To set up the maximization problem with a proportional tax (T), we need to maximize the consumer's utility function subject to income constraints. Let's denote the consumer's consumption as c and the fraction of the day she works as l.
The consumer's budget constraint under a proportional tax is given by:
pcl + (1 - l)w - T = I, where pcl represents the after-tax income from labor, (1 - l)w represents the after-tax non-labor income, and T is the tax.
To set up the maximization problem, we need to form the Lagrangian function. The Lagrangian function is the consumer's utility function minus the budget constraint multiplied by the Lagrange multiplier (λ):
L = u(c, l) - λ(pcl + (1 - l)w - T - I)
Now, let's differentiate the Lagrangian function with respect to c, l, and λ, and set the derivatives equal to zero to find the optimal values:
∂L/∂c = 0.5(1 - l)^0.5 - λp = 0 (Equation 1)
∂L/∂l = -0.5c(1 - l)^(-0.5) + 0.5(1 - l)^0.5 - λw = 0 (Equation 2)
∂L/∂λ = pcl + (1 - l)w - T - I = 0 (Equation 3)
Now, let's proceed to set up the maximization problem assuming a lump-sum tax (S). In this case, every citizen pays a single fixed payment, so the budget constraint becomes:
pcl + (1 - l)w - S = I (Equation 4)
Again, we form the Lagrangian function and differentiate it with respect to c, l, and λ:
L = u(c, l) - λ(pcl + (1 - l)w - S - I)
∂L/∂c = 0.5(1 - l)^0.5 - λp = 0 (Equation 5)
∂L/∂l = -0.5c(1 - l)^(-0.5) + 0.5(1 - l)^0.5 - λw = 0 (Equation 6)
∂L/∂λ = pcl + (1 - l)w - S - I = 0 (Equation 7)
Now, let's move on to comparing the consumer's utility under the lump-sum tax regime (S) and the proportional tax regime (T). We need to show that the utility is higher under the lump-sum tax regime.
We'll assume that G represents the amount the government needs to raise, and in the lump-sum tax regime, S = G. Plug in the numbers for p, w, I, G into Equations 1-3 and Equations 5-7, respectively. Then, solve the equations to find the optimal values of c, l, and λ for both cases.
Finally, substitute the optimal values into the utility function u(c, l), using the solution obtained for each regime. Compare the utility values and show that the utility is higher under the lump-sum tax regime (S) than it is under the proportional tax regime (T).