Principal-agent problems

Transcription

1 Principal-agent Applications of game theory 3 Department of Economics, University of Oslo ECON5200 Fall 2009 How the this topic differs from Adverse selection Adverse selection: Asymmetry of before time of contract Principal-agent : Asymmetry of after time of contract : About put into a job : About how the job should be done and hidden are often combined. Here we will consider each of them in isolation.

2 Principal-agent Examples and outline Examples: Owner Firm Investor Society Insurance company Manufacturer Manager Workers Entrepreneur Criminal Insuree Distributor Outline: The owner s profit Theownerofafirm(theprincipal) wishestohire amanager(theagent) foraonetimeproject. Firm s profit π, whereπ [π, π], depends on the manager s e, wheree E R, and π is stochastically related to e, with conditional pdf f (π e), where f (π e) > 0forallπ [π, π]. Two possible choices: e L < e H. The cdf conditional on e H first-order stochastically dominates the cdf conditional on e L : F (π e H ) F (π e L )forallπ [π, π], with < foranopensetπ [π, π]. πf (π e H )dπ > πf (π e L )dπ. The owner is risk neutral. Zero profit without contract.

3 The manager s utility The manager is an expected utility maximizer with a Bernoulli utility function: u(w, e) u w (w, e) > 0andu ww (w, e) 0forall(w, e) u(w, e L ) > u(w, e H )forallw Special case: u(w, e) =v(w) g(e), where v (w) > 0, v (w) 0, and g(e L ) < g(e H ). The manager is risk averse, except if u ww (w, e) =0forall(w, e) (which in the special case translates to v (w) = 0). Without contract, the manager receives his reservation utility ū. The optimal contract when is observable (1) Step 1: For each e, findcheapestw(π) forwhiche[u] ū. min w(π) w(π)f (π e)dπ s.t. v(w(π))f (π e)dπ g(e) ū FOC: f (π e)+γv (w(π))f (π e) =0 1 v (w(π)) = γ If v (w) < 0, then fixed wage: v(w e ) g(e) =ū or w e = v 1 (ū + g(e)) Step 2: Choose e L or e H to maximize E[π w(π)]. max πf (π e)dπ v 1 (ū + g(e)) e {e L,e H }

4 The optimal contract when is observable (2) In the principal-agent model with observable, an optimal contract specifies that the manager choose the e that maximizes πf (π e)dπ v 1 (ū + g(e)) over {e L, e H }, and pays the manager a fixed wage w e = v 1 (ū + g(e )). This is the uniquely optimal contract if v (w) < 0 for all w. Let ᾱ = πf (π e )dπ v 1 (ū + g(e )). The optimal contract when is unobservable Two cases: 1 A risk-neutral manager Sell the firm to the manager: the manager receives the full marginal returns from his and faces all risk. 2 A risk-averse manager Trade-off between incentives for and insurance against risk

5 Risk-neutral manager In the principal-agent model with unobservable and a risk-neutral manager, an optimal contract generates the same choice and utilities for the owner and the manager as when is observable. Proof. Part 1: There exists an accepted contract where the principal receives ᾱ. Let the principal sell the project for α = πf (π e )dπ g(e ) ū where e =argmax πf (π e)dπ g(e). This contract is accepted: πf (π e )dπ g(e ) α =ū. Also,α =ᾱ. Part 2: No accepted contract where p. receives more. Risk-averse manager (1) Step 1: For each e {e L, e H }, find cheapest w(π) forwhich (PC) E[u(w, e)] ū and (IC) E[u(w, e)] E[u(w, ẽ)]. min w(π) w(π)f (π e)dπ s.t. (PC) v(w(π))f (π e)dπ g(e) ū (IC) e solves max v(w(π))f (π ẽ)dπ g(ẽ) ẽ Step 2: Choose e L or e H to maximize E[π w(π)].

6 Risk-averse manager (2) Implementing e L Fixed wage equal to v 1 (ū + g(e L )) satisfies both (PC) and (IC). Yields principal same utility as when e L is observable. Higher payoff is not feasible when manager chooses e L. Implementing e H (IC) becomes v(w(π))f (π e H )dπ g(e H ) v(w(π))f (π e L )dπ g(e L ) Let γ 0andμ 0 be the multipliers for (PC) &(IC). f (π e H )+γv (w(π))f (π e H )+μ[f (π e H ) f (π e L )]v (w(π)) = 0 [ 1 v (w(π)) = γ + μ 1 f (π e ] L) f (π e H ) Risk-averse manager (3) In the principal-agent model with unobservable and a risk-averse manager, the optimal compensation [ scheme ] 1 for implementing e H satisfies v (w(π)) = γ + μ 1 f (π e L) f (π e H ), gives the manager ū, and involves a larger expected wage payment than is required when is observable. The optimal compensation scheme for implementing e L involves the same fixed wage payment as if were observable. Whenever the optimal level with observable would be e H, nonobservability causes a welfare loss. If f (π e L) f (π e H ),thelikelihood ratio, is monotone, then the optimal compensation scheme for implementing e H is monotone. If e H is optimal with observable, then either implement e H at additional cost due to inefficient risk-sharing, or implement e L.

7 Theownerofafirm(theprincipal) wishestohire amanager(theagent) foraonetimeproject. The manager s e ( [0, )) is fully observable. The manager s cost of exhorting is private. u(w, e,θ)=v(w g(e,θ)) where g(e,θ) measures the disutility of in monetary terms. g(0,θ)=0 { > 0fore > 0 g e (e,θ) =0fore =0 g ee (e,θ) > 0foralle { g θ (e,θ) < 0fore > 0 { =0fore =0 g eθ (e,θ) < 0fore > 0 =0fore =0 Two s: 0 <θ L <θ H λ = Prob(θ = θ H ) (0, 1) The θ is observable max λ [π(e H) w H ]+(1 λ)[π(e L ) w L ] w L,e L,w H,e H 0 s.t. λv(w H g(e H,θ H )) + (1 λ)v(w L g(e L,θ L )) ū v (w H g(e H,θ H)) = v (w L g(e L,θ L)) v(w H g(e H,θ H)) = v(w L g(e L,θ L)) = ū π (e H )=g e(e H,θ H) and π (e L )=g e(e L,θ L) In the principal-agent model with observable, an optimal contract involves an level ei in θ i such that π (ei ) = g e (ei,θ i) and fully insures the manager, setting his wage in each θ i at the level w i such that v(w i g(e i,θ i)) = ū.

8 The θ is observed only by the manager (1) If the owner asks the manager to reveal his type, then with the optimal contract under observable in θ H the manager prefers to report θ L. The manager must be compensated to provide truthful announcement of θ H. (The revelation principle) Θ: The set of possible s. The owner can without loss restrict himself to contracts of the following form: 1 After θ is realized, the manager is asked to reveal. 2 The contract specifies an outcome [w(ˆθ), e(ˆθ)] for each possible announcement ˆθ Θ. 3 In every θ Θ, truthful announcement is optimal. The θ is observed only by the manager (2) Assume infinite managerial risk averse: u =min{u L, u H }. max λ [π(e H) w H ]+(1 λ)[π(e L ) w L ] w L,e L,w H,e H 0 s.t. (PC L ) w L g(e L,θ L ) v 1 (ū) (PC H ) w H g(e H,θ H ) v 1 (ū) (IC L ) w L g(e L,θ L ) w H g(e H,θ L ) (IC H ) w H g(e H,θ H ) w L g(e L,θ H ) 1 (PC H ) can be ignored. 2 (PC L ) binds (holds with equality). 3 e L e L and e H = e H ( (IC L) can be ignored). 4 e L < e L : [π (e L ) g e (e L,θ L )] = λ 1 λ [g e(e L,θ L ) g e (e L,θ H )]

9 The θ is observed only by the manager (3) In the hidden principal-agent model with an infinitely risk-averse manager, the optimal contract sets the level of in θ H at its full observability level eh. The level in θ L is distorted downward from its full observability level el. The manager is inefficiently insured, with u H > ūandu L =ū. The owner s expected utility is below its full observability level, while the manager s utility is at its full observability level (= ū). Other topics: The hidden model with infinite risk aversion can be used to study monopolistic screening, in which a single firm screens workers who, at the time of contracting, have different unobservable productivity levels. Applicable also to regulation, e.g. of a monopolist with unknown cost. Hybrid models of hidden and hidden.