# Household Socio-Demographic Characteristics 1

*** Household Socio-Demographic Characteristics

Let d be a (k1) vector of household socio-demographic characteristics (e.g., household size, household composition in terms of age, sex, etc.). Household preferences are given by the utility function U(x, d), where x = (x1, …, xn)’ is a (n1) vector of consumption goods, where U/x > 0. The household faces the budget constraint I = p’x, where I > 0 is household income, and p = (p1, …, pn)’ > 0 is a (n1) vector of prices for x. The household decisions are given by the maximization problem

V(p/I, d) = maxx{U(x, d): subject to I = p’ x},

where V(p/I, d) is the indirect utility function. Let x*(p/I, d) denote the solution of the above optimization problem, giving the Marshallian demand functions for x.

The associated dual expenditure function is

C(p, d, U) = minx{p’ x: subject to U(x, d) = U},

where C(p, d, U) is the expenditure function. Let xc(p, d, U) be the solution of the minimization problem, giving the Hicksian demand functions.

** The general case

Consider the Lagrangean associated with the utility maximization problem

L(x, , ) = U(x, s) +  [I – p’x],

where  = (p, I, d). Comparative static analysis gives

= - H-1,

where H = 2L/(x, )2 is the (n+1)(n+1) bordered hessian, or

= - H-1.

Let H-1 = , where A is a (nn) symmetric matrix, B is a (n1) vector, and C is a scalar. It follows that

x*/p =  A + B x’,

x*/I = -B,

and

x*/d = - A (2U/xd).

This implies that

x*/d = -(1/)[x*/p + (x*/I) x’](2U/xd),

or using the Slutksy decomposition, xc/p = x*/p + (x*/I) x’,

x*/d = -(1/) (xc/p) (2U/xd).

This shows that the effects of socio-demographic variables on consumption behaviorx*/d are equal to -(1/) times the (nn) substitution matrix (xc/p), times the (nk) matrix of second derivatives (2U/xd).

** Scaling and translating

Consider the special case where

U(x, d)= U[(x1 - T1(d))/S1(d), … , (xn - Tn(d))/Sn(d)],

where T1(d), …, Tn(d) are “translating factors”,

S1(d), …, Sn(d) are “scaling factors”.

* Marshallian demand

Then, household choices are then given by

V(p/I, d) = maxx{U[(x1 - T1(d))/S1(d), … , (xn - Tn(d))/Sn(d)]: p’ x = I}.

Let

Xi = (xi - Ti(d))/Si(d), i = 1, …, n,

or

xi = Xi Si(d) + Ti(d), i = 1, …, n.

It follows that household choices can be alternatively written as

V(p/I, d) = maxX {U(X): subject to pi [Si(d) Xi + Ti(d)] = I},

where X = (X1, …, Xn)’, or

V(p/I, d) = maxX {U(X): subject to pi Si(d) Xi = I - pi Ti(d)}.

This has for solution Xi*[(p1 S1(d))/(I - p’T), … ,(pn Sn(d))/(I - p’ T(d))], i = 1, …, n, where T(d) = (T1(d), …, Tn(d))’ is a (n1) vector of the translating factors. But since xi = Xi Si(d) + Ti(d), if follows that the associated Marshallian demand functions are

xi*(p/I, d) = Si(d) Xi*[(p1 S1(d))/(I - p’T(d)), … ,(pn Sn(d))/(I - p’T(d))] + Ti(d),

i = 1, …, n i = 1, …, n.

Differentiating this expression with respect to d gives

xi*/d = (Si/d) Xi* + Si (Xi*/d) + Ti/d.

Note that

Xi* = (xi* - Ti)/Si,

Xi*/d = (Xi*/ln(pj))(ln(Sj)/d) - (Xi*/I)(pjTj/d).

Thus,

xi*/d = (Si/d)(xi*-Ti)/Si

+ Si(Xi*/ln(pj))(ln(Sj)/d) - Si (Xi*/I)(pjTj/d) + Ti/d.

This decomposes the behavioral effects of socio-demographic variables (xi*/d) into four additive parts: the direct scale effects [(Si/d)(xi*+Ti)/Si], plus the indirect scale effects [Si(Xi*/ln(pj))(ln(Sj)/d)], plus the indirect translating effects [-Si (Xi*/I)(pjTj/d)], plus the direct translating effects (Ti/d). Note that indirect scale effects are proportional to price effects, and that the indirect translating effects are proportional to income effects.

* Hicksian demand

Under scaling and translating, the expenditure function is

C(p, d, U) = minx{p’x: U = U[(x1 - T1(d))/S1(d), … , (xn - Tn(d))/Sn(d)],

which has for solution the Hicksian demand function xc(p, d, U).

Given

Xi = (xi - Ti(d))/Si(d),

or

xi = Xi Si(d) + Ti(d), i = 1, …, n,

the above expenditure function can be written as

C(p, d, U) = minX {pi [Si(d) Xi + Ti(d)]: subject to U = U(X)},

where X = (X1, …, Xn)’, or

C(p, d, U) = pi Ti(d) + minX {pi Si(d) Xi: subject to U = U(X)}

This has for solution Xic[p1 S1(d), … , pn Sn(d), U], i = 1, …, n. But since xi = Xi Si(d) + Ti(d), if follows that the associated Hicksian demand functions are

xic(p, d, U) = Si(d) Xic[p1 S1(d), … , pn Sn(d)] + Ti(d),

i = 1, …, n i = 1, …, n.

Differentiating this expression with respect to d gives

xic/d = (Si/d) Xic + Si (Xic/d) + Ti/d.

Note that

Xic = (xic - Ti)/Si,

Xic/d = (Xic/ln(pj))(ln(Sj)/d).

Thus,

xic/d = (Si/d)(xic-Ti)/Si + Si(Xic/ln(pj))(ln(Sj)/d) + Ti/d.

This decomposes the Hicksian socio-demographic effects (xic/d) into three additive parts: a direct scaling effect [(Si/d)(xic-Ti)/Si], plus an indirect scaling effect [Si(Xic/ln(pj)) (ln(Sj)/d)], plus a translating effect (Ti/d).

* Properties of the expenditure function

Consider the properties of the expenditure function C(p, d, U). Under scaling and translating, using the envelope theorem, we obtain

C(p, d, U)/ln(d) = pi [Ti/ln(d)] + pi [Si/ln(d)] Xic,

= pi [Ti/ln(d)] + pi [Si/ln(d)](xic – Ti)/Si.

This can be written alternatively as

ln(C)/ln(d) = pi [Ti/ln(d)]/C + wic [ln(Si)/ln(d)],

where wic = pi (xic – Ti)/C is an “adjusted cost share” for the i-th commodity, i = 1, …, n.

Now consider a change from in the socio-demographic variables d from dato db. The associated change in the expenditure function is given by

ln[C(p, db, U)/C(p, da, U)] = ln[C(p, db, U)] – ln[C(p, da, U)]

= [ln(C)/ln(d)] dln(d).

Using the above result, it follows that

C(p, db, U)/C(p, da, U) = exp{[ln(C)/ln(d)] dln(d)}.

= exp[{pi [Ti/ln(d)]/C + wic [ln(Si)/ln(d)]} dln(d)].

This shows how the socio-demographic variables d affect the “cost of living” C through the translating factors T and the scaling factors S.