There are 2 firms in the yoghurt market, Dannon and Yoplait. They produce yoghur

Question

There are 2 firms in the yoghurt market, Dannon and Yoplait. They produce yoghurt using milk (M) and capital (K) according to the production function f(M,K)= 2 (MK)^1/2 where q is the quantity of yoghurt produced. (Since labor does not appear in this equation, you can assume that it is fixed in the short run and embedded in the capital measure.) The price of milk is p = 4, and the current price of capital is r = 1.

In the short run, both firms have a fixed plant size, so that capital is fixed. Dannon has plant size K D = 100, while Yoplait has plant size K (Y) = 81.

1) Compute the short run total cost functions of Dannon and Yoplait.

2) Compute the short run average cost functions of Dannon and Yoplait.

3)Compute the short run average variable cost functions of Dannon and Yoplait.

4)Compute the short run marginal cost functions of Dannon and Yoplait.

5) Yannick asserts that because capital is fixed in the short run and variable costs do not include the cost of capital, Dannon and Yoplait should have the same variable costs of producing any given amount of yoghurt. Is Yannick correct? Justify your answer.

6) Do the two firms have the same long run cost function? Justify your answer.

7) Derive the firms’ long run cost function(s) CLR(q) using the Lagrange method. Please show all derivations.

Explanation / Answer

Y = 2 x M0.5K0.5

(1) Total cost, TC = M x price of M + K x price of K = 4M + K

(i) Dannon: K = 100

Y = 2 x M0.5K0.5 = 2 x M0.5(100)0.5 = 2 x M0.5 x 10 = 20 x M0.5

So,

M0.5 = Y / 20

M = Y2 / 400

Substituting in TC:

TC = 4M + K = 4 x (Y2 / 400) + 100 = (Y2 / 100) + 100

(ii) Yoplait: K = 81

Y = 2 x M0.5K0.5 = 2 x M0.5(81)0.5 = 2 x M0.5 x 9 = 18 x M0.5

So,

M0.5 = Y / 18

M = Y2 / 324

Substituting in TC:

TC = 4M + K = 4 x (Y2 / 324) + 81 = (Y2 / 81) + 81

(2) Short run average cost (SAC) = TC / Y

(i) Dannon:

SAC = [(Y2 / 100) + 100] / Y = (Y / 100) + (100 / Y)

(ii) Yoplait:

SAC = [(Y2 / 81) + 81] / Y = (Y / 81) + (81 / Y)

(3) Short run average variable cost (SAVC) = VC / Y, where VC: Part of TC that varies with Y

(i) Dannon:

SAVC = (Y2 / 100) / Y = Y / 100

(ii) Yoplait:

SAVC = (Y2 / 81) / Y = Y / 81

(4) Short run Marginal cost (SMC) = dTC / dY

(i) Dannon:

SMC = 2Y / 100 = Y / 50

(ii) Yoplait:

SMC = 2Y / 81 = Y / 40.5

Note: First 4 questions are answered.

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