efficiency: waste

Frank Cowell: Efficiency-Waste

EFFICIENCY: WASTEMICROECONOMICSPrinciples and Analysis Frank Cowell

Almost essential Welfare and Efficiency

Prerequisites

July 2017


Agenda Build on the efficiency presentation• Focus on relation between competition and efficiency

Start from the “standard” efficiency rules• MRS same for all households• MRT same for all firms• MRS=MRT for all pairs of goods

What happens if we depart from these rules? How to quantify departures from efficiency?

July 2017


Basic model

Applications

Overview

Background

Model with production

Efficiency: Waste

How to evaluate inefficient states

July 2017


The approach Use standard general equilibrium analysis to…• Model price distortion• Define reference set of prices

Use consumer welfare analysis to…• Model utility loss

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

July 2017


A reference point Address the question: how much waste? Need a reference point• where there is zero waste• quantify departures from this point

Any efficient point would do But it is usual to take a CE allocation• gives us a set of prices• we’re not assuming it is the “default” state• just a convenient benchmark

Can characterise inefficiency as price distortion

July 2017


= p1~p1 [1+d]

= p2~p2

= p3~p3

pn

= ……

= pn~

consumerprices

firms' prices

But now we have a distortion

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

What are the implications for MRS and MRT?

Distortion

July 2017


Price distortion: MRS and MRT

Consumption: pjMRSijh = —

pi

For every household marginal

rate of substitution = price ratio

Production:• for commodities 2,3,…,n

pjMRTnj = —

pn

pjMRT3j = —

p3

pjMRT2j = —

p2

pjMRT1j = —

p1

[1+ d]

… … …

• But for commodity 1…

Illustration…

July 2017


x1 0

x2

Consumers

Price distortion: efficiency loss

Production possibilities An efficient allocation Some other inefficient

allocation

How to measure importance of this wedge …

• x

• x*

p*

Producers

At x* producers and consumers face same prices

At x producers and consumers face different prices

Price "wedge" forced by the distortion

July 2017


Waste measurement: a methodTo measure loss we use a reference pointTake this as competitive equilibrium…• …which defines a set of reference prices

Quantify the effect of a notional price change:• Dpi := pi – pi*• This is [actual price of i] – [reference price of i]

Evaluate the equivalent variation for household h :• EVh = Ch(p*,u h) – Ch(p,u h) – [y*h – yh]• This is D(consumer costs) – D(income)

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

July 2017


Basic model

Applications

Overview

Background


Efficiency: Waste

Taking producer prices as constant…

July 2017


x1 0

x2

If producer prices constant… Production possibilities Reference allocation and prices Actual allocation and prices

• x

• x*

p*

Measure cost in terms of good 2

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

Cost of u at prices p

C(p, u)

Cost of u at prices p*

C(p*, u)

Change in valuation of output

u p

DP

L is difference between C(p*, u) C(p, u) and DP

July 2017


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

Use Shephard’s Lemma • xi

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

Take a Taylor expansion to evaluate L:

L is a sum of areas under compensated demand curve July 2017


Basic model

Applications

Overview

Background


Efficiency: Waste

Allow supply-side response…

July 2017


x1 0

x2

Waste measurement: general case Production possibilities Reference allocation and prices Actual allocation and prices

• x*

p*

Measure cost in terms of good 2

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

Cost of u at prices p

C(p, u)

Cost of u at prices p*

C(p*, u)

Change in valuation of output

u p

DP

L is difference between C(p*, u) C(p, u) and DP

• x

July 2017


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

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

July 2017


Basic model

Applications

Overview

Background


Efficiency: Waste

Working out the hidden cost of taxation and monopoly…

July 2017


Application 1: commodity tax Commodity taxes distort prices• Take the model where producer prices are given• Let price of good 1 be forced up by a proportional commodity tax t• Use the standard method to evaluate waste• What is the relationship of tax to waste?

Simplified model:• identical consumers• no cross-price effects • (impact of tax on good 1 does not affect demand for other goods)

Use competitive, non-distorted case as reference

July 2017


A model of a commodity tax

Dp1

compensateddemand curve

p1

p1*

x1h

Dx1h

x1*

revenue raised =

tax x quantity

L

Equilibrium price and quantity The tax raises consumer price… …and reduces demand Gain to the government Loss to the consumer Waste

Waste given by size of triangle Sum over h to get total waste Known as deadweight loss of tax

July 2017


Tax: computation of waste An approximation using Consumer’s Surplus The tax imposed on good 1 forces a price wedge • Dp1 = 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* + Dx1

h and Dx1h < 0

Revenue raised by government from h:• Th = tp1

* x1**= x1

**Dp1 > 0 Absolute size of loss of consumer’s surplus to h is • |DCSh| = ∫ x1

h dp1 ≈ x1** Dp1 − ½ Dx1

hDp1• = Th − ½ t p1

* Dx1

h > Th Use the definition of elasticity• e := p1Dx1

h / x1hDp1< 0

Net loss from tax (for h) is• Lh = |DCSh| − Th = − ½tp1

* Dx1h

• = − ½teDp1x1** = − ½t e Th

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

July 2017


Dp1

compensateddemand curve

p1

p1*

x1h

Dx1h

Size of waste depends upon elasticity

e low: relatively small waste e high: relatively large waste

Redraw previous example

Dp1

p1

p1*

x1h

Dx1h

Dp1

p1

p1*

x1h

Dx1h

July 2017

Dp1

p1

p1*

x1h

Dx1h


Application 1: assessment Waste inversely related to elasticity• Low elasticity: waste is small• High elasticity: waste is large

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

lower elasticities In practice considerations other than waste-minimisation will

also influence tax policy• distributional fairness among households• administrative costs

July 2017


Application 2: monopolyMonopoly power is supposed to be wasteful• but why?

We know that monopolist:• charges price above marginal cost• so equilibrium solution is inefficient

But how inefficient?Take simple version of main model• suppose markets for goods 2, …, n are competitive• good 1 is supplied monopolistically

July 2017


Monopoly: computation of waste (1) Monopoly power in market for good 1 forces a price wedge • Dp1 = 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

* + Dx1h,

• where Dx1h < 0

Same argument as before gives:• loss imposed on household h: −½Dp1Dx1

h > 0• loss overall: − ½Dp1Dx1, where x1 is total output of good 1• using definition of elasticity e, loss equals − ½Dp1

2 e x1*

*/p1

* *

To evaluate this need to examine monopolist’s action…

July 2017


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 / e]

• FOC is therefore p1* * [ 1 + 1 / e] = MC

• hence Dp1= p1* * − MC = − p1

* * / e

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

* * x1

* * / |e|

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

July 2017


Summary Starting point: an “ideal” world• pure private goods• no externalities etc• so CE represents an efficient allocation

Characterise inefficiency in terms of price distortion• in the ideal world MRS = MRT for all h, f and all pairs of goods

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

Extends to more elaborate models • straightforward in principle• but messy maths

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

July 2017

efficiency: waste

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