, but not necessarily when the uninformed agents move first. For the applied economist who studies, e.g., the regulation of credit or insurance markets, the discrepancy between the predictions of the different game-theoretic models presents a fundamental dilemma. On the one hand, it is important that he determine which of the different models is appropriate for the market at hand. On the other hand, this may be impossible to determine because matters like the order in which people make moves, which make all the difference for the game-theoretic specification, may not be observable and may not even be fixed in a given market.
References Bester, H., 1985a, Screening vs. rationing in credit markets with imperfect information, American Economic Review 85G855. Bester, H., 1985b, The role of collateral in credit markets with imperfect information, Discussion paper no. A-30 (Sonderforschungsbereich 303, University of Bonn, Bonn). Cho, I.K. and D. Kreps, 1986, Signalling games and stable equilibria, Discussion paper (Graduate School of Business, Stanford University, Stanford, CA). Grossman, H., 1979, Adverse selection, dissembling, and competitive equilibrium, Bell Journal of Economics 10, 336343. Hellwig, M.F., 1986, A sequential approach to modelling competition in markets with adverse selection, Discussion paper (Sonderforschungsbereich 303, University of Bonn, Bonn) forthcoming. Kohlberg, E. and J.F. Mertens, 1986, On the strategic stability of equilibria, Econometrica 54, 1003-1038. Kreps, D. and R. Wilson, 1982, Sequential equilibria, Econometrica 50, 863-894. Riley, J., 1979, Informational equilibrium, Econometrica 47, 331-359. Rothschild, M. and J.E. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics 90,629-650. Spence, M., 1973, Job market signalling, Quarterly Journal of Economics 87, 355-374. Stiglitz, J.E. and A. Weiss, 1981, Credit rationing in markets with imperfect information, American Economic Review 71, 393410. Wilson, C., 1977, A model of insurance markets with incomplete information, Journal of Economic Theory 16, 167-207.