Question

Which of the following production techs will exhibit a well-behaved solution to profit maximization that satisfies both the first and second order conditions in the short run when capital is fixed at some positive level?

1) q = f(K, L) = K^2 + ln(L)

2) q = f(K, L) = 2K + (1/2)L

3) q = f(K, L) = min (ln(K), ln(L))

4) q = f(K, L) = KL^2

Answer 1

A production technology exhibit a well- behaved solution to profit maximization if the following condition holds:

- Isoquants must be smooth. No kinks should be present

- Diminishing MRTS

Production function given in option 3) does not satisfy the 1st condition. Isoquants are L-shaped and hence have a kink. So, They do not give well-behaved solution to profit maximization. Option 3) is correct

Option 2) is the case of perfect substitute production function. It satisfies the first condition but violate the second one. The MRTS in this case is constant and does not diminish. So, option 2) is also incorrect

1)

q = f(K, L) = K^2 + ln(L)

MRTS = MPL/MPK

= (1/L)/2K

= 0.5(1/LK)

Take the derivative of MRTS wrt L

dMRTS/dL = 0.5 ( -1/L2K) < 0

the second-order derivative is less than 0. So, MRTS is diminishing and the isoquants are drawn as a convex curve which is smooth. Hence, option 1) shows a well-behaved solution to profit maximization

Option 4)

q = f(K, L) = KL^2

MRTS = MPL/MPK

= (2KL/L2)

= (2K/L)

Take the derivative of MRTS wrt L

dMRTS/dL = (-2)*(K/L^2) < 0

Second-order derivative is less than 0. So, MRTS is diminishing and the isoquants are drawn as a convex curve which is smooth. Hence, option 4) shows well-behaved solution to profit maximization.

Hence, option 1) and option 4) are correct

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