Consider a consumer with preferences over R 4 + represented by the utility function u(x1, x2, x3, x4) = min{x1 x2, x3 x4}. All prices are strictly positive.

a. Argue that, at the optimum, x1x2 = x3x4. b. Argue that, at the optimum, the budget constraint is satisfied with equality. c. Argue that there cannot be a corner solution. 1 d. Incorporate the results from (a) and (b) into the consumer's maximization problem. You will be left with a different but equivalent maximization problem. Hint: this new maximization problem has two constraints: the budget constraint and another one. Moreover, the objective function is different than the original one (by part (a)). e. Set up the Lagrangian and derive the Kuhn-Tucker conditions for a solution. Recall the result in (c). f. Find the solution to the consumer's optimization problem.