Advanced Microeconomics - coin.wne.uw.edu.plcoin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/2012_micro... · Introduction Welfare evaluation Aggregate demand Advanced Microeconomics

IntroductionWelfare evaluationAggregate demand

Advanced Microeconomics

Welfare measures and aggregation

Jan Hagemejer

October 30, 2012

Jan Hagemejer Advanced Microeconomics


Introduction

Introduction

The plan:

1 Welfare measures

2 Aggregate demand



The problemEV and CVEV vs. CV

The problem

Example:

1 Our consumer has initial wealth w and is facing the initial set ofmarket prices p0.

2 Now he is faced with another set of market prices p1.

3 How to evaluate changes in consumer's situation?

4 Combinations of prices and incomes will be called projects.

5 For now - let us consider projects with only prices changing (income�xed).




The (likely) solution

For both projects (p0,w) and (p1,w) we could compute the indirectutility function if we knew the form of v : v(p0,w) and v(p1,w)

We could then compare the levels of utility and compare thesituations.

But maybe we do not know much about utility and prefer to onlytalk about money?

Let us take an arbitrary price vectorp̄ - a reference/base pricevector.

We can convert the the problem to monetary terms by computingexpenditure function:e(p̄, v(p0,w)) and e(p̄, v(p1,w)).

This, in fact, is a money-metric utility function:e(p̄, v(p1,w))− e(p̄, v(p0,w)) - strictly increasing in v .

Let us take either p1 or p0 to be our reference price vector.




EV and CV

The welfare measure will be either:

Equivalent Variation (how much would I have to pay at old prices toattain new level of utility)?

EV (p0, p1,w) = e(p0, v(p1,w))− e(p0, v(p0,w))

= e(p0, v(p1,w))− w = p0 · h(p0, v(p1,w))− w

or Compensating Variation (how much would I have to pay at newprices to attain old level of utility) or how much compensation Ineed to attain the old level of welfare.

CV (p0, p1,w) = e(p1, v(p1,w))− e(p1, v(p0,w))

= w − e(p1, v(p0,w)) = w − p1 · h(p1, v(p0,w))




EV and CV

Or:

How much the consumer has to be paid to be exactly as well-o�with old prices as in the new situation (a transfer that is equivalent interms of welfare to the price change).

v(p0,w + EV ) = v(p1,w)

How much the consumer has to be paid to be exactly as well-o�with new prices as in the old situation(the net revenue of a plannerwho must compensate the consumer for the price change).

v(p0,w) = v(p1,w − CV )




EV and CV




Demand functions and EV and CV

Example:

Only one price changes (good 1): p016= p1

1and p0l = p1l = p̄l ∀l 6=1.

We know that: w = e(p0, u0) = e(p1, u1) and h1(p, u) = ∂e(p,u)∂p1

, so:

EV (p0, p1,w) = e(p0, u1)−w = e(p0, u1)−e(p1, u1) =

ˆ p01

p11

h1(p1, p̄−1, u1)dp1

and similarly:

CV (p0, p1,w) =

ˆ p01

p11

h1(p1, p̄−1, u0)dp1




EV, CV and welfare evaluation

They will di�er

They will provide the correct ranking

They will be di�erent from so-called consumer surplus (CS)

CS(p0, p1,w) =

ˆ p01

p11

x1(p1, p̄−1,w)dp1

EV>CS>CV for normal goods. Opposite ordering for inferior goods(income e�ects are negative and in that case Hicksian demands are�atter than Walrasian demand).




Demand functions and EV and CV and CS




A complication

What if 3 projects are to be compared: eg: 1,2 with 0?

Can we use EV?EV (p0, p1,w) = e(p0, u1)− w and EV (p0, p2,w) = e(p0, u2)− wso basically, we will have 2 � 1 if e(p0, u2) > e(p0, u1).EV is OK

Can we use CV?

EV uses the new prices as base CV (p0, p1,w) = w − e(p1, u0) andCV (p0, p2,w) = w − e(p2, u0).So: CV (p0, p1,w)− CV (p0, p2,w) = e(p2, u0)− e(p1, u0)We cannot use CV, due to di�erent price vectors in the expenditurefunction. If prices are di�erent, higher expenditure does not have tomean higher utility.

EV is a money-metric utility function




Consumer surplus

is given by:

CS(p0, p1,w) =L∑

l=1

ˆ p1l

p0l

−xl(p11 , p12 , . . . , p1l−1, τ, p0l+1. . . , p0L,w)dτ

If more than one price changes, the problem may be path dependent (thesequence of integration may matter) - not if preferences are homothethic.




When income is also changing

EV:

EV ((p0,w0), (p1,w1)) = e(p0, v(p1,w1))−w0 = p0·h(p0, v(p1,w1))−w0

CV:

CV ((p0,w0), (p1,w1)) = w1−e(p1, v(p0,w0)) = w1−p1·h(p1, v(p0,w0))

CS

CS((p0,w0), (p1,w1)) =

ˆ w1

w0

dτ

+L∑

l=1

ˆ p1l

p0l

−xl(p11 , p12 , . . . , p1l−1, τ, p0l+1. . . , p0L,w)dτ




Under what conditions use CS?

From Chipman and Moore, 1976. Consider a triple of projects, a baseand two new projects (note that income di�ers in those projects, not onlyprices).

1 Even if preferences are homothetic, CS((p0,w0), (p1,w1)) > 0 doesnot guarantee that (p1,w1) is better than (p0,w1).

2 Fix w0. Consumer surplus correctly ranks the projects for everytriple of projects such that (p,w) : w = w0 if and only if consumerpreferences are homothetic (rescale the prices with income prior tocalculating CS).

3 Fix p0i . Consumer surplus correctly ranks the projects for every tripleof projects such that (p,w) : pi = p0i if and only if consumerpreferences are quasi-linear with respect to commodity i .




Example of problems with CS

(from W. Novshek):

Consider a consumer with demand functions x1 = w2p1

and x2 = w2p2

.

The corresponding indirect utility is: v = w2

4p1p2(not that this is

Cobb-Douglas with α′s = 1/2). Show that using CS with projects(p0

1, p0

2,w0) : (1, 1, 1) and (p1

1, p1

2,w1) : (e, e, 2.2) will lead to a

ranking opposite to that stemming directly from utility.

Correct ranking: v0 = 12

4·1·1 = 1

4> v1 = 2.22

4·e2 = 0.16376, so project 0is in fact better than project 1.

CS: CS =´ p11p01

(− w0

2p1)dp1 +

´ p12p02

(− w0

2p2)dp2 +

´ w1

w0 dτ =

= −(w0

2ln p1

1− w0

2ln p0

1+ w0

2ln p1

2− w0

2ln p0

2) + w1 − w0

= −w0

2ln(

p11p12

p01p02

) + w1 − w0 = − 1

2ln(

p11p12

p01p02

) + w1 − w0 =

− 1

2ln( e2

1) + 2.2− 1 = .2 > 0, so according to CS, project 1 is better

than 0.




Example continued

The function is homothethic, so we could get rid of the incomechange by normalizing prices by income.

In that case, the equivalent projects would be (p01, p0

2,w0) : (1, 1, 1)

and (p11, p1

2,w1) : (e/2.2, e/2.2, 1)

CS =´ p11p01

(− w0

2p1)dp1 +

´ p12p02

(− w0

2p2)dp2 =

= −w0

2ln(

p11p12

p01p02

) = − 1

2ln( e2

2.22 ) ≈ −0.21 < 0.

So after normalizing prices with income, CV provides the correctranking.




A special case (UMP)

Quasi-linear preferences:

u(x) = x0 + φ(x1, . . . , xL)

L = x0 + φ(x1, . . . , xL) + λ(L∑

L=0

plxl − w)

FOC:

1 = λp0

φ′(xl) = λpl

so:φ′(xl) = pl/p0 → xl = (φ′)−1(pl/p0). Walrasian demand DOES NOTdepend on wealth.




A special case (EMP)

L =L∑

L=0

plxl − λ(x0 + φ(x1, . . . , xL)− u)

FOC:

λ = p0

λφ′(xl) = pl

so:φ′(xl) = pl/p0 → xl = (φ′)−1(pl/p0) = hl(p, u). Hicksian demand isTHE SAME as Walrasian demand.



De�nitionsWealth e�ectsThe Gorman form

The problem

Can we use the techniques from previous classes to derive aggregatedemand?

Can we aggregate demands of individual consumers?

Can we only look at aggregate demand ignoring the underlyingconsumer optimization?




De�nitions

De�ne the distribution rule

w1(p,w), . . . ,wI (p,w)

that for every level of aggregate wealth w ∈ R assigns individual wealthsto all consumers 1, . . . , I .

We assume that: ∑i

wi (p,w) = w ∀p,w

and that wi (·, ·) is continuous and homogeneous of degree 1.

Aggregate demand function:

x(p,w) =∑i

xi (p,wi (p,w))

is just a sum of Walrasian demands as described in previous sections(continuous, homogeneous of degree zero, Walras law).




Wealth e�ects

Take x(p,w) =∑

i xi (p,wi (p,w)) and assume that∑

i dwi = 0.

Question: under what conditions the aggregate demand will beirrespective of the distribution of income?

Lets take a derivative:

∂x(p,w)/∂wi =xi (p,wi (p,w))

∂dwi

∂x(p,w)

∂w

∂w

∂widwi =

xi (p,wi (p,w))

∂dwidwi

∂x(p,w)

∂w

∑i

dwi =∑i

xi (p,wi (p,w))

∂dwidwi

0 =∑i

xi (p,wi (p,w))

∂dwidwi

The wealth e�ects have to cancel-out.Jan Hagemejer Advanced Microeconomics



Wealth e�ects

It is equivalent to saying that:

∂xli (p,wi )

∂wi=∂xlj(p,wj)

∂wj

for every l , any two individuals i and j , and all (w1, . . . ,wI ).

We need parallel and straight wealth expansion paths.




Gorman form of preferences

A necessary and su�cient condition for the set of consumers to exhibitparallel, straight wealth expansion paths at any price vector p is thatpreferences admit indirect utility function of the Gorman form with thecoe�cients on wi the same for every consumer i . That is:

vi (p,wi ) = ai (p) + b(p)wi

Proof: Use Roys identity for the general case with bi (p):

xij =∂ai (p)/∂pj + wi∂bi (p)/∂pj

bi (p)=∂ai (p)/∂pj

bi (p)︸︷︷︸shift

+wi∂bi (p)/∂pj

bi (p)︸︷︷︸slope




Examples

Special cases:

preferences need to be homothetic

or preferences need to be quasi-linear.


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Advanced Microeconomics - coin.wne.uw.edu.plcoin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/2012_micro... · Introduction Welfare evaluation Aggregate demand Advanced Microeconomics