Question

The Laffer Curve graphs the amount of government revenue as a function of the tax rate. It is hypothesized to be inverse U-shaped -- first increasing, reaching a maximum, and then decreasing. The maximum of the Laffer curve represents the largest size of government that can be sustained, because beyond it, higher tax rates actually reduce government revenue. Consider a consumer with preferences U = ln(x) + ln(1 − ℓ), where x is consumption and ℓ is labor supply. Let the budget constraint be x = (1 − t)ℓ where t is the tax rate, so that consumers can only consume a dollar amount equal to their after-tax income.

^ Laffer Curve, from Malcomson (1986)

a. Find the optimal labor supply of the consumer (plug x from the budget constraint into the utility function then maximize the resulting utility function with respect to ℓ.)

b. Use the solution for labor supply to calculate tax revenue R, where R is obviously R= t*ℓ.

c. Plot revenue a10s a function of the tax rate. Comment on the form of the function.

Answer 1

a) Optimal labor supply l is found by optimizing utility function, after plugging in the value of x from the budget constraint. Mechanism is given below:

(a) u = ln (x) + ln (1-1) x=(1-tol I plugg in x into U :- Tu= ln [(1-2) 1] + ln (1-1) du - [(lt) l I'+(1-l) a Ol [(lt) I ] - = (1-4 -1 [CFE) IJ 1-1 Setting this to o to find optimal bradle labor supply :- s-teo 1-1-1² - 0 roots of l =-1155 2.

Since negative l is not possible, the answer is l= (-1+2.23)/2= 0.618

b) Revenue is simply t*l

R= 0.618t

c) Graph is plotted below:

Revenue Obit 0.54 € 0.48 F 042 F 0.36 00 0.24 0.18 € 0.12 0.00 THT 0.6 0.7 0.8ogtax rate 0.1 0.2 0.3 0.4 0.5

The function is positively sloping. As tax rate increases, then revenue from tax also increases.