Freddie has preferences over gym equipment represented by the Cobb-Douglas utility function, U(G, E)= G1/5E4/5 where G is gym equipment and E is everything else. Assume Freddie has an income of $200, the price of gym equipment is $100 and the price of a unit of "everything else" is $50.

(a) Set up Freddie's Lagrangian function, take the first order conditions, and solve the system of equations to find Freddie's optimal combination of gym equipment and everything else.

(b) With gym equipment on the x-axis and everything else on the y-axis, draw Freddie's budget constraint. On the same graph, draw the indifference curve passing through the optimal point you found in part A (plot at least 5 points for each curve). Hint: U =G 1/5 E 4/5 where (G , E ) are the optimal values found in part A. Are your results consistent?

(c) Now assume that Freddie's gym offers a new promotion whereby gym equipment is buy 1, get 1 free. While Freddie can resell his goods, he cannot take the free gym equipment out of the packaging and try to sell that. If he wants to resell a piece of gym equipment, he must sell it with the free piece with it. Draw this new budget line. Without solving for the new optimal bundle, can you unequivocally state whether Freddie will be better off? Hint: Are the preferences represented by the Cobb-Douglas utility function monotonic?