Consumer Behavior

9

*** 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 ¹ j, and

the income elasticity ¶ln(xi*)/¶ln(I) = (¶xi*/¶I)(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.

* 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) = mayb {b: u(x – b g) ≥ U, (x – b g) ≥ 0}.

Starting from point x, the benefit function B(x, U) measures the consumer’s willingness-to-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*/¶I)(¶C/¶p),

or, using Shephard’s lemma,

¶xc/¶p = ¶x*/¶p + (¶x*/¶I) 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*/¶I) x*’. In addition, since ¶xc/¶p is a symmetric, negative semi-definite, it follows that the Slutsky matrix in (A1) is also symmetric, negative semi-definite, i.e. that

¶x*/¶p + (¶x*/¶I) x*’ = a symmetric, negative semi-definite.

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*/¶I) xj* = ¶xj*/¶pi + (¶xj*/¶I) xi*, (symmetry restrictions)

for all i ¹ j, i, j = 1, …, n,

and

¶xi*/¶pi + (¶xi*/¶I) 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

(¶xi*/¶pj) pj + (¶xi*/¶I) I = 0,

or

¶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

[pi (¶xi*/¶I)] = 1,

or

[(pi xi*/I)(¶xi*/¶I)(I/xi*)] = 1,

or [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) + l [I – p’x],

where l is the Lagrange multiplier for the budget constraint. The first-order conditions (FOC) are

¶L/¶xi = ¶U/¶xi - l pi = 0, for all i = 1, …, n,

and

¶L/¶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 l*(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

U(x) = bi ln(xi - gi),

where bi > 0, bi = 1, and xi > gi, 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 = bi/(xi - gi) - l pi = 0, for all i = 1, …, n, (FOC1)

and

¶L/¶l = I – p’x = 0. (FOC2)

The (FOC1) condition implies

bi = l pi (xi - gi), for all i = 1, …, n.

Using bi = 1, this yields

l pi (xi - gi) = 1,

or

l = 1/[pi (xi - gi)]

= 1/(I - pi gi) (using the budget constraint).

Substituting this result into (FOC1) yields

bi/(xi - gi) = pi/(I - pi gi), for all i = 1, …, n,

or

xi - gi = (bi/pi)(I - pi gi), for all i = 1, …, n,

giving the Marshallian demand functions.

xi* = gi + (bi/pi)(I - pi gi), for all i = 1, …, n.

This can be alternatively written as

pi xi* = pi gi + bi (I - pi gi), for all i = 1, …, n,

which is called the linear expenditure system (LES). It shows that bi = ¶(pi xi*)/¶I is the marginal propensity to consume out of income, i = 1, …, n. Since bi > 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 > gi, the parameters g are sometimes called “subsistence consumption levels”, and (I - pi gi) the “super-numerary income” (representing income available after the “subsistence expenditures” pi gi are paid).

** 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

a(p) = a0 + ai ln(pi) + ½ aij ln(pi) ln(pj),

b(p) = > 0,

c(p) = gi ln(pi),

where [U + c(p)] ≠ 0.

This includes the Almost Ideal Demand System (AIDS) as a special case when gi = 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

aij = aji, for all i ¹ 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

¶ln(C)/¶ln(pi) = 1,

always.

Using the QAIDS expenditure specification, we have

¶ln(C)/¶ln(pi) = ai + aij ln(pj) - bi × b(p)/[U c(p)] + gi × b/[U + c(p)]2.

Then, the homogeneity restrictions give

[ai + aij ln(pj) - bi × b(p)/[U c(p)] + gi × b/[U + c(p)]2] = 1,

for all p and U. This implies

ai = 1,

aij (= aji) = 0, for all j = 1, …, n,

bi = 0,

and

gi = 0.

These are the homogeneity restrictions.

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 = ai + aij ln(pj) - bi × b(p)/[U + c(p)] + gi × 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* = ai + aij ln(pj) – bi × b(p)/[V + c(p)] + gi × b(p)/[V + c(p)]2,

where –b(p)/[V + c(p)] = ln(I) - a(p),

or

wi* = ai + aij ln(pj) + bi [ln(I) – a(p)] + [gi/b(p)] × [ln(I) - a(p)]2,

for all i = 1, …, n, where a(p) = a0 + ai ln(pi) + ½ aij ln(pi) ln(pj), and b(p) = > 0.

Defining

ln(P) = a0 + ai ln(pi) + ½ aij ln(pi) ln(pj),

this can be alternatively written as

wi* = ai + aij ln(pj) + bi ln(I/P) + [gi/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 gi = 0, i = 1, …, n, giving

wi* = ai + aij ln(pj) + bi ln(I/P),

for all i = 1, …, n.

Note: The specification for ln(P) implies that the AIDS and QAIDS models are non-linear in the parameters (since bi interacts with the a’s in ln(P)). One way to simplify the model is to consider the approximation ln (P) » 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) = bi + 2 [gi/b(p)] × ln(I/P),

¶wi*/¶ln(pk) = aik – [ak + akj ln(pj)] × [βi + 2 [gi/b(p)] × ln(I/P)]

– [gi 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,

= 1 + [bi + 2 [gi/b(p)] × ln(I/P)]/wi*, in the QAIDS.

Note that ¶wi*/¶ln(pk) = wi* × [¶ln(xi*)/¶ln(pk) + dik], where dik is the Kronecker delta satisfying dik = when i k. It follows that the Marshallian price elasticities are

¶ln(xi*)/¶ln(pk) = – dik + [¶wi*)/¶ln(pk)]/wi*, in general,

= – dik + {aik – [ak + akj ln(pj)] × [βi + 2 [gi/b(p)] × ln(I/P)]

– [gi 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