NTU micro econ homework #4

1. Consider the model of delegation to bureaucrats and refer to the figure in the lecture slides. Suppose the legislature’s choice of allowable policies is given by P = [p, p0 ] ? [p 00 , p¯] where p 0 < p00 . (a) As e changes from 0 to a large positive number, specify the policy choice (of agent) and policy outcome as a function of e for successive ranges of e and explain why agent’s policy choice is optimal in each range. (b) Find the expression for the variance in the policy outcome at e = p 0+p 00 2 – a and verify that the variance is lowered by moving p 0 and p 00 closer together. (c) Given the agent’s best response, we have derived the expected utility of a legislator with ideal position li as – Z 8 p¯-a (li – p¯+ e) 2 f(e)de – Z p¯-a p-a (li – a) 2 f(e)de – Z p-a -8 (li – p + e) 2 f(e)de. Apply Leibniz’s Rule to derive the first-order conditions about p and ¯p. 2. Recall the all-pay auction model of electoral contests and the social welfare function associated with imperfect voting parameter q. More precise way of formulating social welfare is to treat a(?1)/?1 and a(?2)/?2 separately – as ?1 and ?2 are separate (but i.i.d.) random variables – and to think about their joint distributions. (a) Modify the social welfare associated with q accordingly to reflect the above consideration. (b) Derive the first-order condition from the modified social welfare function in part (a) and find out the conditions under which imperfect voting is optimal. 1

Homework 4 (Due on 5/26) 1. Consider the model of delegation to bureaucrats and refer to the figure in the lecture slides. Suppose the legislature’s choice of allowable policies is given by P = [p, p0 ] ∪ [p 00 , p¯] where p 0 < p00 . (a) As ε changes from 0 to a large positive number, specify the policy choice (of agent) and policy outcome as a function of ε for successive ranges of ε and explain why agent’s policy choice is optimal in each range. (b) Find the expression for the variance in the policy outcome at ε = p 0+p 00 2 − a and verify that the variance is lowered by moving p 0 and p 00 closer together. (c) Given the agent’s best response, we have derived the expected utility of a legislator with ideal position li as − Z ∞ p¯−a (li − p¯+ ε) 2 f(ε)dε − Z p¯−a p−a (li − a) 2 f(ε)dε − Z p−a −∞ (li − p + ε) 2 f(ε)dε. Apply Leibniz’s Rule to derive the first-order conditions about p and ¯p. 2. Recall the all-pay auction model of electoral contests and the social welfare function associated with imperfect voting parameter q. More precise way of formulating social welfare is to treat α(θ1)/θ1 and α(θ2)/θ2 separately – as θ1 and θ2 are separate (but i.i.d.) random variables – and to think about their joint distributions. (a) Modify the social welfare associated with q accordingly to reflect the above consideration. (b) Derive the first-order condition from the modified social welfare function in part (a) and find out the conditions under which imperfect voting is optimal. 1