Relative Performance Evaluation

Transcription

1 Relative Performance Evaluation Ram Singh Department of Economics March, 205 Ram Singh (Delhi School of Economics) Moral Hazard March, 205 / 3

2 Model I Multiple Agents: Relative Performance Evaluation Relative Performance Evaluations are widely used when individual performances can be observed: Question At School: In grading, ranking etc. At Work: In promotions, hiring and firing, etc. In Sports: In declaring winner, runners-up etc. Is relative performance evaluation efficient? Should the wage/reward be based only on the absolute value of the output, or also on the relative ranking of performances? Consider: One Principal and Two agents and Two outputs Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

3 Model II Two agents produce Two (possibly different) individually observable outputs Principal is Risk-neutral but agents are Risk-averse with CARA preferences The production technology: Q = q + q 2, where q (e, ɛ, ɛ 2 ) = e + ɛ + αɛ 2 q 2 (e 2, ɛ, ɛ 2 ) = e 2 + ɛ 2 + αɛ where ɛ and ɛ 2 are iid with ɛ i N(0, σ 2 ). Three cases: α = 0: Technologically independent outputs α > 0: Positively correlated outputs α < 0: Negatively correlated outputs Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

4 Model III Principal is risk-neutral. V (q, q 2, w) = E[q + q 2 w w 2 ] Agents are risk-averse. u i (w i, e) = e r i (w i ψ i (e)), r i > 0, where r i = u i u i > 0, i.e., CARA, and ψ i (e) = 2 c ie 2 is the (money) cost of effort e by agent i. e is not contractible but q i s are. For simplicity assume: r = r 2 = r, and c = c 2 = c, as a result, ψ (.) = ψ 2 (.) = ψ(.) = 2 ce2 Linear Contracts: w (q, q 2 ) = t + s q + s q 2 w 2 (q, q 2 ) = t 2 + s 2 q 2 + s 2 q Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

5 Model IV Multiple Agents: Relative Performance Evaluation s = 0 and s 2 = 0 will imply no relative performance evaluation. When is it optimum to have s 0 and s 2 0? Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

6 Model V Second Best: The principal will solve max E( q i w i ) s i, s i,t i However, since the agents are assumed to be identical, for each agent the principal solves max s i, s i,t i E(q i w i ) say s.t. max E(q w ) s, s,t E(u (w, e )) = E( e r(w ψ(e )) ) e r( w) = u( w) (IR) e = arg max E( e r(w ψ(e)) ) e (IC) Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

7 Model VI e is the effort chosen by first agent. Let s define Note that: ŵ (e) }{{} certainty equivalent wage e rŵ (e) = E( e r(w ψ (e)) ) =. }{{} expected wage. }{{} effort cost. }{{} risk premium Therefore, w (q, q 2 ) = t + s q + s q 2 = t + s (e + ɛ + αɛ 2 ) + s (e 2 + ɛ 2 + αɛ ) = t + s e + s e 2 + s (ɛ + αɛ 2 ) + s (ɛ 2 + αɛ ) Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

8 Model VII Var[w (q, q 2 )] = Var[s (ɛ + αɛ 2 ) + s (ɛ 2 + αɛ )], i.e., = Var[(s + α s )ɛ + ( s + αs )ɛ 2 ], i.e., = σ 2 [(s + α s ) 2 + ( s + αs ) 2 ]. The two agents will choose efforts independently in a N.E. For given e 2 opted by the second agent, the certainty equivalent payoff of the first agent is a function of his effort level e and is given by ŵ (e) = E(w (q, q 2 )) 2 ce2 rσ2 2 [(s + α s ) 2 + ( s + αs ) 2 ], i.e., in view of w (q, q 2 ) = t + s q + s q 2 ; q (e, ɛ, ɛ 2 ) = e + ɛ + αɛ 2, and q 2 (e 2, ɛ, ɛ 2 ) = e 2 + ɛ 2 + αɛ, we have E(w (q, q 2 )) = t + s e + s e 2. Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

9 Model VIII Therefore, ŵ (e) = t + s e + s e 2 2 ce2 rσ2 2 [(s + α s ) 2 + ( s + αs ) 2 ] () So, given e 2, the agent will solve max {t + s e + s e 2 e 2 ce2 rσ2 2 [(s + α s ) 2 + ( s + αs ) 2 ]} (2) That is, e SB solves the following foc e SB = s c (3) Now from () and (3), we get ŵ (e SB ) = t + s2 2c + s s 2 rσ2 c 2 [(s + α s ) 2 + ( s + αs ) 2 ] (4) Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

10 Model IX Now, in view of e SB = s c, the P s problem can be written as s.t. max { s t, s,s c (t + s2 c + s s 2 c )} ŵ = t + s2 2c + s s 2 rσ2 c 2 [(s + α s ) 2 + ( s + αs ) 2 ] = w (5) Using the value of t from (5) and ignoring w, the P s problem can be rewritten as max{ s s,s c s2 2c rσ2 2 [(s + α s ) 2 + ( s + αs ) 2 ]} Remark: Note: For given s, optimizing the above w.r.t. s is equivalent to solving min{ rσ2 s 2 [(s + α s ) 2 + ( s + αs ) 2 ]} Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

11 Model X Multiple Agents: Relative Performance Evaluation So, for given s, s SB solves the following the following foc 2α = ( + α 2 )s (6) s SB In view of (6), the P s problem reduces to So, the s SB solves the following foc max{ s s c s2 2c rσ2 ( α 2 ) 2 2 s2 ( + α 2 ) } s SB + α 2 = + α 2 + rcσ 2 ( α 2 ) 2 (7) Ram Singh (Delhi School of Economics) Moral Hazard March, 205 / 3

12 Model XI Multiple Agents: Relative Performance Evaluation Remark From (6) note that α = 0 s SB = 0 and α = 0 s SB. That is, +rcσ 2 if the outputs are technologically independent than relative performance evaluation is not optimum. Why? = α > 0 s SB < 0, i.e., an agent is penalized[rewarded] when the other individual s performance is higher[lower]. However, α < 0 s SB > 0. In this case, an agent is compensated[penalized] when the other agent s performance is higher [lower]. Ram Singh (Delhi School of Economics) Moral Hazard March, / 3

13 Model XII Remark From (6), (α ) [ s SB = s SB ] and from (7), (α ) [ssb = ]. When α =, there is a common shock affects the two performances. In this case, the relative performance evaluation allows filtering out of the common shock. Therefore, the FB can be implemented even with risk-averse agents. Question Suppose α =. Can the agents collude and choose e = 0 each? Ram Singh (Delhi School of Economics) Moral Hazard March, / 3