1

13. Expenditure minimization

Econ 494

Spring 2013

2

Agenda

Expenditure minimizationSet up problem and solveRelation to utility maximizationGraphical illustrationSlutsky equation

3

Expenditure minimization

Dual of the utility maximization problemMathematically identical to cost minimizationSuppose we assume that consumers minimize the cost of achieving a given level of utility Hold utility constant (rather than income)

This problem will give us Hicksian, or compensated, demands Useful for welfare analysis

“How much money income can be taken away from an individual to make her as well off after some change (e.g., price decrease) as she was before?” – Compensating variation

4

Expenditure minimization

The expenditure function is:HOD(1) in prices (p1, p2)

increasing in (p1, p2, u)

concave in (p1, p2)

1 2

1 2 1 1 2 2 1 2,

1 1 2 2 1 2

( , , ) s.t. ( , )

( , )

x xE p p u Min p x p x U x x u

p x p x u U x x

L

same as cost min

5

FONC and SOSC

By IFT, solution to FONC are xih(p1, p2, u) and lh(p1, p2, u)

1 1 2 2 1 2( , )p x p x u U x x L

1 1 1 1 2

2 2 2 1 2

1 2

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

FONC

SOS

( , ) 0

( , ) 0

( , ) 0

0

C

0

p U x x

p U x x

u U x x

U U U

BH U U U

U U

L

L

L

L L L

L L L

L L L

6

Compare E-min and U-max

Both E-min and U-max require tangency of indifference curve and budget line.

Difference is in the constraintsFrom the FONC: lm = 1 / lh

1 1 2 2 1 2

1 1 1

2 2

1 2 1 2 1 1 2 2

1 1 1

2 2 2

1 1

1

2 2

1

2 2 2

U-MAX

( , , ) ( ,

E-MIN

( , )

( ) 0

( ) 0

( )

)

( ) 0

( )

0

0

0

h h

h h

m m

m

h

m

m m

h m

h

p x p x u U x x

p U

p U

x x U x x M p x p x

U p

U p

M

U p

U p

u U p x p x

L

L

L L

L

L

LL

7

Compare E-min and U-max

E-MIN

Solution to FONC:xi

h (p1, p2, u)

lh (p1, p2, u)

Find lowest budget line that reaches indifference curve

Holds utility constant

Demand unobservable

Only substitution effect

U-MAX

Solution to FONC:xi

m (p1, p2, M)

lm (p1, p2, M)

Find highest indifference curve that reaches budget line

Holds income constant

Demand is observable

Both substitution and income effects

8

Apply envelope theorem to expenditure function

l represents the marginal cost of utility the additional money required to attain a higher

level of utility

1 2

1 2 1 2

1 2

( , , ) 1 2( , , ) ( , , )( , , )

( , , ) 0h

h h

h

hx p p

x p p p pp p

Ep p u

u u

L

1 2

1 21 2

1 2

( , , ) 1 2( , , )( , , )

( , , )

( , , ) 0h

hh

h

hx p pi i

p px p pi ip p

Ex x p p u

p p

L

Same idea as Shephard’s LemmaDerivative of indirect expenditure function wrt

price yields Hicksian demand function

9

Apply envelope theoremfor comparative statics

By concavity of E (p1, p2, u) in prices:1 1 2 1

1 2 2 2

, , 1 1 1 2

, , 2 1 2 2

1 2

1 2

1 2

2 1

is NSD.

0, 0,

>0 (only for two goods, see cost min notes)

h hp p p p

h hp p p p

h h

h h

E E x p x p

E E x p x p

x x

p p

x x

p p

1 2

1 21 2

1 2

( , , ) 1 2( , , )( , , )

( , , )

By Envelope theorem:

( , , ) 0h

hh

h

hx p pi i

p px p pi ip p

Ex x p p u

p p

L

we get refutable hypotheses, but they are unobservable !!

All e-min comparative statics are same as cost min.

10

E-min vs. U-maxGraphical illustration

If initial price p10, then choose A.

E-min vs. U-maxGraphical illustration

Slope = – p11 / p2

M / p11

C

U1

U0

A

M / p10

x1

x2

M / p2

B

If price falls to p11, then choose B for E-Min,

and choose C for U-max.

Slope = – p10 / p2

11

Hicksian demands

Hicksian, or compensated demands, are the solution to the FONC of the E-min problemProperties of Hicksian demands:HOD(0) in prices

Comparative statics for Hicksian demands produce refutable hypotheses, but are not observableSlutsky equation lets you convert these into

something observable so you can test implications

12

Under what conditions will bundle of goods chosen under U-max and E-min be the same?

Four identities1.E (p1, p2, V (p1, p2, M)) º M

The minimum expenditure necessary to reach utilityV (p1, p2, M) is M.

2.V (p1, p2, E (p1, p2, u)) º u The maximum utility from income E (p1, p2, u) is u.

3.xim (p1, p2, M) º xi

h (p1, p2, V (p1, p2, M)) Marshallian demand at income M is the same as the

Hicksian demand with a utility level V (p1, p2, M) that can be reached with income M.

4.xih (p1, p2, u) º xi

m (p1, p2, E (p1, p2, u)) Hicksian demand at utility u is the same as Marshallian

demand with an income level E (p1, p2, u) that will achieve the same level of utility u.

13

Prove identity 4

U-max problem with income :

( , ) ( , ( , ))

is a vector of prices, is a vector of goods

h mx p u x p E p u

p x

( ) s.t. ( , ) Solution: ( , ( , ))m

xMax U x p x E p u x p E p u

Substitute solution into budget constraint:( , ( , )) ( , )mp x p E u Ep p u

By definition of the expenditure function: s.t. ( ) ( ,( , ) )h

xMin p x U x u p xE p u p u

Therefore:( , )( , ( , )) ( , )

( , ( , )) ( , ) Q.E.D.

m h

m h

p x p E p u p x p u

x p E p u x p

p u

u

E

14

Prove identity 2

By definition of indirect utility function:

( , ( , ))V p E p u u

( , ( , )) ( ) s.t. ( , )

( , ( , )) subst. solution into obj. fctn.

( , ) from identity 4

since ( , ) solves E-min

Q.E.D.

M M

x

m

h

h

V p E p u Max U x p x E p u

U x p E p u

U x p u

u x p u

15

Slutsky equation:Modern derivation

1 2 1 2 1 2( , , ) ( , , ( ,

I

, ))

dentity 4:h mi ix p p u x p p E p p u

Differentiate wrt :h

i

m mi i i

i i i

x x x

p p

E

pM

p

Note !

1 2

1 2 1 2

By the Env. Thm.:

By identity 4 ( , , ( ,

( , )

, ))

,

:

hi

i

mi

E

p

x p

x

p E p p

p p

u

u

h m mi i m

ii

i i

x xx

x

p p M

Can also be done for cross-prices

Unobservable, but a function of observables.

16

Slutsky equation

The Slutsky equation decomposes a change in demand induced by a price change into:substitution effects (move along indiff. curve) income effects (move to new indiff. curve)

You can do the same thing for cross-price effects.

m h mmi i ii

i i

x x xx

p p M

17

Income and substitution effectsfor a decrease in p1

A C

B

x1

x2

U1

U0

x10 x1

h x1m

Subst. effect

Income effect

18

Slutsky equation in elasticity form

Silb. §10.6 has other elasticity formulae.Useful for demand estimation

m h mmi i ii

j j

x x xx

p p M

Multiply by :

m h mmi i ii

j j

m hij ij j

j

i

j j j

i

i

i i

M

p

x

p p M

M

x x xx

p

s

x xp

p

x M

19

Slutsky matrix

We said that because the expenditure function is concave in prices, then:

1 1 1 2

2 1 2 2

, , 1 1 1 2

, , 2 1 2 2

1 1 1 11 2

1 2

2 2 2 21 2

1 2

is NSD.

= is NSD.

h hp p p p

h hp p p p

m m m mm m

m m m mm m

E E x p x pE E x p x p

x x x xx x

p M p M

x x x xx x

p M p M

20

Integrability

How do I know that the demand functions I estimated came from rational (U-max) decisions?They do if Slutsky matrix is NSD and symmetric

Integrating back from demand function to indirect utility functionShould be able to go from:

Indirect utility fctn. xim(•)

xim(•) Indirect utility fctn.