Question

In Country A, the income tax rate is 30%.

John, who is a resident of Country A, earns 10 a year, is going to report his income.

(A) If he underreports his income, it is detected with 10% probability and the penalty is
the square of the underreported income. (If he reports 8 and is detected, he should pay
2² = 4 as penalty.) If John is risk neutral, how much income is he going to report?

(B) The tax office can increase the detection probability to 20% with the cost of x. Compare
the expected tax revenue and penalty income of the tax office. For which range of x, is
it beneficial for the tax office to increase the detection probability?

(C) Suppose the detection probability of underreporting is increasing with the amount of underreporting.

The detection probability is given as 10% + 10% x M, where M is the amount of underreporting.

The penalty, once detected, is the same as the amount of underreporting. If John is risk neutral, how much income is he going to report?

Answer 1

allows the sum he reports = j

at that point his complete utility = p1U1 + p2U2 = 0.1*(- (10-j)^2 + j*0.7) + 0.9*(0.7*j + 10-j) = 0.1*(- 100 - j^2 + 20.7j) + 0.9(10 - 0.3j)

= - 10 - 0.1j^2 + 2.07j + 9 - 0.27j

= - 1 - 0.1j^2 + 1.8j

chance nonpartisan individual will endeavor to boost its utility.

dU/dj = 0

=> - 0.2j + 1.8 = 0

=> j = 1.8/0.2 = 9

Along these lines, He will report 9 as his salary.

b)

Chesoloo's all out utility = 0.2*(- 100 - j^2 + 20.7j) + 0.8(10-0.3j)

= - 20 - 0.2j^2 + 4.14j + 8 - 0.24j

= - 12 - 0.2j^2 + 3.9j

charge division's utility(expexted charge income) = 0.2*( 0.3*j + (10-j)^2 - x) + 0.8*(0.3j - x)

= 0.2(0.3j + 100 + j^2 - 20j - x) + 0.8(0.3j - x)

= 20 - 3.94j +0.2j^2 - 0.2x + 0.24j - 0.8x

= 20 - 3.7j + 0.2j^2 - x

punishment salary = 0.2*(10-j)^2

in the event that Chesoloo endeavor to augment its capacity

dU(chesoloo)/dj = 0

=> - 0.4j + 3.9 = 0

=> j = 3.9/0.4 = 9.75

accepting Expense division mollifies this dimension of hazard profile:

Duty division's capacity's esteem = 20 - 3.7(9.75) + 0.2(9.75)^2 - x = 20 - 36.075 + 19.0125 - x = 2.9375 - x

along these lines, x ought to be lower than 2.9375 for advantage of expense division

punishment pay will be 0.2(10-9.75)^2 = 0.2*0.25*0.25 = 0.0125

c)

All out utility of Chesoloo = (0.2 - M)( M*0.7 + - (10-M)) + (1-(0.2 - M))(M*0.7 + 10-M )

= (0.2-M)(1.7M - 10) + (M+ 0.8)(10-0.3M)

=0.34M - 2 - 1.7M^2 + 10M + 10M - 0.3M^2 + 8 - 0.24M

= 20.1M + 6 - 2M^2

for max utility dU/dM =0

=> 20.1 - 4M =0

=> M = 20.1/4 = 5.025