13. multiproduct firm Ralph uses three inputs (denoted z1 ≥ 0, z2 ≥ 0 and z3 ≥ 0) to produce two products (denoted q1 and q2). Respective factor prices are P1 = 1,P2 = 5 and P3 = 2. Technology requires z1+z3 ≥ q1 and z2 + z3 ≥ q2.

(a) Write down the optimization program to determine Ralph’s cost, C(q;P).

(b) Determine the first product’s marginal cost if q1 < q2.

(c) Determine the first product’s marginal cost if q1 > q2.

(d) Explain your findings intuitively, and in terms of the shadow prices on the technology constraints.