Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Exercise 4.13

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank CowellFrank Cowell

November 2006 November 2006

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(1) Question

purposepurpose: to derive a simple model of monopoly regulation : to derive a simple model of monopoly regulation with a welfare evaluation using CVwith a welfare evaluation using CV

methodmethod: build model up step-by-step through the question : build model up step-by-step through the question partsparts

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(1)

A natural monopoly requires that costs be A natural monopoly requires that costs be subadditivesubadditive

Subadditivity implies the followingSubadditivity implies the following given an integer given an integer mm > 1 > 1 CC(w(w, , qq) < ) < mCmC(w(w, , q/mq/m)) (see Ex 3.1)(see Ex 3.1)

In the present case costs are In the present case costs are CC00 + + cqcq ClearlyClearly m m[[CC00 + + cq/mcq/m] = ] = mCmC00 + + cq cq > > CC00 + + cqcq

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(2) Question

MethodMethod: : Find monopolist’s AR from consumer demand using Find monopolist’s AR from consumer demand using

answer to Ex 4.12. answer to Ex 4.12. Then use standard optimisation procedureThen use standard optimisation procedure

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(2) Monopoly profits

Aggregate demand over Aggregate demand over NN consumers using Exercise 4.12consumers using Exercise 4.12

Rearrange to get AR curve:Rearrange to get AR curve:

Total Revenue is:Total Revenue is: Profits are therefore:

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(2) Maximising profits

FOC (MC = MR) yields:FOC (MC = MR) yields:

So monopolist’s optimal output is:So monopolist’s optimal output is:

From AR curve, price at optimum is:From AR curve, price at optimum is:

Simplify this to:Simplify this to: (clearly price > MC)

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(3) Question

MethodMethod: : Aggregate the CV for each consumer to define Aggregate the CV for each consumer to define LL. . Use marginal cost and monopolist’s equilibrium price to evaluate Use marginal cost and monopolist’s equilibrium price to evaluate LL

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

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

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13(4) Question

MethodMethod: : Add bonus Add bonus BB into the expression for profits into the expression for profits Again use standard optimisation procedureAgain use standard optimisation procedure

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

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:

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

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

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 4.13: Points to note

Aggregate welfare loss is found from individual CV

Unregulated monopoly makes profits smaller than losses top consumer

Regulation causes monopoly to behave like competitive firm