Frank Cowell Microeconomics March 2007 Exercise 11 1

Frank Cowell: Microeconomics March 2007 Exercise 11. 1 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 11. 1(1): Question Frank Cowell: Microeconomics n n purpose: to illustrate and solve the “hidden information” problem method: find full information solution, describe incentive-compatibility problem, then find second-best solution

Ex 11. 1(1): Budget constraint Frank Cowell: Microeconomics n Consumer has income y and faces two possibilities u u n Define a binary variable i: u u n “not buy”: all y spent on other goods “buy”: y F(q) spent on other goods i = 0 represents the case “not buy” i = 1 represents the case “buy” Then the budget constraint can be written u x + i. F(q) ≤ y

Ex 11. 1(2): Question Frank Cowell: Microeconomics method: n First draw ICs in space of quality and other goods n Then redraw in space of quality and fee n Introduce iso-profit curves n Full-information solutions from tangencies

Ex 11. 1(2): Preferences: quality Frank Cowell: Microeconomics §(quality, other-goods) space F x §high-taste type §low-taste type §redraw in (quality, fee) space ta tb tb ta pre fer en ce §IC must be linear in t §t a > t b §Because linear ICs can only intersect once quality q

Ex 11. 1(2): Isoprofit curves, quality Frank Cowell: Microeconomics §(quality, fee) space F §Iso-profit curve: low profits §lso-profit curve: medium profits inc pro reas fit ing §lso-profit curve: high profits P 2 = F 2 C(q) §Increasing, convex in quality P 1 = F 1 C(q) P 0 = F 0 C(q) quality q

Ex 11. 1(2): Full-information Frank Cowell: Microeconomics solution §reservation IC, high type F §Firm’s feasible set for a high type §Reservation IC + feasible set, low type §lso-profit curves taq §Full-information solution, high type • F*a F*b §Full-information solution, low type tbq §Type-a participation constraint taqa Fa ≥ 0 • q*b §Type-b participation constraint tbqb Fb ≥ 0 q*a quality q §Full information so firm can put each type on reservation IC

Ex 11. 1(3, 4): Question Frank Cowell: Microeconomics method: n Set out nature of the problem n Describe in full the constraints n Show which constraints are redundant n Solve the second-best problem

Ex 11. 1(3, 4): Misrepresentation? Frank Cowell: Microeconomics §Feasible set, high type F §Feasible set, low type §Full-information solution taq §Type-a consumer with a type-b deal pre fer e F*a F*b nc e • §Type-a participation constraint taqa Fa ≥ 0 tbq §Type-b participation constraint tbqb Fb ≥ 0 • q*b q*a quality q §A high type-consumer would strictly prefer the contract offered to a low type

Ex 11. 1(3, 4): background to problem Frank Cowell: Microeconomics n Utility obtained by each type in full-information solution is y u u n If a-type person could get a b-type contract u u n n each person is on reservation utility level given the U function, if you don’t consume the good you get exactly y a-type’s utility would then be taq*b F*b +y given that tbq*b F*b = 0… …a-type’s utility would be [ta tb]q*b + y >y So an a-type person would want to take a b-type contract In deriving second-best contracts take account of 1. 2. participation constraints this incentive-compatibility problem

Ex 11. 1(3, 4): second-best problem Frank Cowell: Microeconomics n n Participation constraint for the two types u ta qa Fa ≥ 0 u tb qb Fb ≥ 0 Incentive compatibility requires that, for the two types: u taqa Fa ≥ taqb Fb u tb qb Fb ≥ tb qa Fa Suppose there is a proportion p, 1 p of a-types and b-types Firm's problem is to choose qa, qb, Fa and Fb to max expected profits u p[Fa C(qa)] + [1 p][Fb C(qb)] subject to u u n the participation constraints the incentive-compatibility constraints However, we can simplify the problem u u which constraints are slack? which are binding?

Ex 11. 1(3, 4): participation, b-types Frank Cowell: Microeconomics n First, we must have taqa Fa ≥ tbqb Fb u 1. 2. n This implies the following: u u n if tbqb Fb > 0 (b-type participation slack) then also taqa Fa > 0 (a-type participation slack) But these two things cannot be true at the optimum u u n this is because taqa Fa ≥ taqb Fb (a-type incentive compatibility) and ta > tb (a-type has higher taste than b-type) if so it would be possible for firm to increase both Fa and Fb thus could increase profits So b-type participation constraint must be binding u tbqb Fb = 0

Ex 11. 1(3, 4): participation, a-types Frank Cowell: Microeconomics n If Fb > 0 at the optimum, then qb > 0 u u n This implies taqb Fb > 0 u u n because a-type has higher taste than b-type ta > tb This in turn implies taqa Fa > 0 u u n follows from binding b-type participation constraint tbqb Fb = 0 follows from a-type incentive-compatibility constraint taqa Fa ≥ taqb Fb So a-type participation constraint is slack and can be ignored

Frank Cowell: Microeconomics Ex 11. 1(3, 4): incentive compatibility, a-types n Could a-type incentive-compatibility constraint be slack? u n If so then it would be possible to increase Fa … u u u n could we have taqa Fa > taqb Fb ? …without violating the constraint this follows because a-type participation constraint is slack taqa Fa > 0 So a-type incentive-compatibility must be binding u taqa Fa = taqb Fb

Frank Cowell: Microeconomics Ex 11. 1(3, 4): incentive compatibility, b-types n n Could b-type incentive-compatibility constraint be binding? u tbqa Fa = tbqb Fb ? If so, then qa = qb u u n So, both incentive-compatibility conditions bind only with “pooling” u u n follows from fact that a-type incentive-compatibility constraint is binding ta qa Fa = ta qb Fb which, with the above, would imply [tb ta]qa = [tb ta]qb given that ta > tb this can only be true if qa = qb but firm can do better than pooling solution: increase profits by forcing high types to reveal themselves So the b-type incentive-compatibility constraint must be slack u tbqb Fb > taqb Fb u …and it can be ignored

Ex 11. 1(3, 4): Lagrangean Frank Cowell: Microeconomics n Firm's problem is therefore u u u n max expected profits subject to. . …binding participation constraint of b type …binding incentive-compatibility constraint of a type Formally, choose qa, qb, Fa and Fb to max p[Fa C(qa )] + [1 p][Fb C(qb )] + l[tbqb Fb] + m[taqa Fa taqb +Fb] n Lagrange multipliers are u u l for the b-type participation constraint m for the a-type incentive compatibility constraint

Ex 11. 1(3, 4): FOCs Frank Cowell: Microeconomics n Differentiate Lagrangean with respect to Fa and set result to zero: u u n Differentiate Lagrangean with respect to qa and set result to zero: u u n p m=0 which implies m = p p. Cq(qa) + mta = 0 given the value of m this implies Cq(qa) = ta But this condition means, for the high-value a types: u u marginal cost of quality = marginal value of quality the “no-distortion-at-the-top” principle

Ex 11. 1(3, 4): FOCs (more) Frank Cowell: Microeconomics n Differentiate Lagrangean with respect to Fa and set result to zero: u u n Differentiate Lagrangean with respect to qb and set result to zero: u u u n 1 p l+m=0 given the value of m this implies l = 1 [1 p]Cq(qb) + ltb mta = 0 given the values of l and m this implies Cq (qa ) = ta [1 p]Cq(qb) + tb pta = 0 Rearranging we find for the low-value b-types u marginal cost of quality < marginal value of quality

Ex 11. 1(3, 4): Second-best solution Frank Cowell: Microeconomics §Feasible set for each type F §Iso-profit contours §Contract for low type taq §Contract for high type pre fer Fa Fb en ce • §Low type is on reservation IC, but MRS≠MRT tbq §High type is on IC above reservation level, but MRS=MRT • qb q*a quality q

Ex 11. 1: Points to remember Frank Cowell: Microeconomics n n n Full-information solution is bound to be exploitative Be careful to specify which constraints are important in the second-best Interpret the FOCs carefully