*** Consumer Behavior
** Consumption Theory
Consider a household making consumption decisions. Let U(x) be an ordinal utility
function representing household preferences, where x = (x1, …, xn)’ is a (n1) vector of
consumption goods. Assume that U(x) is increasing and quasi-concave in x. The
household faces the budget constraint
p’x  I,
where p = (p1, …, pn)’ > 0 is a (n1) vector of prices for x, and I > 0 denotes household
income. In consumer theory, each household is assumed to make consumption decisions
as follows
V(p/I) = maxx{U(X): subject to I = p’x},
where V(p/I) is the indirect utility function. Let x*(p/I) be the solution of the above
maximization problem. They are called Marshallian demand functions, representing
household consumption behavior.
The indirect utility function V(p/I) is homogenous of degree zero in (p and I), and a
decreasing and quasi-convex in (p/I).
The Marshallian demand functions x*(p/I) are homogenous of degree zero in prices and
income (p and I). Consumption behavior can be characterized by the properties of
x*(p/I). For the i-th commodity, this includes
the own-price elasticity ln(xi*)/ln(pi) = (xi*/pi)(pi/xi*),
the cross-price elasticities ln(xi*)/ln(pj) = (xi*/pj)(pj/xi*), i  j, and
the income elasticity ln(xi*)/ln(I) = (xi*/)(I/xi*),
i = 1, …, n.
Definition:
The i-th commodity is said to be
an inferior good if ln(xi*)/ln(I) < 0,
a necessity if 0 < ln(xi*)/ln(I) < 1,
a luxury good if ln(xi*)/ln(I) > 1,
i = 1, …, n.
Definition:
The i-th commodity is said to be
a Giffen good if ln(xi*)/ln(pi) > 0,
a normal good with inelastic demand if 0 > ln(xi*)/ln(pi) > -1,
a normal good with elastic demand if ln(xi*)/ln(pi) < -1,
i = 1, …, n.
** Duality
* The expenditure function C(p, U)
C(p, U) = minx{p’x: subject to U = U(x)},
which has for solution the Hicksian demand functions xc(p, U).
The expenditure function C(p, U) is increasing in (p, U), and linear homogenous and
concave in p.
The Hicksian demand functions xc(p, U) are homogenous of degree zero in prices p.
2
* The distance function D(x, U).
The functions D(x, U) = 1 and U(x/D) = U are inverse functions of each other.
The distance function D(x, U) is increasing in x, decreasing in U, and linear homogenous
in x.
* The indirect utility function V(p/I).
The functions C(p, U) = I and V(p/C(p, U)) = U are inverse functions of each other.
* The benefit function B(x, U).
Let g = (g1, …, gn)’ be a reference bundle satisfying g ≥ 0 and g ≠ 0. The benefit function is
defined as
B(x, U) = may {: u(x –  g) ≥ U, (x –  g) ≥ 0}.
Starting from point x, the benefit function B(x, U) measures the consumer’s willingnessto-pay (measured in units of the reference bundle g) to obtain utility level U.
When u(x) is quasi-concave, the benefit function satisfies the duality relationships:
C(p, U) = minx {p’ x – B(x, U) (p’ g), x ≥ 0},
B(x, U) = minp {p’ x – C(p, U): p’ g = 1, p ≥ 0}.
** Envelope results
From the envelope theorem, we have
C(p, U)/pi = xic(p, U), i = 1, …, n, (Shephard’s lemma)
and
-[V(p/I)/pi]/[V(p/I)/I] = xi*(p/I), i = 1, …, n. (Roy’s identity).
** Theoretical implications
* Integrability conditions
The expenditure function C(p, U) is concave in p. This implies that
2C/p2 = a (nn) symmetric, negative semi-definite matrix.
From Shephard’s lemma, C/p = xc’, this gives
xc/p = a symmetric, negative semi-definite.
Bu duality, we have
xc(p, U) = x*(p/C(p, U)).
Differentiating this identity with respect to p yields
xc/p = x*/p + (x*/)(C/p),
or, using Shephard’s lemma,
xc/p = x*/p + (x*/) x*’.
(A1)
Expression (A1) is called the Slutsky matrix. It shows that the Hicksian (compensated)
price slopes xc/p are equal to the Marshallian (uncompensated) price slopes x*/p,
plus an income effect (x*/) x*’. In addition, since xc/p is a symmetric, negative
semi-definite, it follows that the Slutsky matrix in (A1) is also symmetric, negative semidefinite, i.e. that
x*/p + (x*/) x*’ = a symmetric, negative semi-definite.
3
These are the integrability conditions of the Marshallian demand functions x*(p/I). They
must be satisfied for consumer behavior to be consistent with consumer theory (i.e.,
utility maximization subject to the budget constraint). They imply
xi*/pj + (xi*/) xj* = xj*/pi + (xj*/) xi*, (symmetry restrictions)
for all i  j, i, j = 1, …, n,
and
xi*/pi + (xi*/) xi*  0, (negativity restrictions)
for all i = 1, …, n.
The symmetry restrictions involve (n2 – n)/2 restrictions on the Marshallian demand
functions.
* Homogeneity restrictions
The Marshallian demand functions are homogenous of degree zero in (p, I). Using Euler
equation, this gives for the i-th demand
 nj1 (xi*/pj) pj + (xi*/I) I = 0,
or
 nj1 ln(xi*)/ln(pj) + (ln(xi*)/ln(I) = 0,
for all i = 1, …, n. These are n homogeneity restrictions on the Marshallian demand
functions.
* Adding-up restriction
The Marshallian demand functions must satisfy the budget constraint p’x*(p/I)= I.
Differentiating this budget constraint with respect to income (I) gives
n
*
 i
1 [pi (xi /)] = 1,
or
n
*
*
*
 i
1 [(pi xi /I)(xi /)(I/xi )] = 1,
n
*
*
or
 i
1 [wi ln(xi )/ln(I)] = 1,
where wi* = pi xi*/I is the (Marshallian) budget share for the i-th commodity. This is an
adding-up restriction (also called Engel aggregation restriction) on the Marshallian
demand functions.
Note: The symmetry, homogeneity, and adding-up restrictions generate
(n2 – n)/2 + n + 1 = (n2 + n)/2 + 1
theoretical restrictions on the Marshallian demand functions x*. Each of these
restrictions must be satisfied for consumer behavior to be consistent with
consumer theory.
** Direct utility function approach to consumption analysis
Consumer theory gives the following representation of consumption behavior
maxx{U(x): subject to p’x = I}.
the associated Lagrangean is
L = U(x) +  [I – p’x],
4
where  is the Lagrange multiplier for the budget constraint. The first-order conditions
(FOC) are
L/xi = U/xi -  pi = 0, for all i = 1, …, n,
and
L/ = I – p’x = 0.
This is a system of (n+1) equations that can be solved for the Marshallian demand
functions x*(p/I) and the optimal Lagrangean *(p/I). Then knowing U(x) can be used to
generate the associated behavioral demand functions x*(p, I).
* The linear expenditure system (LES)
Consider the Stone-Geary utility function
n
U(x) =  i
1 i ln(xi - i),
n
where i > 0,  i
1 i = 1, and xi > i, for all i = 1, …, n. Note that the Stone-Geary utility
function is strongly separable in the x’s. The associated first-order conditions (FOC) are
L/xi = i/(xi - i) -  pi = 0, for all i = 1, …, n,
(FOC1)
and
L/ = I – p’x = 0.
(FOC2)
The (FOC1) condition implies
i =  pi (xi - i), for all i = 1, …, n.
n
Using  i
1 i = 1, this yields
n
 i1  pi (xi - i) = 1,
or
n
 = 1/[  i
1 pi (xi - i)]
n
= 1/(I -  i
1 pi i) (using the budget constraint).
Substituting this result into (FOC1) yields
n
i/(xi - i) = pi/(I -  i
1 pi i), for all i = 1, …, n,
or
n
xi - i = (i/pi)(I -  i
1 pi i), for all i = 1, …, n,
giving the Marshallian demand functions.
n
xi* = i + (i/pi)(I -  i
1 pi i), for all i = 1, …, n.
This can be alternatively written as
n
pi xi* = pi i + i (I -  i
1 pi i), for all i = 1, …, n,
which is called the linear expenditure system (LES). It shows that i = pi xi*)/I is the
marginal propensity to consume out of income, i = 1, …, n. Since i > 0, this restricts the
marginal propensities to consume (and thus the income elasticities) to be positive. Thus,
the linear expenditure system does not allow for “inferior goods”.
Given xi > i, the parameters  are sometimes called “subsistence consumption levels”,
n
and (I -  i
1 pi i) the “super-numerary income” (representing income available after the
n
“subsistence expenditures”  i
1 pi i are paid).
5
** Expenditure function approach to consumption analysis
This approach consists in three steps:
1/ specify the expenditure function C(p, U),
2/ obtain the Hicksian demand functions from Shephard’s lemma: xc = (C/p)’,and
3/ obtain the Marshallian demand functions by using the duality relationship
x*(p/I) = xc(p, V(p/I)),
where the indirect utility function V(p/I) is obtained from inverting the
expenditure function
C(p, V(p/I)) = I.
* The quadratic almost ideal demand system (QAIDS)
Let the expenditure function be
ln[C(p, U)] = a(p) – b(p)/[U + c(p)],
where
n
n
n
a(p) = 0 +  i
1 i ln(pi) + ½  i1  j1 ij ln(pi) ln(pj),
b(p) =  in1 p i i > 0,
n
c(p) =  i
1 i ln(pi),
where [U + c(p)] ≠ 0.
This includes the Almost Ideal Demand System (AIDS) as a special case when i = 0, i =
1, …, n (i.e., when c(p) = 0).
1. Theoretical restrictions
- symmetry: We have 2ln(C)/ln(p2) = a (nn) symmetric matrix. In the context of the
above AIDS expenditure specification, this implies that
ij = ji, for all i  j, i, j = 1, …, n. (symmetry restrictions).
This generates (n2 – n)/2 symmetry restrictions.
- homogeneity: The expenditure function C(p, U) is linear homogenous on p. From
Euler equation, this implies
n
 i
1 ln(C)/ln(pi) = 1,
always.
Using the QAIDS expenditure specification, we have
ln(C)/ln(pi) = i +  nj1 ij ln(pj) - i  b(p)/[U c(p)] + i  b/[U + c(p)]2.
Then, the homogeneity restrictions give
n
n
2
 i
1 [i +  j1 ij ln(pj) - i  b(p)/[U c(p)] + i  b/[U + c(p)] ] = 1,
for all p and U. This implies
n
 i
1 i = 1,
n
n
 i1 ij (=  i
1 ji) = 0, for all j = 1, …, n,
n
 i1 i = 0,
and
n
 i
1 i = 0.
These are the homogeneity restrictions.
6
2. Empirical implementation
Note that Shephard’s lemma xic = C/pi can be alternatively written as
pi xic/C = (C/pi)(pi/C),
or
wic = ln(C)/ln(pi),
where wic(p, U) = pi xic/C is the i-th Hicksian expenditure share for the i-th commodity, i
= 1, …, n. In the context of the QAIDS specification, this gives
wic = i +  nj1 ij ln(pj) - i  b(p)/[U + c(p)] + i  b/[U + c(p)]2.
for all i = 1, …, n. But this involves “Hicksian behavior”, which is unobservable since U
is typically not directly observable. Thus, we need to transform this expression into the
corresponding “Marshallian behavior” (which is directly observable).
By duality, inverting C(p, V(p/I) = I gives
ln[C(p, V)] = ln(I)
or, using the QAIDS specification,
a(p) – b(p)/[V + c(p)] = ln(I),
or
– b(p)/[V + c(p)] = ln(I) – a(p),
or
V(p, I) = – c(p) – b(p)/[ln(I) – a(p)].
By duality x*(p, I) = xc[p, V(p, I)]. Denote the Marshallian budget share for the i-th
commodity by wi*(p, I) = pi xi*/I. It follows that
wi*(p, I) = wic(p, V(p, I)), for all i = 1, …, n.
In the context of the QAIDS specification, this gives
wi* = i +  nj1 ij ln(pj) – i  b(p)/[V + c(p)] + i  b(p)/[V + c(p)]2,
where –b(p)/[V + c(p)] = ln(I) - a(p),
or
wi* = i +  nj1 ij ln(pj) + i [ln(I) – a(p)] + [i/b(p)]  [ln(I) - a(p)]2,
n
n
n
for all i = 1, …, n, where a(p) = 0 +  i
1 i ln(pi) + ½  i1  j1 ij ln(pi) ln(pj), and b(p) =

 in1 p i i > 0.
Defining
n
n
n
ln(P) = 0 +  i
1 i ln(pi) + ½  i1  j1 ij ln(pi) ln(pj),
this can be alternatively written as
wi* = i +  nj1 ij ln(pj) + i ln(I/P) + [i/b(p)]  [ln(I/P)]2,
for all i = 1, …, n. This is the QAIDS specification, which expresses the observable
Marshallian budget share wi* as a function of prices p and income I. It provides the basis
for an econometric estimation of the associated demand parameters.
Note: The AIDS specification is obtained as a special case when i = 0, i = 1, …, n,
giving
wi* = i +  nj1 ij ln(pj) + i ln(I/P),
for all i = 1, …, n.
7
Note: The specification for ln(P) implies that the AIDS and QAIDS models are nonlinear in the parameters (since i interacts with the ’s in ln(P)). One way to
n
*
simplify the model is to consider the approximation ln (P)   i
1 wi ln(pi). Using,
this approximation, the AIDS specification (but not the QAIDS) becomes linear in
the parameters.
3. Elasticities
From the QAIDS specification, we have
wi*/ln(I) = i + 2 [i/b(p)]  ln(I/P),
wi*/ln(pk) = ik – [k +  nj1 kj ln(pj)]  [βi + 2 [i/b(p)]  ln(I/P)]
– [i βk/b(p)]  [ln(I/P)]2.
Note that wi*/ln(I) = wi*  [ln(xi*)/ln(I) – 1]. It follows that the income elasticities are
ln(xi*)/ln(I) = 1 + [wi*/ln(I)]/wi*, in general,
i + 2 [i/b(p)]  ln(I/P)]/wi*, in the QAIDS.
Note that wi*/ln(pk) = wi*  [ln(xi*)/ln(pk) + ik], where ik is the Kronecker delta
1 
 
satisfying ik =   when i   k. It follows that the Marshallian price elasticities are
0 
 
*
ln(xi )/ln(pk) = – ik + [wi*)/ln(pk)]/wi*, in general,
– ik + {ik – [k +  nj1 kj ln(pj)]  [βi + 2 [i/b(p)]  ln(I/P)]
– [i βk/b(p)]  [ln(I/P)]2}/wi*, in the QAIDS.
Finally, note that xic/pk = xi*/pk + xi*/I  xk*. This implies that the Hicksian price
elasticities are
ln(xic)/ln(pk) = ln(xi*)/ln(pk) + [ln(xi*)/ln(I)]  wk*, in general,
– ik + {ik – [k +  nj1 kj ln(pj)]  [βi + 2 [i/b(p)]  ln(I/P)]
– [i βk/b(p)]  [ln(I/P)]2}/wi*
+ {i + 2 [i/b(p)]  ln(I/P)]/wi*}  wk*, in the QAIDS.
Note: The elasticities for the AIDS model are obtained as a special case when i = 0, i =
1, …, n.
4. Estimation
Consider adding an error to the above QAIDS specification. This yields
wi* = i +  nj1 ij ln(pj) + i ln(I/P) + i [1/b(p)] [ln(I/P)]2 + ei
where ei is a random variable with mean zero and finite variance, for all i = 1, …, n. This
is the standard econometric specification for the QAIDS model, consisting of a system of
n equations. (Again, the AIDS model is a special case when i = 0, i = 1, …, n).
n
*
Note that, by definition,  i
1 wi = 1. Thus, the dependent variables are linearly
dependent, implying the singularity of the variance of the e’s. This singularity problem
8
can be handled by dropping one equation, thus estimating the remaining (n-1) equations.
The parameters from the equation dropped can be recovered from the homogeneity
restrictions. Finally, the parameter estimates are invariant to the equation dropped if the
maximum likelihood estimation method is used.
5. Testing consumer theory
The testable theoretical restrictions are
- the symmetry restrictions:
ij = ji for all i  j, i, j = 1, …, n,
- the homogeneity restrictions:
 nj1 ij = 0, for i = 1, …, n.
These can be tested using standard statistical tests.
** Indirect utility function approach to consumption analysis
This approach consists in the following steps:
1/ specify the indirect utility function V(p, I)
2/ obtain the Marshallian demand functions from Roy’s identity: xi* = (V/pi)/(V/I, for all i = 1, …, n.
3/ estimate econometrically the resulting system of equations.
Note: Roy’s identity xi* = - (V/pi)/(V/I) can also be written as
wi* = pi xi*/I = -[ln(V)/ln(pi)]/[ln(V)/ln(I)] for all i = 1, …, n.
This is a convenient formulation when a logarithmic specification is used for V.
This is the case for the translog indirect utility function
n
n
n
ln(V) = a0 +  i
1 ai ln(pi/I) + ½  i1  j1 aij ln(pi/I) ln(pj/I),
implying that
wi* = [ai +  nj1 aij ln(pj/I)]/[  nk1 ak +  nk1  nj1 akj ln(pj/I)],
for all i = 1, …, n.
This is the indirect-translog specification of consumer behavior. It provides a
system of n equations that can be estimated econometrically…
** Benefit function approach to consumption analysis
Let g = (g1, …, gn)’ be some reference bundle satisfying g  0 and g  0. Following
Luenberger (1992), the benefit function is defined as
B(x, U) = may {: u(x –  g) ≥ U, (x –  x) ≥ 0}.
The benefit function B(x, U) measures the number of units of the bundle g the
consumer is willing to give up to move from point x to obtain utility level U. In the case
where the bundle g has a unit price, this provides a measure of consumer willingness-topay for x. When B(x, U) is differentiable in x, it follows that the marginal benefit B(x,
U)/x is a measure of the marginal willingness-to-pay for x. In general, B(x, U) is nonincreasing in U. And B(x, U) = 0 is an implicit form of the utility function u(x). When
u(x) is quasi-concave, Luenberger (1992) showed that B(x, U) is concave in x (and thus
exhibits diminishing marginal benefits). And for any , it satisfies
9
B(x +  g, U) =  + B(x, U).
Proof: B(x +  g, U) = may {: u(x +  g –  g) ≥ U, (x +  g –  x) ≥ 0}
= may { + : u(x –  g) ≥ U, (x –  x) ≥ 0,  = β – }
=  + may {: u(x –  g) ≥ U, (x –  x) ≥ 0}
=  + B(x, U).
Differentiating this expression with respect to  gives
(B/x) g = 1,
and
(2B/x2) g = 0.
Thus, the matrix (2B/x2) is in general symmetric and negative semi-definite (from the
concavity of B in x).
From Luenberger (1992), when u(x) is quasi-concave, the benefit function B(x,
U) is dual to the expenditure function C(p, U) = minx {p’ x: u(x)  U, x ≥ 0}, with
B(x, U) = minp {p’ x – C(p, U): p’ g = 1, p ≥ 0}.
Denote the solution of this minimization problem by pL(x, U), which is a
compensated price-dependent inverse demand, holding utility U constant. Under
differentiability, applying the envelope theorem to this minimization problem gives
∂B(x, U)/∂x = pL(x, U),
showing that price-dependent inverse demands pL(x, U) measure marginal benefits. Since
U is typically unobservable, compensated inverse demands pL(x, U) are not observed.
Using duality, the corresponding observable (uncompensated) price-dependent demands
are
p*(x) = pL(x, u(x)).
The function p*(x) gives uncompensated inverse demands.
Consider the case where
B(x, U) = a(x) – [U b(x)]/[1 – U c(x)],
where b(x) > 0 and [1 – U c(x)] ≠ 0. Then, we have
pL(x) = ∂B(x, U)/∂x
= (∂a/∂x) – (∂b/∂x) U/[1 – U c(x)] – (∂c/∂x) b(x) [U/[1 – U c(x)]]2.
Since B(x, U) = 0 is equivalent to U = u(x), we obtain a(x)/b(x) = U/[1 – U c(x)], and
p*(x) = pL(x, u(x)),
= ∂a/∂x – (∂b/∂x) a(x)/b(x) – (∂c/∂x) a(x)2/b(x).
After an appropriate parametric choice for a(x), b(x) and c (x), this gives price-dependent
demands that are empirically tractable, flexible, and consistent with consumer theory.
Assume that
a(x) =  Mj1 aj xj +  Mj1  Mk 1 ½ ajk xj xk
where ajk = akj when k ≠ j,
b(x) = exp( Mj1 bj xj) > 0,
and
c(x) =  Mj1 cj xj.
Then, the price-dependent demand for the i-th commodity becomes
pi*(x) = ai +  Mj1 aij xj – bi a(x) – ci a(x)2/b(x).
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1. Theoretical restrictions
- symmetry: The symmetry restrictions are 2B/xjxk = 2B/xkxj, or
ajk = akj, for all j ≠ k.
This generates [n(n-1)/2] restrictions.
-adding-up: (B/x) g = 1 for all (x, U) implies that (∂a/∂x) g = 1, (∂b/∂x) g = 0, and
(∂c/∂x) g = 0, or
 Mj1 aj gj = 1,
 Mj1 aij gj (=  Mj1 aji gj) = 0, for all i,
 Mj1 bj gj = 0,
 Mj1 cj gj = 0.
As discussed below, when g = (0, 0, …, 0, 1), imposing these restrictions is equivalent to
normalizing the n-th price to be equal to 1, with B/xn = pnL = 1.
-negative semi-definiteness: (2B/x2) = a negative semi-definite matrix (from the
concavity of B(x, U) in x).
2. Empirical implementation
The price-dependent inverse demand for the i-th good is given by
piL(x, U) = /pi.
When B(x, U) = a(x) – [U b(x)]/[1 – U c(x)], a(x) =  Mj1 aj xj +  Mj1  Mk 1 ½ ajk xj xk, b(x) =
exp( Mj1 bj xj), and c(x) =  Mj1 cj xj, this gives the compensated inverse demands
piL(x, U) = ∂a/∂xi – (∂b/∂xi) U/[1 – U c(x)] – (∂c/∂xi) b(x) [U/[1 – U c(x)]]2,
= ai +  nj1 aij xj – bi  b(x)  U/[1 – U c(x)] – ci  b(x)  [U/[U – c(x)]]2,
for all i = 1, …, n. The associated uncompensated inverse demands are
pi*(x) = piL(x, u(x)),
= ∂a/∂xi – (∂b/∂xi) a(x)/b(x) – (∂c/∂xi) a(x)2/b(x),
= ai +  nj1 aij xj – bi  a(x) – ci  a(x)2/b(x),
i = 1, …, n. This provides a parametric specification of inverse demands that can be
estimated…
3. Evaluating welfare effects
The above specification can be used to evaluate the welfare effects of consumption, as
measured by the effects of x on the benefit function B(x, U).
It also provides a basis to evaluate own and cross-commodity effects of compensated
inverse demands, piL/xj, as well as uncompensated inverse demands pi*/xj…
4. Estimation
Consider the case where g = (0, 0, …, 0, 1), i.e. where the reference bundle g is chosen
to be one unit of the n-th good. Then, (B/x) g = 1 implies that pnL = B/xn = 1, i.e.,
that the shadow price of the n-th good is set equal to 1: pnL = pn* = 1. When prices p = (p1,
…, pn)’ are observed, this is equivalent to normalizing all prices with respect to the price
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of the n-th good: pi*  pi/pn, i = 1, …, n. Then, the uncompensated inverse demands can
be specified as
pi*  pi/pn = ai +  nj1 aij xj – bi  a(x) – ci  a(x)2/b(x) + ei,
where pi*  pi/pn is the i-th normalized price, and ei is an error term distributed with mean
zero, i = 1, …, n.
Consider the n-th equation. It satisfies pn*  pn/pn = 1 for all x, implying that an = 1, anj =
0 for all j, bn = 0, and cn = 0. It means that the n-th equation can be dropped and the
econometric estimation can focus on the system of (n-1) equations
pi*  pi/pn = ai +  nj1 aij xj – bi  a(x) – ci  a(x)2/b(x) + ei,
i = 1, …, n-1.
5. Testing consumer theory
The testable theoretical restrictions are the symmetry restrictions:
jk = kj for all j  k, j, k = 1, …, n,
These can be tested using standard statistical tests.
** Direct specification of consumption behavior
Another possibility is to consider a direct specification for the Marshallian demand
functions x*(p, I). A simple and convenient approach is to focus on specifications that
are linear in the parameters:
h(xi*) = i0 +  nj1 ij h(pj) + iI h(I), for all i = 1, …, n
where the ’s are parameters to be estimated. This includes as special cases
- the linear model where h(z) = z, implying that marginal price and income effects are
constant, (with (xi*)/pj = ij as marginal price effect, and (xi*)/ = iI as
marginal income effect).
and
- the dougle-log model where h(z) = ln(z), implying that price and income elasticities are
constant (with ln(xi*)/ln(pj) = ij as price elasticity, and ln(xi*)/ln(I) = iI as
income elasticity).
Note that, in general, these specifications provide a first-order local approximation to any
demand functions. As such, they are “locally flexible”. However, they are typically not
globally consistent with consumer theory.
* The Rotterdam model
This is another example of directly specified demand functions.
1/ Model specification
Consider the differential of xi*(p, I):
dxi* =  nj1 (xi*/pj) dpj + (xi*/) dI, for all i = 1, …, n,
or, using dzi/zi = dln(zi),
xi* dln(xi*) =  nj1 (xi*/pj)pj dln(pj) + (xi*/)I dln(I), for all i = 1, …, n.
Multiplying by (pi/I) yields
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[(pi xi*)/I] dln(xi*) =  nj1 (xi*/pj)(pj pi/I) dln(pj) + pi (xi*/) dln(I), for all i = 1,
…, n.
Denoting the i-th budget share by wi* = (pi xi*)/I, and using the Slutsky equation, xi*/pj
= xic/pj – (xi*/I) xj*, this gives
wi* dln(xi*) =  nj1 (xic/pj)(pj pi/I) dln(pj) -  nj1 (xi*/)(pi wj*) dln(pj)
+ pi (xi*/) dln(I),
for all i = 1, …, n.
Define
ij = (xic/pj)(pj pi/I)
and
i = pi (xi*/) = the marginal propensity to consume the i-th commodity.
The above model then becomes
wi* dln(xi*) =  nj1 ij dln(pj) -  nj1 i wj* dln(pj) + i dln(I), or
wi* dln(xi*) =  nj1 ij dln(pj) + i [dln(I) -  nj1 wj* dln(pj)],
for all i = 1, …, n. Treating the ’s as parameters, this is the standard specification of the
Rotterdam model, giving a system of n equations that can be estimated econometrically.
Note: The total differential of I = p’x gives
dI =  nj1 pj dxj +  nj1 xj dpj,
or, using dln(z) = dz/z and dividing by I,
dln(I) =  nj1 (pj xj/I) dln(xj) +  nj1 (pj xj/I) dln(pj), or
dln(I) =  nj1 (pj xj/I) dln(xj) +  nj1 (pj xj/I) dln(pj),
or, given wj = pj xj/I,
dln(I) -  nj1 wj dln(pj) =  nj1 wj dln(xj).
Substituting this equality into the above model yields the following alternative
specification for the Rotterdam model
wi* dln(xi*) =  nj1 ij dln(pj) + i  nj1 wj* dln(xj),
for all i = 1, …, n.
2/ Theoretical restrictions
Recall that ij = (xic/pj)(pj pi/I) and i = pi (xi*/).
symmetry restrictions: The symmetry of the Slutsky matrix (xic/pj = xjc/pi) implies
that
ij = ji
for all i  j, i, j = 1, …, n.
This gives (n2 – n)/2 symmetry restrictions.
adding-up restriction: The adding-up restriction (or Engel aggregation restriction) implies
n
*
 i
1 pi (xi /I) = 1, or
n
 i
1 i = 1.
This gives one Engel aggregation restriction.
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homogeneity restrictions: The Hicksian demand function xic(p, U) being homogenous of
degree zero in p, Euler equation implies that
 nj1 (xic/pj) pj = 0, i = 1, …, n, or
 nj1 ij = 0, for all i = 1, …, n.
This gives n homogeneity restrictions.
3/ Estimation
Consider using time series data. To make the Rotterdam model empirically tractable, at
time t let
wi* = (wit + wi,t-1)/2,
where wit = pitxit/It is the i-th commodity budget share at time t,
dln(xi*) = ln(xit) – ln(xi,t-1)
where xit is the consumption level of the i-th good at time t,
dln(pi) = ln(pit) – ln(pi,t-1) = ln(pit/pi,t-1)
where pit is the price of the i-th good at time t,
dln(I) = ln(It) – ln(It-1) = ln(It/It-1)
where It is the consumption level of the i-th good at time t,
dln(It) = ln(It) – ln(It-1) = ln(It/It-1).
Substituting these into the Rotterdam specification gives a system of n equations that can
be estimated econometrically after adding an additive error term to each equation.
Note: The alternative specification is
wi* dln(xi*) =  nj1 ij dln(pj) + i  nj1 wj* dln(xj).
Summing across all commodities i gives
n
n
n
n
n
*
*
*
 i
1 wi dln(xi ) =  j1  i1 ij dln(pj) +  i1 i  j1 wj dln(xj),
or
n
n
*
*
*
 i
1 wi dln(xi ) =  j1 wj dln(xj),
n
n
since  i
1 ij = 0 (from the homogeneity restrictions), and  i1 i = 1 (from the
adding-up restriction). This is an identity, implying a linear dependency in the
dependent variables. This generates a singular variance of the error terms. The
model can then be estimated after dropping an equation, the estimates being
invariant to the equation dropped if the maximum likelihood estimation method is
used.