Question

What is the​ long-run cost function if the production function is qequals=33Lplus+22​K? Let the cost of each unit of L be w and the cost of each unit of K be r. The​ long-run cost function is A. ​C(q)equals=wplus+r. B. ​C(q)equals=​(w/33​)q if ​(w/33​)less than or equals≤​(r/22​), and ​(r/22​)q otherwise. C. ​C(q)equals=q. D. ​C(q)equals=wq if wless than or equals≤​r, and rq otherwise. E. ​C(q)equals=​(w/33​)q if ​(w/33​)greater than or equals≥​(r/22​), and ​(r/22​)q otherwise.

Answer 1

C(q)equals=​(w/33​)q if ​(w/33​)greater than or equals≥​(r/22​), and ​(r/22​)q otherwise.

Explanation:

The production function is q = 33L + 22​K

Cost function is then, C = wL+ rK

Minimising the functions, we get:

σ = wL+ rK+ λ(q -33L -22K)

dσ/dL = w - λ33 = 0 ----------(1)

dσ/dK = r - λ22 = 0 -----------(2)

dσ/dλ = q -33L -22K ---------(3)

From equation (1) and (2)

w/r ⇒ λ33 / λ22 ⇒ ³³/₂₂

Or, w = r(³³/₂₂)

Substitute for 'w' in C

C = r(³³/₂₂)L + rK

C = r [(³³/₂₂)L + K]

Or, (³³/₂₂)L + K = C / r

So, C/r = q or r = C/q

recall that w = r(³³/₂₂)

∴ w = (³³/₂₂)C/q

So, cost function C = wL+ rK

Substitute for 'w' and 'r' in the equation