Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Welfare and Efficiency
Frank Cowell: Microeconomics
November 2006
Efficiency: Waste
MICROECONOMICS
Principles and Analysis
Frank Cowell
Agenda
Frank Cowell: Microeconomics
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Build on the efficiency presentation
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Start from the “standard” efficiency rules
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Focus on relation between competition and efficiency
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?
Efficiency: Waste
Overview...
Frank Cowell: Microeconomics
Background
How to evaluate
inefficient states
Basic model
Model with
production
Applications
The approach
Frank Cowell: Microeconomics
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Use standard general equilibrium analysis
to...
Model price distortion
 Define reference set of prices
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Use consumer welfare analysis to…
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Model utility loss
Use standard analysis of household budgets
to…
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Model change in profits and rents
A reference point
Frank Cowell: Microeconomics
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Address the question: how much waste?
Need a reference point
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Any efficient point would do
But it is usual to take a CE allocation
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where there is zero waste
quantify departures from this point
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
A model of price distortion
Frank Cowell: Microeconomics
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Assume there is a competitive equilibrium
If so, then everyone pays the same prices
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But now we have a distortion
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Distortion
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What are the implications
for MRS and MRT?
p1
p2
p3
consumer
prices
...
pn
~
= p1 [1+d]
~
= p2
~
= p3
= ...
~
= pn
firms'
prices
Price distortion: MRS and MRT
For every household marginal rate of
substitution = price ratio
Frank Cowell: Microeconomics

Consumption:

Production:
 for commodities 2,3,...,n

MRSij
MRT1j
But for commodity 1...
MRT2j
h
pj
= —
pi
pj
= — [1+ d]
p1
pj
= —
p2
pj
MRT3j = —
p3
... ... ...
pj
MRTnj = —
pn
Illustration....
Price distortion: efficiency loss
Frank Cowell: Microeconomics
Production possibilities
An efficient allocation
Some other inefficient allocation
x2
 At x* producers and
consumers face same prices.

 At x producers and
x

x*
consumers face different
prices.
Producers
 Price "wedge" forced by the
distortion.
How to measure
importance of this
wedge ....
p*
Consumers
0
x1
Waste measurement: a method
Frank Cowell: Microeconomics
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To measure loss we use a reference point
Take this as competitive equilibrium...
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Quantify the effect of a notional price change:
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Dpi := pi – pi*
This is [actual price of i] – [reference price of i]
Evaluate the equivalent variation for household h :
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...which defines a set of reference prices
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…
Efficiency: Waste
Overview...
Frank Cowell: Microeconomics
Background
Taking producer
prices as
constant…
Basic model
Model with
production
Applications
If producer prices constant…
Frank Cowell: Microeconomics
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 value of output at
consumer prices
 Measure cost in terms of


C(p*,u)
good 2.
x
 Losses to consumers are
C(p*,u)  C(p,u)

x*
p
0
p*
 L is difference between
C(p*,u)  C(p,u) and DP
u
x1
Model with fixed producer prices
Frank Cowell: Microeconomics
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Waste L involves both demand and supply responses.
Simplify by taking case where production prices constant.
Then waste is given by:
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Use Shephard’s Lemma
h
hi
h
h
h
 xi = H (p,u ) = Ci (p,u )
Take a Taylor expansion to evaluate L:
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L is a sum of areas under compensated demand curve.

Efficiency: Waste
Overview...
Frank Cowell: Microeconomics
Background
Allow supply-side
response…
Basic model
Model with
production
Applications
Waste measurement: general case
Frank Cowell: Microeconomics
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 value of output at
consumer prices

 Measure cost in terms of
C(p*,u)

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
0
x1
Model with producer price response
Frank Cowell: Microeconomics
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Adapt the L formula to allow for supply responses.
Then waste is given by:
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where qi (∙) is net supply function for commodity i
Again use Shephard’s Lemma and a Taylor expansion:
Efficiency: Waste
Overview...
Frank Cowell: Microeconomics
Background
Working out the
hidden cost of
taxation and
monopoly…
Basic model
Model with
production
Applications
Application 1: commodity tax
Frank Cowell: Microeconomics
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Commodity taxes distort prices
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Simplified model:
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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?
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
Frank Cowell: Microeconomics
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
L
Dp1
size of triangle
Sum over h to get total
waste
Commonly known as
deadweight loss of tax.
p1*
x1 *
Dx1h
x1 h
Tax: computation of waste
Frank Cowell: Microeconomics
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The tax imposed on good 1 forces a price wedge
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h’s demand for good 1 is lower with the tax:
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e := p1Dx1h / x1hDp1< 0
Net loss from tax (for h) is
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DCSh = ∫ x1h dp1 ≈ x1** Dp1 − ½ Dx1hDp1
= Th + ½Dx1hDp1= Th − ½ t p1* Dx1h > Th
Use the definition of elasticity
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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
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Dp1 = tp1* > 0 where is p1* is the untaxed price of the good
Lh = DCSh − Th = − ½tp1* Dx1h
= − ½teDp1x1** = − ½t e Th
Overall net loss from tax (for h) is
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½ |e| tT
uses the assumption that all consumers are identical
Size of waste depends upon elasticity
Frank Cowell: Microeconomics
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
Application 1: assessment
Frank Cowell: Microeconomics
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Waste inversely related to elasticity
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Suggests a policy rule
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Low elasticity: waste is small
High elasticity: waste is large
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
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distributional fairness among households
administrative costs
Application 2: monopoly
Frank Cowell: Microeconomics
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Monopoly power is supposed to be wasteful…
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We know that monopolist…
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but why?
charges price above marginal cost
so it is inefficient …
…but how inefficient?
Take simple version of main model
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suppose markets for goods 2, …, n are competitive
good 1 is supplied monopolistically
Monopoly: computation of waste (1)
Frank Cowell: Microeconomics
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Monopoly power in market for good 1 forces a price wedge
* * − p * > 0 where
 Dp1 = p1
1
**
 p1 is price charged in market
 p1* is marginal cost (MC)
h’s demand for good 1 is lower under this monopoly price:
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x1** = x1* + Dx1h,
where Dx1h < 0
Same argument as before gives:
h
 loss imposed on household h: −½Dp1Dx1 > 0
 loss overall: − ½Dp1Dx1, where x1 is total output of good 1
2
* */p * *
 using definition of elasticity e, loss equals − ½Dp1 e x1
1
To evaluate this need to examine monopolist’s action…
Monopoly: computation of waste (2)
Frank Cowell: Microeconomics
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Monopolist chooses overall output
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Evaluate MR in terms of price and elasticity:
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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:
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use first-order condition
MR = MC:
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
Frank Cowell: Microeconomics
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Starting point: an “ideal” world
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Characterise inefficiency in terms of price distortion
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fine for individual
OK just to add up?
Extends to more elaborate models
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in the ideal world MRS = MRT for all h, f and all pairs of goods
Measure waste in terms of income loss
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pure private goods
no externalities etc
so CE represents an efficient allocation
straightforward in principle
but messy maths
Applications focus on simple practicalities
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elasticities measuring consumers’ price response
but simple formulas conceal strong assumptions