consumer: welfare - london school of...

Frank Cowell: Consumer Welfare

CONSUMER: WELFARE

MICROECONOMICSPrinciples and Analysis

Frank Cowell

July 2017 1

Almost essential

Consumer Optimisation

Prerequisites


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


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


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

July 2017 4


The two aspects of the problem

July 2017 5

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


Interpreting the Lagrange multiplier (1)

July 2017 6

▪ 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…


Interpreting the Lagrange multiplier (2)

July 2017 7

▪ 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


A useful connection

July 2017 8

▪ 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 *


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


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


Overview

July 2017 11

Utility and

income

CV and EV

Consumer’s

surplus

Consumer welfare

Exact money

measures of welfare


The problem of valuing utility change

July 2017 12

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?


Approaches to valuing utility change

July 2017 13

▪ 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...


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


The compensating variation

July 2017 15

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


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

July 2017 16


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'

17

new utility level reached at

original prices p

original utility level at prices p

new utility level at prices p'

July 2017


The equivalent variation

July 2017 18

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


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

July 2017 19

remember: cost decreases mean welfare increases.


Welfare change as – (cost)

July 2017 20

▪ 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


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


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

July 2017 21

Example of

cost-of-living

index


Cost-of-living indices

July 2017 22

▪ 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?


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)


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


Overview

July 2017 23

Utility and

income

CV and EV

Consumer’s

surplus

Consumer welfare

A simple, practical

approach?


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


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'


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


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…


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


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


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

July 2017 29

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