Prerequisites
Almost essential
Welfare and Efficiency
MICROECONOMICS
Principles and Analysis
Frank Cowell
Frank Cowell: Efficiency-Waste July 2015

 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?
Frank Cowell: Efficiency-Waste July 2015

How to evaluate inefficient states
Efficiency: Waste
Background
Basic model
Model with production
Applications
Frank Cowell: Efficiency-Waste July 2015

 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 July 2015

 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 July 2015


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?
p
1 p p
2
3
~
= p
1
~
= p
2
~
= p
3 consumer prices
…
=
… p n
~
= p n
[1
+d]
Distortion firms' prices
Frank Cowell: Efficiency-Waste July 2015


Consumption:

Production:
• for commodities 2,3,…, n
• But for commodity 1…
July 2015
For every household marginal rate of substitution = price ratio
ij h p
p i j
MRT
MRT
1 j
2 j p j
= — p
1 p j
= — p
2
MRT
3 j p j
= — p
3
… … …
MRT n j p j
= — p n
Frank Cowell: Efficiency-Waste
[1+ d
]
Illustration …

x
2
• x
• x* Producers

Production possibilities

An efficient allocation
 Some other inefficient allocation
 At x * producers and consumers face same prices
 At x producers and consumers face different prices
 Price "wedge" forced by the distortion p*
0
July 2015
Consumers
How to measure importance of this wedge … x
1
Frank Cowell: Efficiency-Waste

 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:
•
D p i
:= p i
– p i
*
• This is [actual price of i ] – [reference price of i ]
 Evaluate the equivalent variation for household h :
•
• EV h = C h (p*, u h ) – C h (p, u h ) – [ y* h –
This is
D
(consumer costs) –
D
(income) y h ]
 Aggregate over agents to get a measure of loss,
L
• We do this for two cases…
Frank Cowell: Efficiency-Waste July 2015

Taking producer prices as constant…
Efficiency: Waste
Background
Basic model
Model with production
Applications
Frank Cowell: Efficiency-Waste July 2015

C ( p , u
)
 x
2
DP
 Production possibilities
 Reference allocation and prices

Actual allocation and prices
 Cost of u at prices p
 Cost of u at prices p*
 Change in valuation of output
C ( p* , u
)

• x
July 2015
0
• x* p p*
 Measure cost in terms of good 2
 Losses to consumers are
C ( p* , u
)

C ( p , u
)

L is difference between
C ( p* , u
)

C ( p , u
) and DP u x
1
Frank Cowell: Efficiency-Waste

 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
• x i h = H hi ( p , u h ) = C i h ( p , u h )
Take a Taylor expansion to evaluate
L
:

L is a sum of areas under compensated demand curve
Frank Cowell: Efficiency-Waste July 2015

Allow supply-side response…
Efficiency: Waste
Background
Basic model
Model with production
Applications
Frank Cowell: Efficiency-Waste July 2015

C ( p , u
)
 x
2
DP
 Production possibilities
 Reference allocation and prices

Actual allocation and prices
 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
• x* p p* u
 Losses to consumers are
C ( p* , u
)

C ( p , u
)
 L is difference between
C ( p* , u
)

C ( p , u
) and DP
July 2015
0 x
1
Frank Cowell: Efficiency-Waste

 Adapt the
L formula to allow for supply responses
 Then waste is given by:

• where q i
(∙) is net supply function for commodity i
Again use Shephard’s Lemma and a Taylor expansion:
Frank Cowell: Efficiency-Waste July 2015

July 2015
Working out the hidden cost of taxation and monopoly…
Efficiency: Waste
Background
Basic model
Model with production
Applications
Frank Cowell: Efficiency-Waste

 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 July 2015

p
1 revenue raised = tax x quantity
D p
1 p
1
* compensated demand curve
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
D x
1 h x
1
* x
1 h
Frank Cowell: Efficiency-Waste July 2015



An approximation using Consumer’s Surplus






The tax imposed on good 1 forces a price wedge
•
D p
1 is the untaxed price of the good h’ s demand for good 1 is lower with the tax:
•
•
= tp
1
* > 0 where is p
1
* x
1
** rather than x where x
1
** = x
1
*
1
*
+ D x
1 h and
D x
1 h < 0
Revenue raised by government from h :
• x
1
** = x
Absolute size of loss of consumer’s surplus to h is
•
•
T h = tp
1
*
1
**
D p
1
|D
CS h
| = ∫ x
1 h d p
1
≈ x
= T h − ½ t p
1
**
1
*
> 0
D p
D x
1
1 h
− ½ D x
> T h
1 h
D p
1
Use the definition of elasticity
• e
:= p
1
D x
1 h / x
1 h
D p
1
< 0
Net loss from tax (for h ) is
•
•
L h =
|D
CS h
| − T h = − ½ tp
= − ½ t eD p
1 x
1
** = − ½
1
* t e
D x
T h
1 h
Overall net loss from tax (for h ) is
•
• ½
| e| tT uses the assumption that all consumers are identical
July 2015 Frank Cowell: Efficiency-Waste

p
1 p
1 compensated demand curve
 Redraw previous example
 e low: relatively small waste
 e high: relatively large waste
D p
1 p
1
* x
1 h
D p p
1
1 p
1
*
D x
1 h
D p
1 p
1
* p
1 x
1 h
D p
1 p
1
*
D x
1 h
D x
1 h x
1 h x
1 h
Frank Cowell: Efficiency-Waste July 2015
D x
1 h

 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 July 2015

 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 July 2015



Monopoly power in market for good 1 forces a price wedge
•
•
•
D p
1 p
1
** p
1
*
= p
1
* * − p
1
*
> 0 where is price charged in market is marginal cost (MC) h’ s demand for good 1 is lower under this monopoly price:
•
• x
1
**
= x
1 where
D x
*
1 h
+ D x
< 0
1 h ,


Same argument as before gives:
•
•
• loss imposed on household h : loss overall:
− ½ D p
1
− ½ D p
1
D x
1 h
D x
1
, where x
1
> 0 is total output of good 1 using definition of elasticity e
, loss equals
− ½ D p
1
2 e x
1
* *
/ p
1
* *
To evaluate this need to examine monopolist’s action…
Frank Cowell: Efficiency-Waste July 2015

 Monopolist chooses overall output
• use first-order condition
• MR = MC:
 Evaluate MR in terms of price and elasticity:
•
•
• p
1
* * [ 1 + 1 / e
]
FOC is therefore p
1
* * hence
D p
1
= p
1
* * −
[ 1 + 1 /
MC = − p e
] = MC
1
* * / e
 Substitute into triangle formula to evaluate measurement of loss:
• ½ p
1
* * x
1
* * / | 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 July 2015

 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 July 2015