Cost Function + Marginal vs Unit Costs

Assignment 1:

Construction of cost function – two products

1.a:

Question:

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

Answer:

To determine the optimization program used to determine Ralph’s cost, we use equation 3.4:

Cq;P≡minz1,z2,z3 ≥0i=13Pizi

s.t.

q ∈f(z)

The cost function is equal to the sum of the price of the factors multiplied by the number of units used. We can rewrite the general cost function when we know how many different input factors there are and their respective prices, P1 = 20, P2 = 100 and P3 = 60.

Cq;P≡ minz1,z2,z3 ≥020z1+100z2+60z3

s.t.

z1, z2, z3 ≥0

0.5 z1+z3≥ q1

z2+z3≥q2

To optimize Ralph’s cost he therefor has to minimize the cost function subject to the three constraints.

1.b:

Question:

Determine the marginal cost of q1 if q1 < q2.

Answer:

If q2 were much bigger than q1 then you would choose to produce q2 by using the third factor, because it is the cheapest, and because the second factor, z2, and the third factor, z3, are perfect substitutes.

The marginal cost of q1, when q2 is much bigger than q1, is therefor 0, because you will get the first product, q1, as a byproduct of the large production of q2, because one unit of the third factor, z3, can produce both one unit of the first product, q1, and one unit of the second product, q2.

1.c:

Question:

Determine the marginal cost of q1 if q1 > q2.

Answer:

If q1 were much bigger than q2, then you would choose to produce q1 by using the first factor, z1, because it is the cheapest and because the first factor, z1, and the third factor, z3, are perfect substitutes.

The marginal cost of q1, when q1 is much bigger than q2, is therefor 40, because it takes 2 units of the first factor, z1, to produce a unit of the first product, q1.

1.d:

Question:

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

Answer:

Because the prices of the input factors are different and because the available two input factors are...