Effect of changing MC on monopolists' maximised profits

Question

Let a monopolist have constant MC = c, with no fixed costs. If $p(c)$ and $q(c)$ are the profit maximising price and output as a function of MC and the market demand schedule is given by $q = D(p)$, how does the monopolists' profit vary with a small change in MC? This is what I have so far:

Maximised profits is given by:

$$ π = p(c)q(c) - cq(c) $$

Differentiating w.r.t c:

$$ \frac{dπ(c)}{dc} = q(c)[p'(c) - 1] + q'(c)[p(c) - c] $$

The first term $q(c)[p'(c) - 1]$ is clearly positive because $q(c) > 0$ and for a monopolist $[p'(c) - 1]$ is also positive because:

$$ p(c) = \frac{c}{1+\frac{1}{Ɛ}} $$

$$ p'(c) = \frac{1}{1+\frac{1}{Ɛ}} $$

and for there to be a profit maximising level of output, $Ɛ < -1$

$$ p'(c) = \frac{1}{1+\frac{1}{Ɛ}}> 1 $$

For the second term, $q'(c)$ is clearly negative since higher MC leads to higher price and therefore lower quantity. $[p(c) - c] > 0$ since for a monopolist $p > MC = c$.

At this stage, the effect on profits of a rise in MC is ambiguous since both terms have opposing effects (one is positive, other is negative) and how profits change depends on the magnitude of the two terms. How can we quantify the effects or at least know the relative magnitudes of both terms to find how profits will change to an increase in MC?

Answer 1

If MC increases, profits need to go down. Let $\pi (q(c))$ and $\pi' (q(c'))$ denote profits before and after the cost increase. Observe that producing $q(c')$ was feasible when the marginal cost was low and, therefore, it must be the case that

$$ \pi (q(c))=[p(q(c))-c]q(c)\geq[p(q(c'))-c]q(c').$$

In addition, we have that

$$[p(q(c'))-c]q(c')>[p(q(c'))-c']q(c')=\pi' (q(c')),$$

which gives us $\pi (q(c))>\pi' (q(c')).$