Question

Evan Evader earns $5000 in income. Income is taxed at 20%. Evan can underreport his income to the tax collecting ministry and pay taxed only on the amount that he reports, but should he be audited, the ministry will impose a surcharge of 100% on the unpaid taxes; that is, he will have to pay 40% of any unreported income if he is audited. Evan realizes that the probability is 0.4 that he will be audited. His Bernoulli utility function is u(x) = ln x and he is an expected utility maximizer. Assume that Evan can never overreport.

(a)(10%) Draw Evan's budget set in a diagram where on the x-axis, label Evan's final income when he is audited, while on the y-axis, label his final income when he is not audited. What i s the slope of the budget line?

(b)(10%) Is Evan's indifference curve convex to the origin? Explain.

(c)(10%) How much will Evan report to ministry?

Answer 1

Answer:

A)

given utility funtion U(x)=ln(x)

Audited Income=80%*(5000-underreported income)+60%*underreported income

Unaudited income=80%*(5000-underreported income)+Underreported income

For Underreported income =1000

Audited income=80%*(5000-1000)+60%*1000=$3800

Unaudited inocme=80%*(5000-1000)+1000=$4200

Slope of line is: -1

B) Evan's indifference curve is not convex to origin because it is dependent on only one variable i.e. under reported income.

Expected utility for Evan=0.6*U(unaudited income)+0.4*U(audited income)

EU=0.6*ln(80%(5000-y)+y)+0.4*ln(80%*(5000-y)+60%*y)

C)Let Evan under reported $y  

Expected utility for Evan=0.6*U(unaudited income)+0.4*U(audited income)

EU=0.6*ln(80%(5000-y)+y)+0.4*ln(80%*(5000-y)+60%*y)

For maximum utility let differentiate EU with respect to  y

dEU/dy=0.6*{0.2/(80%(5000-y)+y)} + 0.4*{-0.2/(80%*(5000-y)+60%*y)}=0

So Solving for y we get

y=$4000

EU=0.6*ln(4800)+0.4*ln(3200)=8.314