Question

If tax revenue is given by the following function: REV(t)=t×w×(h-l(t)), where t is the labor tax rate, w is the wage rate, h is the maximum amount of time available to the household, and l(t) is leisure as an increasing function of the tax rate i.e. if the labor tax rate t increases, leisure increases, so that individuals work less. Assume that l(t)=min[h,t²]. This simply means that leisure cannot exceed h which is the maximum amount of time available. Find the tax rate that maximizes the tax revenue when h=1, w=$5. Be precise and report 4 digits after the decimal point.

Answer 1

Rev(t) = t*w*[h-l(t)], we have l(t) = min[h,t²], since l(t) is the min function so we can state that l(t) can take value either h or t². We know that leisure cannot exceed h, so l(t) = t², substituting this value in the revenue function along with the values h = 1 and w =$5 gives,

Rev(t) = t*5*[1-t²] = 5t - 5t³. Now we differentiate it w.r.t 't' (we will get the MR) and equate it equal to 0,

So MR = 5 - 15t² = 0 which gives t = square root of 1/3 = 0.577350.

Therefore, the tax rate which maximizes the tax revenue is equal to 0.5773.