By Günter Bamberg, Klaus Spremann (auth.), Prof. Dr. Günter Bamberg, Prof. Dr. Klaus Spremann (eds.)
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Extra resources for Agency Theory, Information, and Incentives
The constraint a rent r which has the level r at least. Given the response (32) and the reservation constraint (33), the principal's welfare is v E[Y - (r + sy + tz)] s2 s+t 2 t2 2 -2- - m - 1l(2aa e - 1) - 1l(2aa E (34 ) - 1) + + ~ _ (s+t)2 2 2 This welfare, taken as a function of s and of t is, for small, concave and will be maximized for s* (35) t* - 1 1 -t a sufficient t* 2 2aa e - s* 1 + 2aa 2E 1 The linear system (35) has the explicit solution s* 1 2 1 + 2aa e + a~/a~ (36) t* 1 2 2 1 + 2aa 2 + a/ae E Equations (36), together with (33), provide the reward scheme (r*,s*,t*) selected.
Is the set of contracts 1,2 working as a self-selection device? The answer is yes. To prove this answer one has to consider the choice between contracts 1,2 for each type of individuals. Firstly, contract 2 was constructed in such a way that type-O agents are indifferent between contracts 1,2. So increase the rent of contract 2 by one dollar or so to induce type-O agents definitely decide for contract 2. Secondly, type-1 agents still prefer contract 1 when having the choice among contracts 1,2.
We proceed by analyzing the preference maps of a representative principal and an agent. Starting with the agent's situation we can derive following proposition: convex with a following dF da I G=- and show the In an a - F space the agent's indifference unique turning point (minimum) at Ga = O. curve is To see this, note the properties of the indifference (iso-G) curve: %/G F I d2F = -G aa /G F da 2 G =- ]J + Aa02~0, = +A0 2 ( 1 3a) ( 13b) )0. As to the indifference curve of the principal we show the following proposition: In an a- F space the principal's indifference curve is concave with a unique turning point (maximum) at Ha = O.