Frank Cowell: Consumer Welfare
CONSUMER: WELFARE
MICROECONOMICSPrinciples and Analysis
Frank Cowell
July 2017 1
Almost essential
Consumer Optimisation
Prerequisites
Frank Cowell: Consumer Welfare
Using consumer theory
▪ Consumer analysis is not just a matter of consumers' reactions to prices
▪ Examine effects of budget changes on consumer's welfare
• changes in incomes
• changes in prices
▪ Useful in the design of economic policy
• tax structure
• subsidies
▪ We can use tools that have become standard in applied microeconomics
• price indices
• cost-benefit analysis
July 2017 2
Frank Cowell: Consumer Welfare
Overview
July 2017 3
Utility and
income
CV and EV
Consumer
welfare
Interpreting the outcome
of the optimisation in
problem in welfare terms
3
Consumer’s
surplus
Frank Cowell: Consumer Welfare
How to measure a person's “welfare”?
▪ We can re-use some concepts
▪ Assume that people know what's best for them
• preferences revealed through behaviour
• preference map can be used as a guide to welfare
▪ Re-examine the concept of “maximised utility”
• use the indirect utility function
• use the cost (expenditure) function
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Frank Cowell: Consumer Welfare
The two aspects of the problem
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x1
x2
x*
▪ Primal: Max utility subject to
the budget constraint
▪ Dual: Min cost subject to a
utility constraint
Interpretation
of Lagrange
multipliersx1
x2
x*
▪ What effect on max-utility of
an increase in budget?V(p, y) C(p,u)
V
▪ What effect on min-cost of
an increase in target utility?
C
Frank Cowell: Consumer Welfare
Interpreting the Lagrange multiplier (1)
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▪ Differentiate with respect to y:
Vy(p, y) = Si Ui(x*) Di
y(p, y)
+ m* [1 – Sipi Diy(p, y) ]
▪ The solution function for the primal:
V(p, y) = U(x*)
= U(x*) + m* [y – Si pixi* ]
Second line follows because,
at the optimum, either the
constraint binds or the
Lagrange multiplier is zero
Make use of the ordinary
demand functions
Optimal value of
Lagrange multiplier
Optimal value of
demands
▪ Rearrange:
Vy(p, y) = Si[Ui(x*)–m*pi]D
iy(p,y)+m*
Vy(p, y) = m*
The Lagrange multiplier in the
primal is just the marginal
utility of money!Vanishes because of FOC
Ui(x*) = m *pi
All sums are
from 1 to n
We find that the same trick can be worked with the solution to the dual…
Frank Cowell: Consumer Welfare
Interpreting the Lagrange multiplier (2)
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▪ Differentiate with respect to u:
Cu(p, u) = SipiHiu(p, u)
– l* [Si Ui(x*) Hi
u(p, u) – 1]
▪ The solution function for the dual:
C(p, u) = Sipi xi*
= Sipi xi* – l* [U(x*) – u]
Once again, at the optimum,
either the constraint binds or
the Lagrange multiplier is zero
Make use of the conditional
demand functions
▪ Rearrange:
Cu(p, u) = Si [pi–l*Ui(x*)] Hi
u(p, u)+l*
Cu(p, u) = l*
Lagrange multiplier in the dual
is the marginal cost of utilityVanishes because of
FOC l*Ui(x*) = pi
Again we have an application of the general envelope theorem
Frank Cowell: Consumer Welfare
A useful connection
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▪ the underlying solution can be
written this way...
y = C(p, u)
Mapping utility into income
▪ the other solution this way.
u = V(p, y)
Mapping income into utility
▪ Putting the two parts together...
y = C(p, V(p, y))
We can get fundamental results
on the person's welfare...
Constraint
utility in
the dual
Maximised utility in
the primal
Minimised budget
in the dual
Constraint
income in
the primal
▪ Differentiate with respect to y:
1 = Cu(p, u) Vy(p, y)1 .
Cu(p, u) = ———Vy(p, y)
A relationship between the
slopes of C and V.
marginal cost (in
terms of utility) of a
dollar of the budget
= l *
marginal cost of
utility in terms of
money = m *
Frank Cowell: Consumer Welfare
Utility and income: summary
▪ This gives us a framework for welfare evaluations:
• impact of marginal changes of income
• interpretation of the Lagrange multipliers
▪ The Lagrange multiplier:
• for primal problem is the marginal utility of income
• for dual problem is the marginal cost of utility
▪ But does this give us all we need?
July 2017 9
Frank Cowell: Consumer Welfare
Utility and income: limitations
▪ This gives us some useful insights but is limited:
1.We have focused only on marginal effects
• infinitesimal income changes
2.We have dealt only with income
• not the effect of changes in prices
▪ We need a general method of characterising impact of budget changes:
• valid for arbitrary price changes
• easily interpretable
▪ For the essence of the problem re-examine the basic diagram
July 2017 10
Frank Cowell: Consumer Welfare
Overview
July 2017 11
Utility and
income
CV and EV
Consumer’s
surplus
Consumer welfare
Exact money
measures of welfare
Frank Cowell: Consumer Welfare
The problem of valuing utility change
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x1
x*
▪Take the consumer's equilibrium
▪Allow a price to fall...
▪Obviously the person is better off
•
u'
x**•
x2
u▪But how much better off?
How do we
quantify this gap?
Frank Cowell: Consumer Welfare
Approaches to valuing utility change
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▪ Three things that are not much use:
1. u' – u
2. u' / u
3. d(u', u)
depends on the units of the U function
depends on the origin of the U
function
Utility
differences
depends on the cardinalisation of the
U function
Utility ratios
some distance
function
▪ A more productive idea:
1. Use income not utility as a measuring rod
2. To do the transformation we use the V function
3. We can do this in (at least) two ways...
Frank Cowell: Consumer Welfare
Story number 1 (CV)
▪Price of good 1 changes
• p: original price vector
• p': vector after price change
▪This causes utility to change
• u = V(p, y)
• u' = V(p', y)
▪Value this utility change in money terms:
• what change in income would bring a person
back to the starting point?
• Define the Compensating Variation:
• u = V(p', y – CV)
▪Amount CV is just sufficient to undo
effect of going from p to p'
14
original utility level restored
at new prices p'
original utility level at prices p
new utility level at prices p'
July 2017
Frank Cowell: Consumer Welfare
The compensating variation
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x1
x**
x*
u
▪ The fall in price of good 1
▪ Reference point is original utility level
Original
prices
new
price
▪ CV measured in terms of good 2
x2
••
CV
Frank Cowell: Consumer Welfare
CV assessment
▪ The CV gives us a clear and interpretable measure of welfare change
▪ It values the change in terms of money (or goods)
▪ But the approach is based on one specific reference point
▪ The assumption that the “right” thing to do is to use the original utility level
▪ There are alternative assumptions we might reasonably make
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Frank Cowell: Consumer Welfare
Story number 2 (EV)
▪Price of good 1 changes
• p: original price vector
• p': vector after price change
▪This causes utility to change
• u = V(p, y)
• u' = V(p', y)
▪Value this utility change in money terms:
• what income change would have been needed
to bring the person to the new utility level?
• Define the Equivalent Variation:
• u' = V(p, y + EV)
▪Amount EV is just sufficient to mimic
effect of going from p to p'
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new utility level reached at
original prices p
original utility level at prices p
new utility level at prices p'
July 2017
Frank Cowell: Consumer Welfare
The equivalent variation
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x1
x**
x*
u'
▪ Price fall is as before
▪ Reference point is the new utility level
Original
prices
new
price
▪ EV measured in terms of good 2
x2
••
EV
Frank Cowell: Consumer Welfare
CV and EV
▪ Both definitions have used the indirect utility function
• But this may not be the most intuitive approach
• So look for another standard tool
▪ As we have seen there is a close relationship between the
functions V and C
▪ So we can reinterpret CV and EV using C
• consumer’s cost (or expenditure) function
• gives a result measured in monetary terms
▪ The result will be a welfare measure
• the change in cost of hitting a welfare level
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remember: cost decreases mean welfare increases.
Frank Cowell: Consumer Welfare
Welfare change as – (cost)
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▪ Equivalent Variation as –(cost):
EV(pp') = C(p, u' ) – C(p', u' )
▪ Compensating Variation as –(cost):
CV(pp') = C(p, u) – C(p', u)
(–) change in cost of hitting
utility level u. If positive we
have a welfare increase
(–) change in cost of hitting
utility level u'. If positive we
have a welfare increase
Prices
after
Prices
before
▪ Using these definitions we also have
CV(p'p) = C(p', u' ) – C(p, u' )
= – EV(pp')
Looking at welfare change in
the reverse direction, starting
at p' and moving to p
Reference
utility level
Frank Cowell: Consumer Welfare
Welfare measures applied...
▪The concepts we have developed are regularly put to work in practice.
▪Applied to issues such as:
• Consumer welfare indices
• Price indices
• Cost-Benefit Analysis
▪Often this is done using some (acceptable?) approximations
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Example of
cost-of-living
index
Frank Cowell: Consumer Welfare
Cost-of-living indices
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▪ An approximation:Si p'i xiIL = ———Si pi xi
ICV .
▪ An index based on CV:
C(p', u)ICV = ———
C(p, u)
What's the proportionate change in cost
of hitting the base welfare level u?
What's the proportionate change in cost
of buying base consumption bundle x?
This is the Laspeyres index (the basis for
the Consumer Price Index)
= C(p, u)
All summations
are from 1 to n.
▪ An index based on EV:
C(p', u')IEV = ————
C(p, u')
▪ An approximation:Si p'i x'iIP = ———Si pi x'i
IEV .
C(p', u)
What's the proportionate change in cost
of hitting the new welfare level u' ?
= C(p', u')
C(p, u')What's the proportionate change in cost
of buying the new consumption bundle x'?
This is the Paasche index
Frank Cowell: Consumer Welfare
Overview
July 2017 23
Utility and
income
CV and EV
Consumer’s
surplus
Consumer welfare
A simple, practical
approach?
Frank Cowell: Consumer Welfare
Another (equivalent) form for CV
July 2017 24
▪ Assume that the price of good 1
changes from p1 to p1' while other
prices remain unchanged. Then we
can rewrite the above as:
CV(pp') = C1(p, u) dp1
▪ Use the cost-difference definition:
CV(pp') = C(p, u) – C(p', u)(–) change in cost of hitting
utility level u. If positive we
have a welfare increase
Using the definition of a definite
integral
Prices
after
Prices
before
▪ Further rewrite as:
CV(pp') = H1(p, u) dp1 Using Shephard’s lemma again
Hicksian (compensated)
demand for good 1
Reference
utility level
▪ So CV can be seen as an area under
the compensated demand curve
p1
p1'
p1
p1'
Frank Cowell: Consumer Welfare
Compensated demand and the value of a price
fall
July 2017 25
Compensating
Variation
compensated (Hicksian)
demand curve
pri
ce
fall
x1
p1
H1(p, u)
*x1
▪ The initial equilibrium
▪ price fall: (welfare increase)
▪ value of price fall, relative to
original utility level
initial price
level
original
utility level
▪The CV provides an exact
welfare measure
▪ But it’s not the only approach
Frank Cowell: Consumer Welfare
Compensated demand and the value of a price
fall (2)
July 2017 26
x1
Equivalent
Variation
pri
ce
fall
x1
p1
**
H1(p, u )
compensated
(Hicksian)
demand curve
▪ As before but use new utility
level as a reference point
▪ price fall: (welfare increase)
▪ value of price fall, relative
to new utility level
new
utility level
▪The EV provides another exact
welfare measure.
▪ But based on a different
reference point
▪Other possibilities…
Frank Cowell: Consumer Welfare
Ordinary demand and the value of a price fall
July 2017 27
x1
pri
ce
fall
x1
p1
***x1
D1(p, y)
▪The initial equilibrium
▪Price fall: (welfare increase)
▪An alternative method of
valuing the price fall?
Consumer's
surplus
ordinary (Marshallian)
demand curve
▪CS provides an approximate
welfare measure
Frank Cowell: Consumer Welfare
Three ways of measuring the benefits of a
price fall
July 2017 28
x1
pri
ce
fall
x1
p1
**
H1(p,u )
*x1
H1(p,u)
D1(p, y)
▪Summary of the three approaches
▪Illustrated for normal goods
▪For normal goods: CV CS EV
CS EVCV CS ▪For inferior goods: CV > CS > EV
Frank Cowell: Consumer Welfare
Summary: key concepts
▪ Interpretation of Lagrange multiplier
▪ Compensating variation
▪ Equivalent variation
• CV and EV are measured in monetary units.
• In all cases: CV(pp') = – EV(p'p)
▪ Consumer’s surplus
• The CS is a convenient approximation
• For normal goods: CV CS EV
• For inferior goods: CV > CS > EV
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