Game Theory, Information, Incentives

Transcription

1 Game Theory, Information, Incentives Ronald Wendner Department of Economics Graz University, Austria Course # : Analytical Methods (part 6)

2 The Moral Hazard Problem Moral hazard as a problem of hidden action (effort) Choice between two effort levels Effort as continuous variable first-order approach Applications incentives for managers rationing in the credit market introduction of know-how in technology transfer contracts c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 2/27

3 The Moral Hazard Problem Effort Timing A s behavior not observable contract labor contracts, insurance contracts c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 3/27

4 The Moral Hazard Problem A RA, P RN complete information (no hidden action): w o (x i ) = w o (x j ) A chooses e MIN, as utility = n p i(e) u(w o ) v(e) = u(w o ) v(e) P anticipates this choice and offers w MIN = u 1 (U + v(e MIN )) inefficiency due to lack of incentives: e MIN < e o, w MIN < w o alternative to fixed-wage contract: franchise franchise solves incentive problem but A is RA: A does not pay P too much to accept risk inefficiency trade off b/w efficiency and incentives (no efficient outcome, opt. contract in b/w fixed-wage & franchise) c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 4/27

5 Subgame perfect equilibrium 3 rd (final) stage: A chooses e { n e arg max p i (e ) u(w(x i )) v(e ) } incentive compatibility constraint 2 nd stage: A accepts or rejects contract n p i (e) u(w(x i )) v(e) U participation constraint c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 5/27

6 Subgame perfect equilibrium 1 st stage: P designs contract max e,{w(x i )},...,n s.t. n n p i (e) B(x i w(x i )) p i (e) u(w(x i )) v(e) U e arg max { n p i (e ) u(w(x i )) v(e ) } c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 6/27

7 Choice b/w two effort levels A RA, P RN if A RN, P RA: franchise is optimal, as in symmetric info context (why?) e {e H, e L }, v(e L ) < v(e H ) x 1 < x 2 <... < x n p L i p i (e L ) > 0, p H i p i (e H ) > 0, i = 1, 2,..., n first order stochastic dominance of p H k ph i < k pl i, k = 1, 2,..., n 1 c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 7/27

8 Digression: first order stochastic dominance P H (x), P L (x) cdf, p H (x), p L (x) pdf P H (x) fosd P L (x) if 1 P H (x) > 1 P L (x) for each x X for every x i, prob of getting at least x i higher under P H (x) than under P L (x) P H (x) < P L (x) for all x X p H (x)dx < p L (x)dx for all x X (*) discrete framework condition (*) is: k 1=1 ph i < k 1=1 pl i for k = 1,..., n 1 for any k < n, probability that x i > x k is higher under e H than under e L c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 8/27

9 Choice b/w two effort levels cont. ed Suppose for P, e o = e L incentive compatibility constraint satisfied u(w L ) v(e L ) > u(w L ) v(e H ) symm. inf contract continues to be optimal (no incentive probl., no efficiency probl.) Suppose for P, e o = e H incentive compatibility constraint n ph i u(w(x i )) v(e H ) n pl i u(w(x i )) v(e L ) w o = w o (x i ), & needs to increase sufficiently c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 9/27

10 [ n n ] L({w(x i )}, λ, µ) = pi H [x i w(x i )] + λ pi H u(w(x i )) v(e H ) U [ n ] + µ [pi H pi L ] u(w(x i )) [v(e H ) v(e L )] necessary foc µ > 0 p H i u (w(x i )) = λ ph i + µ (p H i p L i ), i = 1,..., n λ = n p H i u (w(x i )) > 0 [ 1 u (w(x i )) = λ + µ 1 pl i pi H ], i = 1,..., n ( ) c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 10/27

11 1/u (w(x i )) = λ + µ [ 1 p L i /p H i ] ( ) as µ > 0: w(x i ) changes in p L i /p H i fixed-wage contract not optimal, w o (x i ) depends on result tradeoff b/w incentives and efficiency Monotone likelihood ratio: p L i /p H i decreases in i x i p L i /p H i RHS ( ) LHS ( ) u (w(x i )) w(x i ) u (w(x i )) = 1/ [ λ + µ (1 p L i /p H i ) ] w(x i ) = u 1 [ 1/ ( λ + µ (1 p L i /p H i ) )] w o depends on x i to influence A s effort c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 11/27

12 Effort as continuous variable: first order approach e [0, 1] incentive compatibility constraint n p i(e)u(w(x i )) v (e) = 0 P designs contract max e,{w(x i )},...,n s.t. n n n p i (e) (x i w(x i )) p i (e) u(w(x i )) v(e) U p i(e)u(w(x i )) v (e) = 0 c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 12/27

13 foc w.r.t. w(x i ) 1 u (w(x i )) = λ + µ p i(e) p i (e) µ > 0 fosd; p i(e)/p i (e) rises in i w (x i ) > 0 rise in e rises p i the more the higher i foc w.r.t. e n p i(e) (x i w(x i )) = µ [ n p i (e) u(w(x i )) v (e) ] increase in expected profit = increase in expected cost (via incentive compatibility constraint) c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 13/27

14 Applications I: incentives for managers Symmetric information P = shareholders, A = manager A: U (w, e) = u(w) v(e), usual properties P: B(x, w) = p x c x w(x), sales x random var. with density f (x; e) max w(x) s.t. X X [p x c x w(x)]f (x; e)dx u(w(x)) f (x; e)dx v(e) U no incentive constraint (due to symmetric information) c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 14/27

15 Applications I: incentives for managers, cont. ed [ ] L = [p x c x w(x)]f (x; e)dx + λ u(w(x)) f (x; e)dx v(e) U X X = [px cx]f (x; e)dx w f (x; e)dx X + λ [ u(w) X f (x; e)dx v(e) u ] X L = [px cx]f (x; e)dx w + λ [u(w) v(e) u] X L w = 1 + λu (w) = 0 λ = 1 u (w) > 0 from PC w o = u 1 (U + v(e)) x i fixed wage contract if e observable c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 15/27

16 Applications I: incentives for managers,effort P takes into account: w = u 1 (U + v(e)), λ = 1 u (w), L e = 0 X [p x c x]f e(x; e)dx = v (e)/u (w) expected marginal revenue of e = expected marginal cost of e c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 16/27

17 Hidden action/effort max w(x) s.t. X X X [p x c x w(x)]f (x; e)dx u(w(x)) f (x; e)dx v(e) U u(w(x)) f e(x; e)dx v (e) = 0 foc 1 u (w(x)) = λ + µ f e(x; e) f (x; e) if f e(x; e)/f (x; e) increases in x so does w(x): bonus according to sales sufficient: x = e + ɛ, ɛ normally distributed c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 17/27

18 Applications II: Rationing in the credit market Firm (A): choice b/w two risky investment projects, a, b, requiring I payoff X i = X i with prob p i (0 with prob 1 p i ), i = a, b p a X a > p b X b > I, 1 > p a > p b > 0, X b > X a U (R, i) = p i (X i R) Bank (P): Π(R, i) = p i R I, i = a, b Credit market: N firms, I investment per firm, L total amount of loans I L < N I credit rationing: firm cannot obtain all the money it wants symmetric information (P observes choice of (a, b)) R = X a, U (X a, a) = 0, no credit rationing (no firm makes positive profits with credits) c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 18/27

19 Moral hazard: P cannot observe choice of A (a, b) Subgame perfect equilibrium step 3 (incentive compatibility) A chooses project {a, b}, depending on R } { a pa (X a R) p b (X b R) if b p a (X a R) < p b (X b R) R ˆR R > ˆR, ˆR p a X a p b X b p a p b. step 2 (participation): U (R, i) = p i (X i R) 0 if R > ˆR, R X b c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 19/27

20 . step 1 (contract design by principal) P chooses R so to maximize expected profit Π (R) = { pa R I if 0 R ˆR p b R I if ˆR < R X b as p i > 0, R = ˆR if p a ˆR > pb X b, otherwise R = X b c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 20/27

21 Credit rationing problem under moral hazard Case 1: R = X b as firms profits U (X b, b) = 0, firm is indifferent b/w loaning money or not (no credit rationing) Case 2: R = ˆR U ( ˆR, a) = p a (X a ˆR) > 0 all firms ask for a loan, but L < N I consequences of moral hazard market interest rate changes (R X a ) banks may decide not to increase interest rate, even if firms would be willing to pay a higher interest rate c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 21/27

22 Applications III: Know-how in technology contracts Monopolist (A) buying technology, T, from a research laboratory (P) p = a Q technology, T, may be purchased with/without know how: K {0, k} T is observable, while K is not, C(T, K), C(0, 0) > C(T, 0) > C(T, k) A: Π(Q, T, K) = (a Q)Q [C(T, K) + V K ]Q F K Research laboratory (P) B(Q, T, K) = F K + V K Q d(k) P s effort not observable c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 22/27

23 Subgame perfect equilibrium stage 4: A considers profit max Q ˆQ(T, K, V ) = arg max Q Π(Q, T, K) = [a C(T, K) V K ]/2 stage 3: P decides on K = k F k + V k ˆQ(T, k, V k ) d(k) F k + V k ˆQ(T, 0, V k ) K = k V k 2d/[C(T, 0) C(T, k)]... or K = 0 F 0 + V 0 ˆQ(T, 0, V 0 ) F 0 + V 0 ˆQ(T, k, V 0 ) d(k) K = 0 V 0 2d/[C(T, 0) C(T, k)] c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 23/27

24 . stage 2: A s participation constraints Π(Q, T, k) Π(Q, 0, 0) F k [(a C(T, k) V k )/2] 2 [(a C(0, 0))/2] 2 Π(Q, T, 0) Π(Q, 0, 0) F 0 [(a C(T, 0) V 0 )/2] 2 [(a C(0, 0))/2] 2 stage 1: P designs contract (F k, V k ) or (F 0, V 0 ) depending on which yields higher payoff c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 24/27

25 Case 1. (F 0, V 0 ) is optimal contract consider ˆQ, participation constraint binds optimal contract: max V 0 [(a C(T, 0) V 0 )/2] 2 [(a C(0, 0))/2] 2 + V 0 [a C(T, 0) V 0 ]/2 = V 0 /2 < 0 V 0 = 0 V 0 = 0, F 0 = [(a C(T, 0) V 0 )/2] 2 [(a C(0, 0))/2] 2 if K o = 0, moral hazard does not impose cost in terms of incentives P prefers not to distort A s MC! c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 25/27

26 Case 2. (F k, V k ) is optimal contract V k = 2 d F k = C(T, 0) C(T, k) > 0 [ a C(T, k) V k 2 ] 2 [ ] 2 a C(0, 0) > 0 2 only a royalty payment induces P to transmit K together w/ T (otherwise K not effectively transferred) c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 26/27

27 Specifics of Application III unverifiable behavior on P s part(!), not on A s part no uncertainty in outcome tradeoff b/w incentives and efficiency does not imply distortion in optimal distribution of risk c Ronald Wendner Games & Information Economics - Ph.D. 6 v1.4 27/27