Question

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 ve. 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 a of future output that will be kept by the worker. After observing a, the worker gets to choose his effort level e. Effort is costly to the worker. Given a and e, the landlord's utility is v(a; e) = (1 - a) ve (the output less the worker's share), and the worker's utility (which could, in principle, be negative) is u(a; e) = ave-e (her share of output minus her effort cost). Assume that 0 sas 1 and Oses 1. (a) Use backward induction to find the level of a 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 va; e) + u(a; 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 a. What level of a will the social planner set?

Answer 1

Answer vca, e) =(1-adre. - Landlords utility worker's cetility - uca, elzarene. Assoming that osas I and osess. the landlord a leader and everler is follower Worker problem. em no (a,e)= avere. édla % 0. Landard's problem Man VLCX, e) = Cindore substituting too eusing o we get amare Ci-d) (al). (1-x) - -0 7. by di fferentratin. a=% le=%. abs social plannel problem Mar 1₂ (dney+Vw Ca,e). el-a) retare-e. rece!

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