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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

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

Ex 4.13(2) Question

Method:

- Find monopolist’s AR from consumer demand using answer to Ex 4.12.
- Then use standard optimisation procedure

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:

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)

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

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

Ex 4.13(4) Question

Method:

Add bonus B into the expression for profits

Again use standard optimisation procedure

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:

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:

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

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