3 An absentee landlord owns a farm and hires a laborer to work it. The output of the farm as a function of the effort level, e, of the worker is x/E....

An absentee landlord owns a farm and hires a laborer to work it. The output of the farm as a function of the effort level, e, of the worker is √. The landlord cannot directly observe the effort level supplied by the worker, but she does get to write a contract ahead of time, specifying the share α of future output that will be kept by the worker. After observing α, the worker gets to choose his effort level e. Effort is costly to the worker. Given α and e, the landlord's utility is v(α; e) = (1 - α) √ (the output less the worker's share), and the worker's utility (which could, in principle, be negative) is u(α; e) = α√- e (her share of output minus her effort cost). Assume that 0 ≤ α ≤ 1 and 0≤e≤ 1. (a) Use backward induction to find the level of α that the landlord will set, and the effort level e that this will induce. (b) Suppose that a social planner can set e, the effort level of the worker. Suppose that the planner aims to maximize total utility v(α; e) + u(α; e). What level of e will the social planner choose? (c) Suppose now that the social planner still wants to maximize total utility but that she cannot specify e (perhaps because she too cannot observe effort). Instead, the social planner only gets to set α. What level of α will the social planner set?

3— An absentee landlord owns a farm and hires a laborer to work it. The output of the farm as a function of the effort level, e, of the worker is x/E. The landlord cannot directly observe the effort level supplied bythe worker, but she does get to write a contract ahead of time, specifying the share (1 of future output thatwill be kept by the worker. After observing 0., the worker gets to choose his effort level e. Effort is costly to the worker. Given 0. and e, the landlord’s utility is v(0t; e) : (1 — 0.) J? (the output less the worker's share), and the worker's utility (which could, in principle, be negative) is u(0.; e) : (NE— E (her share ofoutput minus her effort cost). Assume that 0 s 0L 5 1 and Uses 1. (a) Use backward induction to find the level of (1 that the landlord will set, and the effort level e thatthis will induce. (b) Suppose that a social planner can set 0, the effort level of the worker. Suppose that the planneraims to maximize total utility v(0.; e) + u(0t; c). What level of e will the social planner choose?{c} Suppose now that the social planner still wants to maximize total utility but that she cannot specify e (perhaps because she too cannot observe effort). Instead, the social planner only gets toset a. What level of or will the social planner set?