#### Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics March 2007 Exercise 11.1 MICROECONOMICS Principles and Analysis Frank Cowell Ex 11.1(1): Question Frank Cowell: Microeconomics 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 Consumer has income y and faces two possibilities Define a binary variable i: “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 x + iF(q) ≤ y Ex 11.1(2): Question Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics (quality, other-goods) space F x high-taste type low-taste type redraw in (quality, fee) space ta tb ta tb IC must be linear in t ta > tb Because linear ICs can only intersect once q quality Ex 11.1(2): Isoprofit curves, quality Frank Cowell: Microeconomics (quality, fee) space F Iso-profit curve: low profits lso-profit curve: medium profits lso-profit curve: high profits P2 = F2 C(q) Increasing, convex in quality P1 = F1 C(q) P0 = F0 C(q) q quality 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 • Type-b participation constraint tbqb Fb ≥0 q q*b q*a quality Full information so firm can put each type on reservation IC Ex 11.1(3,4): Question Frank Cowell: Microeconomics 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? Frank Cowell: Microeconomics Feasible set, high type F Feasible set, low type Full-information solution taq Type-a consumer with a type-b deal • F*a F*b Type-a participation constraint taqa Fa ≥0 tbq Type-b participation constraint tbqb Fb ≥0 • q q*b q*a quality 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 Utility obtained by each type in full-information solution is y If a-type person could get a b-type contract 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 Participation constraint for the two types taqa Fa ≥ 0 b b b t q F ≥ 0 Incentive compatibility requires that, for the two types: 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 taqa Fa ≥ taqb Fb tbqb Fb ≥ tbqa Fa 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 Frank Cowell: Microeconomics First, we must have taqa Fa ≥ tbqb Fb 1. 2. 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 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 tbqb Fb = 0 Ex 11.1(3,4): participation, a-types Frank Cowell: Microeconomics If Fb > 0 at the optimum, then qb > 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 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 Could a-type incentive-compatibility constraint be slack? If so then it would be possible to increase Fa … 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 taqa Fa = taqb Fb Frank Cowell: Microeconomics Ex 11.1(3,4): incentive compatibility, b-types Could b-type incentive-compatibility constraint be binding? 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” tbqa Fa = tbqb Fb ? 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 Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Differentiate Lagrangean with respect to Fa and set result to zero: Differentiate Lagrangean with respect to qa and set result to zero: pm=0 which implies m = p 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) Frank Cowell: Microeconomics Differentiate Lagrangean with respect to Fa and set result to zero: Differentiate Lagrangean with respect to qb and set result to zero: 1pl+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 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 • Fa Low type is on reservation IC, but MRS≠MRT tbq High type is on IC above reservation level, but MRS=MRT Fb • q qb q*a quality Ex 11.1: Points to remember Frank Cowell: Microeconomics Full-information solution is bound to be exploitative Be careful to specify which constraints are important in the second-best interpret the FOCs carefully