14.8 More on the welfare analysis of quality choice An alternative way to study the welfare properties of a monopolist’s choices is to assume the existence of a utility function for the customers of the monopoly of the form utility = U(Q, X ), where Q is quantity consumed and X is the quality associated with that quantity. A social planner’s problem then would be to choose Q and X to maximise social welfare as represented by SW = U(Q, X ) − C(Q, X ).

a.

What are the irst-order conditions for a welfare maximum?

b.

The monopolist’s goal is to choose the Q and X that maximise π = P(Q, X ) · Q − C(Q, X ). What are the irst-order conditions for this maximisation?

c.
d.
Use your results from parts

(a) and

(b) to show that, at the monopolist’s preferred choices, ∂SWQ /∂Q > 0. That is, as we have already shown, prove that social welfare would be improved if more were produced. Hint: Assume that �U/�Q = P.
Show that, at the monopolist’s preferred choices, the sign of �SW/�X is ambiguous – that is, it cannot be determined (on the sole basis of the general theory of monopoly) whether the monopolist produces either too much or too little quality.