14. multiproduct firm Ralph has grown up and now manages a two product firm. The

Question

14. multiproduct firm Ralph has grown up and now manages a two product firm. The technology requires a mixture of capital and labor to produce each product. Capital is shared, while labor is specific to each of the two products. For the first product, capital [K 0) and labor (L10 must satisfy the following, in order to produce q1 units: q1 KL1 Likewise, producing q2 units of the second product requires capital (K) and labor (now L2 0) such that: q2 KL2. In addition, total capital is limited to a maximum of 200 units. (So K 200.) Naturally, K. L1 and L2 are all required to be non-negative. Capital costs 100 per unit, labor for the first product costs 150 per unit and labor for the second product costs 175 per unit. The first product sells for 275 per unit, and the second sells for 300 per unit. (a) Initially suppose only the first product is present. Determine and interpret Ralph's optimal production plan. (b) Next suppose only the second product is present. Determine and interpret Ralph's optimal production plan. (c) Now assume both products are present. Determine and interpret Ralph's optimal production plan. (d) Repeat (a), (b) and (c) assuming the first product sells for 200 per unit. (e) Fill in the following table q1 q 2 M C1 (q: P) M C2 (q: P) 50 50 50 100 50 150 100 50 100 100 100 150 150 50 150 100 150 150 (f) Write a short paragraph interpreting your findings.

Explanation / Answer

Using the production functions and MRTS for the two production functions we see that

q1 = 6, q2 = 12

K1L1 = 36 K2L2 = 144

From MRTS we know that

K1/L1 = 4 K2 = 3L2

So

4L1^2 = 36 3L2^2 = 144

L1 = 3 and L2 = 6.9

K1 = 12 and K2 = 2.3

K1 + K2 < 15.

Marginal cost for product 1 = 10q1 + 2q2 + 10q1 = 22q1 = 132 => Unit cost is $22

Marginal cost for product 2 = 245.5, unit cost is 49.5

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