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

Frank Cowell: Microeconomics Ex 11.1(1): Question purpose: to illustrate and solve the “hidden information” problem purpose: to illustrate and solve the “hidden information” problem method: find full information solution, describe incentive-compatibility problem, then find second-best solution method: find full information solution, describe incentive-compatibility problem, then find second-best solution

Frank Cowell: Microeconomics Ex 11.1(1): Budget constraint Consumer has income y and faces two possibilities 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  : Define a binary variable  :   = 0 represents the case “not buy”   = 1 represents the case “buy” Then the budget constraint can be written Then the budget constraint can be written  x +  F(q) ≤ y

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

Frank Cowell: Microeconomics Ex 11.1(2): Preferences: quality quality q x bb aa    a >  b   IC must be linear in    Because linear ICs can only intersect once quality q F bb aa   (quality, other-goods) space   low-taste type   high-taste type   redraw in (quality, fee) space preference

Frank Cowell: Microeconomics Ex 11.1(2): Isoprofit curves, quality   (quality, fee) space   lso-profit curve: medium profits   Increasing, convex in quality   Iso-profit curve: low profits  2 = F 2  C(q) q F quality  1 = F 1  C(q)  0 = F 0  C(q) increasing profit   lso-profit curve: high profits

Frank Cowell: Microeconomics Ex 11.1(2): Full-information solution   reservation IC, high type   Full-information solution, high type   Full information so firm can put each type on reservation IC   Firm’s feasible set for a high type   lso-profit curves q F bqbq aqaq quality q*aq*a q*bq*b F*bF*b F*aF*a   Reservation IC + feasible set, low type   Full-information solution, low type   Type-a participation constraint  a q a  F a ≥0   Type-b participation constraint  b q b  F b ≥0

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

Frank Cowell: Microeconomics Ex 11.1(3,4): Misrepresentation?   Full-information solution   A high type-consumer would strictly prefer the contract offered to a low type   Feasible set, high type   Feasible set, low type   Type-a consumer with a type-b deal   Type-a participation constraint  a q a  F a ≥0   Type-b participation constraint  b q b  F b ≥0 q F bqbq aqaq quality q*aq*a q*bq*b F*bF*b F*aF*a preference

Frank Cowell: Microeconomics Ex 11.1(3,4): background to problem Utility obtained by each type in full-information solution is y 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 If a-type person could get a b-type contract  a-type’s utility would then be   a q *b  F *b +y  given that  b q *b  F *b = 0…  …a-type’s utility would be [  a   b ]q *b + y >y So an a-type person would want to take a b-type contract So an a-type person would want to take a b-type contract In deriving second-best contracts take account of In deriving second-best contracts take account of 1. participation constraints 2. this incentive-compatibility problem

Frank Cowell: Microeconomics Ex 11.1(3,4): second-best problem Participation constraint for the two types Participation constraint for the two types   a q a  F a ≥ 0   b q b  F b ≥ 0 Incentive compatibility requires that, for the two types: Incentive compatibility requires that, for the two types:   a q a  F a ≥  a q b  F b   b q b  F b ≥  b q a  F a Suppose there is a proportion , 1   of a-types and b-types Suppose there is a proportion , 1   of a-types and b-types Firm's problem is to choose q a, q b, F a and F b to max expected profits Firm's problem is to choose q a, q b, F a and F b to max expected profits   [F a  C(q a )] + [1   ][F b  C(q b )] subject to  the participation constraints  the incentive-compatibility constraints However, we can simplify the problem However, we can simplify the problem  which constraints are slack?  which are binding?

Frank Cowell: Microeconomics Ex 11.1(3,4): participation, b-types First, we must have ≥ First, we must have  a q a  F a ≥  b q b  F b  this is because  ≥ (a-type incentive compatibility) and   a q a  F a ≥  a q b  F b (a-type incentive compatibility) and  (a-type has higher taste than b-type)   a >  b (a-type has higher taste than b-type) This implies the following: This implies the following:  if > 0 (b-type participation slack)  if  b q b  F b > 0 (b-type participation slack)  then also > 0 (a-type participation slack)  then also  a q a  F a > 0 (a-type participation slack) But these two things cannot be true at the optimum But these two things cannot be true at the optimum  if so it would be possible for firm to increase both and  if so it would be possible for firm to increase both F a and F b  thus could increase profits So b-type participation constraint must be binding So b-type participation constraint must be binding  = 0   b q b  F b = 0

Frank Cowell: Microeconomics Ex 11.1(3,4): participation, a-types If > 0 at the optimum, then > 0 If F b > 0 at the optimum, then q b > 0  follows from binding b-type participation constraint  = 0   b q b  F b = 0 This implies > 0 This implies  a q b  F b > 0  because a-type has higher taste than b-type    a >  b This in turn implies > 0 This in turn implies  a q a  F a > 0  follows from a-type incentive-compatibility constraint  ≥   a q a  F a ≥  a q b  F b So a-type participation constraint is slack and can be ignored 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? Could a-type incentive-compatibility constraint be slack?  could we have > ?  could we have  a q a  F a >  a q b  F b ? If so then it would be possible to increase … If so then it would be possible to increase F a …  …without violating the constraint  this follows because a-type participation constraint is slack  > 0   a q a  F a > 0 So a-type incentive-compatibility must be binding So a-type incentive-compatibility must be binding  =   a q a  F a =  a q b  F b

Frank Cowell: Microeconomics Ex 11.1(3,4): incentive compatibility, b-types Could b-type incentive-compatibility constraint be binding? Could b-type incentive-compatibility constraint be binding?   b q a  F a =  b q b  F b ? If so, then q a = q b If so, then q a = q b  follows from fact that a-type incentive-compatibility constraint is binding   a q a  F a =  a q b  F b  which, with the above, would imply [  b   a ]q a = [  b   a ]q b  given that  a >  b this can only be true if q a = q b So, both incentive-compatibility conditions bind only with “pooling” 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 So the b-type incentive-compatibility constraint must be slack   b q b  F b >  a q b  F b  …and it can be ignored

Frank Cowell: Microeconomics Ex 11.1(3,4): Lagrangean Firm's problem is therefore 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 q a, q b, F a and F b to max Formally, choose q a, q b, F a and F b to max   [F a  C(q a )] + [1   ][F b  C(q b )]  + []  + [  b q b  F b ]  +  [ ]  +  [  a q a  F a   a q b +F b ] Lagrange multipliers are Lagrange multipliers are   for the b-type participation constraint   for the a-type incentive compatibility constraint

Frank Cowell: Microeconomics Ex 11.1(3,4): FOCs Differentiate Lagrangean with respect to and set result to zero: Differentiate Lagrangean with respect to F a and set result to zero:    = 0  which implies  =  Differentiate Lagrangean with respect to and set result to zero: Differentiate Lagrangean with respect to q a and set result to zero:    C q () +  = 0    C q (q a ) +  a = 0  given the value of  this implies C q () =  given the value of  this implies C q (q a ) =  a But this condition means, for the high-value a types: 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

Frank Cowell: Microeconomics Ex 11.1(3,4): FOCs (more) Differentiate Lagrangean with respect to and set result to zero: Differentiate Lagrangean with respect to F a and set result to zero:  1    = 0  given the value of  this implies = 1 Differentiate Lagrangean with respect to and set result to zero: Differentiate Lagrangean with respect to q b and set result to zero:  [1   ]C q () +  = 0  [1   ]C q (q b ) + b   a = 0  given the values of  and  this implies C q () =  given the values of  and  this implies C q (q a ) =  a  [1   ]C q () +   = 0  [1   ]C q (q b ) +  b   a = 0 Rearranging we find for the low-value b-types Rearranging we find for the low-value b-types  marginal cost of quality < marginal value of quality

Frank Cowell: Microeconomics Ex 11.1(3,4): Second-best solution   Contract for high type   Low type is on reservation IC, but MRS≠MRT   Feasible set for each type   Iso-profit contours   Contract for low type bqbq aqaq q F quality q*aq*a qbqb FbFb FaFa preference   High type is on IC above reservation level, but MRS=MRT

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