1 Dynamic Moral Hazard • Intertemporal risk-sharing • Better information (output, actions, consumption) • Larger games (action spaces) • Generic complexity (?spot contracting) Simple M (separable): t =1, 2. a ∈ A, #Q = n, Pr(qt i = qi|a = at)= pi(at) > 0. Agent: u(c) − ψ(a) (in each t), limc↓¯c u(c)= −∞. Principal: V (q − w). Contracting: t =1: {a1,w1(q1 i ),a2(q2 i ),w2(q1 i ,q2 j )}. (RP) • No savings o r borrowing. Principal chooses: wi,wij; Agent: α, ai. maxwi,wij P i pi(α) h V (q1 i − wi)+ P j pj(ai)V (q2 i − wij) i , s.t. α, ai ∈ arg max AG(α, ai,wi,wij),and IR. Euler equation: V 0(q1 i − wi) u0(wi) = X j pj(ai) " V 0(q2 i − wij) u0(wij) # When V 0 = const, we have “smoothing” 1 u0(wi) = X j pj(ai) " 1 u0(wij) # . Two observations: (1) Optimal contract has memory, No memory wo uld imply RHS is constant for all i, perfect insurance in period 1, wrong incentives.(2) Agent wants to save (and so the contract is “front- • Free savings. loaded”). Example: Effort in t =2, consumption in both periods ∂EU P 0(0(∂s = j pj(ai)u wij) − u wi) ≥ 0 (Jensen’s in-(borrowing in the first period) equality). a ∈ {H, L}, ψ(H)=1,ψ(L)=0. • Monitored savings q ∈ {0, 1},pH = p1(H) >pL > 0. Suppose a ∗ = H. Contract (w0,w1). Add ti, si (principal, agent)’s savings. Let cj be consumption with planned j = H, L. The above contract can be achieved without history-cj ∈ arg maxcu(c)+pju(w1− c)+(1− pj)u(w0− c).dependent wages, and, so, is spot-implementable. We have Set: cij = wij = wj + si, wi = ci − si. u(cH)+pHu(w1− cH)+(1− pH)u(w0− cH) − 1= = u(cL)+pLu(w1− cL)+(1− pL)u(w0− cL)Problem separates to: incentive p rovision and consump->u(cH)+pLu(w1− cH)+(1− pL)u(w0− cH)tion smoothing. Thus ICH2 is slack. Room for ren egotiation (unless CARA)1.1 T-period Problem Subcases: • Repeated Output (better statistical inference) • Repeated Actions (multitask in time) • Repeated Consumption (consumption smoothing) • Repeated Actions and Output (consumption at the end) • Infinitely repeated Actions, Output, and