Problem 3 (50 points)

Consider a simple model of tax evasion. The taxpayer has a true income equal to y and chooses to report 0 < 1 y to the tax authority. The income reported by the taxpayer is taxed at a proportional rate t (0,1). If * < y, we say that the taxpayer is evading: the amount of under-reported income is y- a and the unpaid tax is t (y - x). The government audits tax returns with probability p (0,1): if the taxpayer is caught evading, all income is taxed and a fine at rate F is levied on the tax that has been evaded.

Assume that the taxpayer's utility function is u with u' > 0 and u" < 0. The taxpayer chooses the amount of reported income to maximize his expected utility:

max EU (x) = (1 - 1) u (YNA) + pu (YA) (1)

where YNA and YA denote the taxpayer's final income depending on whether he is audited (A) or not (NA).

1. Show that

YNA = y-tx,

YA = (1-t)y - Ft(y-x).

Substitute this in (1) and compute the first-order condition for the optimal choice of x.

1. Illustrate with a graph the problem of the taxpayer. In particular, draw a graph with YNA on the horizontal axis and YA on the vertical axis. Draw the certainty line. Show in the graph the two extreme cases of x = y, when the taxpayer reports all his income (no evasion), and x = 0, when the minimum declaration is made (total evasion). Represent graphically the budget constraint of the taxpayer, i.e. the set of all options available to him.
2. With the help of the graph drawn in (2), compute the (constant) slope of the budget constraint. What is the slope of an indifference curve evaluated at a generic allocation (YNA, YA)?
3. What condition guarantees that evasion will occur, i.e. that the optimal solution is x* < y? Hint: you can answer this question either analytically using the first-order condition derived above or with the help of the graph.
4. Consider now the level of revenues raised by the government. Government revenues consist of taxes paid plus the money received from fines. What is the expected revenue raised by the government from a taxpayer with income y?