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Formal derivation of demand systems

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IV.I.1 Formal derivation of demand systems

The objective in non-probabilistic consumer behavior is to maximize utility such that a budget constraint is satisfied. If U(q) is twice continuously differentiable (then the Hessian of U(q) is symmetric and) then it is possible to obtain a Marshallian demand function q = q(m, p) (which is dependent on the known budget m and a known price vector p (h*1)) where q (h*1) contains the consumption quantities for a set of h commodities such that U(q) reaches a maximum subject to p'q = m:

Formal derivation of demand systems

(IV.I.1-1)

The optimization problem can be formulated as a Lagrange function

(IV.I.1-2)

The first order conditions are

(IV.I.1-3)

Remark that it is implicitly assumed that U(q) is stationary in its optimum and that all marginal utilities are positive.

Since the marginal utilities and (obviously also) prices are strictly positive, the Lagrange multiplier (in (IV.I.1-3)) must be positive as well.

The second order condition must be satisfied in the maximum:

(IV.I.1-4)

It can be shown that under strict quasi-concavity of utility (c.q. strict convexity of the iso-utility curves), that (IV.I.1-4) is satisfied.

It can also be shown that a unique solution exists for q = q(m, p), and the Lagrange multiplier, for which the solution follows from (IV.I.1-3) as

(IV.I.1-5)

where q* = q(m, p).

Obviously (IV.I.1-3) can be written as a differential

(IV.I.1-6)

or

(IV.I.1-7)

where H* is the Hessian matrix of the utility function U(q*).

On noting that we already defined

(IV.I.1-8)

and if we define

(IV.I.1-9)

it is possible to write

(IV.I.1-10)

or equivalently

(IV.I.1-11)

We interpret qm as the vector of marginal income sensitivities of consumption, and Qp as the matrix of marginal price sensitivities of consumption. Remark that the diagonal elements of Qp are the price sensitivities, whereas the off-diagonal elements are the cross-price sensitivities.

By combining (IV.I.1-7) and (IV.I.1-11) we find

(IV.I.1-12)

It is possible to prove that q = q(m, p) is continuously differentiable if and only if

(IV.I.1-13)

so that the property of differentiability holds if the Hessian of the utility function is non-singular.

It can be shown that if U(q) is assumed to have a decreasing marginal rate of substitution (MRS) that H* is non-singular. Therefore if U(q) has a diminishing MRS, the demand equations are continuously differentiable.

Since we know that H* is a symmetric matrix, it follows that

(IV.I.1-14)

is also symmetric.

On substituting (IV.I.1-14) into (IV.I.1-12) we find

(IV.I.1-15)

In (IV.I.1-15) it is quite obvious to see that

(IV.I.1-16)

Furthermore we can rewrite (IV.I.1-14) as

(IV.I.1-17)

thus

(IV.I.1-18)

and

(IV.I.1-19)

and

(IV.I.1-20)

and

(IV.I.1-21)

Now it follows from (IV.I.1-18) and (IV.I.1-19) that

(IV.I.1-22)

which implies that Z is symmetric (since H* is symmetric). Above this the inverse of the Hessian matrix has negative diagonal elements which is equivalent to saying that the indifference curves are convex (see (IV.I.1-4)).

Equation (IV.I.1-20) implies homogeneity meaning that substitution effects are in fact a reallocation of goods without changing the overall level of consumption. Therefore only the relative price changes are important for substitution effects.

In (IV.I.1-21) the adding up property is illustrated. All prices multiplied by their respective propensity to consume sum up to one (i.e. a weighted sum of budget elasticities of demand).

From (IV.I.1-11) we find quite easily

(IV.I.1-23)

Partial substitution of (IV.I.1-15) into (IV.I.1-23) yields

(IV.I.1-24)

From (IV.I.1-16) equation (IV.I.1-24) becomes

(IV.I.1-25)

Substituting (IV.I.1-22) into (IV.I.1-25) gives

(IV.I.1-26)

which is the structural form of our (general) demand system where the first term is attributed to the income effect whereas the second term takes the substitution effect into account.

The substitution effect, due to Hicks and Allan, is given by

.

This can be shown quite easily by using (IV.I.1-15) and (IV.I.1-16)

(IV.I.1-27)

which is just a "compensated price effect" (due to Hicks). In micro economics the Hicksian demand function is equal to the Marshallian function which is "compensated" for the change in real income (such that the original utility level is attained).

Thus from (IV.I.1-27) and (IV.I.1-22) we obtain

(IV.I.1-28)

and thus the second term of (IV.I.1-26) represents the substitution effect.

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