1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 4.13
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. Ex 4.13(1) Question
purpose: to derive a simple model of monopoly regulation with a welfare evaluation using CV
method: build model up step-by-step through the question parts

3. Ex 4.13(1) A natural monopoly requires that costs be subadditive
Subadditivity implies the following
given an integer m > 1
C(w, q) < mC(w, q/m)
(see Ex 3.1)
In the present case costs are C0 + cq
Clearly m[C0 + cq/m] = mC0 + cq > C0 + cq

4. Ex 4.13(1): “Natural monopoly”AC c q

5. Ex 4.13(2) Question
Method:
Find monopolist’s AR from consumer demand using answer to Ex 4.12.
Then use standard optimisation procedure

6. Ex 4.13(2) Monopoly profits
Aggregate demand over N consumers using Exercise 4.12
Rearrange to get AR curve:
Total Revenue is:
Profits are therefore:

7. Ex 4.13(2) Maximising profits
FOC (MC = MR) yields:
So monopolist’s optimal output is:
From AR curve, price at optimum is:
Simplify this to:
(clearly price > MC)

8. Ex 4.13(3) Question
Method:
Aggregate the CV for each consumer to define L.
Use marginal cost and monopolist’s equilibrium price to evaluate L

9. Ex 4.13(3) Evaluating loss Use definition of CV with p1' = c:
Evaluate L at p1 = 2c:
Firm’s profits are:
Clearly L > profits

10. Ex 4.13(4) Question
Method:
Add bonus B into the expression for profits
Again use standard optimisation procedure

11. Ex 4.13(4) Evaluating profits (again)
Profits including bonus are:
Value of bonus is:
Use demand curve to express this in terms of q:
So profits can now be expressed as:

12. Ex 4.13(4) Evaluating profits (again)
Take the expression for profits including bonus
FOC for a maximum is again MR = MC:
Rearranging we get the value of optimal output for the regulated monopolist:
Use demand curve to find:
Clearly the regulated price = MC:

13. Ex 4.13: Points to note
Aggregate welfare loss is found from individual CV
Unregulated monopoly makes profits smaller than losses to consumer
Regulation causes monopoly to behave like competitive firm