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Hidden Information Problem in Microeconomics

This exercise illustrates and solves the hidden information problem in microeconomics. It explores the concept of full-information solutions, incentive-compatibility problems, and finding second-best solutions.

kathleenrutledge
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Hidden Information Problem in Microeconomics

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

  2. 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

  3. 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

  4. 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

  5. F ta tb q quality Ex 11.1(2): Preferences: quality • (quality, other-goods) space x • 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

  6. F 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)

  7. 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

  8. 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

  9. Ex 11.1(3,4): Misrepresentation? • Feasible set, high type F • Feasible set, low type • Full-information solution • Type-a consumer with a type-b deal taq • preference F*a • 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

  10. 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

  11. 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?

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. F q quality Ex 11.1(3,4): Second-best solution • Feasible set for each type • Iso-profit contours • Contract for low type • Contract for high type taq 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

  20. 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

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