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A consumer with utility function U = Y½ determines the amount of income to declare to the tax authority.
a. Denoting the probability of detection by p, the tax rate by t, and the fine by F, provide an expression for the optimal value of X.
b. For F = ½ and p = ½ show that the declaration X is an increasing function of t.
c. Assume that the revenue authority aims to maximize the sum of tax revenue plus fines less the cost of auditing. If the latter is given by c(p) = p2, graph the income of the revenue authority as a function of p for Y = 10, F = ½, and t = 1/3 . Hence derive the optimal value of p.