Problem 2 A consumer chooses x, and x, to maximize [(x, x,) = log x, + log x,, subject to the budget constraint: px, + x, =y (note that the price of x, has been normalized to equal one). 1. Solve the budget constraint for x, as a function of x, . Using the substitution method, convert the consumer's optimization problem into a one -dimensional maximization problem, and find the consumer's optimal level of x , as a function of p and y. Do the same for X,. 2. Now solve the consumer's optimization problem using the Lagrange multiplier method. Interpret the first-order conditions in terms of the slope of budget constraint and indifference curves. Show your results are exactly the same as in (a)