MICROECONOMICS Principles and Analysis Frank Cowell Exercise 11.1 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

Ex 11.1(1): Question 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 Consumer has income y and faces two possibilities “not buy”: all y spent on other goods “buy”: y  F(q) spent on other goods Define a binary variable i: i = 0 represents the case “not buy” i = 1 represents the case “buy” Then the budget constraint can be written x + iF(q) ≤ y

Ex 11.1(2): Question method: First draw ICs in space of quality and other goods Then redraw in space of quality and fee Introduce iso-profit curves Full-information solutions from tangencies

Ex 11.1(2): Preferences: quality tb ta (quality, other-goods) space high-taste type low-taste type redraw in (quality, fee) space ta tb preference IC must be linear in t ta > tb Because linear ICs can only intersect once q quality

Ex 11.1(2): Isoprofit curves, quality (quality, fee) space Iso-profit curve: low profits lso-profit curve: medium profits lso-profit curve: high profits increasing profit P2 = F2  C(q) Increasing, convex in quality P1 = F1  C(q) P0 = F0  C(q)

Ex 11.1(2): Full-information 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 Full-information solution, low type • F*a tbq Type-a participation constraint taqa  Fa ≥0 • F*b Type-b participation constraint tbqb  Fb ≥0 Full information so firm can put each type on reservation IC q q*b q*a quality

Ex 11.1(3,4): Question method: Set out nature of the problem Describe in full the constraints Show which constraints are redundant Solve the second-best problem

Ex 11.1(3,4): Misrepresentation? F Feasible set, high type Feasible set, low type Full-information solution taq Type-a consumer with a type-b deal • F*a preference Type-a participation constraint taqa  Fa ≥0 tbq Type-b participation constraint tbqb  Fb ≥0 • F*b A high type-consumer would strictly prefer the contract offered to a low type q q*b q*a quality

Ex 11.1(3,4): background to problem Utility obtained by each type in full-information solution is y each person is on reservation utility level given the U function, if you don’t consume the good you get exactly y If a-type person could get a b-type contract 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 participation constraints this incentive-compatibility problem

Ex 11.1(3,4): second-best problem Participation constraint for the two types taqa  Fa ≥ 0 tbqb  Fb ≥ 0 Incentive compatibility requires that, for the two types: taqa  Fa ≥ taqb  Fb tbqb  Fb ≥ tbqa  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 p[Fa  C(qa)] + [1  p][Fb  C(qb)] subject to the participation constraints the incentive-compatibility constraints However, we can simplify the problem which constraints are slack? which are binding?

Ex 11.1(3,4): participation, b-types First, we must have taqa  Fa ≥ tbqb  Fb this is because taqa  Fa ≥ taqb  Fb (a-type incentive compatibility) and ta > tb (a-type has higher taste than b-type) This implies the following: 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 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 tbqb  Fb = 0

Ex 11.1(3,4): participation, a-types If Fb > 0 at the optimum, then qb > 0 follows from binding b-type participation constraint tbqb  Fb = 0 This implies taqb  Fb > 0 because a-type has higher taste than b-type ta > tb This in turn implies taqa  Fa > 0 follows from a-type incentive-compatibility constraint taqa  Fa ≥ taqb  Fb So a-type participation constraint is slack and can be ignored

Ex 11.1(3,4): incentive compatibility, a-types Could a-type incentive-compatibility constraint be slack? could we have taqa  Fa > taqb  Fb ? If so then it would be possible to increase Fa … …without violating the constraint this follows because a-type participation constraint is slack taqa  Fa > 0 So a-type incentive-compatibility must be binding taqa  Fa = taqb  Fb

Ex 11.1(3,4): incentive compatibility, b-types Could b-type incentive-compatibility constraint be binding? tbqa  Fa = tbqb  Fb ? If so, then qa = qb follows from fact that a-type incentive-compatibility constraint is binding taqa  Fa = taqb  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 So, both incentive-compatibility conditions bind only with “pooling” 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 tbqb  Fb > taqb  Fb …and it can be ignored

Ex 11.1(3,4): Lagrangean Firm's problem is therefore 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] Lagrange multipliers are l for the b-type participation constraint m for the a-type incentive compatibility constraint

Ex 11.1(3,4): FOCs Differentiate Lagrangean with respect to Fa and set result to zero: p  m = 0 which implies m = p Differentiate Lagrangean with respect to qa and set result to zero:  pCq(qa) + mta = 0 given the value of m this implies Cq(qa) = ta But this condition means, for the high-value a types: marginal cost of quality = marginal value of quality the “no-distortion-at-the-top” principle

Ex 11.1(3,4): FOCs (more) Differentiate Lagrangean with respect to Fa and set result to zero: 1  p  l + m = 0 given the value of m this implies l = 1 Differentiate Lagrangean with respect to qb and set result to zero: [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 marginal cost of quality < marginal value of quality

Ex 11.1(3,4): Second-best solution q F quality Feasible set for each type Iso-profit contours Contract for low type taq Contract for high type preference • Low type is on reservation IC, but MRS≠MRT Fa tbq High type is on IC above reservation level, but MRS=MRT • Fb qb q*a

Ex 11.1: Points to remember Full-information solution is bound to be exploitative Be careful to specify which constraints are important in the second-best Interpret the FOCs carefully