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please  only do: if you can teach explain steps of how to solve each part how was formula retrieve? why? Utility from reporting full tax = (1-t)π why this U(w) = w Expected utility (from reporting less profit) what is this? = Auditable profit*(after-tax income with audit) + Auditable profit*(after-tax income without audit) Expected utility from reporting less profit = (π - tr - (t+f)(π-r)+(1-p)(π-tr) Utility from reporting full tax = (1-t)π p(π - tr - (t+f)(π-r)+(1-p)(π-tr) < (1-t)π pπ - ptr - p(t+f)(π-r)+(π-tr)-p(t+f)(π-r)+(π-tr)< π-p(t+f)(π-r)-tr < -tπ p*>t(π-r)/(t+f)(π-r)   As a result, the value of p* should be at least equal to t/(t+f)

please  only do: if you can teach explain steps of how to solve each part how was formula retrieve? why? Utility from reporting full tax = (1-t)π why this U(w) = w Expected utility (from reporting less profit) what is this? = Auditable profit*(after-tax income with audit) + Auditable profit*(after-tax income without audit) Expected utility from reporting less profit = (π - tr - (t+f)(π-r)+(1-p)(π-tr) Utility from reporting full tax = (1-t)π p(π - tr - (t+f)(π-r)+(1-p)(π-tr) < (1-t)π pπ - ptr - p(t+f)(π-r)+(π-tr)-p(t+f)(π-r)+(π-tr)< π-p(t+f)(π-r)-tr < -tπ p*>t(π-r)/(t+f)(π-r)   As a result, the value of p* should be at least equal to t/(t+f)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please  only do: if you can teach explain steps of how to solve each part how was formula retrieve?

why? Utility from reporting full tax = (1-t)π
why this
U(w) = w

Expected utility (from reporting less profit) what is this? = Auditable profit*(after-tax income with audit) + Auditable profit*(after-tax income without audit)

Expected utility from reporting less profit = (π - tr - (t+f)(π-r)+(1-p)(π-tr)

Utility from reporting full tax = (1-t)π

p(π - tr - (t+f)(π-r)+(1-p)(π-tr) < (1-t)π

pπ - ptr - p(t+f)(π-r)+(π-tr)-p(t+f)(π-r)+(π-tr)< π-p(t+f)(π-r)-tr < -tπ

p*>t(π-r)/(t+f)(π-r)

 

As a result, the value of p* should be at least equal to t/(t+f)

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.
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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