Finance:Integrability of demand

Short description: Problem in microeconomics

In microeconomic theory, the problem of the integrability of demand functions deals with recovering a utility function (that is, consumer preferences) from a given walrasian demand function. The "integrability" in the name comes from the fact that demand functions can be shown to satisfy a system of partial differential equations in prices, and solving (integrating) this system is a crucial step in recovering the underlying utility function generating demand.

Mathematical formulation

Given consumption space [math]\displaystyle{ X }[/math] and a known walrasian demand function [math]\displaystyle{ x: \mathbb{R}_{++}^{L} \times \mathbb{R}_{+} \rightarrow X }[/math], solving the problem of integrability of demand consists in finding a utility function [math]\displaystyle{ u: X \rightarrow \mathbb{R} }[/math] such that

[math]\displaystyle{ x(p, w) = \operatorname{argmax}_{x \in X} \{u(x) : p \cdot x \leq w\} }[/math]

That is, it is essentially "reversing" the consumer's utility maximization problem.

Sufficient conditions for solution

There are essentially two steps in solving the integrability problem for a demand function. First, one recovers an expenditure function [math]\displaystyle{ e(p, u) }[/math] for the consumer; then with it, one constructs an at-least-as-good set [math]\displaystyle{ V_u = \{x \in \mathbb{R}^L_+: u(x) \geq u\} }[/math], which is equivalent to finding a utility function [math]\displaystyle{ u(x) }[/math]. If the demand function [math]\displaystyle{ x(p, w) }[/math] is homogenous of degree zero, satisfies Walras' Law, and has a negative semi-definte substitution matrix [math]\displaystyle{ S(p, w) }[/math], then it is possible to follow those steps to find a utility function [math]\displaystyle{ u(x) }[/math] that generates [math]\displaystyle{ x(p, w) }[/math].

Proof: if the first two conditions (homogeneity of degree zero and Walras' Law) are met, then duality between the expenditure minimization problem and the utility maximization problem tells us that

[math]\displaystyle{ x(p, w) = h(p, v(p, w)) }[/math]

where [math]\displaystyle{ v(p, w) = u(x(p, w)) }[/math] is the consumers' indirect utility function and [math]\displaystyle{ h(p, u) }[/math] is the consumers' hicksian demand function. Fix a utility level [math]\displaystyle{ u_0 = v(p, w) }[/math] . From Shephard's lemma, and with the identity above we have

[math]\displaystyle{ \frac{\partial e(p)}{\partial p} = x(p, e(p)) }[/math]

(1)

where we omit the fixed utility level [math]\displaystyle{ u_0 }[/math] for conciseness. (1) is a system of PDEs in the prices vector [math]\displaystyle{ p }[/math], and Frobenius' theorem can be used to show that if the matrix

[math]\displaystyle{ D_p x(p, w) + D_w x(p, w) x(p, w) }[/math]

is symmetric, then it has a solution. Notice that the matrix above is simply the substitution matrix [math]\displaystyle{ S(p, w) }[/math], which we assumed to be symmetric firsthand. So (1) has a solution, and it is (at least theoretically) possible to find an expenditure function [math]\displaystyle{ e(p) }[/math] such that [math]\displaystyle{ p \cdot x(p, e(p)) = e(p) }[/math].

For the second step, by definition,

[math]\displaystyle{ e(p) = e(p, u_0) = \min \{p \cdot x : x \in V_{u_0}\} }[/math]

where [math]\displaystyle{ V_{u_0} = \{x \in \mathbb{R}^L_+: u(x) \geq u_0\} }[/math]. By the properties of [math]\displaystyle{ e(p, u) }[/math], it is not too hard to show that [math]\displaystyle{ V_{u_0} = \{x \in \mathbb{R}^L_+: p \cdot x \geq e(p, u_0)\} }[/math]. Doing some algebraic manipulation with the inequality [math]\displaystyle{ p \cdot x \geq e(p, u_0) }[/math], one can reconstruct [math]\displaystyle{ V_{u_0} }[/math] in its original form with [math]\displaystyle{ u(x) \geq u_0 }[/math]. If that is done, one has found a utility function [math]\displaystyle{ u: X \rightarrow \mathbb{R} }[/math] that generates consumer demand [math]\displaystyle{ x(p, w) }[/math].