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An alternative way to study the welfare properties of a monopolists 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 planners problem then would be to choose Q and X to maximize social welfare as represented by SW = U(Q ,X) C(Q ,X).a. What are the first-order conditions for a welfare maximum?b. The monopolists goal is to choose the Q and X that maximize = P(Q ,X) Q C(Q ,X). What are the first-order conditions for this maximization?c. Use your results from parts (a) and (b) to show that, at the monopolists preferred choices, SW/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.d. Show that, at the monopolists preferred choices, the sign of SW/X is ambiguousthat 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.

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