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### Efficiency: Waste

Almost essential

Welfare and Efficiency

MICROECONOMICS

Principles and Analysis

Frank Cowell

November 2006

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 them?

How to quantify departures from them?

AgendaOverview...

Background

How to evaluate inefficient states

Basic model

Model with production

Applications

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

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

= p1

~

= p2

~

= p3

consumer

prices

firms' prices

~

= pn

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

- But now we have a distortion

Distortion

- What are the implications for MRS and MRT?

p1

[1+d]

p2

p3

...

= ...

pn

Price distortion: MRS and MRT

For every household marginal rate of substitution = price ratio

pj

MRSijh= —

pi

- Consumption:

- Production:
- for commodities 2,3,...,n

pj

MRT1j = —

p1

[1+ d]

- But for commodity 1...

pj

MRT2j = —

p2

pj

MRT3j = —

p3

... ... ...

Illustration....

pj

MRTnj = —

pn

Price distortion: efficiency loss

- Production possibilities

- An efficient allocation

- Some other inefficient allocation

x2

- At x* producers and consumers face same prices.

- At x producers and consumers face different prices.

- x

Producers

- x*

- Price "wedge" forced by the distortion.

How to measure importance of this wedge ....

p*

Consumers

x1

0

Waste measurement: a method

- To measure loss we use a reference point
- Take 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…

Overview...

Background

Taking producer prices as constant…

Basic model

Model with production

Applications

If producer prices constant…

- Production possibilities

C(p, u)

x2

- Reference allocation and prices

- Actual allocation and prices

DP

- Cost of u at prices p.

- Cost of u at prices p*.

- Change in value of output at consumer prices

- Measure cost in terms of good 2.

- x

C(p*,u)

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

- x*

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

p

p*

u

x1

0

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
- xih = Hhi(p,uh) = Cih(p,uh)

- Take a Taylor expansion to evaluate L:
- L is a sum of areas under compensated demand curve.

Overview...

Background

Allow supply-side response…

Basic model

Model with production

Applications

Waste measurement: general case

- Production possibilities

C(p, u)

x2

- Reference allocation and prices

- Actual allocation and prices

DP

- Cost of u at prices p.

- Cost of u at prices p*.

- Change in value of output at consumer prices

- Measure cost in terms of good 2.

C(p*,u)

- x

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

- x*

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

p*

p

u

x1

0

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:

Overview...

Background

Working out the hidden cost of taxation and monopoly…

Basic model

Model with production

Applications

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:

A model of a commodity tax

p1

- Equilibrium price and quantity

- The tax raises consumer price...

compensated

demand curve

- ...and reduces demand

- Gain to the government

- Loss to the consumer

- Waste

revenue raised =

tax x quantity

- Waste measured by size of triangle
- Sum over h to get total waste
- Commonly known as deadweight loss of tax.

L

Dp1

p1*

x1*

x1h

Dx1h

Tax: computation of waste

- 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* + Dx1h and Dx1h < 0

- Revenue raised by government from h:
- Th = tp1*x1**

- Loss of consumer’s surplus to h is
- DCSh = ∫ x1hdp1 ≈ x1**Dp1−½Dx1hDp1
- = Th + ½Dx1hDp1= Th−½ t p1* Dx1h > Th

- Use the definition of elasticity
- e := p1Dx1h / 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

p1

p1

compensated

demand curve

Dp1

p1*

x1h

Dx1h

Dp1

p1

p1

p1*

Dp1

Dp1

p1*

p1*

x1h

Dx1h

x1h

x1h

Dx1h

Dx1h

Size of waste depends upon elasticity- Redraw previous example

- e low: relatively small waste

- e high: relatively large waste

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

Application 2: monopoly

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

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

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

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:−½Dp1Dx1h > 0
- loss overall:− ½Dp1Dx1, wherex1 is total output of good 1
- using definition of elasticity e, loss equals −½Dp12e x1**/p1**

- To evaluate this need to examine monopolist’s action…

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

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

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