The Lump Sum Principle In the 1980s Margaret Thatcher
was the British Prime Minister. She was very popular and e�ective until
she pushed for the idea of a so called lump-sum tax. The lump-sum tax is
one in which every citizen pays a single �xed payment.
Consider the scenario, in which a consumer chooses consumption and
the fraction of the day she works subject to income constraints, given taxes
and prices p;w. Let non-labor income be I and suppose the consumers
preferences are given by:
u(c; l) = c:5(1 l):5 (1)
(a.) Set up the maximization problem assuming a proportional tax, T (i.e.
at tax, T constant, : : :, the one we did in class)
b.) Set up the maximization problem assuming a lump-sum tax, S.
c.) Suppose that the government must raise G < I pounds. (That is
wTl = G or S = G) Show that the consumer's utility is higher under
the lump-sum tax regime than it is under the
at tax regime(Plug in
numbers for everything but c and l if you want.
First, I don't understand your utility function; I think the Jiskha site modified or eliminated some "special" characters you may have been using.
Second, I don't see a question.
That said, a proportional income tax should lower the marginal utility from working, while a lump sum tax should have no direct impact on a person's leisure/labor choice.
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).