Lecture 5 Adverse Selection.

Lecture 5 Adverse Selection

Contents The Market for Lemons
Market Failure and Asymmetric Information (Insurance) Two-Part Tariffs, Screening and Adverse Selection

A. Market for Lemons Quality Uncertainty Assumptions
Assumptions There are two types of used cars good and bad "Lemons".

Assumptions There are two types of used cars good and bad "Lemons". Only the sellers know whether the car is good or bad.

There are two types of used cars good and bad "Lemons". Only the sellers know whether the car is good or bad. Buyers only know the general shares of good and bad. (There are lots of buyers and sellers who meet in a perfectly competitive market for cars. There is only one price for cars.)

There are two types of used cars good and bad "Lemons". Only the sellers know whether the car is good or bad. Buyers only know the general shares of good and bad. There is only one price for used cars.

The supply of bad used cars
Q

The supply of good used cars lies above the supply of bad ones (less are supplied at each price).
Q

The total supply of good and bad cars is the horizontal sum of good and bad cars supply.
Q

The demand for used cars depends on the proportions of good and bad cars. If all cars are bad, then demand is low. P Q

If buyers think there is a better chance of a good car then demand increases.
P Q

If buyers think all cars are good then demand is maximized.
P Q

Suppose the buyers believe all the cars on the market are good, then in the SR mainly bad cars are sold at price P’ P’

As buyers learn that cars are mainly bad the demand shifts in
As buyers learn that cars are mainly bad the demand shifts in. Further decreasing average quality and price P’’

In the limit only bad cars are sold and buyers know this.
P’’’

A. Market for Lemons Summary In extreme cases, no good cars are sold.
Sellers of good cars would like to be able to convince buyers of this but cannot, because anything they say could also be said by low-quality sellers. That is, there is no credible way of distinguishing your good car from someone else’s bad car. Low quality squeezes out high quality. This is called Adverse Selection. It is also present in financial markets, where sellers may be selling because the stock is bad and buyers may be buying because the stock is good. This leads to Bid-Ask Spreads: sellers get offered lower prices to protect the market from those people who are selling because the news is bad.

B. Market Failure and Asymmetric Info
Here we consider a second example of a perfectly competitive market with asymmetric information. By this we mean one side of the market knows more about the product than the other. (Buyers know something sellers don’t or vice versa). In this case we will talk about insurance markets – not used cars. The result is that although the market is competitive the market is inefficient, because not all trade that ought to occur does occur – this is called market failure. Later we will talk about 2 kinds of remedies for this inefficiency: Signalling (where players send credible signals about what they know). Screening (where players choose contracts that only certain types will accept).

The Insurance Market and Adverse Selection The Buyers of Insurance: Consumers come in many different types. Each consumer’s type, 0 <  < 1, is their riskiness of having an accident: p :=Pr(This consumer has an accident).

Assumption : Suppose that there is a continuum of customers with types that are distributed uniformly on the interval between 0 and 1. p :=Pr(This consumer has an accident) 1 p 1

The Insurance Market The Buyers of Insurance: Consumers come in many different types. Each consumer’s type is p :=Pr(This consumer has an accident) Each consumer has wealth = w An accident costs each consumer = L Each has a Utility of wealth = w1/2 Expected utility of type p consumer: = (1-p)w1/2+ p(w-L)1/2

Insurance firms (Suppliers of Insurance): There are many perfectly-competitive firms selling insurance – they compete the profit of selling insurance down to zero. Insurance is sold at price = P If a consumer buys insurance they can claim back their loss when an accident occurs. Expected profit from selling to type p = Revenue – Expected Costs = P- p L

Suppose that each consumer’s riskiness (p) is known to the firms. Then, the firms will offer insurance to the consumer with riskiness  at a price Pp at which they just breakeven: Pp = p L (Perfect competition implies that firms make zero expected profit.) Conclusion: Every consumer gets insurance at the customized price for them. The outcome is Pareto efficient because everyone gets insurance.

Let us check that the consumer wants to buy insurance at this price If they don’t buy they run the risk of making a loss and get the expected payoff. Utility from No Insurance: (1-) w1/2 +  (w-L)1/2 If they buy insurance at the price Pp = p L they have no risk but a lower wealth. Utility from Fair Insurance: (w- p L)1/2

Let us first plot the individual’s utility function: w1/2 w

Now let us plot the utility from making a loss and from buying insurance: w1/2 (w-  L)1/2 (w-L)1/2 w w-L w-L w

We can also plot on this figure the expected payoff from not being insured: w1/2 w1/2 (w-  L)1/2 (1-)w1/2 +(w-L)1/2 (w-L)1/2 w w-L w-L w

As utility is concave being insured is better than not being insured. w1/2 w1/2 (w-  L)1/2 (1-)w1/2 +(w-L)1/2 (w-L)1/2 w w-L w-L w

If the consumer’s riskiness is private (Adverse Selection) There will be only one price for insurance not customised prices, because firms cannot tell customers apart. The price of insurance will reflect the average probability of a claim being made by the buyers of insurance. This is the average cost of supplying the insurance. The consumers buy (or not) because of their individual probability of a claim.

An individual buys if: Expected Value(Insurance) > Expected Value(No Insurance)

An individual buys if: Expected Value(Insurance) > Expected Value(No Insurance) (w-P)1/2 > p(w-L)1/2 + (1-p)w1/2

An individual buys if: Expected Value(Insurance) > Expected Value(No Insurance) (w-P)1/2 > p(w-L)1/2 + (1-p)w1/2 (dividing) (1-(P/w))1/2 > p(1-(L/w))1/2 + (1-p)

An individual buys if: Expected Value(Insurance) > Expected Value(No Insurance) (w-P)1/2 > p(w-L)1/2 + (1-p)w1/2 (dividing) (1-(P/w))1/2 > p(1-(L/w))1/2 + (1-p) p[1-(1-(L/w))1/2 ] > 1- (1-(P/w))1/2

An individual buys if: Expected Value(Insurance) > Expected Value(No Insurance) (w-P)1/2 > p(w-L)1/2 + (1-p)w1/2 (dividing) (1-(P/w))1/2 > p(1-(L/w))1/2 + (1-p) p > 1- (1-(P/w))1/2 1-(1-(L/w))1/2

An individual buys if they have high risk: p> h(P) = 1- (1-(P/w))1/2 1-(1-(L/w))1/2

An individual buys if they have high risk: p> h(P) = 1- (1-(P/w))1/2 1-(1-(L/w))1/2 Whether individuals buy or not also depends on the price: P=0 then h(P)=0/(1-(1-(L/w))1/2) everyone buys When P=L then h(P)=1. no-one buys.

This is what the consumers’ types look like: Distribution of consumer’s types. 1 p 1

The price determines the h(P) threshold: h(P) 1 p 1

The consumers with riskiness above this threshold buy insurance h(P) Not Buy Buy 1 p 1

The total demand is determined by the number of consumers above the threshold: h(P) Not Buy Buy 1 p 1

This is an integral/area: Demand = 1-h(P) h(P) Not Buy Buy 1 p 1

Supply is determined by average riskiness of the buying customers. h(P) Not Buy Buy 1 p 1

This is the average of the types in the blue region. Average Buyer Riskiness h(P) Not Buy Buy 1 p 1

Firms costs are determined by average riskiness of a customer. 0.5[1+h(P)] Not Buy Buy 1 p h(P) 1

B. Market Failure and Asymmetric Information
Price is determined by the average riskiness of a customer. Profit from an average customer = P – L[1+h(P)]/2 Perfect competition gives zero profit L[1+h(P)]/2 – P=0 or L[1+h(P)]/2 = P This is the equation we will solve to find the equilibrium.

Solving L[1+h(P)]/2 = P P

Solving L[1+h(P)]/2 = P P P

Solving L[1+h(P)]/2 = P P P L

Solving L[1+h(P)]/2 = P* P L P=0 L[1+h(0)]/2=L/2 P L

Solving L[1+h(P)]/2 = P P=L P L= L[1+1]/2 L/2 P L

Joining dots and using the fact that h(P) is increasing L[1+h(P)]/2 = P P L L[1+h(0)]/2 P L

IN this case the equilibrium (where L[1+h(P)]/2 = P) occurs where no-one gets insurance and P=L. P L L[1+h(0)]/2 P L

But there could be circumstances in which some (but not all) people get insurance (L[1+h(P)]/2 = P). P L L[1+h(0)]/2 P L

B. Market failure and Asymmetric Information
In the most extreme form of the equilibrium the two lines cross at the highest possible insurance type and only the worst possible customer get’s insurance. This is most famously described in the market for lemons model of Akerlof. There is a massive inefficiency here because almost nobody gets insurance although everybody wants to buy it. There are two kinds of solution to this problem. The first is to use signalling – to allow customers to tell insurance suppliers (credibly) their type. The second is screening. Insurance companies offering products that only low risk types want to buy.

C. Two-Part Tariffs and Adverse Selection
Here we show how a monopolist can offer contracts that screen out different customer types. Let us start by considering a monopolist selling to one customer and setting a two-part tariff. The customer has an income y. If she buys x units of the monopolist’s good at total cost F she gets total value, or utility = y(x)-F+y = Value from good + money left We will suppose that y(0)=0, y`(.)>0, y``(.)<0 (Utility is concave.)

Suppose the monopolist sold the good at a price p. How many units would the consumer purchase? Maximizex y(x)-px+y 0 = y`(x)-p or p = y`(x*(p)) The customer buys x*(p) units and would equate the marginal benefit from another unit of x to its marginal cost.

y is a concave function so it’s derivative decreases y`(x) x

y`(x) P x

y`(x) = Demand Curve for good x P x x*(p)

Suppose the monopolist sold the good at a price p and also charges a fixed fee F. How many units would the consumer purchase? Maximize y(x)-px –F +y

Suppose the monopolist sold the good at a price p and also charges a fixed fee F. How many units would the consumer purchase? Maximize y(x)-px –F +y 0 = y`(x)-p or p = y`(x*)

Suppose the monopolist sold the good at a price p and also charges a fixed fee F. How many units would the consumer purchase? Maximize y(x)-px –F +y 0 = y`(x)-p or p = y`(x*) The customer buys x*(p) units. There is a fixed cost of F of buying the good, so we must check that the fixed costs are worth paying

Suppose the monopolist sold the good at a price p and also charges a fixed fee F. How many units would the consumer purchase? Maximize y(x)-px –F +y 0 = y`(x)-p or p = y`(x*) The customer buys x*(p) units. There is a fixed cost of F of buying the good, so we must check that the fixed costs are worth paying y(x*)-px* –F +y > y(0) +y = y

Suppose the monopolist sold the good at a price p and also charges a fixed fee F. How many units would the consumer purchase? Maximize y(x)-px –F +y 0 = y`(x)-p or p = y`(x*) The customer buys x*(p) units. There is a fixed cost of F of buying the good, so we must check that the fixed costs are worth paying: y(x*)-px* –F +y > y(0) +y = y or y(x*)-px* > F

Summary: The customer buys: x*(p) if y(x*)-px* >F 0 if y(x*)-px* < F That is, the customer either buys the quantity where marginal benefit equals marginal cost (when the value it receives is greater than the fixed cost) or nothing. Another way of saying this is that if the fixed fee is less than the total value the customer obtains then she is prepared to buy the good.

y(x*) is the integral of the y`(.) function. y`(x) P x x*(p)

y(x*) is the integral of the y`(.) function so the blue area equals y(x*) the customer’s value from x*. y`(x) P x x*(p)

y(x*) =F+px* y`(x) P x x*(p)

The fixed fee is the area below the demand curve and above the per unit sales. y`(x) F P px* x x*(p)

How should the monopolist set p and the fixed fee F to maximize profit? The customer will buy as long as y(x*)-px* >F, so the fee F should be set to exhaust grab all the customer’s value. y(x*)-px* = F

y(x*)-px* = F What price maximizes profit then? Profit = Revenue – Cost = px*+F – cx*

y(x*)-px* = F What price maximizes profit then? Profit = Revenue – Cost = px*+F – cx* = px*+ y(x*)-px* – cx*

y(x*)-px* = F What price maximizes profit then? Profit = Revenue – Cost = px*+F – cx* = px*+ y(x*)-px* – cx* = y(x*) – cx*

y(x*)-px* = F What price maximizes profit then? Profit = y(x*) – cx* To maximize this we want y`(x*)= c. That is sell the good at marginal cost and recoup all the profit from the fixed fee. Hence full solution is: p=c, y`(x*)=c and F=y(x*)-cx*.

The price for per unit sales should equal MC and the fixed fee is the area below the demand curve and above the per unit sales. y`(x) F P=MC=c px* x x*(p)

Comments: This is a general principle for two part tariffs. Set the price at marginal cost set the fixed fee to grab all the customer’s value.

Comments: This is a general principle for two part tariffs. Set the price at marginal cost set the fixed fee to grab all the customer’s value. Disney Land/Salad Bars….

Comments: This is a general principle for two part tariffs. Set the price at marginal cost set the fixed fee to grab all the customer’s value. Disney Land/Salad Bars…. Notice finding the monopolist’s solution here can be thought of in terms of finding x* and the fixed fee.

Comments: This is a general principle for two part tariffs. Set the price at marginal cost set the fixed fee to grab all the customer’s value. Disney Land/Salad Bars…. Notice finding the monopolist’s solution here can be thought of in terms of finding x* and the fixed fee. Once you know x* the price is always going to have to be determined by the demand function y`(x*).

Comments: This is a general principle for two part tariffs. Set the price at marginal cost set the fixed fee to grab all the customer’s value. Disney Land/Salad Bars…. Notice finding the monopolist’s solution here can be thought of in terms of finding x* and the fixed fee. Once you know x* the price is always going to have to be determined by the demand function y`(x*). This approach will be taken in the next section.

At last adverse selection and screening! Suppose customers come in two types: Type a and Type b. Share f and 1- f in the population.

At last adverse selection and screening! Suppose customers come in two types: Type a and Type b. Share f and 1- f in the population. Type a customers like good x more: Value of a type a = ay(x)+y

At last adverse selection and screening! Suppose customers come in two types: Type a and Type b. Share f and 1- f in the population. Type a customers like good x more: Value of a type a = ay(x)+y Type b customers like good x less: Value of a type b = by(x)+y b< a.

At any given price the type a’s will buy more and get more value. ay`(x) by`(x) P x

Adverse selection is present, because the monopolist would like to offer to sell the good at marginal cost and take all the value.

Adverse selection is present, because the monopolist would like to offer to sell the good at marginal cost and take all the value. (But if it did this it would have to offer a lower fixed fee to the low-value customers and a higher fixed fee to the high-value customers.) But: All the high-value customers would then pretend to be low-value customers!

Adverse selection is present, because the monopolist would like to offer to sell the good at marginal cost and take all the value. But: All the high-value customers would then pretend to be low-value customers! We will now treat the monopolist’s problem as setting: A quantity xa for the a customers and a total payment they must make Fa.

Adverse selection is present, because the monopolist would like to offer to sell the good at marginal cost and take all the value. But: All the high-value customers would then pretend to be low-value customers! We will now treat the monopolist’s problem as setting: A quantity xa for the a customers and a total payment they must make Fa. A quantity xb for the b customers and a total payment they must make Fb.

To get the a customers to buy these must satisfy: ay(xa)> Fa This is called an “individual rationality constraint”.

To get the a customers to buy these must satisfy: ay(xa)> Fa This is called an “individual rationality constraint”. To get the a types not to pretend to be b customers must satisfy ay(xa)- Fa > ay(xb)- Fb This is called an “incentive compatibility constraint”.

To get the b customers to buy these must satisfy by(xb)> Fb Another “individual rationality constraint”.

To get the b customers to buy these must satisfy by(xb)> Fb Another “individual rationality constraint”. To get the b types not to pretend to be a customers must satisfy: by(xb)- Fb > by(xa)- Fa Another “incentive compatibility constraint”.

Combining all the constraints: ay(xa) – Fa > 0 IR’s by(xb) – Fb > 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa Actually some of these are redundant:

Some of these are redundant: Consider the individual rationality constraint for the high value types, ay(xa) – Fa > 0,

Some of these are redundant: Consider the individual rationality constraint for the high value types, ay(xa) – Fa > 0, This is always true because the a’s do not want to pretend to be b’s and they can get positive value when they do this:

Some of these are redundant: Consider the individual rationality constraint for the high value types, ay(xa) – Fa > 0, this is ensured because the a’s do not want to pretend to be low value types and they can get positive value when they do this: ay(xa)- Fa > ay(xb)- Fb (IC for a’s)

Some of these are redundant: Consider the individual rationality constraint for the high value types, ay(xa) – Fa > 0, this is ensured because the a’s do not want to pretend to be low value types and they can get positive value when they do this: ay(xa)- Fa > ay(xb)- Fb (IC for a’s) > by(xb)- Fb ( a>b)

Some of these are redundant: Consider the individual rationality constraint for the high value types, ay(xa) – Fa > 0, this is ensured because the a’s do not want to pretend to be low value types and they can get positive value when they do this: ay(xa)- Fa > ay(xb)- Fb (IC for a’s) > by(xb)- Fb ( a>b) > by(xb) – Fb > 0 (IR for b’s)

Some of these are redundant: Consider the individual rationality constraint for the high value types, ay(xa) – Fa > 0, this is ensured because the a’s do not want to pretend to be low value types and they can get positive value when they do this: ay(xa)- Fa > ay(xb)- Fb (IC for a’s) > by(xb)- Fb ( a>b) > by(xb) – Fb > 0 (IR for b’s) Hence the IR constraint for the the a’s is guaranteed by the other constraints.

So we only need to consider the constraints: IR’s by(xb) – Fb > 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa But we are still not done: Two of these constraints hold with equality and this will allow us to substitute in from the constraints.

IR’s by(xb) – Fb > 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa First consider increasing Fa Fb by the same amount (this is good for the monopolist). The green constraints still hold. Eventually (as Fb increases) the blue constraint will bind. So it is enough to consider the blue constraint as binding.

IR’s by(xb) – Fb = 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa The monopolist would still like to make Fa as big as possible. This might violate this constraint

IR’s by(xb) – Fb = 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa The monopolist would still like to make Fa as big as possible. This might violate this constraint.

IR’s by(xb) – Fb = 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa The monopolist still would like to make Fa as big as possible. This might violate this constraint. But would never harm this constraint.

IR’s by(xb) – Fb = 0 ay(xa)- Fa = ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa The monopolist still would like to make Fa as big as possible. This might violate this constraint. But would never harm this constraint. So we can assume the middle constraint holds with equality ay(xa)- Fa = ay(xb)- Fb.

The firm’s wants to maximize its profit under these four constraints, so it has the maximization problem: Maximize: f(Fa – cxa)+ (1-f)(Fb – cxb) Subject to: ay(xa) - Fa > 0 IR’s by(xb) - Fb > 0 ay(xa)- Fa > ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa

We have shown that one of these constraints is redundant and two others are equalities Maximize: f(Fa – cxa)+ (1-f)(Fb – cxb) Subject to: by(xb) - Fb = 0 ay(xa)- Fa = ay(xb)- Fb IC’s by(xb)- Fb > by(xa)- Fa We will substitute the equality constraints to solve this problem…

If by(xb) – Fb = 0, and ay(xa)- Fa = ay(xb)- Fb We can solve for the fees: Fb = by(xb)

If by(xb) – Fb = 0, and ay(xa)- Fa = ay(xb)- Fb We can solve for the fees: Fb = by(xb) Fa = ay(xa)- ay(xb)+ Fb

If by(xb) – Fb = 0, and ay(xa)- Fa = ay(xb)- Fb We can solve for the fees: Fb = by(xb) Fa = ay(xa)- ay(xb)+ Fb Fa = ay(xa)- ay(xb)+ by(xb)

If by(xb) – Fb = 0, and ay(xa)- Fa = ay(xb)- Fb We can solve for the fees: Fb = by(xb) Fa = ay(xa)- ay(xb)+ Fb Fa = ay(xa)- ay(xb)+ by(xb) Fa = ay(xa)+(b- a)y(xb)

If by(xb) – Fb = 0, and ay(xa)- Fa = ay(xb)- Fb We can solve for the fees: Fb = by(xb) Fa = ay(xa)- ay(xb)+ Fb Fa = ay(xa)- ay(xb)+ by(xb) Fa = ay(xa)+(b- a)y(xb) The monopolist’s profit then becomes: Profit = f(Fa – cxa)+ (1-f)(Fb – cxb)

If by(xb) – Fb = 0, and ay(xa)- Fa = ay(xb)- Fb We can solve for the fees: Fb = by(xb) Fa = ay(xa)- ay(xb)+ Fb Fa = ay(xa)- ay(xb)+ by(xb) Fa = ay(xa)+(b- a)y(xb) The monopolist’s profit then becomes: Profit = f(Fa – cxa)+ (1-f)(Fb – cxb) = fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb

The firm wants to maximize its profit under these two constraints, so it has the maximization problem: Maximize: fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb Subject to: by(xb)- Fb > by(xa)- Fa

The firm wants to maximize its profit under these two constraints, so it has the maximization problem: Maximize: fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb Subject to: by(xb)- Fb > by(xa)- Fa by(xb)- by(xb) > by(xa)- ay(xa)-(b- a)y(xb)

The firm wants to maximize its profit under these two constraints, so it has the maximization problem: Maximize: fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb Subject to: by(xb)- Fb > by(xa)- Fa by(xb)- by(xb) > by(xa)- ay(xa)-(b- a)y(xb) 0 > (b- a)[y(xa) - y(xb)]

The firm wants to maximize its profit under these two constraints, so it has the maximization problem: Maximize: fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb Subject to: by(xb)- Fb > by(xa)- Fa by(xb)- by(xb) > by(xa)- ay(xa)-(b- a)y(xb) 0 > (b- a)[y(xa) - y(xb)] y(xa) > y(xb)

The firm wants to maximize its profit under these two constraints, so it has the maximization problem: Maximize: fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb Subject to: by(xb)- Fb > by(xa)- Fa by(xb)- by(xb) > by(xa)- ay(xa)-(b- a)y(xb) 0 > (b- a)[y(xa) - y(xb)] y(xa) > y(xb) xa > xb

The firm wants to maximize its profit under these two constraints, so it has the maximization problem: Maximize: fay(xa)+(b- fa)y(xb)- fcxa – (1-f)cxb Subject to: xa > xb Lagrangean= fay(xa)+(b- fa)y(xb)- fcxa– (1-f)cxb + l(xa – xb) L/xa = fay`(xa)- fc + l = 0 L/xb = (b- fa)y`(xb) - (1-f)c - l = 0

Suppose the constraint binds l > 0 and xa = xb Then, the first order conditions become: fay`(xa)- fc + l = 0, (b- fa)y`(xa) - (1-f)c - l = 0.

Suppose the constraint binds l > 0 and xa = xb Then, the first order conditions become: fay`(xa)- fc + l = 0, (b- fa)y`(xa) - (1-f)c - l = 0. Adding these gives by`(xa) - c = 0 Or by`(xa) = c.

Suppose the constraint binds l > 0 and xa = xb Then, the first order conditions become: fay`(xa)- fc + l = 0, (b- fa)y`(xa) - (1-f)c - l = 0. Adding these gives by`(xa) - c = 0 Or by`(xa) = c. Substitution back into the first of these then gives

Suppose the constraint binds l > 0 and xa = xb Then, the first order conditions become: fay`(xa)- fc + l = 0, (b- fa)y`(xa) - (1-f)c - l = 0. Adding these gives by`(xa) - c = 0 Or by`(xa) = c. Substitution back into the first of these then gives f(a-b)y`(xa) + l = 0,

Suppose the constraint binds l > 0 and xa = xb Then, the first order conditions become: fay`(xa)- fc + l = 0, (b- fa)y`(xa) - (1-f)c - l = 0. Adding these gives by`(xa) - c = 0 Or by`(xa) = c. Substitution back into the first of these then gives f(a-b)y`(xa) + l = 0, Which cannot hold unless l < 0 a contradiction.

Thus this constraint does not bind and l = 0. Hence the monopolist will screen the two types of customer by choosing contracts: L/xa = fay`(xa)- fc = 0 L/xb = (b- fa)y`(xb) - (1-f)c = 0 Hence the monopolist chooses ay`(xa)=c The high value types are sold the bundle they most want.

At any given price the type a’s will buy more and get more value. ay`(.) by`(x) c=MC x ay`(xa)

For the low-value types: L/xb = (b- fa)y`(xb) - (1-f)c = 0 Hence the monopolist chooses by`(xb) = c (1-f)/[1- f(a/b)] > c As a/b>1 the low types are sold the bundle at above marginal cost.

At any given price the type a’s will buy more and get more value. ay`(.) by`(.) c(1-f)/[1-f(a/b)] c x by`(xb) ay`(xa)

What Fees do they Pay? Recall we substituted for the fees: Fb = by(xb) Fa = ay(xa)+(b- a)y(xb) First Fb = by(xb), so the B-types pay a fee that exhausts all their value. (So as we suspected traveling economy makes you wish you’d not bothered.)

ay`(.) by`(.) c(1-f)/[1-f(a/b)] c x by`(xb) ay`(xa)

What Fees do they Pay? Fb = by(xb) Fa = ay(xa)+(b- a)y(xb) Hence Fa = ay(xa)-(a- b)y(xb) The A-types pay less than their total value by the amount (a- b)y(xb). This is the gap between ay(xb) - the A’s value from having B’s bundle.

ay`(.) by`(.) c(1-f)/[1-f(a/b)] c x by`(xb) ay`(xa)

The A-types pay less than their total value by the amount (a- b)y(xb). This is the gap between ay(xb) - the A’s value from having B’s bundle. And B’s value from their bundle: by(xb)

(a-b)y(xb) by`(.) c(1-f)/[1-f(a/b)] c x by`(xb) ay`(xa)

Hence if the A-types pay less than their total value by the amount ay(xa) - (a- b)y(xb). This is the area

ay(xa)-(a-b)y(xb) by`(.) c(1-f)/[1-f(a/b)] c x by`(xb) ay`(xa)

Summary: The right way to solve the adverse selection problem in this case to efficiently serve the high-value customers and treat the low-value types suboptimally. How would this relate to air-travel?

Lecture 5 Adverse Selection.

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Lecture 5 Adverse Selection.

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Presentation on theme: "Lecture 5 Adverse Selection."— Presentation transcript:

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