*** DEMAND ANALYSIS

** A REVIEW OF CONSUMER THEORY:

1/ Marshallian demands:

Let U(y) denote the (direct) utility function of the household over the (n´1) vector of consumption goods. We assume that U(y) is an increasing, quasi-concave function of y. The consumption decision is then represented by the optimization problem:

V(p/I) = maxy [U(y): p'y £ I],

where p is the (n´1) vector of prices for y, I > 0 denotes household income, and V(p/I) is the indirect utility function. The above optimization problem can be alternatively expressed in terms of the Lagrangean: L = U(y) + l (I-p'y), where l is the Lagrange multiplier associated with the budget constraint (p'y < I) and measuring the marginal utility of income. The first order necessary conditions are:

L/yi = U/yi - l pi = 0, i = 1, 2, ..., n,

L/l = I - p'y = 0.

These first order conditions imply that, at the optimum, the marginal rate of substitution, MRSij = (U/yi)/(U/yj), equals the price ratio, pi/pj, for all i ¹ j. The solution of the above optimization problem gives the Marshallian (uncompensated) demand functions y*(p/I). The Marshallian demand functions characterize consumption behavior for a rational household. As such, it is commonly used in positive analysis of household behavior.

2/ Hicksian demands:

Consider the optimization problem:

C(p, u) = Miny [p'y: U(y) ³ u]

where u is a reference utility level, and C(p, u) is the (indirect) expenditure function. The solution of this optimization problem gives Hicksian demand functions yc(p, u). Since it holds household welfare level at some constant level u, the above optimization problem is commonly used as the basis for compensation tests which are at the heart of applied welfare analysis.

Note 1: (Shephard's lemma): From the envelope theorem (and assuming differentiability), the expenditure function C(p, u) satisfies the property:

C/pi = yic(p, u), i = 1, 2, ..., n.

Note 2: The direct utility function U(x), the indirect utility function V(p/I), and the expenditure function C(p, u) are dual to each other: each one can be obtained from any of the others. For example, C(p, u) = I and V(p/I) = u are inverse functions of each other.

 

3/ Properties of the Marshallian demands:

The Marshallian demands y*(p/I):

. are homogenous of degree zero in (p, I),

. satisfies y*[p/C(p,u)] º yc(p, u), which implies

.y*/p + (y*/I) (C/p) º yc/p,

where y*/p and yc/p are respectively (n´n) matrices of Marshallian and Hicksian price slopes, and (y*/I) is a (n´1) vector of Marshallian income effects. Using Shephard's lemma (C/p = yc) along with the identity y*[p/C(p,u)] º yc(p, u), it follows that

y*/p + (y*/I) y*' º yc/p,

or

y*/p º yc/p - (y*/I) y*'.

This is the (n´n) Slutsky matrix which decomposes of the Marshallian price effects (y*/p) into two additive terms: a substitution effect (yc/p), and an income effect [-(y*/I) y*']. The substitution effect corresponds to the Hicksian price effects (yc/p), where the matrix (yc/p) can be shown to be symmetric, negative semi-definite, implying that (yic/pj) = (yjc/pi) for all i ¹ j (from symmetry), and (yic/pi) £ 0 for all i (from negative semi-definiteness). Thus, the Hicksian demand functions are downward sloping with respect to their own price (yic/pi £ 0 for all i).

Commodities can be classified according to their Marshallian income elasticities, ·i º ln yi*/ln I = (yi*/I)/(yi*/I), i = 1, 2, ..., n, and their Marshallian price elasticities, eij º lnyi*/ln pj = (yi*/pj)/(yi*/pj) for all i, j.

 

Definition 1: The i-th commodity is defined to be:

. an inferior good if ·i < 0,

. a necessity if 0 £ ·i £ 1,

. a luxury good if ·i > 1.

 Definition 2: The i-th commodity is defined to be a Giffen good if eii > 0.

From the Slutsky equation, we have:

yi*/pi º yic/pi - (yi*/I) yi*,

where the substitution effect is always non positive (yic/pi £ 0), i = 1, 2, ..., n. It follows that a sufficient condition for the i-th commodity to be a non-Giffen good (eii £ 0, or yi*/pi £ 0) is that it is a non-inferior good (·i ³ 0, or yi*/I ³ 0), i = 1, 2, ..., n. In other words, any commodity that exhibits a non-negative income elasticity (·i ³ 0) is necessarily a non-Giffen good exhibiting a downward sloping demand function with respect to its own price (eii £ 0, or yi*/pi £ 0).

 

** HOUSEHOLD PRODUCTION THEORY: Deaton and Muellbauer, chap. 10.

Assume that household utility is generated by non-market goods. These non-market goods are in turn produced by combining the market goods, time, and human capital. Then, the household decision making process can be represented by:

. the household utility function U(z, L1), where L1 denotes leisure time, and z is a vector of non-market goods (besides leisure) affecting household welfare.

(Note that, if household members enjoy working within the household, this specification could be easily generalized to U(z, L1, L2) where L2 is the labor time used in household production. However, this would affect the validity of some of the arguments presented below).

. the household production function represented by the implicit production function:

f(z, y, L2, K) = 0,

where y is a vector of market goods, L2 is labor time used in household production, and K denotes human capital. This reflects the household production technology, where z is a vector of outputs produced from the following set of inputs: market goods (y), household time (L2), and human capital (K).

. the household budget constraint:

I + w L3 = p'y,

where I denotes exogenous income, w is the wage rate, L3 is the amount of wage labor, p is the price vector of y, and y is the vector of purchased market goods (or sold market goods if y < 0).

. the time constraint:

T = L1 + L2 + L3,

where T denotes total time available, L1 is leisure time, L2 is time spent in household production activities, and L3 is the amount of wage labor.

1/ Household behavior:

The household makes decisions according to the following optimization problem:

Maxz,L,y [U(z, L1): I + w L3 p'y; T = L1 + L2 + L3; f(z, y, L2, K) = 0],

where L = (L1, L2, L3). The corresponding optimal choice functions are the Marshallian functions:

z*(p/I, w/I, K),

L*(p/I, w/I, K),

and

y*(p/I, w/I, K).

Note that the above optimization problem can be written as:

Maxz,L,y [U(z, L1): I + w L3 = p'y; T = L1 + L2 + L3; f(z, y, L2, K) = 0]
= Maxz,L,y [U(z, L1): I + w (T-L1-L2) = p'y; f(z, y, L2, K) = 0]

= Maxz,L,y [U(z, L1): I+wT = p'y + w (L1+L2); f(z, y, L2, K) = 0],

where (I + wT) is called "full income" measuring the maximum potential earning that could be obtained if all the household time available was allocated to wage labor. Consider decomposing the above optimization problem into two stages. In a first stage, choose y and L2 (conditional on z and L1). In a second stage, choose z and L1. (Note that, if household members enjoy working within the household, then U = U(z, L1, L2) and the L2 variable would become a "second-stage decision variable" in the analysis presented below).

 

* First stage: As long as marginal utility of income is positive, the first stage takes the form of the following cost minimization problem:

c(p,w,z,K) = Miny,L2 [p'y + w L2: f(z, y, L2, K) = 0],

where c(p, w, z, K) is an indirect cost function. In other words, utility maximization in the presence of market goods is fully consistent with cost minimizing behavior with respect to y and L2. The solution of the first stage optimization are the cost minimizing demand function denoted by:

y+(p, w, z, K),

and

L2+(p, w, z, K).

* Second stage: The second stage then becomes:

Maxz,L1 [U(z, L1): I+wT = c(p,w,z,K) + w L1],

which has for solution

z*(p/I, w/I, K)

and

L1*(p/I, w/I, K)

since the two-stage decomposition obviously gives the same solution to the original problem. This second-stage optimization problem can be formulated using the Lagrangean L = U(z, L1) + l [I+wT - c(p,w,z,K) - w L1], where l is the Lagrange multiplier associated with the budget constraint [I+wT = c(p,w,z,K) + w L1] and measuring the marginal utility of income. The first order necessary conditions for the second stage problem are:

U/zi = l c/zi for all i,

and

U/L1 = l w.

The first order conditions with respect z imply:

(U/zi)/(U/zj) = (c/zi)/(c/zj), for all i ¹ j.

The left hand side is the marginal rate of substitution between two non market goods, zi and zj. The right hand side is the ratio of the marginal costs of zi and zj. The marginal cost c/zi can be interpreted as the shadow price of zi, or alternatively as the hedonic price of zi, i = 1, 2, ... Then, the above expression states that, at the optimum, the marginal rate of substitution between any two non market goods must be equal to the ratio of their corresponding shadow prices.

Combining the two stage decomposition results yields the following results:

y*(I,p,w,K) = y+(p,w, z*(I,p,w,K), K) (1a)

and

L2*(I,p,w,K) = L2+(p,w, z*(I,p,w,K), K). (1b)

2/ The expenditure function:

Using L3 = T - L1 - L2, define the expenditure function as:

C(p,w,K,u) = Minz,L,y [p'y - w L3: T = L1+L2+L3; U(z, L1) ³ u; f(z, y, L2, K) = 0]
= Minz,L1 MinL2,z [p'y - w(T-L1-L2): U(z, L1) ³ u; f(z, y, L2, K) = 0]

= - w T + Minz,L1 [w L1 + c(p,w,z,K): U(z, L1) ³ u],

where c(p,w,z,K) = MinL2,z [p'y + w L2: f(z,y,L2,K)] is the cost function defined in "stage 1" above. Let the solution of the above expenditure minimization problem be the (compensated) Hicksian demand functions:

zc(p,w,K,u),

Lc(p,w,K,u),

and

yc(p,w,K,u).

By duality, we have:

zc(p,w,K,u) = z*[C(p,w,K,u), p,w,K], (2a)

Lc(p,w,K,u)= L*[C(p,w,K,u), p,w,K], (2b)

and

yc(p,w,K,u)= y*[C(p,w,K,u), p,w,K]. (2c)

Differentiating these expressions with respect to (p, w) gives the following Slutsky decomposition:

(zc, Lc, yc)/(p, w) = (z*, L*, y*)/(p, w) + [(z*, L*, y*)/I] [C/(p, w)],
= (z*, L*, y*)/(p, w) + [(z*, L*, y*)/I] [y*, L3*]', (3)

where we used Shephard's lemma (C/p = yc, and C/w = L3c = T - L1c - L2c). The left-hand side of (3) gives the Hicksian (compensated) price effects. The first term on the right-hand side of (3) is the Marshallian price effects, while the second term reflects income effects. As in the traditional model of consumer demand, the matrix of Hicksian price effects can be shown to be symmetric, negative semi-definite.

3/ Some special cases:

a/ traditional consumer theory: If the household production function take the form y = z, then, the non-market goods z are in fact market goods with market price p. In this case, the household production model becomes identical with traditional consumer theory.

b/ no market exists: If y = 0 and L3 = 0 are the only feasible points for y and L3, then the household does not participate in any market activity. This is the "autarky case", where the household produces only for its own consumption.

c/ z is a vector of quality factors: Examples of quality factors include health, nutrition, characteristics of differentiated products, etc.

A simple illustration is given by the linear characteristic model proposed by Lancaster. In the Lancasterian model, the household production function f(z, y, L2, K) = 0 is assumed to take the linear form:

zj = Si bij yi,

where zj is the j-th characteristic (or quality factor) treated as a non-market good,

yi is the i-th market good purchased at a price pi

bij is a fixed parameter measuring the quantity of the j-th non-market good contained in one unit of market good i.

An example could the case of human nutrition where the y's include food consumption, the z's include nutrient intake, and the bij's measure the nutritional composition of the various food items.

(Note: The Lancasterian model implicitly assumes that the characteristics are in fixed proportions in each market good, and are perfect substitutes across market goods).

d/ marketed surplus: Consider an agricultural household which produces and also consumes part of its production. Let y = (y1, y2, y3) and z = (z1, z2). Define:

Y1 = quantity of outputs produced by the household,

z1 = quantity of the household outputs consumed by the household,

(Y1 - z1) = "marketed surplus" = quantity of outputs sold on the market place at price p1, where y1 º -(Y1 - z1).

y3 is the vector of purchased inputs used in household production,

f(Y1, y3, L2, K) is the household production function,

z2 º y2 = quantity of purchased consumer goods.

Letting Y = (Y1, y3), the household production model then takes the form:

Maxz,L,Y [U(z, L1): I+wL3 = -p1(Y1-z1)+p2z2+p3y3; T = L1+L2+L3; f(Y1, y3, L2, K) = 0]
= Maxz,L,Y [U(z, L1): I+wT+p1Y1-p3y3-wL2 = p1z1+p2z2+wL1; f(Y1, y3, L2, K) = 0],

which has for solution z*(I,p,w,K), L*(I,p,w,K) and Y*(I,p,w,K).

Using a two-stage decomposition, we can choose (L2, Y) in a first stage (conditional on (z, L1), and then choose (z, L1) in a second stage.

Given positive marginal utility of income, the first stage problem becomes the profit maximization problem:

p (p1, p3, w, K) = MaxL2,Y [p1Y1 - p3y3 - wL2: f(Y1, y3, L2, K) = 0],

which has for solution the profit maximizing supply/demand functions: L2+(p1,p3,w,K) and Y+(p1,p3,w,K). This shows that profit maximizing behavior with respect to household labor L2, marketed surplus Y1, and purchased inputs y3 is fully consistent with household behavior. It also shows that the production decisions are separable from the consumption decisions since they can be evaluated without any information on household preferences.

The second stage problem then becomes:

Maxz,L1 [U(z, L1): I+wT+p (p1,p3,w,K) = p1z1+p2z2+wL1],

which has for solution z*(I,p,w,K) and L1*(I,p,w,K). And combining the two stages gives the identities:

L2+(p1,p3,w,K) = L2*(I,p,w,K), (4a)

and

Y+(p1,p3,w,K) = Y*(I,p,w,K), (4b)

where Y+ = (Y1+, y3+). Differentiating these identities with respect to I and p2 yields:

(L2*, Y*)/(I, p2) = 0, (5)

which implies no effect of income (I) or consumer prices (p2) on production decisions (Y = (Y1, y3) and L2). This is an empirically testable implication of the separability of production decisions from the consumption decisions.

 

Note that, treating p as given and equal to some level p 0, the above problem can be alternatively written as:

Maxz,L1 [U(z, L1): I+wT+p 0 = p1z1+p2z2+wL1],

which has for solution za(I+wT+p 0,p1,p2,w) and L1a(I+wT+p 0,p1, p2,w). This is a traditional model of consumer behavior where "income" (I+wT+p 0) is treated as "exogenous". But letting p 0 = p (p1,p3,w,K), the following identities must necessarily hold:

z*(I,p,w,K) = za(I+wT+p (p1,p3,w,K),p1,p2,w), (6a)

and

L1*(I,p,w,K) = L1a(I+wT+p (p1,p3,w,K),p1,p2,w). (6b)

Differentiating the identity (6a) with respect to p = (p1, p2, p3) gives:

z1*/p1 = z1a/p1 + [z1a/I] [¶p /p1]
= z1a/p1 + [z1a/I] Y1*, (7a)

z1*/p2 = z1a/p2, (7b)

z1*/p3 = z1a/p3 + [z1a/I] [¶p /p3]

= -[z1a/I] y3* ¹ 0, given z1a/p3 = 0, (7c)

where we used the envelope theorem: ¶p /p1 = Y1+ and ¶p /p3 = - y3+. Expressions (7) present the effects of market prices on consumer demand for the commodities z1 (which are both produced and consumed by the household). Expression (7b) indicate that the price effects of a change in p2, the price of consumer goods, is the same as in the traditional model of consumer theory. Perhaps more importantly, expressions (5a) and (5c) decompose Marshallian price effects into two additive terms: 1/ the first term on the right hand side is the classical Marshallian price effect of traditional consumer theory where income (I+wT+p 0) is treated as given; 2/ the second term is a "profit effect" which takes into consideration the effect of prices (p1, p3) on farm profit which can then be spend (through the effect of a higher income) on consumer goods. This second term indicates the non-separability of the consumption decisions from the production decisions: production decisions affect consumption decisions through the income effect generated by the profit from the production activities. If Y1* > 0 and z1 is a non-inferior good (z1a/I ³ 0), then equation (7a) implies:

z1*/p1 ³ z1a/p1 < 0,

This shows that, as compared to the household production model, the traditional consumer demand model (which treats profit as "exogenous") would give a biased estimate of the effect of an increase in price p1 on household demand for the commodity z1.

Given that marketed surplus is defined as (Y1 - z1), the (Marshallian) elasticity of marketed surplus with respect to price p1 is:

ln(Y1*-z1*)/lnp1 = [(Y1*-z1*)/p1] [p1/(Y1*-z1*)]
= [(ln Y1*/ln p1)(Y1*/p1) - (ln z1*/ln p1)(z1*/p1)] [p1/(Y1*-z1*)]

= {(ln Y1*/ln p1) [Y1*/(Y1*-z1*)]} - {(ln z1*/ln p1) [z1*/(Y1*-z1*)]},

This shows that the (Marshallian) elasticity of marketed surplus with respect to price p1 is a weighted difference between the elasticity of output (ln Y1*/ln p1) minus the elasticity of consumption (ln Y1*/ln p1), where the weights involve the ratios of production or consumption with respect to marketed surplus.