Question

3. A taxpayer has utility function U(x, L) = x ^1/2 − L where L is hours of labour supply and x is consumption. The taxpayer earns a wage of $4 per hour worked (which is fixed throughout the analysis).

(a) Suppose that the government imposes a proportional (percentage) tax at rate τ on labour income, so that the taxpayer’s budget constraint is x = (1 − τ )4L. Solve for the optimal labour supply (L) and consumption (x) as a function of τ .

(b) What is the taxpayer’s maximized level of utility (i.e., the indirect utility function), as a function of τ?

(c) How much revenue, which is the tax rate τ times labour income 4L, or R = τ4L is raised by the tax? (Note: Use the expression for the optimal labour supply from a) here; R will be a function of τ )?

(d) Say the government wants to maximize the amount of revenue it generates from the tax. It does this by choosing τ such that dR/dτ = 0 and solving for τ . Do this, and determine the revenue maximizing τ and the amount of revenue raised. With this tax rate, what is the maximized level of utility based on the indirect utility function you determined in part c) (this will now be an actual number)?

(e) Now instead, suppose that the government imposes a fixed lumpsum tax T so that the taxpayer’s budget constraint is now x = 4L−T. Solve for the optimal labour supply (L) and consumption (x) as a function of T in this case. Calculate the maximized level of utility (the indirect utility function) as a function of T.

(f) Say that the government chooses T so as to generate the same amount of revenue it raised in part d). What is the maximized 2 level of utility in this case (this will be an actual number), and how does it compare to d)?

Answer 1

According to the question:

$U (x,L) = \sqrt x -L \\ subject \, to \\ x = (1-\tau)4L$

$\textit{H} = \sqrt x - L -\lambda (x - (1- \tau)4L) \\ \frac{\partial H}{\partial x} = \frac{1}{2 \sqrt x} - \lambda = 0 \, (1) \\$

$\frac{\partial H}{\partial L} = 1 - \lambda ((1-\tau) 4) = 0 \, (2) \\$

$\frac{\partial H}{\partial L} = x - ((1-\tau) 4L) = 0 \, (3) \\$

Using (1) and (2)

$\frac{1}{2 \sqrt x} = \lambda \\$

$1 = \lambda ((1-\tau) 4)$

$1 = \frac{1}{2 \sqrt x} ((1-\tau) 4) \\ \sqrt x = 2(1- \tau)\\ x = 4(1-\tau)^2$

$x = ((1-\tau) 4L) \\ 4(1- \tau)^2 = ((1-\tau) 4L)\\ 1-\tau = L$

b) Taxpayer's maximized utility -

$x = 4(1-\tau)^2 \\ 1-\tau = L \\ U (x,L) = \sqrt x - L \\ U (x,L) = \sqrt (4(1-\tau)^2) - (1-\tau)\\$

$U (x,L) =2 (1-\tau) - (1-\tau) \\ U(x,L) = 2 - 2 \tau -1 + \tau \\ U(x,L) = 1 - \tau$

c) Revenue :

$R = \tau 4L \\ R = \tau (4 (1 - \tau))\\ R = 4 \tau (1-\tau)$

d) Maximized value of R :

$R = 4 \tau (1-\tau) \\ differentiating \, this \, with \, respect \, to \, \tau \\ \frac{dR}{d\tau} = 4 - 8 \tau = 0 \\ 4 = 8 \tau \\ \tau = \frac{1}{2}$

$R = 4 \tau (1-\tau) \\ \tau = \frac{1}{2} \\ R = 4 * \frac{1}{2} * \frac{1}{2}= 1$

$U(x,L) = 1 - \tau \\ \tau = \frac{1}{2} \\ U(x,L) = 1 - \frac{1}{2} \\ U(x,L) = \frac{1}{2} \\$

e)

$U (x,L) = \sqrt x -L \\ subject \, to \\ x = 4L -T$

$\textit{H} = \sqrt x - L -\lambda (x - 4L + T) \\ \frac{\partial H}{\partial x} = \frac{1}{2 \sqrt x} - \lambda = 0 \, (1) \\$

$\frac{\partial H}{\partial L} = 1 - 4 \lambda = 0 \, (2) \\$

$\frac{\partial H}{\partial L} = x - 4 L +T = 0 \, (3) \\$

$\frac{1}{2 \sqrt x} = \lambda \\$

$\frac{1}{4} = \lambda$

$\frac{1}{2 \sqrt x} =\frac{1}{4} \\ \sqrt x = 2 \\ x =4$

$4 = 4 L - T \\ 4 -T = 4L \\ 1- \frac{T}{4} = L$

$U (x,L) = \sqrt 4 - (1- \frac{T}{4}) \\ U (x,L) = 2 - (1- \frac{T}{4}) \\ U (x,L) = 1 + \frac{T}{4} \\$

f) With proportional tax, the government raises the Revenue of $1. Now, with lumpsum tax, if the government has to raise the same amount then they have to charge lumpsum tax 1 $ ; T= $1

Now utility becomes :

$U (x,L) = 1 + \frac{1}{4} \\ U (x,L) = \frac{5}{4} \\$

Taxpayer is better of with lumpsum tax as his utility increases from 1/2 to 5/4.