Answer:
To find the values of x1 and x2 that maximize U subject to the budget constraint, we need to use Lagrange multipliers. The problem can be set up as follows:
Maximize U = alpha ln(x1) + beta ln(x2)
Subject to the budget constraint:
P1x1 + P2x2 = M
where P1 is the price of good x1, P2 is the price of good x2, and M is the consumer's income.
To use Lagrange multipliers, we first set up the Lagrangian function:
L = alpha ln(x1) + beta ln(x2) + λ(M - P1x1 - P2x2)
where λ is the Lagrange multiplier.
We then take the partial derivatives of L with respect to x1, x2, and λ, and set them equal to zero:
dL/dx1 = alpha/x1 - λP1 = 0
dL/dx2 = beta/x2 - λP2 = 0
dL/dλ = M - P1x1 - P2x2 = 0
Solving these equations simultaneously, we get:
x1 = (alpha/lambda)P1
x2 = (beta/lambda)P2
M = P1x1 + P2x2 = (alpha/lambda)P1^2 + (beta/lambda)P2^2
To solve for the Lagrange multiplier, we substitute the expressions for x1 and x2 into one of the partial derivative equations and solve for λ:
alpha/x1 - λP1 = 0
alpha/((alpha/lambda)P1) - λP1 = 0
lambda = alpha/P1
Substituting this value of λ back into the expressions for x1 and x2, we get:
x1 = (alpha/lambda)P1 = alpha
x2 = (beta/lambda)P2 = beta
Therefore, the values of x1 and x2 that maximize U subject to the budget constraint are x1 = alpha and x2 = beta.