9. shadow prices We find Ralph studying cost, and how cost depends on the way a choice problem is framed. Ralph now produces two products. Let x and y, respectively, denote the quantities of the two products that are produced and sold. Any nonnegative quantities satisfying the following constraints can be produced: (1) x + y ≤ 400; and (2)

x + 2y ≤ 500. Ralph’s revenue is given by 40x + 42y, and his cost is given by 30x + 30y. (Though this is clearly a short-run story, as capacity is fixed, we suppress the fixed cost and measure "profit" as 10x + 12y.)

(a) Determine an optimal solution for Ralph.

(b) Determine the shadow prices on the two constraints.

(c) In what sense are the shadow prices on the two constraints opportunity costs?