ECN611 Handout10

ECN611 Handout10 - E c o n o m ic s 6 1 1 H a n d o u t # 1...

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Unformatted text preview: E c o n o m ic s 6 1 1 H a n d o u t # 1 0 T H E E C O N O M IC S O F A S Y M M E T R IC IN F O R M A T IO N Chap. 14 Cont’d: Asymmetric info after the time of contracting: Moral Hazard Hidden Information (vs. hidden action) Assumptions: Two values of è, èH (with prob ë) and èL 0<ë<1 èH > èL e￿0 ð(0) = 0, ð￿(e) > 0, ð￿(e) < 0 Reservation utility: u* > v(0) v￿ > 0; v￿ < 0 g(0, è) = 0 Partial derivatives of disutility of effort: ge(e, è) > 0 if e > 0 gee(e, è) > 0 for all e ge(e, è) = 0 if e = 0 geè(e, è) < 0 if e > 0 gè(e, è) < 0 for all e geè(e, è) = 0 if e = 0 [[ Example: ð(e) = e½ ; v(w – g(e, è)) = (w – g(e, è))½ ; g(e, è) = e2/è ]] 611.10- 1 First: Treat related game where è is also observable (after the contract) by the owner. Contract: { (wH, eH), (wL, eL) } Owner: With instruments wH, eH, wL, eL, maximize expected profit ë[ð(eH) – wH] + (1 – ë)[ð(eL) – wL] subject to: wH, wL, eH, eL all ￿ 0, and ëv(wH – g(eH, èH)) + (1 – ë)v(wL – g(eL, èL)) ￿u ￿ 611.10- 2 At the optimum contract, the constraint will be satisfied with equality. Lagrangian: ￿ = ë[ð(eH) – wH] + (1 – ë)[ð(eL) – wL] – ã[ëv(wH – g(eH, èH)) + (1 – ë)v(wL – g(eL, èL)) – u)] ￿ Karush-Kuhn-Tucker conditions: (1) – ë – ãëv￿(wH – g(eH, èH)) ￿ 0 ; wH ￿ 0 ; {– ë – ãëv￿(wH – g(eH, èH))}{wH} = 0 (2) – (1 – ë) – ã(1 – ë)v￿(wL – g(eL, èL)) ￿ 0 ; wL ￿ 0 ; {– (1 – ë) – ã(1 – ë)v￿(wL – g(eL, èL))}{wL} = 0 (3) ëð￿(eH) – ãëv￿(wH – g(eH, èH))[– ge(eH, èH)] ￿ 0 ; eH ￿ 0 ; {ëð￿(eH) – ãëv￿(wH – g(eH, èH))[– ge(eH, èH)]}{eH} = 0 (4) (1 – ë)ð￿(eL) – ã(1 – ë)v￿(wL – g(eL, èL))[– ge(eL, èL)] ￿ 0 ; eL ￿ 0 ; {(1 – ë)ð￿(eL) – ã(1 – ë)v￿(wL – g(eL, èL))[– ge(eL, èL)]}{eL} = 0 (5) ëv(wH – g(eH, èH)) + (1 – ë)v(wL – g(eL, èL)) – u = 0 ￿ Suppose eH = 0. ð￿(0) > 0 and g(0, èH) = 0, contrary to (3). Therefore eH > 0. Similarly, eL > 0 via (4). Exercise: Show these conditions also imply: wH > 0 and wL > 0. 611.10- 3 So: (1￿) – ë – ãëv￿(wH – g(eH, èH)) = 0 (2￿) – (1 – ë) – ã(1 – ë)v￿(wL – g(eL, èL)) = 0 (3￿) ëð￿(eH) – ãëv￿(wH – g(eH, èH))[– ge(eH, èH)] = 0 (4￿) (1 – ë)ð￿(eL) – ã(1 – ë)v￿(wL – g(eL, èL))[– ge(eL, èL)] = 0 (5￿) ëv(wH – g(eH, èH)) + (1 – ë)v(wL – g(eL, èL)) – u = 0 ￿ From (1￿) and (2￿), ã = – ë / [ëv￿(wH – g(eH, èH))] = – (1 – ë) / [(1 – ë)v￿(wL – g(eL, èL))] Hence v￿(wH – g(eH, èH)) = v￿(wL – g(eL, èL)). Since v￿ < 0, then, v(wH – g(eH, èH)) = v(wL – g(eL, èL)). Substituting into the constraint, v(wH – g(eH, èH)) = u and so also ￿ v(wL – g(eL, èL)) = u ￿ ð￿(eH*) = ge(eH*, èH) ð￿(eL*) = ge(eL*, èL) 611.10- 4 Next: è is NOT known by the owner (not known by either at the time of the contract) A. Problem with previous contract B. Possible new forms of contract The revelation principle (Special case): If optimal contracts exist, one can be found having the following properties: (i) [revelation mechanism] After nature selects è, the manager announces a value è*(è); (ii) For each possible è*, the contract specifies a particular wage, effort pair: [w(è*), e(è*)]. (iii) [Incentive compatibility] For every possible è, the manager is induced to tell the truth: è*(è) = è. 611.10- 5 Special case: infinite risk aversion for manager Manager announces èH or èL and With instruments wH, eH, wL, eL, the owner maximizes expected profit ë[ð(eH) – wH] + (1 – ë)[ð(eL) – wL] subject to: wH, wL, eH, eL all ￿ 0, and: v(wH – g(eH, èH) ￿ u ￿ (i) v(wL – g(eL, èL) ￿ u ￿ (ii) wH – g(eH, èH) ￿ wL – g(eL, èH) (iii) wL – g(eL, èL) ￿ wH – g(eH, èL) (iv) Proposition 14.C.3 The solution {(wH, eH), (wL, eL)} to this problem has the properties (1) eH = eH*; (2) eL < eL*; (3) For the owner, expected profit is lower than for the case where è is observable; (4) For the manager, utility at èH is greater than u* but at èL, utility equals u*, so the payoff (with infinite risk aversion) is u* just as in the case where è is 611.10- 6 ...
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This note was uploaded on 01/13/2012 for the course ECN 611 taught by Professor Kelly,j during the Fall '08 term at Syracuse.

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