Question 8.3.3: A monopolist is faced with the inverse demand function P(Q) ......
A monopolist is faced with the inverse demand function P(Q) denoting the price when output is Q. The monopolist has a constant average cost k per unit produced.
(a) Find the profit function π(Q), and prove that the first-order condition for maximal profit at Q^{∗}> 0 is
P(Q^{*})+Q^{*}P^{\prime}(Q^{*})=k (∗)
(b) By implicit differentiation of (∗) find how the monopolist’s choice of optimal production is affected by changes in k.
(c) How does the optimal profit react to a change in k?
(a) The profit function is π(Q) = QP(Q) − kQ, so π^{\prime}(Q) = P(Q) + QP^{\prime}(Q) − k. In order for Q^{∗}> 0 to maximize π(Q), one must have π^{\prime}(Q^{∗}) = 0, or equivalently (∗).
(b) Assuming that Eq. (∗) defines Q^{∗} as a differentiable function of k, we obtain
Solving for dQ^{∗}/dk gives
{\frac{\mathrm{d}Q^{*}}{\mathrm{d}k}}={\frac{1}{Q^{*}P^{\prime\prime}(Q^{*})+2P^{\prime}(Q^{*})}}(c) Because π(Q^{∗}) = Q^{∗}P(Q^{∗}) − kQ^{∗}, differentiating w.r.t. k gives
{\frac{\mathrm{d}\pi(Q^{*})}{\mathrm{d}k}}={\frac{\mathrm{d}Q^{*}}{\mathrm{d}k}}P(Q^{*})+Q^{*}P^{\prime}(Q^{*}){\frac{\mathrm{d}Q^{*}}{\mathrm{d}k}}-Q^{*}-k{\frac{\mathrm{d}Q^{*}}{\mathrm{d}k}}But the three terms containing dQ^{∗}/dk all cancel because of the first-order condition (∗). So dπ^{∗}/dk = −Q^{∗}. Thus, if the cost increases by one unit, the optimal profit will decrease by approximately Q^{∗}, the optimal output level.