1. Using the Lagrange Method, maximize U = X1/2Y1/2 subject to X + Y = 4.

a) Write the objective function, U(X,Y), the constraint function, L(X,Y), and the Lagrangian function, L(X,Y,), where is the Lagrangian multiplier.

b) Find the first-order conditions (F.O.C.) for the Lagrangian (and solve for l). Then, find Y in terms of X (from L/X and L/Y).

c) What is the budget constraint and optimal condition? (Use previous answers.) So, what is the optimal consumption?

d) For the dual of optimization problem above, write the objective, constraint, and the Lagrangian functions - aka the "dual." (No calculations required.)