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# Consider a monopolist who sells his product to a group of consumers

Subject: Economics    / General Economics
Question

Consider a monopolist who sells his product to a group of consumers, all of whom have the same individual demand curve, which is p = a°bx. Both a and b are positive constants,p is the per-unit price of consumption, and x represents the amount of product purchased.The monopolist faces no costs of production at all (and so wants to maximise revenue), and while he must charge the same to every consumer, he is able to charge both a Öxed fee,F, and a price per-unit of consumption, p. For any given value of p, and resulting level of consumption, x, the Öxed fee cannot exceed the value of consumer surplus corresponding to(F; p). Therefore, the monopolistís problem is to choose F and p such that his revenue per consumer,R(F; p) = F + px, is maximised, subject to the chosen (F; p) pair being feasible.The graphical space that you should use for this problem has F on the vertical axis and p on the horizontal.

1. Find the value of consumer surplus for any given choice of per-unit price p.

2. Write the set notation description of the feasible set.

3. Find the equation, F(p), of the upper boundary of the feasible set, and represent itgraphically as accurately as you can.

4. Is the feasible set convex? Provide as much detail as you can.

5. Now analyse the contours of the objective function, by Önding the marginal rate ofsubstitution between F and p using the implicit function theorem.

6. Draw a set of contours of the objective function, again as accurately as you can.

7. Finally, use your graph to identify the optimal choice, (F§; p§). Provide a clear explanationas to why the point you identify is in fact the optimum.