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) 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.

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

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

Explanation / Answer

1. Input cost for Dannon = r.K + p.M = 1.K + 4.M

Short run cost = 100 + 4.M

Input cost for Yoplait = r.K + p.M = 1.K + 4.M

Short run cost = 81 + 4.M

Since both firm have same production function, so Inputs (K,M) required will be same.

and since M is the onlly variable input , so both will have same variable cost = 4M

2. Yes,  the two firms will have the same long run cost function, as production function is same for both the firms and there are no fixed costs/variable in the long run, so K r capital will also be variable in the long run.

3.   

Total Cost = r*K + p*M

Minimize cost r*K + p*M

subject to Q = f(M,K)= 2 (MK)^1/2 , where Q is output

So cost minimization for the given level of output wiil be found by setting up lagrange function

L =r*K + p*M -   [2 (MK)^1/2 - Q ]

Now finding

dL/dM = p - *(K/M)^1/2

dL/dK = r - *(M/K)^1/2

dL/d = 2 (MK)^1/2 - Q

Now putting dL/dX1 = dL/dX2 =dL/d = 0

dL/dM = p - *(K/M)^1/2 = 0

dL/dK = r - *(M/K)^1/2 = 0

dL/d = 2 (MK)^1/2 - Q = 0

Solving eq 1 and 2

[(K/M)^1/2]/[(M/K)^1/2 = p/r

putting p =4 , and r = 1

K/M = 4

K = 4M

Putting in production function

Q = 2 (M*4M)^1/2 = 2*2M = 4M

M = Q/4

K = 4M = 4*Q/4 = Q

Putting in cost function

S0, the cost funcion is

C = 1*K + 4*M

= 1*Q + 4*Q/4

C = 2Q

is the cost function

If you don't understand anything, then comment, I will revert back on the same.

And If you liked the answer then please do review the same. Thanks :)

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