Integrability of demand

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 $X$ and a known walrasian demand function $x:\mathbb {R} _{++}^{L}\times \mathbb {R} _{+}\rightarrow X$, solving the problem of integrability of demand consists in finding a utility function $u:X\rightarrow \mathbb {R}$ such that

$x(p,w)=\operatorname {argmax} \{u(x):p\cdot x\leq w,\;x\in X\}$

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 $e(p,u)$ for the consumer; then with it, one constructs an at-least-as-good set $V_{u}=\{x\in \mathbb {R} _{+}^{L}:u(x)\geq u\}$, which is equivalent to finding a utility function $u(x)$. If the demand function $x(p,w)$ is homogenous of degree zero, satisfies Walras' Law, and has a negative semi-definte substitution matrix $S(p,w)$, then it is possible to follow those steps to find a utility function $u(x)$ that generates $x(p,w)$.

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

$x(p,w)=h(p,v(p,w))$

where $v(p,w)=u(x(p,w))$ is the consumers' indirect utility function and $h(p,u)$ is the consumers' hicksian demand function. Fix a utility level $u_{0}=v(p,w)$ . From Shephard's lemma, and with the identity above we have

${\frac {\partial e(p)}{\partial p}}=x(p,e(p))$ 1

where we omit the fixed utility level $u_{0}$ for conciseness. (1) is a system of PDEs in the prices vector $p$, and Frobenius' theorem can be used to show that if the matrix

$D_{p}x(p,w)+D_{w}x(p,w)x(p,w)$

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

For the second step, by definition,

$e(p)=e(p,u_{0})=\min\{p\cdot x:x\in V_{u_{0}}\}$

where $V_{u_{0}}=\{x\in \mathbb {R} _{+}^{L}:u(x)\geq u_{0}\}$. By the properties of $e(p,u)$, it is not too hard to show that $V_{u_{0}}=\{x\in \mathbb {R} _{+}^{L}:p\cdot x\geq e(p,u_{0})\}$. Doing some algebraic manipulation with the inequality $p\cdot x\geq e(p,u_{0})$, one can reconstruct $V_{u_{0}}$ in its original form with $u(x)\geq u_{0}$. If that is done, one has found a utility function $u:X\rightarrow \mathbb {R}$ that generates consumer demand $x(p,w)$.