Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency: Waste

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

Almost essential

Welfare and Efficiency

Almost essential

Welfare and Efficiency

PrerequisitesPrerequisites

November 2006November 2006

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Agenda

Build on the efficiency presentationBuild on the efficiency presentation Focus on relation between competition and efficiencyFocus on relation between competition and efficiency

Start from the “standard” efficiency rulesStart from the “standard” efficiency rules MRS same for all householdsMRS same for all households MRT same for all firmsMRT same for all firms MRS=MRT for all pairs of goodsMRS=MRT for all pairs of goods

What happens if we depart from them?What happens if we depart from them? How to quantify departures from them?How to quantify departures from them?

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Basic model

Applications

Overview...Background

Model with production

Efficiency: Waste

How to evaluate inefficient states

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

The approach

Use standard general equilibrium analysis Use standard general equilibrium analysis to...to... Model price distortionModel price distortion Define reference set of pricesDefine reference set of prices

Use consumer welfare analysis to…Use consumer welfare analysis to… Model utility lossModel utility loss

Use standard analysis of household budgets Use standard analysis of household budgets to…to… Model change in profits and rentsModel change in profits and rents

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

A reference point

Address the question: how much waste?Address the question: how much waste? Need a reference pointNeed a reference point

where there is zero wastewhere there is zero waste quantify departures from this pointquantify departures from this point

Any efficient point would doAny efficient point would do But it is usual to take a CE allocationBut it is usual to take a CE allocation

gives us a set of pricesgives us a set of prices we’re not assuming it is the “default” statewe’re not assuming it is the “default” state just a convenient benchmarkjust a convenient benchmark

Can characterise inefficiency as price distortionCan characterise inefficiency as price distortion

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

= p1~p1 [1

= p2~p2

= p3~p3

pn

= ......

= pn~

consumerprices

firms' prices

But now we have a distortionBut now we have a distortion

A model of price distortion Assume there is a competitive equilibriumAssume there is a competitive equilibrium If so, then everyone pays the same pricesIf so, then everyone pays the same prices

What are the What are the implications for MRS implications for MRS and MRT?and MRT?

Distortion Distortion

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Price distortion: MRS and MRT

Consumption:Consumption: pjMRSij h = —

pi

For every household marginal rate of

substitution = price ratio

For every household marginal rate of

substitution = price ratio

Production:Production: for commodities 2,3,...,for commodities 2,3,...,nn

pj

MRTnj = —pn

pj

MRT3j = —p3

pj

MRT2j = —p2

pj

MRT1j = —p1

[1+ ]

... ... ...

But for commodity 1...But for commodity 1...

Illustration....Illustration....

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

x1 0

x2

ConsumersConsumers

Price distortion: efficiency loss Production possibilitiesAn efficient allocationSome other inefficient allocation

How to measure importance of this

wedge ....

How to measure importance of this

wedge ....

x

x*

p* p*

ProducersProducers

At x* producers and consumers face same prices.

At x producers and consumers face different prices.

Price "wedge" forced by the distortion.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Waste measurement: a method To measure loss we use a reference pointTo measure loss we use a reference point Take this as competitive equilibrium...Take this as competitive equilibrium...

...which defines a set of reference prices...which defines a set of reference prices

Quantify the effect of a notional price change:Quantify the effect of a notional price change: ppii := := ppii – – ppii** This is [actual price of This is [actual price of ii] ] – – [reference price of [reference price of ii]]

Evaluate the equivalent variation for household Evaluate the equivalent variation for household hh : : EVEVhh = = CChh(p*,(p*,hh) – ) – CChh(p,(p,hh) – [) – [y*y*h h – – yyhh]] This is This is (consumer costs) (consumer costs) – – (income) (income)

Aggregate over agents to get a measure of loss, Aggregate over agents to get a measure of loss, We do this for two cases…We do this for two cases…

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Basic model

Applications

Overview...Background

Model with production

Efficiency: Waste

Taking producer prices as constant…

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

x1 0

x2

If producer prices constant… Production possibilitiesReference allocation and pricesActual allocation and prices

x

x*

p* p*

Measure cost in terms of good 2.

Losses to consumers are C(p*,) C(p,)

Cost of at prices p.

C(p, )C(p, )

Cost of at prices p*.

C(p*,)C(p*,)

Change in value of output at consumer prices

p p

is difference between C(p*,) C(p,) and

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Model with fixed producer prices Waste Waste involves both demand and supply responses. involves both demand and supply responses. Simplify by taking case where production prices constant.Simplify by taking case where production prices constant. Then waste is given by:Then waste is given by:

Use Shephard’s Lemma Use Shephard’s Lemma xi

h = Hhi(p,h) = Cih(p,h)

Take a Taylor expansion to evaluate Take a Taylor expansion to evaluate ::

is a sum of areas under compensated demand curve. is a sum of areas under compensated demand curve.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Basic model

Applications

Overview...Background

Model with production

Efficiency: Waste

Allow supply-side response…

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

x1 0

x2

Waste measurement: general caseProduction possibilitiesReference allocation and pricesActual allocation and prices

x*

p* p*

Measure cost in terms of good 2.

Losses to consumers are C(p*,) C(p,)

Cost of at prices p.

C(p, )C(p, )

Cost of at prices p*.

C(p*,)C(p*,)

Change in value of output at consumer prices

p p

is difference between C(p*,) C(p,) and

x

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Model with producer price response Adapt the Adapt the formula to allow for supply responses. formula to allow for supply responses. Then waste is given by:Then waste is given by:

where qi (∙) is net supply function for commodity i Again use Shephard’s Lemma and a Taylor expansion:Again use Shephard’s Lemma and a Taylor expansion:

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Basic model

Applications

Overview...Background

Model with production

Efficiency: Waste

Working out the hidden cost of taxation and monopoly…

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Application 1: commodity tax Commodity taxes distort pricesCommodity taxes distort prices

Take the model where producer prices are givenTake the model where producer prices are given Let price of good 1 be forced up by a proportional commodity tax Let price of good 1 be forced up by a proportional commodity tax

tt Use the standard method to evaluate wasteUse the standard method to evaluate waste What is the relationship of tax to waste?What is the relationship of tax to waste?

Simplified model:Simplified model: identical consumersidentical consumers no cross-price effects… no cross-price effects… ……impact of tax on good 1 does not affect demand for other goodsimpact of tax on good 1 does not affect demand for other goods

Use competitive, non-distorted case as reference:Use competitive, non-distorted case as reference:

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

A model of a commodity tax

p1

compensateddemand curve

p1

p1*

x1h

x1h

x1*

revenue raised =

tax x quantity

revenue raised =

tax x quantity

Equilibrium price and quantityThe tax raises consumer price...

...and reduces demand

Gain to the government Loss to the consumer

Waste

Waste measured by size of triangleSum over h to get total wasteCommonly known as deadweight loss of tax.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Tax: computation of waste The tax imposed on good 1 forces a price wedge

p1 = tp1*> 0 where is p1

* is the untaxed price of the good h’s demand for good 1 is lower with the tax:

x1** rather than x1

* where x1

** = x1*x1

h and x1h < 0

Revenue raised by government from h: Th = tp1

*x1**

Loss of consumer’s surplus to h is CSh = ∫ x1

h dp1 ≈ x1** p1− ½ x1

hp1 = Th½x1

hp1= Th− ½ t p1*

x1h > Th

Use the definition of elasticity := p1x1

h / x1hp1< 0

Net loss from tax (for h) is h = CSh − Th = − ½tp1

*x1h

= − ½tp1x1** = − ½t Th

Overall net loss from tax (for h) is ½ |tT uses the assumption that all consumers are identical

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

p1

compensateddemand curve

p1

p1*

x1h

x1h

Size of waste depends upon elasticity

low: relatively small waste

high: relatively large waste

Redraw previous example

p1

p1

p1*

x1h

x1h

p1

p1

p1*

x1h

x1h

p1

p1

p1*

x1h

x1h

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Application 1: assessment

Waste inversely related to elasticityWaste inversely related to elasticity Low elasticity: waste is smallLow elasticity: waste is small High elasticity: waste is large High elasticity: waste is large

Suggests a policy ruleSuggests a policy rule suppose required tax revenue is givensuppose required tax revenue is given which commodities should be taxed heavily?which commodities should be taxed heavily? if you just minimise waste – impose higher taxes on commodities if you just minimise waste – impose higher taxes on commodities

with lower elasticities.with lower elasticities.

In practice considerations other than waste-minimisation In practice considerations other than waste-minimisation will also influence tax policywill also influence tax policy distributional fairness among householdsdistributional fairness among households administrative costsadministrative costs

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Application 2: monopoly

Monopoly power is supposed to be wasteful…Monopoly power is supposed to be wasteful… but why?but why?

We know that monopolist…We know that monopolist… charges price above marginal costcharges price above marginal cost so it is inefficient …so it is inefficient … ……but how inefficient?but how inefficient?

Take simple version of main modelTake simple version of main model suppose markets for goods 2, …, suppose markets for goods 2, …, nn are competitive are competitive good 1 is supplied monopolisticallygood 1 is supplied monopolistically

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Monopoly: computation of waste (1)

Monopoly power in market for good 1 forces a price wedge p1 = p1

* * − p1

* > 0 where p1

** is price charged in market p1

* is marginal cost (MC)

h’s demand for good 1 is lower under this monopoly price:

x1** x1

*x1h,

where x1h < 0

Same argument as before gives: loss imposed on household h: −½p1x1

h > 0 loss overall:− ½p1x1, where x1 is total output of good 1 using definition of elasticity , loss equals −½p1

2 x1*

*p1

* *

To evaluate this need to examine monopolist’s action…

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Monopoly: computation of waste (2)

Monopolist chooses overall output use first-order condition MR = MC:

Evaluate MR in terms of price and elasticity: p1

* * [ 1 + 1 / ]

FOC is therefore p1*

* [ 1 + 1 / ] = MC

hence p1= p1*

* − MC = − p1

* * /

Substitute into triangle formula to evaluate measurement of loss: ½ p1

* * x1

* * / |

Waste from monopoly is greater, the more inelastic is demand Highly inelastic demand: substantial monopoly power Elastic demand: approximates competition

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Summary Starting point: an “ideal” worldStarting point: an “ideal” world

pure private goodspure private goods no externalities etcno externalities etc so CE represents an efficient allocationso CE represents an efficient allocation

Characterise inefficiency in terms of price distortionCharacterise inefficiency in terms of price distortion in the ideal world MRS = MRT for all in the ideal world MRS = MRT for all hh, , ff and all pairs of goods and all pairs of goods

Measure waste in terms of income lossMeasure waste in terms of income loss fine for individualfine for individual OK just to add up?OK just to add up?

Extends to more elaborate models Extends to more elaborate models straightforward in principlestraightforward in principle but messy mathsbut messy maths

Applications focus on simple practicalitiesApplications focus on simple practicalities elasticities measuring consumers’ price responseelasticities measuring consumers’ price response but simple formulas conceal strong assumptions but simple formulas conceal strong assumptions