21. In Italy, firms pay tax on reported profits at a constant proportionate rate t€ (0, 1). If the firm's profit is, the owner of the firm can choose to report any amount of profit r where 0 ≤r≤, and thus pay tr in tax. However, if the firm is audited, it must pay additional tax on unreported profit -r at a rate of t+f, where 0 < f <1-t. Thus if the firm is audited, it pays tr+(t+f)(n-r) in tax. The probability of being audited is p. Assume that the owner of each firm maximizes expected utility with a strictly increasing von Neumann-Morgenstern utility function that depends only on after-tax profit. (a) What is the smallest auditing probability p* for which a risk neutral owner is willing to report the firm's full profit ? Solution: A risk neutral owner maximizes the expected after-tax profit, which is given by p(ntnf(nr)) + (1 − p)(a − tr) = π- tpa - fpx + (fp-(1-p)t)r. Given that r ≤, this is marimized when r = if and only if fp-(1-p)t ≥ 0, that is, if and only if p≥=P*. (b) If the auditing probability is equal to p* from part (a), will a risk averse owner report the full profit, less than the full profit, or is it impossible to determine? Solution: When p = p², the expected value of after-tax profit does not depend on the reported profit r. Since r= involves no uncertainty and r < involves uncertainty, a risk averse uner prefers the certain lottery giving the same expected value. Therefore, she will report the full profit.
21. In Italy, firms pay tax on reported profits at a constant proportionate rate t€ (0, 1). If the firm's profit is, the owner of the firm can choose to report any amount of profit r where 0 ≤r≤, and thus pay tr in tax. However, if the firm is audited, it must pay additional tax on unreported profit -r at a rate of t+f, where 0 < f <1-t. Thus if the firm is audited, it pays tr+(t+f)(n-r) in tax. The probability of being audited is p. Assume that the owner of each firm maximizes expected utility with a strictly increasing von Neumann-Morgenstern utility function that depends only on after-tax profit. (a) What is the smallest auditing probability p* for which a risk neutral owner is willing to report the firm's full profit ? Solution: A risk neutral owner maximizes the expected after-tax profit, which is given by p(ntnf(nr)) + (1 − p)(a − tr) = π- tpa - fpx + (fp-(1-p)t)r. Given that r ≤, this is marimized when r = if and only if fp-(1-p)t ≥ 0, that is, if and only if p≥=P*. (b) If the auditing probability is equal to p* from part (a), will a risk averse owner report the full profit, less than the full profit, or is it impossible to determine? Solution: When p = p², the expected value of after-tax profit does not depend on the reported profit r. Since r= involves no uncertainty and r < involves uncertainty, a risk averse uner prefers the certain lottery giving the same expected value. Therefore, she will report the full profit.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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please only do: if you can teach explain steps of how to solve each part
a) how to derive
b) r= p?
Transcribed Image Text:21. In Italy, firms pay tax on reported profits at a constant proportionate rate t€ (0, 1). If the
firm's profit is, the owner of the firm can choose to report any amount of profit r where
0 ≤r ≤, and thus pay tr in tax. However, if the firm is audited, it must pay additional tax
on unreported profit -r at a rate of t+f, where 0<f<1-t. Thus if the firm is audited,
it pays tr+(t+f)(n-r) in tax. The probability of being audited is p. Assume that the owner
of each firm maximizes expected utility with a strictly increasing von Neumann-Morgenstern
utility function that depends only on after-tax profit.
(a) What is the smallest auditing probability p* for which a risk neutral owner is willing to
report the firm's full profit ?
Solution: A risk neutral owner maximizes the expected after-tax profit, which is given
by
p(n-tn-f(nr)) + (1 − p)(x − tr) = π- tpx − ƒp + (fp-(1-p)t)r.
Given that r ≤, this is maximized when r = if and only if fp-(1-p)t≥ 0, that is,
if and only if p≥t=P².
(b) If the auditing probability is equal to p* from part (a), will a risk averse owner report
the full profit, less than the full profit, or is it impossible to determine?
Solution: When p = p, the expected value of after-tax profit does not depend on the
reported profit r. Since r = involves no uncertainty and r < involves uncertainty, a
risk averse owner prefers the certain lottery giving the same expected value. Therefore,
she will report the full profit.
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