Consumer theory The important questions concern the effect of changes in relative prices and real income. Any change in prices changes also the real income. The aim is to decompose the effects of price changes into two parts, the price effect and the income effect. The way to do this is to imagine that when prices change the consumer’s income is changed so that s/he can still reach the utility level that s/he enjoyed before price change. The change from the original situation to the situation with new prices AND new income is called the substitution effect, due to price change. The remainder is called the income effect, due to the change in real income. The formalisation of this in derivative form, i.e. for small changes, is called the Slutsky equation.
Consumer theory
Theorem (Slutsky equation). Consider the Marshalian demand x(p, y ) and denote the utility level achieved at prices p and income y by u ∗ . Then ∂ xi (p, y ) ∂ xih (p, u ∗ ) ∂ xi (p, y ) = − xj (p, y ) ∂ pj ∂ pj ∂y for i, j ∈ {1, ..., n}.
Consumer theory Proof. Denote the utility maximising bundle at (p, y ) by x ∗ and let u ∗ = u (x ∗ ). Differentiate identity xih (p, u ∗ ) = xi (p, e (p, u ∗ )) to get ∂ xih (p, u ∗ ) ∂ xi (p, e (p, u ∗ )) ∂ xi (p, e (p, u ∗ )) ∂ e (p, u ∗ ) = + ∂ pj ∂ pj ∂y ∂ pj Since u ∗ = v (p, y ) and since e and v are inverses we get e (p, u ∗ ) = y . Also the partial of e w.r.t. pj is the Hicksian demand; but evaluated at point u ∗ = v (p, y ) this must equal the Marshalian demand at (p, y ). Inserting these data into expression yields ∂ xih (p, u ∗ ) ∂ xi (p, y ) ∂ xi (p, y ) = + xj (p, y ) ∂ pj ∂ pj ∂y
Consumer theory
What happens in the proof? We take an identity connecting the Hicksian and Marshalian demands. Then we fix the utility level, i.e., the indifference curve things are evaluated at; this, however, is arbitrary so that the results holds generally. Then we use the fact that the cost minimising bundle and utility maximising bundle are the same for the income that allows the consumer to get to the fixed indifference curve.
Consumer theory We should like to understand the effect of price changes to the Marshalian, observable, demand. This seems to depend on the Hicksian, unobservable, demand. But theory still tells something about it. Theorem For i ∈ {1, ..., n} the own price effect on Hicksian demand is negative
∂ xih (p,u ∗ ) ∂ pi
≤ 0.
Proof. The expenditure function e is concave in p, i.e. its matrix of second derivatives is negative semidefinitive. Law of demand: For a normal good the own price effect on demand is negative.
Consumer theory
Theorem For i ∈ {1, ..., n}
∂ xih (p,u) ∂ pj
=
∂ xjh (p,u) . ∂ pi
Proof. ∂ xih (p,u) ∂ pj
=
∂ 2 e(p,u) ∂ pi ∂ pj
=
∂ 2 e(p,u) ∂ pj ∂ pi
=
∂ xjh (p,u) . ∂ pi
Consumer theory Theorem Let x h (p, u) be the Hicksian demand (vector). The substitution matrix ∂ x1h (p,u) ∂ xnh (p,u) . . . ∂ p1 ∂ p1 . . σ (p, u) = . . . . ∂ xnh (p,u) ∂ p1
. . .
is symmetric and negative semidefinitive. Proof. Same as for Theorem.
∂ xnh (p,u) ∂ pn
Consumer theory Theorem Let x(p, y ) be the Marshalian demand (vector), and denote ) ) . The Slutsky matrix sij = ∂ x∂i (p,y + xj (p, y ) ∂ xi∂(p,y pj y s11 . . . s1n . . . s(p, y ) = . . . sn1 . . . snn is symmetric and negative semidefinitive. Proof. sij =
∂ xnh (p,u) ∂ p1
.
Consumer theory
Finally we have a result that says something about the observable parts of the theory. Theory of the consumer has the following testable implications: Homogeneity of demand, budget balancedness, symmetricity and negative semidefiniteness of the Slutsky matrix. Are there other testable implications?
Consumer theory
ABOUT DUALITY Whenever we are given a function that has the properties of the indirect utility function or the expenditure function has, it is an indirect utility function or an expenditure function. This means that from these functions one can always ’recover’ the underlying preferences, or utility function, that generates exactly the given indirect utility or expenditure functions. We bypass the details of this development but present one result.
Consumer theory Normalise prices so that income y = 1, and consider the indirect utility function v (p, 1) = maxx u(x) subject to px = 1. Proposition. Given v (p, 1) the utility function is given by u(x) = minp v (p, 1) subject to px = 1. We utilise this to construct the inverse demand functions. Normalise prices so that income y = 1. Now u(x) = v (p(x), 1), where p(x) is the vector of inverse demands, and p(x)x = 1. The Lagrangean associated with the programme for u is given by L(p, λ ) = −v (p, 1) − λ [px − 1]. By the envelope theorem
∂ u(x) ∂ xi
= −λ ∗ pi∗ where p ∗ = p(x).
Multiply the partial by xi and sum over all indeces to get ∂ u(x) ∑ni=1 xi ∂ xi = −λ ∗ ∑ni=1 pi∗ xi = −λ ∗ .
Consumer theory
Inserting −λ ∗ above and solving gives pi (x) =
∂ u(x) ∂ xi ∂ u(x) n ∑j=1 xj ∂ xj
Consumer theory
Not all implications are independent. Theorem If x(p, y ) satisfies budget balancedness and its Slutsky matrix is symmetric, it is homogeneous of degree zero in (p, y ).
Consumer theory Proof. Differentiate the budget constraint (equality) at point (tp, ty ) where ∂ x (tp,ty ) ∂ x (tp,ty ) t > 0 to get ∑nj=1 tpj j ∂ pi = 1. = −xi (p, y ) and ∑nj=1 tpj j ∂ y Consider function fi (t) = xi (tp, ty ). Pay attention to indeces! 0 ) ) Differentiate to get fi (t) = ∑nj=1 pj ∂ xi (tp,ty + y ∂ xi (tp,ty . Dividing ∂ pj ∂y the budget balance tpx(tp, ty ) = ty by t yields px(tp, ty ) = y which is equivalent to ∑nj=1 pj xj (tp, ty ) = y . Using this h i 0 ) ∂ xi (tp,ty ) fi (t) = ∑nj=1 pj ∂ xi (tp,ty and the term in the + x (tp, ty ) j ∂ pj ∂y brackets is the entry sij in the Slutsky matrix. As this is symmetric one can swap the h indeces to get i 0 ∂ xj (tp,ty ) ∂ xj (tp,ty ) n fi (t) = ∑j=1 pj + x (tp, ty ) = i ∂ pi ∂y h i h i ∂ xj (tp,ty ) ∂ xj (tp,ty ) n n 1 1 tp + x (tp, ty ) tp . The first ∑j=1 j ∂ y t ∑j=1 j t i ∂ pi bracket equals −xi (tp, ty ) and second bracket equals unity.
Consumer theory
The only testable implications of the theory are budget balancedness, symmetry of the Slutsky matrix and negative semidefiniteness of the Slutsky matrix.
Consumer theory
Theorem (Integrability theorem). A continuously differentiable function n x : Rn+1 ++ → R+ is the demand function generated by an increasing, quasiconcave utility function if (and only if) it satisfies budget balancedness, and its Slutsky matrix is symmetric and negative semidefinite.
Consumer theory
Proof. (Idea) Consider x(p, y ) that satisfies the conditions of the theorem. Try to find e such that ∂ e(p,u) = xi (p, e(p, u)). If this partial ∂ pi differential equation has a solution then ∂ xi (p,y ) ) ∂ 2 e(p,u) + xj (p, y ) ∂ xi∂(p,y where we have utilised ∂ pi ∂ pj = ∂ pj y Shephard’s lemma and denoted y = e(p, u). This is a necessary condition for the existence of solution. Frobenius’s theorem states that this is also sufficient. But the right hand side is exactly the sij term of the Slutsky matrix. The solution is an expenditure function if it has the properties of theorem. This one can show.
Consumer theory Remember section about WARP. There we showed that the Slutsky matrix of the associated choice function is homogeneous of degree zero and negative semidefinite. If one can show symmetry consumer’s demand theory and choice based on revealed preference are equivalent theories. But given that choices satisfy WARP there is not necessarily a utility function that generates the choices expect in the two-good-case. Symmetry (and the utility function that generates the choices) can be achieved by postulating that the choices satisfy the strong axiom of revealed preference (SARP): For each sequence of distinct bundles x 0 , x 1 , ..., x k if x i−1 is revealed preferred to x i then x k is not revealed preferred to x 0 .
