Alan Turing can crack codes using two inputs, labor and machines. In the short r

Question

Alan Turing can crack codes using two inputs, labor and machines. In the short run, Alan has a fixed amount of 4 machines. Cracking codes has production function f(L, M) = 4L1/2M1/2. If the cost of labor is $4 per unit and the cost of machines is $4 per unit, what is the short-run total cost of cracking 48 codes?

In the long run, when the quantity of machines is variable, how would Alan change the amount of labor and machines used to crack 48 codes. With the long run choice of labor and machines, how much is total cost?

Explanation / Answer

Production function, Q = 4L1/2M1/2

w = 4, r = 4, M = 4

Q = 4 x L1/2 x 2 = 8L1/2

If Q = 48, then

8L1/2 = 48

L1/2 = 6

L = 36

Total cost, TC = wL + rM = 4 x 36 + 16 = 160

(b)

Production function, Q = 4L1/2M1/2

w = 4, r = 4

Total cost, TC = wL + rM = 4L + 4M

First, let us calculate the Marginal Rate of Substitution (MRS), defined as (MPL / MPM).

MPL = dQ / dL = 4 x (1/2) x (M / L)1/2

MPM = dQ / dM = 4 x (1/2) x (L / M)1/2

So, MRS = MPL / MPM = M / L

Cost-minimizing optimality requires that, MRS = Price ratio = w / r

Or, (M / L) = 4 / 4 = 1

L = M

So, Q = 4L1/2M1/2 = 4 x L = 4L [Or, Q = 4M]

If Q = 48 (As given in question), then

4L = 48

L = 12

M = 12 [Cost minimizing combination]

TC = wL + rM

= $4 x 12 + $4 x 12 = $96

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