Exercise 4 13
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Exercise 4.13. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006 . 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) .

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Exercise 4.13

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Exercise 4 13

Exercise 4.13

MICROECONOMICS

Principles and Analysis

Frank Cowell

November 2006


Ex 4 13 1 question

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

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 1 natural monopoly

Ex 4.13(1): “Natural monopoly”

AC

c

q


Ex 4 13 2 question

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

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

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

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

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

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

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 again1

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

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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