9.2 Adverse Selection under Certainty: Lemons I and II The - PowerPoint PPT Presentation

9.2 Adverse Selection under Certainty: Lemons I and II  The principal contracts to buy from the agent a car whose quality is noncontractible despite the lack of uncertainty.  The Basic Lemons Model Players ð a buyer and a seller r

The order of play ð ) 0 Nature chooses quality type for the seller F ) according to the distribution ( ). ) The seller knows , but while the buyer knows , he does not know the of ) F the particular seller he faces. 1 The buyer offers a price . P The seller accepts or rejects. 2

Payoffs ð If the buyer rejects the offer, both players receive payoffs of zero. r Otherwise, 1 œ ( ) )  and 1 œ  ( ), ) r V P P U buyer seller where and will be defined later. V U The payoffs of both players are normalized to zero ð if no transaction takes place. The payoff functions show changes from that base. r

Competition between buyers ð It will often be convenient to discuss the game as if it had many sellers . ð There is a population of sellers of different types, r one of whom is drawn by Nature to participate in the game.  A theme running through all four Lemons models is that when quality is unknown to the buyer, less trade occurs .

 Lemons I: Identical Tastes, Two Types of Sellers Specific functional forms ð ) − quality type {2,000, 6,000} r F ( ) puts probability 0.5 on the first type, ) ) œ 2,000, r ) œ and probability 0.5 on the second type, 6,000. A payoff profile of (0, 0) will represent the status quo , r in which the buyer has $50,000 and the seller has the car. the players' valuations for a car of quality ) r 1 œ )  1 œ  ) and r P P buyer seller

If he could observe quality at the time of his purchase, ð the buyer would be willing to pay $6,000 for a good car and $2,000 for a lemon. The buyer cannot observe quality . ð Assume that he cannot enforce a contract based on his discovery r once the purchase is made. The buyer would be willing to pay $4,000, r a price equal to the average quality of cars offered for sale, for a car of unknown quality if all cars were on the market.

The buyer would refuse to pay more than $2,000. r Half of the cars are traded in equilibrium, all of them lemons . r The outcome that half the cars are held off the market is interesting ð since half the cars do have genuinely higher quality.

 Lemons II: Identical Tastes, a Continuum of Types of Sellers Specific functional forms ð The quality types are uniformly distributed r between 2,000 and 6,000. ( ) ) œ [( )  2,000) 4,000] Î ( ) )  ( ), ) r F I I ∞ [2,000, 6,000] (6,000, ) where Z † ( ) is the indicator function of a set I Z the players' valuations for a car of quality ) r 1 œ )  1 œ  ) and r P P buyer seller

The probability density function of the continuous uniform distribution ð is: œ Î  Ÿ Ÿ f x ( ) 1 ( b a ) for a x b   0 for x a or x b . The cumulative distribution function of the uniform distribution is: œ  ( ) 0 for F x x a  Î  Ÿ Ÿ ( ) ( ) for x a b a a x b  1 for . x b The mean of the uniform distribution is: œ  Î E X ( ) ( a b ) 2.

_ ) œ The average quality is 4,000. ð The buyer would be willing to pay $4,000 r for a car of unknown quality if all cars were on the market. the average quality of cars offered for sale r The unravelling continues until the price reaches its equilibrium level ð of $2,000. But at P œ 2,000, the number of cars on the market is r infinitesimal. The market is completely collapsed! r

Figure 9.2 ð the price of used cars on the vertical axis r P _ the average quality of cars offered for sale on the ) horizontal axis r _ ) Each price leads to a different average quality, ( ), and r P _ the slope of ( ) is greater than one: ) P _ ) œ  Î ( ) (2,000 ) 2. P P If the price rises, the quality of the marginal car offered for sale r equals the new price, but the average quality of cars offered for sale is much lower .

The buyer would be willing to pay a price equal to r the average quality of cars offered for sale: _ _ ) œ ) P ( ) . In equilibrium, the average quality must equal the price , and r the quality of the marginal car offered for sale must equal the price . the players' valuations for a car of quality ) r At the intersection of the two lines (or curves), r these equilibrium conditions are met.

The only intersection is the point ($2,000, 2,000). r o The equilibrium lies on the 45 line through the origin. r  There is no efficiency loss in either Lemons I or Lemons II. Since all the players have identical tastes, ð it does not matter who ends up owning the cars.

9.3 Heterogeneous Tastes: Lemons III and IV  Lemons III: Buyers Value Cars More Than Sellers Specific functional forms ð The quality types are uniformly distributed r between 2,000 and 6,000. ( ) ) œ [( )  2,000) 4,000] Î ( ) )  ( ), ) r F I I ∞ [2,000, 6,000] (6,000, ) where Z † ( ) is the indicator function of a set I Z

Sellers value their cars at exactly their qualities , r but buyers have valuations 20 percent greater . the players' valuations for a car of quality ) r 1 œ 1.2 )  and 1 œ  ) r P P buyer seller The buyers outnumber the sellers. r

Figure 9.3 ð _ ) ( ) œ (2,000  ) 2 Î r P P The buyer would be willing to pay a price equal to r 1.2 times the average quality of cars offered for sale: _ _ P ( ) ) œ 1.2 . ) In equilibrium, 1.2 times the average quality must equal the price , r and the quality of the marginal car offered for sale must equal the price .

quality ) the players' valuations for a car of r At the intersection of the two lines (or curves), r these equilibrium conditions are met. _ They intersect only at ( , ) ) P œ (2,500, 3,000). r

Because buyers are willing to pay a premium, ð we only see partial adverse selection. The equilibrium is partially pooling. ð In equilibrium, the sellers will capture the gains from trade. ð The outcome is inefficient . ð In a world of perfect information, all the cars would be owned r by the "buyers," who value them more . Under adverse selection, the buyers only end up owning r the low-quality cars.

 Lemons IV: Sellers' Valuations Differ Specific functional forms ð We model sellers as consumers whose valuations of quality r have changed since they bought their cars. quality ) the players' valuations for a car of r 1 œ )  and 1 œ  (1  % ) ) r P P buyer seller The random disturbance % can be either positive or negative and r has an expected value of zero. The buyers outnumber the sellers. r

Figure 9.4 ð the average quality of cars offered for sale at price r P _ ) ( ) œ ( (1 ) l  % ) ) Ÿ ) P E P If P 6,000, some car owners would be reluctant to sell, r because they received positive disturbances to their valuations. The average quality of cars on the market is less than 4,000 r P œ even at 6,000.

P œ Even if 2,000, some sellers with low-quality cars and r negative realizations of the disturbance do sell, so the average quality remains above 2,000. _ _ P ( ) ) œ ) r In equilibrium, the average quality must equal the price , and r the marginal seller's valuation (1  % ) m ) for his car offered for sale must equal the price . the players' valuations for a car of quality ) r

At the intersection of the two curves, r these equilibrium conditions are met.  A theme running through all four Lemons models is that when quality is unknown to the buyer, less trade occurs .

 More Sellers Than Buyers Lemons III ð 1 œ 1.2 )  and 1 œ  ) r P P buyer seller The buyers outnumber the sellers. r A buyer would offer a higher price to purchase a car. r The sellers earn producer surplus. r The market clears. r

The sellers outnumber the buyers. ð ) œ If there were enough sellers with quality 2,000, r each buyer would pay P œ $2,000 for a car worth 2,400 to him, acquiring a surplus of 400. If there were fewer sellers , r the equilibrium price would be higher and some sellers would receive producer surplus.

 Heterogeneous Buyers: Excess Supply Lemons III ð 1 œ 1.2 )  and 1 œ  ) r P P buyer seller The buyers outnumber the sellers. r A buyer would offer a higher price to purchase a car. r The sellers earn producer surplus. r The market clears. r If buyers have different valuations for a car of quality ) , ð then the market might not clear.

 Risk Aversion Lemons III ð 1 œ )  1 œ  ) 1.2 and r P P buyer seller The buyers outnumber the sellers. r A buyer would offer a higher price to purchase a car. r The sellers earn producer surplus. r The market clears. r