Public Finance, 2nd Edition

SOCIAL WELFARE AND CONSUMPTION: THE

JORGENSON ANALYSIS

Dale Jorgenson provided an important extension of Atkinson's social welfare

analysis shortly after Atkinson's work appeared. He developed a method for

linking measures of social welfare to people's consumption patterns rather

than their incomes.

10

Econometric demand analysis of aggregate consumption data was well

established by the mid±1970s, including estimation of the aggregate con-

sumption function and major categories such as food, clothing, transporta-

tion, and so forth. Panel data sets that permit more microeconometric

demand analysis were not yet available. Jorgenson's idea was to meld econo-

metric demand analysis with the social welfare function by using the esti-

mated demand equations for the major consumption categories to track

changes in social welfare over time. His approach has three distinct steps:

1. Posit individual utility functions deWned over a set of consumer goods

and derive demand equations for the goods from the utility functions.

2. Estimate the demand equations in a manner that allows for recovery

of the unknown parameters of the utility functions.

3. Use the estimated utility functions as the arguments of a Xexible-form

social welfare function that registers society's aversion to inequality,

and track changes in social welfare over time.

The Estimating Share Equations

Jorgenson begins by assuming that each household, h, has an indirect utility

function, V

h

,deWned over three sets of arguments: the prices of the various

consumer goods,

~

P

k

; the household's income, M

h

; and a vector of household

characteristics,

~

A

h

, such as family size, age of the head of household, and

where the household resides:

V

h

 V

h

(

~

P

k

;M

h

;

~

A

h

)h 1, ...,H (4:17)

The parameters of V

h

are assumed to be equal for all households and

constant over time. That is, households have identical, unchanging tastes.

They also face the same vector of consumer prices. Therefore, the diVerences

in households' utilities are due entirely to diVerences in their circumstancesÐ

that is, their incomes and characteristics.

Jorgenson employed the transcendental logarithmic (translog) indirect

utility function to approximate the true indirect utility function. The translog

is a second-order Taylor series expansion in the logs of the independent

variables around their means (each independent variable is scaled by dividing

10

D. Jorgenson, ``Aggregate Consumer Behavior and the Measurement of Social Welfare,''

Econometrica, September 1990.

4. THE SOCIAL WELFARE FUNCTION IN POLICY ANALYSIS 119

by its own mean, so that the log at each variable's mean is zero.). For

example, the translog approximation of V

h

assuming N prices and a single

characteristic A

h

,is

lnV

h



P

N

i1

a

i

lnP

i

 a

M

lnM

h

 a

A

lnA

h

 1/2

P

N

i1

P

N

j1

b

ij

lnP

i

lnP

j



P

N

i1

b

iM

lnP

i

lnM

h



P

N

i1

b

iA

lnP

i

lnA

h

 1/2b

MA

lnM

h

lnA

h

 1/2b

MM

(lnM

h

)

2

 1/2b

AA

(lnA

h

)

2

(4:18)

The estimating equations are obtained by taking log derivatives of the

translog function with respect to each of the prices and income:

qlnV

h

qlnP

k



qV

h

qP

k

P

k

V

h

k  1, ...N(4:19)

From Roy's identity, (qV

h

/qP

k

)  l

h

X

hk

, where l

h

is the marginal utility of

income for household h, and X

hk

is the consumption of good k by household

h. Therefore,

qlnV

h

qlnP

k

 l

h

P

k

X

hk

V

h

k  1, ...,N (4:20)

Similarly,

qlnV

h

qlnM

h



qV

h

qM

h

M

h

V

h



l

h

M

h

V

h

(4:21)

Dividing Eq. (4.20) by (4.21) yields:

qlnV

h

qlnP

k



qlnV

h

qlnM

h





P

k

X

hk

M

h

k  1, ...,N (4:22)

the expenditure share of good k for household h. The expenditure shares

become the dependent variables in the demand estimation. The advantage of

using the expenditure shares is that the researcher does not have to worry

about separating out prices from quantities.

Next, write out the price and income derivatives of the translog function

to see the full system of estimating equations:

qlnV

h

qlnP

k

 a

k



P

N

i1

b

ik

lnP

i

 b

kM

lnM

h

 b

kA

lnA

h

k  1, ...,N

(4:23)

120 SOCIAL WELFARE AND CONSUMPTION: THE JORGENSON ANALYSIS