CHAPTER 10. CORPORATE GOVERNANCE The value of the firm is assumed to be a function v(x,6)=(6-x/2)x of the amount invested >0 and a random variable 8, uniformly distributed on an interval, M, which can be interpreted as the profitability of invest ment The manager's preferences are represented by a utility function u(a, 0)=v(a, 0+a)=(0+a-a/2)c, where a>0 A Pigovian tax t(a)=-am achieves the first best. We assume that no such schemes are available 10.2.1 Delegation without Commitment Delegation without commitment is a special case of the "cheap talk"game ntroduced by Crawford and Sobel(1982). A strategy for the manager is a function f:0,n→[0,n and the shareholders' strategy is a function 9: 0, M-R+. The shareholders beliefs are represented by a function 1:[0,M→△0.,M, where△0,M] denotes the set of probability distribu tions on 0, M. Then u(m)is the shareholders' probability distribution over possible values of 0 when the manager announces m. The equilibrium con ditions require that each player is choosing a best response and that beliefs are consistent with Bayes rule wherever possible. (i)g(m)∈ ∫(6-x/2)xp(m); (ii)f(e)E arg max(0+a-g(m)/2)g(m) (iii)u(m)=unif f-(m), for almost all m If G is the range of the function g, then the manager is effectively choosing the level of investment from the set G and condition (ii) merely requires the choose optimally from G for each value of 0 concave the manager's objective function implies that the set f-() is convex for every a E G. Furthermore, the number of these sets must be finite, as the next lemma shows Lemma 1 Suppose that a and a belong to G and are chosen in equilibrium and Then +a<
6 CHAPTER 10. CORPORATE GOVERNANCE The value of the firm is assumed to be a function v(x, θ)=(θ − x/2)x. of the amount invested x ≥ 0 and a random variable θ, uniformly distributed on an interval[0, M], which can be interpreted as the profitability of investment. The manager’s preferences are represented by a utility function u(x, θ) ≡ v(x, θ + a)=(θ + a − x/2)x, where a > 0. A Pigovian tax t(x) = −ax achieves the first best. We assume that no such schemes are available. 10.2.1 Delegation without Commitment Delegation without commitment is a special case of the “cheap talk” game introduced by Crawford and Sobel (1982). A strategy for the manager is a function f : [0, M] → [0, M] and the shareholders’ strategy is a function g : [0, M] → R+. The shareholders beliefs are represented by a function µ : [0, M] → ∆[0, M], where ∆[0, M] denotes the set of probability distributions on [0, M]. Then µ(m) is the shareholders’ probability distribution over possible values of θ when the manager announces m. The equilibrium conditions require that each player is choosing a best response and that beliefs are consistent with Bayes’ rule wherever possible. (i) g(m) ∈ arg max R M 0 (θ − x/2)xdµ(m); (ii) f(θ) ∈ arg max(θ + a − g(m)/2)g(m); (iii) µ(m) = unif f −1(m), for almost all m. If G is the range of the function g, then the manager is effectively choosing the level of investment from the set G and condition (ii) merely requires the manager to choose optimally from G for each value of θ. The concavity of the manager’s objective function implies that the set f −1(x) is convex for every x ∈ G. Furthermore, the number of these sets must be finite, as the next lemma shows. Lemma 1 Suppose that x and x0 belong to G and are chosen in equilibrium and x<x0 . Then x + a<x0
10. 2. BENEFITS OF MANAGERIAL INDEPENDENCE Without loss of generality, we can identify the manager's strategy with a finite list of intervals (0k, 0k+)k, where 01=0 and 0K+1=M, such that all manager types 0 E( 0k, 0k+1) send the same signal, which causes hareholders to choose an investment level k Theorem 2 Let [ (Ok, k)K be a sequence satisfying 01=0 and Ok 0k+1 and the following conditions ()xk=(6k+bk+1)/2, for k=1,., K, where 0K+1= M (i)(6k+1+a)=(xk+xk+1)/2,fork=1,…,K-1 Then there exists a perfect Bayesian equilibrium (, g, u) such that(0k, 0k+1)C mk) and g(mk)=ak, for k= 1,..,K. Conversely, for any perfect Bayesian equilibrium(, g, p), there erists a sequence [ (Ok, xk))k satisfying conditions(i)and (ii) and such that(0k, 0k+1)Cf-(mk)and g(mk fork=1,…,K 10.2.2 Delegation with Commitment By the revelation principle, we can restrict attention to direct revelation mechanisms. A direct revelation mechanism is a function g: 0, MI where g()is the investment specified by the shareholders when the manager reports his type to be 8. The manager will report his type truthfully if the mechanism is incentive-compatible and the optimal (incentive-compatible mechanism maximizes the shareholders'payoff E[0-g0)/2)g(0) subject t the manager is effectively choosing an element from the range of g, @ at the incentive-compatibility constraint. As in the case without commitment 9(10, M). It will be convenient to use this representation of the mechanism in our analysis. To avoid pathological cases, we assume that g is a closed set Lemma 3 If g is an optimal, incentive-compatible mechanism, the graph G is an interval Theorem 4 If*: 0, M-R+ is an optimal incentive-compatible mech- anism for the shareholders, then for some value of T1 <M, the mechanism has the form g(0)=min 0+a, 21, V0E0, M
10.2. BENEFITS OF MANAGERIAL INDEPENDENCE 7 Without loss of generality, we can identify the manager’s strategy with a finite list of intervals {(θk, θk+1)}K k=1, where θ1 = 0 and θK+1 = M, such that all manager types θ ∈ (θk, θk+1) send the same signal, which causes shareholders to choose an investment level xk. Theorem 2 Let {(θk, xk)}K k=1 be a sequence satisfying θ1 = 0 and θk < θk+1 and the following conditions: (i) xk = (θk + θk+1)/2, for k = 1, ..., K, where θK+1 = M; (ii) (θk+1 + a)=(xk + xk+1)/2, for k = 1, ..., K − 1. Then there exists a perfect Bayesian equilibrium (f, g, µ) such that (θk, θk+1) ⊂ f −1(mk) and g(mk) = xk, for k = 1, ..., K. Conversely, for any perfect Bayesian equilibrium (f, g, µ), there exists a sequence {(θk, xk)}K k=1 satisfying conditions (i) and (ii) and such that (θk, θk+1) ⊂ f −1(mk) and g(mk) = xk, for k = 1, ..., K 10.2.2 Delegation with Commitment By the revelation principle, we can restrict attention to direct revelation mechanisms. A direct revelation mechanism is a function g : [0, M] → R+, where g(θ) is the investment specified by the shareholders when the manager reports his type to be θ. The manager will report his type truthfully if the mechanism is incentive-compatible and the optimal (incentive-compatible) mechanism maximizes the shareholders’ payoff E[(θ−g(θ)/2)g(θ)] subject to the incentive-compatibility constraint. As in the case without commitment, the manager is effectively choosing an element from the range of g, G = g([0, M]). It will be convenient to use this representation of the mechanism in our analysis. To avoid pathological cases, we assume that G is a closed set. Lemma 3 If g is an optimal, incentive-compatible mechanism, the graph G is an interval. Theorem 4 If g∗ : [0, M] → R+ is an optimal incentive-compatible mechanism for the shareholders, then for some value of x1 ≤ M, the mechanism has the form g∗ (θ) = min{θ + a, x1}, ∀θ ∈ [0, M]