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## Efficiency: Waste

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### Efficiency: Waste

Almost essential

Welfare and Efficiency

MICROECONOMICS

Principles and Analysis

Frank Cowell

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 them?
- How to quantify departures from them?

Overview…

Efficiency: Waste

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

- But for commodity 1…

[1+ d]

pj

MRT2j = —

p2

pj

MRT3j = —

p3

… … …

Illustration…

pj

MRTnj = —

pn

Price distortion: efficiency loss

- Production possibilities

- An efficient allocation

x2

- Some other inefficient allocation

- 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…

Efficiency: Waste

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 valuation of output

- 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

0

x1

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…

Efficiency: Waste

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 valuation of output

C(p*, u)

- Measure cost in terms of good 2

- x

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

- x*

p*

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

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…

Efficiency: Waste

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

L

Dp1

p1*

x1*

x1h

Dx1h

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* + Dx1h 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| = ∫ x1hdp1 ≈ x1**Dp1−½Dx1hDp1
- = 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

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, where x1 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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