Question

For each of the following production functions below, find the

 LR total cost as a function of Q: TC(Q)

 LR average total cost as a function of Q: ATC(Q)

 How does LR average total cost vary with Q? (Hint, find the derivative: dATC(Q)/dQ)

 Does this cost function exhibit economies of scale, diseconomies of scale, or constant returns to scale?

A. Q = 4L^2

B. Q = 2L

C. Q = L1/3

Answer 1

Total cost, TC = w*L, where w is wage rate per hour of labor (which is a constant)

Average total cost, ATC = TC(Q)/Q,

partial ATC(Q)/partial Q depicts how average total cost changes with change in Q. If it's negative, this means there exists a negative relation between ATC and Q, that is, with increase in Q, ATC decreases (similarly, positive sign implies a positive relation between the two, and 0 implies no impact of change in Q on ATC that is change in quantity doesn't effect average cost).

Also, economies of scale is achieved when with increase in scale of operation (i.e., production, which Q), cost per unit (which is average cost) decreases. In similar way, diseconomies of scale is achieved when with increase in scale of operation, cost per unit also increases, and constant returns to scale is seen when change in scale of operation doesn't impact cost per unit in any way; then:

A) Q = 4L2

L = (Q/4)1/2 = 0.5*Q0.5

So with, TC = w*L

TC(Q) = w*(0.5*Q0.5) (on substitution)

TC(Q) = 0.5w*Q0.5

ATC(Q) = TC(Q)/Q = 0.5w*Q0.5/Q = 0.5w/Q0.5

partial ATC(Q)/partial Q = -0.5*(0.5w)Q-1.5 = -0.25w/Q1.5

Clearly, with increase in quantity Q, keeping other things unchanged, the average total cost is decreasing (notice the negative sign). This means that this cost function exhibits economies of scale.

B) Q = 2L

L = Q/2 = 0.5*Q

So with, TC = w*L

TC(Q) = w*(0.5*Q) (on substitution)

TC(Q) = 0.5w*Q

ATC(Q) = TC(Q)/Q = 0.5w*Q/Q = 0.5w

partial ATC(Q)/partial Q = 0

Clearly, with increase in quantity Q, keeping other things unchanged, the average total cost is unchanged or constant (since, partial ATC(Q)/partial Q = 0). This means that this cost function exhibits constant returns to scale.

C) Q = L1/3

L = Q3

So with, TC = w*L

TC(Q) = w*(Q3) (on substitution)

TC(Q) = w*Q3

ATC(Q) = TC(Q)/Q = w*Q3/Q = w*Q2

partial ATC(Q)/partial Q = 2*w*Q= 2wQ

Clearly, with increase in quantity Q, keeping other things unchanged, the average total cost is increasing (notice that Q directly increases ATC by a factor of 2w). This means that this cost function exhibits diseconomies of scale.