ECON 321 Assignment 1- Consider a monopolist who sells

Subject: Economics    / General Economics
Question
ECON 321 Assignment 1, 2017.

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 fixed fee,
F , and a price per-unit of consumption, p. For any given value of p, and resulting level of
consumption, x, the fixed fee cannot exceed the value of consumer surplus corresponding to
(F, p). Therefore, the monopolist’s problem is to choose F and p such that his revenue perconsumer, 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 it
graphically 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 finding the marginal rate of
substitution 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 explanation as to why the point you identify is in fact the optimum.