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# 2. Audrey recently lost her job. If she exerts her effort by enduring disutility h (1-p), she expects to find a new job with probability 1 -p. With probability p, however, she will still remain jobless. If unemployed, Audrey will be paid b as unemployment benefits (UI), which is financed via lump-sum tax 7 from wage w of the employed. Thus her expected utility can be written as

EU = -h (1-p)+(1-p) u (w-7)+p.u (b),

where u (c) = log (c) and h (1-p) = 0.5 x (1 - p)²

(a) Suppose the government satisfies the budget balance. Compute the equilibrium tax rate.

(b) Suppose the government can fully monitor Audrey's job search effort 1 - p. What is the optimal unemployment insurance bFB? What is the tax rate under the optimal UI? Explain how the consumption level varies by labor market status.

(c) We now consider the case where the government cannot directly control Audrey's job search effort. What is Audrey's optimal job-finding rate 1 - p*?

(d) Define the second-best problem for the government which maximizes Audrey's ex-ante welfare, and derive its optimality condition (a.k.a. the optimal UI formula) using the elasticity Ep,b = (dp/db) b/p

(e) Solve for the optimal size of benefit as a function of elasticity Ep,b. Does your answer differ from your finding in (b)? Explain.