Efficiency: Waste
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Transcript of Efficiency: Waste
Frank Cowell: Efficiency-Waste
EFFICIENCY: WASTEMICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Welfare and Efficiency
Prerequisites
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Frank Cowell: Efficiency-Waste
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?
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Frank Cowell: Efficiency-Waste
Basic model
Applications
Overview
Background
Model with production
Efficiency: Waste
How to evaluate inefficient states
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
= 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
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Frank Cowell: Efficiency-Waste
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…
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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…
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Frank Cowell: Efficiency-Waste
Basic model
Applications
Overview
Background
Model with production
Efficiency: Waste
Taking producer prices as constant…
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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
Frank Cowell: Efficiency-Waste
Basic model
Applications
Overview
Background
Model with production
Efficiency: Waste
Allow supply-side response…
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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:
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Frank Cowell: Efficiency-Waste
Basic model
Applications
Overview
Background
Model with production
Efficiency: Waste
Working out the hidden cost of taxation and monopoly…
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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
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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
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Frank Cowell: Efficiency-Waste
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
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Dp1
p1
p1*
x1h
Dx1h
Frank Cowell: Efficiency-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
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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…
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Frank Cowell: Efficiency-Waste
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
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Frank Cowell: Efficiency-Waste
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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