***Welfare Analysis and Index Numbers
** Welfare Analysis
* Firm Welfare
Consider the welfare analysis of a profit maximizing firm choosing the (n1) netput vector x =
(x1, …, xn), (where xi > 0 (< 0) corresponds to an output (an input)), and facing the technology
represented by the production function f(x) = 1. Let p be the (n1) vector of prices for x. The
production decisions are made as follows
(p) = maxx{p’x: f(x) = 1},
which gives the profit maximizing functions x*(p). From Hotelling’s lemma, we have
(p)/pi = xi*(p), for all i = 1, …, n.
Consider a change in price from pa to pb. We have
b
(pb) – (pa) = ppa
n
 i
1 (p)/pi dpi.
Using Hotelling’s lemma, this gives
b
(pb) – (pa) = ppa
n
*
 i
1 xi (p) dpi.
This expression measures the change in profit needed to keep the firm at a given welfare level
while facing the price change from pa to pb.
When a single output price is changing, this implies that firm welfare change can be evaluated
using the area “to the left of the output supply curve and between the two prices”. Define
producer surplus as the area “to the left of the supply curve and below the market price”. It
follows that firm welfare can be measured by the change in producer surplus.
When a single input price is changing, this implies that firm welfare change can be evaluated
using the area “to the left of the input demand curve and between the two prices”. Again, it
follows that firm welfare can be measured by the change in the area “to the left of the input
demand curve and below the market price”.
* Household welfare
Consider the welfare analysis of a household with a preference function U(x), and facing the
budget constraint I = p’x, where I > 0 is household income, and p is a (n1) vector of prices. Its
consumption decisions are made as follows
V(p, I) = maxx{U(x): subject to p’x = I},
which gives the Marshallian demand function x*(p, I). The associated expenditure minimization
problem is
C(p, U) = minx{p’x: subject to U = U(x)},
where C(p, U) = p’ xc(p, U) is the expenditure function, and xc(p, U) is the Hicksian demand
function. From Shephard’s lemma, we have
C(p, U)/pi = xic (p, U), for all i = 1, …, n.
Consider a change in price from pa to pb. We have
b
C(pb, U) – C(pa, U) = ppa
n
 i
1 C(p, U)/pi dpi.
Using Shephard’s lemma, this gives
b
C(pb, U) – C(pa, U) = ppa
n
c
 i
1 xi (p, U) dpi.
This expression measures the change in expenditure (or income) needed to keep the household at
a given utility level U while facing the price change from pa to pb. When a single price is
changing, this implies that household welfare change can be evaluated using the area “to the left
of the Hicksian demand curve and between the two prices”.
2
This suggests two possible welfare measures, depending on the choice of U. Two obvious
candidates for U are: Ua = V(pa, I), anf Ub = V(pb, I). This gives the “compensating variation”
measure (using Ua as the reference welfare level)
b
CV = C(pb, Ua) – C(pa, Ua) = ppa
n
c
 i
1 xi (p, Ua) dpi,
and the “equivalent variation” measure (using Ub as the reference welfare level)
b
EV = C(pb, Ub) – C(pa, Ub) = ppa
n
c
 i
1 xi (p, Ub) dpi.
In the presence of income effects (i.e. x*/I  0), in general CV  EV. In addition, neither CV
nor EV are directly measurable since U is typically not observed. This has motivated the
following “consumer surplus” measure
b
CS = ppa
n
*
 i
1 xi (p, I) dpi,
which is obtained simply by replacing the Hicksian demands xc by the Marshallian demands x* in
the above formulas. CS has the advantage of being directly estimable from observed behavior
x*(p, I).
We know from the Slutsky equation that price effects are the same for Marshallian demand and
Hicksian demand in the absence of income effects (xi*/I = 0). This means that if the price
changes are for commodities with zero income effects, (xi*/I = 0), then
CS = EV = CV.
In this case, all three measures CS, EV, and CV are identical. When a single price is changing,
this implies that household welfare change can be evaluated using the area “to the left of the
Marshallian demand curve and between the two prices”. In other words, household welfare can
be measured using consumer surplus, where welfare change is measured by the change in the area
“to the left of the Marshallian demand curve and above the market price”.
This suggests that, as long as income effects are “small”, the above result may still apply
“approximately”. Thus, under “small” income effects,
CS  EV  CV.
The Stutsky equation gives xic/pi = xi*/pi + (xic/I) xi*  0. Under positive income effects
(xi*/I > 0), this implies that 0  xic/pi > xi*/pi, or |xic/pi| < |xi*/pi|. For price increases,
this yields
EV  CS  CV.
** Index Numbers
Index numbers are commonly used for two reasons:
1/ as relative welfare measures (e.g., cost of living index, standard of living index); and
2/ as means of generating price and quantity index for a commodity group (e.g., food,
capital, etc.).
Although these two motivations are quite different, they are unified by their common objective of
“summarizing economic information about a group of commodities”. For relative welfare
measures, the group of commodities includes all commodities relevant to an economic agent. For
price and quantity index, the group of commodities involves only a subset of all the commodities
relevant to a decision-maker (e.g., food, capital, etc.).
Note: In the case of a subset of commodities, the theoretical justification for using a price index
requires imposing weak separability restrictions on the cost function, profit function, or
expenditure function. For example, a weakly separable expenditure function (or cost
function) implies the existence of a “price aggregator function” that summarizes all the
3
information related to the prices of commodities within the given subset. Then, a price
index is simply an empirical measurement of the corresponding price aggregator function
over the subset of commodities in the expenditure-cost function.
Similarly, in the case of a subset of commodities, the theoretical justification for using a
quantity index requires imposing weak separability restrictions on the utility function,
distance function, or production function. Indeed, a weakly separable utility function (or
distance function, or production function) implies the existence of a “quantity aggregator
function” that summarizes all the information related to the quantities of commodities
within the given subset. Then, a quantity index is simply an empirical measurement of
the corresponding quantity aggregator function over the subset of commodities in the
relevant utility-distance-production function.
Note that justifying the existence of both a price aggregator function and a quantity
aggregator function for a subset of commodities requires the weak separability of both
the expenditure-cost function, and the utility-distance-production function. This implies
stronger restrictions than just the weak separability of the utility-distance-production
function. Indeed, it requires the homothetic separability of the utility-distance-production
function (a function is said to be homothetically separable if it is weakly separable and its
aggregator functions are homothetic). Note that this may appear rather restrictive…
* Price index
For simplicity, we discuss the price index in the context of household consumption (with the
understanding that all the arguments could be presented similarly in the context of firm
production). Consider the expenditure function
C(p, U) = p’xc(p, U) = minx{p’x: U = f(x)},
where x = (x1, …, xn)’ is a (n1) vector of commodities, and p = (p1, …, pn)’ is a (n1) vector of
corresponding prices.
If we want to evaluate a cost of living index, then U = f(x) is the household utility function.
Alternatively, if we want to obtain a price index for a subset of commodities, then U = f(x) is an
aggregator function for the subset of commodities, aggregator function that is an argument of the
household utility function under weak separability.
We consider a situation where the prices p change from pa to pb.
1/ The true price index
Definition: The true price index (P) is
P(pa, pb, U) = C(pb, U)/C(pa, U).
This shows that a price index is a unit-free measure of relative cost-expenditure change, reflecting
the effects of changing prices that keep household utility U constant. It satisfies P > (=, or <) 1
whenever the price change tends to increase (maintain, or decrease) cost or expenditure. It means
that (P-1)100 is the percentage change in cost or expenditure due to changing prices, keeping
utility U constant.
Definition: The Divisia price index (DP) is
DP(pa, pb, U) = exp[

ln(p b )
ln(p a )
n
c
 i
1 wi (p, U) dln(pi)],
or
ln(DP) =

ln(p b )
ln(p a )
n
c
 i
1 wi (p, U) dln(pi),
4
where wic(p, U) = pi xic(p, U)/C(p, U) = the i-th Hicksian expenditure share.
Proposition 1: The true price index P and the Divisia price index DP are equivalent:
DP(pa, pb, U) = P(pa, pb, U).
Proof: The definition of P gives
ln(P) = ln C(pb, U) – ln C(pa, U).
But
ln C(pb, U) – ln C(pa, U) =
=

ln(p b )
ln(p a )

pb
pa
n
 i
1 (ln C(p, U)/pi) dpi,
n
 i
1 (ln C(p, U)/ln(pi)) dln(pi).
Shephard’s lemma gives
ln C/ln(pi) = wic(p, U) = pi xic(p, U)/C(p, U) = the i-th Hicksian cost share.
It follows that
b
ln(P) = ln C(pb, U) – ln C(pa, U) = ppa
n
c
 i
1 wi (p, U) dln(pi) = ln(DP).
2/ The Laspeyres price index
Definition: The Laspeyres price index (LP) is
LP(pa, pb, xa) = (pb’xa)/(pa’xa).
Note: The Laspeyres price index can be alternatively written as
n
n
b a
a a
LP(pa, pb, xa) = (  i
1 pi xi )/(  i1 pi xi )
n
n
a a
a a
b a
=  i
1 {[pi xi )/(  j1 pj xj )] (pi /pi )}
n
a
a
b a
=  i
1 [wi(p , x ) (pi /pi )]
where wi(p, x) = pi xi/(  nj1 pj xj) is the “i-th commodity share”.
The consumer price index (CPI) is a Laspeyres price index over all commodities consumed by
households. Note that it does not require information about the quantities xb observed after the
price change.
Note: For general price increases (pb  pa) the Laspeyres price index LP gives an upward biased
estimate of the true price index P. Indeed, we have
P – 1 = [C(pb, U) – C(pa, U)]/C(pa, U)
=[
=[

pb

pb
pa
pa
n
a
 i
1 (C(p, U)/pi) dpi]/C(p , U)
n
c
a
 i
1 xi (p, U) dpi]/C(p , U), (from Shephard’s lemma)
n
c a
b
a
a
[  i
1 xi (p , U) (pi – pi )]/C(p , U), (since Hicksian demands slope downward)
= [pb’ xa – pa’ xa]/[pa’ xa] if U = f(xa)
= LP – 1.
However, there are two situations where this upward bias vanishes:
1/ when all prices increase proportionally, in which case they have no effect on demand
since the Hicksian demands xc(p, U) are homogenous of degree zero in prices p;
5
2/ when all goods are consumed in fixed proportions with zero Allen elasticities of
substitution, implying the absence of price effects on the Hicksian demands xc(p, U).
3/ The Paasche price index
Definition: The Paasche price index (PP) is
PP(pa, pb, xb) = (pb’xb)/(pa’xb).
Note: The Paasche price index can be alternatively written as
n
n
b b
a b
PP(pa, pb, xb) = (  i
1 pi xi )/(  i1 pi xi )
n
n
b b
b b
b a
=  i
1 {[pi xi )/(  j1 pj xj )] (pi /pi )}
n
b
b
b a
=  i
1 [wi(p , x ) (pi /pi )]
where wi(p, x) = pi xi/(  nj1 pj xj) is the “i-th commodity share”.
Note that the Paasche price index requires information about the quantities xb observed after the
price change. To the extent that price information is easier to obtain than quantity information,
the Paasche price index is typically more difficult to implement empirically than the Laspeyres
price index.
4/ The Fisher price index
Definition: The Fisher price index (FP) is
FP(pa, pb, xa, xb) = [LP(pa, pb, xa) PP(pa, pb, xb)]1/2
= {[(pb’xa)(pb’xb)]/[(pa’xa)(pa’xb)]}1/2.
Note that the Fisher price index requires information about both quantities xa and xb (i.e., both
before and after the price change).
5/ The Tornquist-Theil price index
Definition: The Tornquist-Theil price index (TP) is
n
c a
c b
b
a
TP(pa, pb, U) = exp{  i
1 (1/2)[wi (p , U) + wi (p , U)][ln(pi ) – ln(pi )]},
or
n
c a
c b
b
a
ln(TP) =  i
1 (1/2)[wi (p , U) + wi (p , U)][ln(pi ) – ln(pi )],
where wic(p, U) = pi xic(p, U)/C(p, U) = the i-th Hicksian expenditure share.
Note that the Tornquist-Theil price index TP can be interpreted as an approximation to the
Divisia price index DP. Also, calculating TP requires information about expenditures both before
and after the price change.
* Quantity index
For simplicity, we discuss the quantity index in the context of household consumption (with the
understanding that all the arguments could be presented similarly in the context of firm
production). Consider the distance function D(x, U) that is dual to the household utility function
f(x), where D(x, U) = 1 is the implicit solution to f(x/D) = U. The distance function D(x, U) is
linear homogenous and increasing in x. This distance function plays the same role for quantity
index as the expenditure function for price index (as discussed above). In a way similar to
Shephard’s lemma, it satisfies
6
D(x, U)/xi = pic(x, U),
where pic(x, U) is the “Hicksian shadow price” of the i-th commodity.
From the linear homogeneity of D(x, U) in x, Euler equation gives
n
 i
1 (D/xi) xi = D(x, U) = 1,
or
n
c
 i
1 pi (x, U) xi = 1.
Let x = (x1, …, xn)’ is a (n1) vector of commodities, and p = (p1, …, pn)’ is a (n1) vector of
corresponding prices.
If we want to evaluate a standard of living index, then U = f(x) is the household utility function.
Alternatively, if we want to obtain a quantity index for a subset of commodities, then U = f(x) is
an aggregator function for the subset of commodities, aggregator function that is an argument of
the household utility function under weak separability.
We consider a situation where the quantities x change from xa to xb.
1/ The true quantity index (also called the Malmquist index)
Definition: The true quantity index (Q) is
Q(xa, pb, U) = D(xb, U)/D(xa, U).
This shows that a quantity index is a unit-free measure of relative distance, reflecting the effects
of changing quantities that keep household utility U constant. It satisfies Q > (=, or <) 1
whenever the quantity change tends to increase (maintain, or decrease) distance. It means that (1
– Q)100 is the percentage change in distance due to changing quantities, keeping utility U
constant.
Definition: The Divisia quantity index (DQ) is
DQ(xa, xb, U) = exp[

ln(x b )
ln(x a )
n
c
 i
1 wi (x, U) dln(xi)],
or
ln(DQ) =

ln(x b )
ln(x a )
n
c
 i
1 wi (x, U) dln(pi),
n
c
where wic(x, U) = pic(x, U) xi/D(x, U) = the i-th Hicksian share (since  i
1 pi (x, U) xi = D = 1).
Proposition 1: The true quantity index Q and the Divisia quantity index DQ are equivalent:
DQ(xa, xb, U) = Q(xa, xb, U).
Proof: The definition of TQ gives
ln(Q) = ln D(xb, U) – ln D(xa, U).
But
ln D(xb, U) – ln D(xa, U) =
=

ln(x b )
ln(x a )

xb
xa
n
 i
1 ln D(x, U)/xi dxi,
n
 i
1 ln D(x, U)/ln(xi) dln(xi).
Given D(x, U)/xi = pic(x, U), we have
ln D/ln(xi) = wic(x, U) = pic(x, U) xi/D(x, U) = the i-th Hicksian share.
It follows that
7
ln(TQ) = ln D(xb, U) – ln D(xa, U) =

xb
xa
n
c
 i
1 wi (x, U) dln(xi) = ln(DQ).
2/ The Laspeyres quantity index
Definition: The Laspeyres quantity index (LQ) is
LQ(xa, xb, pa) = (pa’xb)/(pa’xa).
Note: The Laspeyres quantity index can be alternatively written as
n
n
a b
a a
LQ(xa, xb, pa) = (  i
1 pi xi )/(  i1 pi xi )
n
n
a a
a a
b a
=  i
1 {[pi xi )/(  j1 pj xj )] (xi /xi )}
n
a
a
b a
=  i
1 [wi(p , x ) (xi /xi )]
where wi(p, x) = pi xi/(  nj1 pj xj) is the “i-th commodity share”.
3/ The Paasche quantity index
Definition: The Paasche quantity index (PQ) is
PQ(xa, xb, pb) = (pb’xb)/(pb’xa).
Note: The Paasche quantity index can be alternatively written as
n
n
b b
b a
PQ(xa, xb, pb) = (  i
1 pi xi )/(  i1 pi xi )
n
n
b b
b b
b a
=  i
1 {[pi xi )/(  j1 pj xj )] (xi /xi )}
n
b
b
b a
=  i
1 [wi(p , x ) (xi /xi )]
where wi(p, x) = pi xi/(  nj1 pj xj) is the “i-th commodity share”.
4/ The Fisher quantity index
Definition: The Fisher quantity index (FQ) is
FQ(xa, xb, pa, pb) = [LQ(xa, xb, pa) PQ(xa, xb, pb)]1/2
= {[(pa’xb)(pb’xb)]/[(pa’xa)(pb’xa)]}1/2.
5/ The Tornquist-Theil quantity index
Definition: The Tornquist-Theil quantity index (TQ) is
n
c a
c b
b
a
TQ(xa, xb, U) = exp{  i
1 (1/2)[wi (x , U) + wi (x , U)][ln(xi ) – ln(xi )]},
or
n
c a
c b
b
a
ln(TQ) =  i
1 (1/2)[wi (x , U) + wi (x , U)][ln(xi ) – ln(xi )],
where wic(x, U) = pic(x, U) xi /D(x, U) = the i-th Hicksian expenditure share.
Note that the Tornquist-Theil quantity index TQ can be interpreted as an approximation to the
Divisia quantity index DQ.
* Superlative index numbers
Definition: A superlative price (quantity) index is a true index associated with a flexible
expenditure (distance) function (i.e. a function that does not impose a priori restrictions
on the Allen elasticities of substitution).
8
Examples of expenditure (or distance) functions that are “flexible” are the quadratic, the
“generalized Leontief”, and the translog. It also includes the class of functions that are
“geometric mean of order k”.
Definition: A “geometric mean function of order k” is f(z) = [(zk/2)’ A (zk/2)]1/k, for some k  0,
where z is a (n1) vector, and A is a (nn) symmetric matrix.
Note: Under homothetic preferences, the expenditure function is C(p, U) = h(u) c(p). Then, if
c(p) is “geometric mean of order k”, it gives
c(p) = [(pk/2)’ A (pk/2)]1/k = a “flexible form” for any k  0,
= a translog cost function as k  0
= (p1/2)’ A (p1/2) = a “generalized Leontief” cost function if k = 1,
= [(p)’ A (p)]1/2 if k = 2.
Proposition: The Tornquist-Theil price (quantity) index is a superlative price (quantity) index
because it is a true price (quantity) index associated with a translog expenditure
(distance) function.
Proof: We present the proof only for a price index associated with the translog expenditure
function
n
n
n
ln[C(p, U)] = a0 +  i
(B1)
1 ai ln(pi) + au ln(U) + (1/2)  j1  i1 aij ln(pi) ln(pj)
n
2
+  i
1 aiu ln(pi) ln(U) + auu [ln(U)] .
For a change in prices from pa to pb, the associated true price index P satisfies
ln(P) = ln[C(pb, U)] – ln[C(pa, U)]
n
n
n
b
a
b
b
=  i
1 ai [ln(pi ) – ln(pi )] + (1/2)  i1  j1 aij ln(pi ) ln(pj )
n
n
n
a
a
b
a
– (1/2)  i
1  j1 aij ln(pi ) ln(pj ) +  i1 aiu ln(U) [ln(pi ) - ln(pi )].
(B2)
Shephard’s lemma gives
ln(C)/ln(pi) = pi xic(p, U)/C(p, U) = wic(p, U) = the i-th Hicksian cost share. (B3)
In the context of the translog expenditure function (B1), we have
ln(C)/ln(pi) = ai +  nj1 aij ln(pj) + aiu ln(U).
(B4)
Let Ck = C(pk, Uk), where k = a, b. Using the above results, the log of the TornquistTheil price index TP is
n
c b
b
c a
a
b
a
ln(TP) =  i
1 (1/2)[wi (p , U ) + wi (p , U )][ln(pi ) – ln(pi )]
n
b
b
a
a
b
a
= (1/2)  i
1 [ln(C )/ln(pi ) + ln(C )/ln(pi )][ln(pi ) – ln(pi )], (from (B3))
n
n
n
b
b
a
= (1/2)  i
1 [ai +  j1 aij ln(pj ) + aiu ln(U ) + ai +  j1 aij ln(pj )
+ aiu ln(Ua)][ln(pib) – ln(pia)], (from (B4))
n
n
n
b
a
b
b
=  i
1 ai [ln(pi ) – ln(pi )] + (1/2)  i1  j1 aij ln(pj ) ln(pi )
n
n
a
a
- (1/2)  i
1  j1 aij ln(pj ) ln(pi )
n
b
a
b
a
+ (1/2)  i
1 aiu [ln(U ) + ln(U )][ln(pi ) – ln(pi )]
n
n
n
b
a
b
b
=  i
1 ai [ln(pi ) – ln(pi )] + (1/2)  i1  j1 aij ln(pi ) ln(pj )
n
n
a
a
– (1/2)  i
1  j1 aij ln(pi ) ln(pj )
n
*
b
a
*
a
b 1/2
+  i
1 aiu ln(U ) [ln(pi ) - ln(pi )], where U = (U U ) ,
9
= ln(P) (from (B2) evaluated at U* = (Ua Ub)1/2).
Proposition: The Fisher price (quantity) index is a superlative index because it is a true index
associated with an expenditure (distance) function that is a “quadratic mean of order
two”.
Proof: Again, we present a proof only for the price index. Consider the case of homothetic
preferences where the expenditure function is C(p, U) = h(U) c(p), and c(p) = (p’ A p)1/2
is a quadratic mean of order two (k = 2). Then, Shephard’s lemma gives
C/p = xc = (p’ A p)-1/2 A p,
(C1)
or
A p = xc (p’ A p)1/2,
(C2)
c
where x = xc(p, U) is the Hicksian demand function.
For a change in price from pa to pb, we have
P = C(pb, U)/C(pa, U)
= c(pb)/c(pa) (under homothetic preferences)
= {[(pb)’ A (pb)]/[(pa)’ A (pa)]}1/2, (where c(p) = (p’ A p)1/2)
= {[(pb)’ xcb ((pb)’ A pb)1/2]/ [(pa)’ xca ((pa)’ A pa)1/2]}1/2, (from (C2)
= {[(pb)’ xcb (pb)’ xca]/ [(pa)’ xca (pa)’ xcb)1/2]}1/2, (from (C1)
= [(LP) (PP)]1/2
= FP.
* Implicit index
In general, an index (either price or quantity) can be interpreted as a ratio of expenditures. If so,
when applied to the same group of commodities, we expect the following relationship to hold
(price index)(quantity index) = [(pb)’ xb]/[(pa’) xa].
Note that this relationship always holds for the Fisher index since
(FP)(FQ) = [(LP) (PP)]1/2 [(LQ) (PQ)]1/2 = [(pb)’ xb]/[(pa’) xa].
However, it does not always hold for the other indexes. In this case, the above relationship can
be treated as an identity. Then, if only one of the two indexes (either price or quantity) is known,
the other one can always be obtained from this identity. Obtaining an index in this manner
generates an implicit index. A useful rule of thumb is to calculate a “direct index” for the
variables (either prices or quantities) that tend to vary “more proportionally”, and calculate an
“implicit index” for the variables that tend to vary “less proportionally”.