Question

In TaxLand, income tax rate is 30%. Cheolsoo, who is a resident of TaxLand and 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
2*2 = 4 as penalty.) If Cheolsoo is risk neutral, how much income is he going to report?
(b) 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 underreport is increasing with the amount of un-derreporting. The detection probability is given as 10% + 10% - M, where M is the amount of underreporting. The penalty, once detected, is the same as the amount of underreporting. If Cheolsoo is risk neutral, how much income is he going to report?

Answer 1

let's the amount he reports = j

then his total 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

risk neutral individual will try to maximize its utility.

dU/dj = 0

=> -0.2j + 1.8 = 0

=> j = 1.8/0.2 = 9

So, He will report 9 as his income.

b)

Chesoloo's total 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

tax department's utility(expexted tax revenue) = 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

penalty income = 0.2*(10-j)^2

if Chesoloo try to maximize its function

dU(chesoloo)/dj = 0

=> -0.4j + 3.9 = 0

=> j = 3.9/0.4 = 9.75

assuming Tax department assuages this level of risk profile:

Tax department's function's value = 20 -3.7(9.75) + 0.2(9.75)^2 -x = 20 -36.075 + 19.0125 -x = 2.9375 - x

so, x should be lower than 2.9375 for benefit of tax department

penalty income will be 0.2(10-9.75)^2 = 0.2*0.25*0.25 = 0.0125

c)

Total 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