Economics of markets and organizations PeneΜ lope, a baker, sells apple pies at a price of π a pie in the market. She has two employees, Antonio and Anna, who produce the pies. Suppose Antonio and Anna both make effort costs of πΆ(ππ) = ππ2 πΎ if they bake ππ apple pies a day (πΎ > 0). All costs are sunk, apart from the cost of Antonio and Anna’s efforts. a) How many apple pies does an efficient contract induce Antonio and Anna to bake every day? An efficient contract maximizes the value V(Q) generated within the Principal-Agent relationship. The value V(Q) as a function of Antonio and Anna’s output equals π(π) = π(π1 + π2 ) − πΆ(π1 ) − πΆ(π2 ) = π(π1 + π2 ) − π1 2 π2 2 − πΎ πΎ Value is maximized if π ′ (ππ ) = π − 2 ππ =0 πΎ or, equivalently, if ππ = ππΎ 2 This result applies to both Antonio and Anna. b) Construct a linear contract for both Antonio and Anna that implements the above efficient contract. Suppose that the linear contract specifies a base wage w plus a bonus ο’ for each unit of output. Antonio and Anna’s individual utility U(qi) from this contract then equals π(ππ ) = π€ + π½ππ − πΆ(ππ ) = π€ + π½ππ − ππ 2 πΎ Antonio and Anna choose their effort levels such that it maximizes their individual utility. The firstorder condition of the corresponding maximization problem yields π ′ (ππ ) = π½ − 2 ππ =0 πΎ or, equivalently ππ = π½πΎ 2 1 (negative quantities do not make sense). When we compare this effort level with the valuemaximizing effort, we find that the optimal bonus is given by π½=π The level of Antonio and Anna’s fixed wage is not relevant as long as all parties are willing to accept the contract. c) Suppose that Antonio and Anna now own this bakery together, and that they agree to share the revenues equally. Show that such a team bonus that is equally shared between Antonio and Anna yields sub-optimal individual effort levels. Give an intuitive explanation. Antonio maximizes his utility π(π1 + π2 ) π(π1 + π2 ) π1 2 π(π1 ) = − πΆ(π1 ) = − 2 2 πΎ The first-order condition of the corresponding maximization problem yields π ππ −2 =0 2 πΎ π ′ (ππ ) = or, equivalently, ππ = ππΎ ππΎ < 4 2 This result applies to Anna as well. So, the individual effort levels are indeed suboptimal. Intuitive explanation: See section 5.3 of the book. An efficient contract requires that the agent becomes the residual claimant of the fruits of his efforts. In the case of revenue sharing, the agent only obtains a fraction (1/2 in this case) of the fruits of his efforts, which leads to effort underprovision. d) Show how a tournament can solve the issue of c) and determine what the prize for winning the tournament should be if one pie sells at €2 and πΎ = 2. You can assume that the probability of winning the tournament is proportional to the quantities baked, i.e., ππ = ππ ππ + ππ Following section 5.3 and page 69 specifically, the utility function of e.g., Antonio now becomes: π(ππ ) = π£ππ − πΆ(ππ ) = π£ ππ ππ 2 − ππ + ππ πΎ 2 Solving this function w.r.t. ππ yields the following first-order condition: π£ ππ (ππ + ππ ) 2 − 2ππ =0 πΎ Since the game is symmetric for both Antonio and Anna, we can assume that both will choose the same quantity in the Nash equilibrium: ππ = ππ = π Substituting this in the first-order condition gives: π£ π 2π − =0 (π + π)2 πΎ Or equivalently: π£πΎ 8 π=√ Setting this equal to the efficient quantity ππ = ππΎ 2 , it must be that π£πΎ ππΎ = 8 2 π=√ which is equivalent to π£ = 2π2 πΎ Plugging in π = 2 and πΎ = 2 gives that π£ = 16. Setting the tournament prize to this level solves the potential free rider problem of c) and makes sure that the efficient quantities are produced. 3
