Suppose a consumer has a utility function over two goods, denoted U(Q1,Q2), whic

Question

Suppose a consumer has a utility function over two goods, denoted U(Q1,Q2), which is increasing in both goods . The consumer has income W and faces prices P1 and P2. State the utility maximization problem and the Lagrange function. Solve the first order condition and discuss the results In terms of a tangency between the utility function and the price ratio. Suppose the utility has Cobb-Douglas form: U(Q1,Q2) = (Q1)alpha(Q2)beta where alpha,beta > 0 and alpha + beta = 1. Solve the optimal quantity (Q1*, Q2*) for general prices (P1, P2). Show a graph for Qi1 vs Q2 with the optimal quantity, indifference curve for that quantity, and the budget constraint. Suppose the consumer has income W=10, prices are P1=1/2 and P2=1, and linear utility: U(Q1, Q2) = Q1 + Q2. Solve the optimal quantity (Q1*, Q2*). Show a graph for Qx 1 vs Q2 with the optimal quantity, indifference cun/rve, and budget constraint. Suppose income W=10, prices are P1=1l/2 and P2=1, and Leontiff utility: U(Q1,VQ2) = min(Q1,Q2). Solve the optimal quantity (Q1*,Q2*). Show a graph for Q1 vs Q2 with the optimal quantity, indifference curve, and budget constraint.

Explanation / Answer

Given a utility function U(x,y) and budget constraint p1x+p2Y = W, then the utility maximization problem becomes:

6)            Max U(x,y) subject to p1x+p2y = W

The lagrangian becomes:

                Lagrange = U(w,y) + G(W-p1x-p2y)

7) First order conditions will become:

dU/dx

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