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.

(b) The MC function is given as . The total cost would be or , where F1 and F2 are integral constants, and in this case, are the fixed costs.

The variable cost will would be . The AVC would be or .

(c) The graph is as below.

(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 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.

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 and , suppose we have the same supply function. Hence, we would have the supply function as and being the same. In that case, solving for both these equation, we would have , which contradicts our assumption that and are different. Thus, it is not possible to obtain a different MC for the same supply function.