Question

Marginal costs of production are given by the following function: MC(Q) = 4 − Q Q ≤ 2 ,2Q − 2 Q > 2 (a) Plot the marginal cost curve. (b) Give the expressions for V C(Q) and AV C(Q). (c) Plot AV C(Q) on the plot from (a). (d) Give the expression for the supply curve of this firm. (e) Is it possible to find a different MC(Q) function that gives rise to the same supply curve? If yes, give an example. If no—prove it.

Answer 1

(a) The graph would be as below.

10 MC 8- 6- MC 4- 2 Q 10 -CO- CO- -2-

(b) The MC function is given as
4-Q for Q<2 MC 2Q-2 for Q> 2
. The total cost would be
MCdQ =f2Q - 2)dQ_for Q>2 TC
or
TC(4Q-0.5Q)+Fi for Q2 (2-20)F for Q>2
, where F1 and F2 are integral constants, and in this case, are the fixed costs.

The variable cost will would be
(4Q 0.50) for Q2 VC= (2-2Q) for Q>2
. The AVC would be
10-0.50 for QS2 AVC= for Q2
or
4 0.5Q for Q2 Q-2 for Q 2 AVC=
.

(c) The graph is as below.

10 8- 6- MC AVC 2 Q 10 -CO- CO- MC,AVC

(d) The supply curve of the firm would be the MC above the AVC. As can be seen, the MC<AVC for Q less than or equal to 2, while MC>AVC for Q greater than 2. The supply function would be hence for or
P20 2
for .

(e) The argument is a bit conditional. It depends on what part of MC is being altered. For example, for the MC being , would give the same supply curve as before, which is different than before only for Q less than or equal to 2, but is same for Q greater than 2.

for Q 2 2Q-2 for Q> 2 2 MC2

Considering the MC being considered as the part of MC where MC>AVC, it is not possible to obtain the same supply function with different MC. Proof is that, for two different MC being
MC
and
MC
, suppose we have the same supply function. Hence, we would have the supply function as
PMC
and
MC
being the same. In that case, solving for both these equation, we would have , which contradicts our assumption that
MC
and
MC
are different. Thus, it is not possible to obtain a different MC for the same supply function.