Asymmetric Information

Transcription

1 Asymmetric Information Econ 235, Sring Wilson [1980] What haens when you have adverse selection? What is an equilibrium? What are we assuming when we define equilibrium in one of the ossible ways? 1.1 Setu A version of the Akerlof [1970] model Two tyes of agents: Buyers and Sellers Sellers observe quality but Buyers don t Buyers Buyer references are u (c, q; t) = c + tq where c is consumtion of numeraire q is the quality of a car if you consume a car. You can only consume 0 or 1 cars. If you consume zero cars we set q = 0 t is the agent s tye - how much you like cars c and q could also reresent consumtion in different eriods and t could be a discount rate - we ll see models like this later H (t) is the measure of Buyers with tye lower than t 1

2 Heterogeneity across Buyers is not essential, it s only role is to create a downward sloing demand even though everyone buys just one unit. An alternative way to do it is to have homogenous Buyers and an intensive margin for demand. Alternative version: continuum of identical Buyers with references: u (c, q, d) = c + G (d) q where d is the number of cars that he buys. These versions are equivalent for some uroses and slightly different for others. Buyers have no endowment of cars but have unlimited endowments of numeraire we ll see models where they have limited wealth later. In those models we don t need to restrict consumtion of cars to be zero or one Sellers Seller references are u (c, q; t) = c + q and is constant across sellers - we ll see models with heterogeneity in later Gains from trade com from < t They have an endowment of one car and no numeraire The measure of Sellers who own cars of quality less than q is F (q) 1.2 Walrasian Equilibrium Everyone takes rices as given Buyers understand what samle of cars is sold No one makes conjectures about what cars would be sold if rices were different Sellers solve max c + q (1 s) c,s {0,1} s.t. c = s 2

3 Buyers solve max c + t qd c,d {0,1} s.t. c = e d where q is the average quality of cars that are sold In the alternative version, they solve maxc + G (d) q c,d s.t. c = e d Imlicity, we are assuming that any Buyer gets a car at random, and because they are risk-neutral with resect to quality, they only care about the average quality q Definition 1. A Walrasian equilibrium is {, q, s (q), d (t)} such that 1. s(q) solves Seller roblem taking as given 2. d (t) solves Buyer roblem taking and q as given 3. markets clear ˆ ˆ s (q) df (q) = d (t) dh (t) 4. the average quality is q = qs (q) df (q) s (q) df (q) Suly and Demand Suly decisions are simle: s (q) = I ( > q) and therefore for a given rice we can find suly and average quality Suly ( ) S () = F Average quality q () = q qdf (q) F ( ) 3

4 Note the selection effect: d q d = f ( ) 1 F ( qdf (q) q ) ( f ) 1 [ ( F )] 2 = f ( ) 1 [ ( F )] 2 ˆ ( ) F qdf (q) = f ( ) 1 [ F ( )] 2 ˆ q ( q ) df (q) > 0 q The average quality is better at higher rices This is because the marginal seller is the highest-quality seller in the samle Demand decisions are: d (t) = I [ < t q] and therefore we can find a demand function D () = H ( t) H ( ) q () In the alternative version, demand is given by the FOC + G (d) q = 0 ) d = (G ) 1 ( q so if (G ) 1 (x) = H ( t) H (x) then demand is the same under both models In both cases, demand deends on () q () is the rice er unit of quality, which can be increasing or decreasing in Equilibrium condition: S () = D ( ()) 4

5 1.2.2 Possible Multilicity Demand can be uward sloing in if () is decreasing over some range. What are the conditions on F that will make demand be uward sloing? Possible Market Breakdown Let the lowest ossible quality be q When q then S () 0 q () q Demand becomes ( ) q D () H ( t) H = H ( t) H () as long as q > 0 q so as long as q > 0 and t > (i.e. the highest valuation Buyer likes cars more than Sellers, which is a natural assumtion), then D ( q ) > S ( q ) so market clearing must occur at a higher rice and trade will not break down entirely In this model trade can only break down if q = 0 More generally the breakdown of trade requires that there be some lowest quality such that there are no gains from trade, and sufficient mass near that quality Is the equilibrium reasonable? Consider the case where: Equilibrium is V ( ) > V ( ) for some > Non-Walrasian deviations by Buyers 5

6 1.3 Buyer s Equilibrium Instead of a Walrasian definition, consider the following game: 1. Buyers simultaneously announce a single rice at which they are willing to buy one car 2. Starting from the highest rice that anyone has announced, Sellers simultaneously decide whether to offer their cars at that rice (a) If demand exceeds or equals suly, all offered cars are allocated at random across Buyers (b) If suly exceeds demand, there is random rationing 3. The rocess is next reeated at the second-highest rice that some Buyer has announced, and so on until the lowest rice that any Buyer offered In the intensive version, Buyers announce a quantity of cars they are willing as well as a rice Definition 2. A Buyer s equilibrium is a Nash equilibrium of that game Seller decisions Offer to sell at every rice until either: You manage to sell The rice reaches = q Notice that the average quality that is offered at a rice is the same q () as in the Walrasian equilibrium. Why? Buyer decisions Given how Sellers behave, it must be that there is a cutoff such that Buyers exect to buy for sure for any rice above and not to be able to buy at any rice below Hence no one announces rices below The rice that each Buyer announces solves (t) arg max t q ( ) In the alternative version, this roblem is identical for all Buyers, but they might still announce different rices because there could be more than one solution 6

7 1.3.3 Equilibrium Let W be the highest Walrasian equilibrium If t q ( ) < t q ( ) W W for all > W and for all t, then the unique Buyer s equilibrium coincides with the Walrasian equilibrium Otherwise, the equilibrium will involve: some Buyers offering rices above the Walrasian equilibrium; rationing at those rices (maybe) some Buyers offering a lower rice at which Sellers can sell for sure 1.4 Seller s equilbrium Now consider an alternative game Sellers announce a rice at which they offer their car Buyers choose at which of the ossible rices they will buy At rices where there is excess suly, there is rationing Buyers commit to a rice: if they do not sell at the rice they announce they cannot try again at a different rice This could be because they commit or because of a true technological imediment to osting a new rice Sellers solve max η () + (1 η ()) q where η () is the robability of selling at a given rice (which is endogenous but the Seller takes as given) If the set of rices that somebody announces is P, Buyers solve max P t qe ( ) where q e ( ) describes beliefs about the average quality that will be available at a given rice In the alternative version, the fact that Buyers can choose from which market to buy imlies that () = q e () is equated across all the markets with ositive demand. Definition 3. A Sellers equilibrium consists of { (q), (t), η (), q e ()} such that: 7

8 (q) solves the Seller roblem (t) solves the Buyer roblem η () = H({t:(t)=}) 1 F ({q:(q)=}) q e () = {q:(q)=} qdf (q) F ({q:(q)=}) for all P This is like a signaling game, with otentially many equilibria (What do you infer from the fact that a Seller osted a articular rice?) The last requirement is the tyical Bayes rule on-equilibrium condition of a PBE, which can be strengthened with refinements The aer was written before a lot of the theory of signaling games and their refinements was develoed General structure of searating equilibria: Sellers announce different rices that erfectly reveal their tyes There is rationing at all these rices Sellers of higher quality cars are willing to tolerate lower robability of selling in exchange for higher rice to a greater extent than owners of lower-quality cars (single crossing) The rice-quality schedule is such that at the margin Buyers are indifferent between buying at different rices. This makes them willing to buy just enough units from the high-rice market to make the robability of selling the correct one that will searate Sellers. 2 Stiglitz and Weiss [1981] Alication to credit markets Imlicitly, uses the Buyer s equilibrium, without calling it that. 1 This is a bit imrecise in two different ways. First, we are assuming that the ratio will be less or equal than 1; otherwise we need to secify how buyers will be rationed if there was excess demand. Second, we have to be measure-theoretically recise for the case where the two measures are zero but the ratios are still well defined. 8

9 2.1 Setu Heterogeneous entrereneurs. Tye θ is unobservable Entrereneur θ has access to a roject that will ay y F (y θ) All entrereneurs have A but need to borrow I A = 1 E (y θ) is the same for all θ Higher θ means a more risky roject in a SOSD sense Contracts are ure debt contracts with some contractually-romised ayment r and actual reayment d l d l = min {r, y} entrereneur gets d e = max {y r, 0} Payoffs for lender are concave while ayoffs for borrower are convex Lenders have an uward-sloing suly of loans (as a function of the exected return R = E (d)): S (R) Buyer (i.e. lender) equilibrium : lenders roose interest rate and the amount they are willing to lend and entrereneurs accet/reject 2.2 Adverse Selection Entrereneur finds the deal attractive if E [d e (r, y)] > A Because d e is a convex function of y, then more risk makes the deal more attractive to entrereneurs. For an interior case, there will be a cutoff tye θ (r) such that entrereneurs take deal iff θ > θ (r) ˆ (y r) df ( y θ ) = A y r 9

10 Adverse selection overall: only the risky tyes borrow How does θ change with r? ˆ y r ( 1) df ( y θ ) [ y r (y r) df ( y θ )] + θ θ r = 0 θ r = 1 F ( r θ) [ y r (y r)df(y θ)] θ > 0 Adverse selection at the margin: if you increase the contractual interest rate r, then marginal tyes dro out and the remaining ool is riskier! Key assumtion: asymmetric information about risk but NOT about E (y). Assume there is heterogeneity in E (y), and it s unobservable Entrereneurs with higher E (y) are MORE willing to borrow at any given rate AND give higher returns to lenders This would give FAVOURABLE selection rather than adverse selection! 2.3 Credit rationing Comute the exected return for lenders as a function of the contractual interest rate: R (r) = E ( min {r, y} θ θ (r) ) This function could be nonmonotonic! Deending on the shae of S (R), there could be credit rationing in a Buyer s equilibrium 2.4 The Riley [1987] critique Assume there is some observable characteristic x of entrereneurs, which is informative about θ (even if very weakly) Lenders could condition lending terms on x Comute R (r, x) = E ( min {r, y} θ θ (r), x ) This function will be slightly different for different grous 10

11 Original aer has different grous heterogeneous in E (y x), but heterogeneity in the conditional distribution F (θ x) leads to the same conclusion Only at most one grou will face ure rationing Redlining rationing? 2.5 The Arnold and Riley [2009] critique Actually, in the Stiglitz and Weiss [1981] model, the maximum of the function R (r) must be reached at the oint in which the riskiest tye has zero demand Because at that oint the lender catures the entire surlus! Related: the whole roblem would go away if we had equity contracts. Equilibrium cannot involve rationing in the sense of inability to borrow at any interest rate There can be rationing at a low rate, but market clearing at a higher rate See grah 3 Prices as markets Alternative way to define equilibrium Walrasian in style (no game, agents are small and take things as given) Catures the off-equilibrium-rice considerations of the game-theoretic formulation. Each rice defines a market No market clearing in a traditional sense Rational exectations about what qualities are available in each market See Gale [1992, 1996], Guerrieri et al. [2010], Guerrieri and Shimer [2012a,b], Chang [2011], Kurlat [2012] 11

12 References George A. Akerlof. The market for lemons : Quality uncertainty and the market mechanism. The Quarterly Journal of Economics, 84(3): , doi: / Lutz G. Arnold and John G. Riley. On the ossibility of credit rationing in the stiglitz-weiss model. American Economic Review, 99(5): , December Briana Chang. Adverse selection and liquidity distortion in decentralized markets. Discussion Paers 1513, Northwestern University, Center for Mathematical Studies in Economics and Management Science, August Douglas Gale. A walrasian theory of markets with adverse selection. Review of Economic Studies, 59(2):229 55, Aril Douglas Gale. Equilibria and areto otima of markets with adverse selection. Economic Theory, 7(2): , Veronica Guerrieri and Robert Shimer. Dynamic adverse selection: A theory of illiquidity, fire sales, and flight to quality. Working Paer 17876, National Bureau of Economic Research, March 2012a. Veronica Guerrieri and Robert Shimer. Markets with multidimensional rivate information. University of Chicago Working Paer, 2012b. Veronica Guerrieri, Robert Shimer, and Randall Wright. Adverse selection in cometitive search equilibrium. Econometrica, 78(6): , November Asset markets with heterogeneous information. Stanford University Working Paer, John G Riley. Credit rationing: A further remark. American Economic Review, 77(1):224 27, March Joseh E Stiglitz and Andrew Weiss. Credit rationing in markets with imerfect information. American Economic Review, 71(3): , June Charles Wilson. The nature of equilibrium in markets with adverse selection. The Bell Journal of Economics, 11(1): ,