Indirect Utility and Expenditure Functions : Some Examples

i Fixed Coefficients

If the utility function isu(x) = min(ax1, bx2)

then the Marshallian demands arex1 =

b

p1b+ p2aM

andx2 =

a

p1b+ p2aM

so thatax1 = bx2 =

ab

p1b+ p2aM

which means that the indirect utility function is

v(p1, p2,M) =ab

p1b+ p2aM

A check on Roy’s Identity : from the above indirect utility function,

∂v

∂M=

ab

p1b+ p2a

∂v

∂p1= − ab2

(p1b+ p2a)2

∂v

∂p2= − a2b

(p1b+ p2a)2

so thatxi

∂v

∂M= − ∂v

∂pi

for goods i = 1, 2.What level of expenditure would give the person the utility level u. If she had E dollars to spend, then

she would get utility of

u =ab

p1b+ p2aE

if the prices she faced were p1 and p2. That means that the expenditure E required to get the utility levelu when prices are p1 and p2 is

E(p1, p2, u) =p1b+ p2a

abu

which is the person’s expenditure function. The person’s Hicksian, or compensated, demands for the goodsare the partial derivatives of the expenditure function with repsect to the prices :

E1(p1, p2, u) =b

abu

E2(p1, p2, u) =a

abu

Notice that here the compensated demands for each good do not depend on the price of the good : withL–shaped indifference curves, the person will always locate at the kink on the indifference curve, regardlessof prices.

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ii Perfect Substitutes

Ifu(x) = a1x1 + a2x2 + · · · anxn

then ifa1/p1 > a2/p2 ≥ · · · an/pn

the person will consume only good 1 :

x1 =M

p1

and xi = 0 for every i > 1. Her level of utility would then be a1Mp1

. More generally, she consumes only thegood(s) for which ai/pi is at a maximum. So

v(p1, p2, . . . , pn,M) = max (a1M

p1,a2M

p2, · · · , anM

pn)

[ left to the reader : Roy’s Identity holds here. ] How much money E would she need to get the utilitylevel u? She would spend all her money on the good i for which ai/pi is lowest. And to get utility u fromconsuming good i means consuming u/ai units of the good, which will cost piu/ai. That means that

E(p1, p2, · · · , pn, u) = min (p1u

a1,p2u

a2, · · · , pnu

an)

which makes the person’s compensated demand for good i u/ai if good i yields the highest level of ai/pi,and 0 otherwise.

iii Quasi–Linear Preferences

iiia u(x1, x2, x3) = x1 + 2√x2 + lnx3

In this case, the demand functions arex2 = (

p1

p2)2 (2′′)

x3 =p1

p3(3′′)

x1 =M

p1− p1

p2− 1

if the person’s income M is high enough so that M > (p1)2/p2 − p1.Substituting into the direct utility function, the person’s utility is

[M

p1− p1

p2− 1] + 2

√(p1

p2)2 + ln (

p1

p3)

so thatv(p1, p2, p3,M) =

M

p1+p1

p2+ ln p1 − ln p3 − 1

The expenditure function E(p1, p2, p3, u) must ( always ) satisfy the condition

v(p1, p2, p3, E[p1, p2, p3, u]) = u

so that here

u =E(p1, p2, p3, u)

p1+p1

p2+ ln p1 − ln p3 − 1

2

or

E(p1, p2, p3, u) = p1u−(p1)2

p2− p1 ln p1 + p1 ln p3 + p1

which means that the compensated ( Hicksian ) demand functions are

E1(p1, p2, p3, u) = u+ 1− 2p1

p2− ln p1 + ln p2

E2(p1, p2, p3, u) = (p1

p2)2

E3(p1, p2, p3, u) =p1

p3

In this case, the Marshallian demand functions for goods 2 and 3 are the same as the Hicksian demandfunctions. This will always be the case with quasi–linear preferences : if the income elasticity of demandfor a good is 0, then the Slutsky equation says that the compensated and uncompensated demand functionswill be the same.

iiib u(x1, x2, x3) = x1 + lnx2 + lnx2 + x3)

Here the Marshallian demand functions are

x2 =p1

p2 − p3(2′′)

x3 =p1(p2 − 2p3)p3(p2 − p3)

(3′′)

x1 =M

p1− 2

when M > 2p1 and when p2 > 2p3.Substituting in the definition of the direct utility function, and simplifying,

v(p1, p2, p3,M) =M

p1+ 2 ln p1 − ln (p2 − p3)− ln p3 − 2

This equation then implies that

E(p1, p2, p3, u) = p1u+ 2p1 − 2p1 ln p1 + p1 ln (p2 − p3) + p1 ln p3

giving Hicksian demand functions

E1(p1, p2, p3, u) = u− 2 ln p1 + ln (p2 − p3) + ln p3

E2(p1, p2, p3, u) =p1

p2 − p3

E3(p1, p2, p3, u) =p1

p3− p1

p2 − p3=p1(p2 − 2p3)p3(p2 − p3)

so that again the Marshallian demand functions for goods 2 and 3 are the same as the Hicksian demandfunctions.

iv Cobb–Douglas Preferences

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u(x1, x2, . . . , xn) = xa11 xa2

2 · · ·xannthen

xi =ai

a1 + a2 + · · ·+ an

M

pii = 1, 2, · · · , n

is the Marshallian demand function for good i. That means that

v(p1, p2, . . . , pn,M) = A(p1)−a1p−a22 · · · p−ann Ma

whereA ≡ (a1)a1(a2)a2 · · · (an)an

anda = a1 + a2 + · · ·+ an

( Note that the particular representation of the direct utility function matters for the indirect utility function.Taking a monotonically increasing transformation of u(x), such as taking U(x) = ln (u(x)), will also meantaking that transformation of the indirect utility function, getting V (p,M) = ln [v(p,M)] for example. )

The expendtiure function in this case is

E(p1, p2, . . . , pn, u) = B(p1)b1(p2)b2 · · · (pn)bnu1/A

whereB ≡ A1/a

andbi ≡

aia

The Hicksian demand functions are

Ei(p1, p2, . . . , pn, u) = bi(p1)b1(p2)a2 · · · (pi−1)ai−1(pi)ai−1(pi+1)ai+1 · · · (pn)anu1/A

which shows that, when preferences are Cobb–Douglas, goods are net substitutes even though the Marshalliandemand for good i does not depend on any other prices j.

v CES Preferences

Done in the textbook.

vi Stone–Geary Preferences

U(x) = b1 ln (x1 − s1) + b2 ln (x2 − b2) + · · ·+ bn ln (xn − sn)

withb1 + b2 + · · ·+ bn = 1

The Marshallian demands are

xi = si +bipi

[M −n∑j=1

pjsj ]

which gives rise to an indirect utility function

v(p1, . . . , pn,M) = ln (M − p · s) +n∑i=1

bi ln bi −n∑i=1

bi ln pi

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( where s is the vector (s1, s2, . . . , sn) ). Taking the exponsnets of both sides

eu = (M − p · s)B̃(p1)−b1(p2)−b2 · · · (pn)bn

whereB̃ ≡ (b1)b1(b2)b2 · · · (bn)bn

and where I have used the facts that ea+b = eaeb and that

ea ln b = ba

ThereforeE(p1, p2, . . . , pn, u) = (eu)

1B̃

(p1)b1(p2)b2 · · · (pn)bn + p · s

leading to Hicksian demand functions

Ei(p1, p2, · · · , pn, u) = bi(eu)1B̃

(p1)b1(p2)a2 · · · (pi−1)ai−1(pi)ai−1(pi+1)ai+1 · · · (pn)an + si

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