Economics: Lagrangian Problem?

Question

When Theo was a kid, he used to get a fixed amount of income as allowance. With his income, he'd go to the local store and buy either boxes hot tamales (candy) denoted C or popsicles denoted S. Suppose his preferences are represented using the utility function C^(1/2) * S^(1/2). In an

effort to promote better health, the government taxes Theo and all other kids for each box of candy they purchase. Suppose the per unit tax is T .

(a) Use the Lagrangian method to solve for the optimal number of boxes of candy and theoptimal number of popsicles.

(b) Find partial derivative of S and C with respect to  . Can you determine the sign of the derivatives? Also, interpret each derivative.

Just wanted to check my work since I can't find this exercise in the book =\

My function would be f(x)=C^(1/2) * S^(1/2) and my constraint

would be that C + S = Y - CT (where Y is income)

This should give me a lagrangian of

L = C^(1/2) * S^(1/2) + lambda[ Y - CT - C -S]

Is this correct ?? thanks =)

Answer 1

You are right . But you need to assume some prices for the two products. If the prices are P1 and P2 respectively, the budget constraint would be:

Y -(P1+ t)*C + P2*S = 0 whete t is the tax per unit of candy box collected from the children (assume the market to be competitive enough so that the tax does not shift to the producers of candy).

So, the lagrantial expression to be maximized would be:

L = C^(1/2) * S^(1/2) + lambda[ Y -(P1+ t)*C + P2*S]