Prerequisites
Almost essential
Welfare and Efficiency
EFFICIENCY: WASTE
MICROECONOMICS
Principles and Analysis
Frank Cowell
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 them?
 How to quantify departures from them?
Frank Cowell: Efficiency-Waste
Overview…
Efficiency: Waste
Background
How to evaluate
inefficient states
Basic model
Model with
production
Applications
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
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
Frank Cowell: Efficiency-Waste
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
What are the implications
~
for MRS and MRT?
Distortion
p1 = p1 [1+d]
~
p2 = p2
~
p3 = p3
consumer
prices
… = …
~
pn = p n
Frank Cowell: Efficiency-Waste
firms'
prices
Price distortion: MRS and MRT
For every household
marginal rate of
substitution = price ratio
 Consumption:
MRSij
 Production:
• for commodities 2,3,…,n
• But for commodity 1…
MRT1j
MRT2j
h
pj
= —
pi
pj
= — [1+ d]
p1
pj
= —
p2
pj
MRT3j = —
p3
… … …
pj
MRTnj = —
pn
Frank Cowell: Efficiency-Waste
Illustration…
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
•x*
Producers
 Price "wedge" forced by the
distortion
p*
Consumers
0
x1
How to measure
importance of this
wedge …
Frank Cowell: Efficiency-Waste
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…
Frank Cowell: Efficiency-Waste
Overview…
Efficiency: Waste
Background
Taking producer
prices as constant…
Basic model
Model with
production
Applications
Frank Cowell: Efficiency-Waste
If producer prices constant…
C(p, u)
Production possibilities
Reference allocation and prices
Actual allocation and prices
Cost of u at prices p
Cost of u at prices p*
x2

DP
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*
p
0
 L is difference between
C(p*, u)  C(p, u) and DP
p*
u
x1
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
• xih = Hhi(p,uh) = Cih(p,uh)
 Take a Taylor expansion to evaluate L:
 L is a sum of areas under compensated demand curve
Frank Cowell: Efficiency-Waste
Overview…
Efficiency: Waste
Background
Allow supply-side
response…
Basic model
Model with
production
Applications
Frank Cowell: Efficiency-Waste
Waste measurement: general case
C(p, u)
Production possibilities
x2
Reference allocation and prices
Actual allocation and prices
Cost of u at prices p
Cost of u at prices p*

DP
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
p*
u
0
 L is difference between
C(p*, u)  C(p, u) and DP
x1
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:
Frank Cowell: Efficiency-Waste
Overview…
Efficiency: Waste
Background
Working out the
hidden cost of
taxation and
monopoly…
Basic model
Model with
production
Applications
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:
Frank Cowell: Efficiency-Waste
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
L
Dp1
Sum over h to get total waste
Known as deadweight loss of tax
p1*
x1 *
Dx1h
x1 h
Frank Cowell: Efficiency-Waste
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| = ∫ x1h dp1 ≈ x1** Dp1 − ½ Dx1hDp1
•
= 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
Frank Cowell: Efficiency-Waste
Size of waste depends upon elasticity
p1
p1
Redraw previous example
compensated
demand curve
e low: relatively small waste
e high: relatively large waste
Dp1
p1*
x1h
Dpp 1
Dx1h
p1
1
p1 *
Dp1
Dp1
p1*
p1*
x1h
Dx1h
Dx1h
x1 h
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
Frank Cowell: Efficiency-Waste
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
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: −½Dp1Dx1h > 0
• loss overall: − ½Dp1Dx1, where x1 is total output of good 1
• using definition of elasticity e, loss equals − ½Dp12 e x1* */p1* *
 To evaluate this need to examine monopolist’s action…
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
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
Frank Cowell: Efficiency-Waste