Integrability of demand

Last updated February 15, 2024

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.^[1] 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} _{x\in X}\{u(x):p\cdot x\leq w\}

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 the properties of expenditure functions, one can construct 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 demand $x(p,w)$ .^[4]

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 $u_{0}=v(p,w)$ ^{[nb 1]}. 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 ^[4] 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 $x(p,w)$ .

Notes

↑ This arbirtrary choice is valid because utility levels are meaningless since preferences are preserved under monotone transformations of utility functions

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References

↑ https://core.ac.uk/download/pdf/14705907.pdf
↑ Samuelson, Paul (1950). "The Problem of Integrability in Utility Theory". Economia. 17 (68): 355–385. doi:10.2307/2549499.
↑ Hurwicz, Leonid; Uzawa, Hirofumi (1971). "Chapter 6: On the integrability of demand functions". In Chipman, John S.; Richter, Marcel K.; Sonnenschein, Hugo F. (eds.). Preferences, utility, and demand: A Minnesota symposium. New York: Harcourt, Brace, Jovanovich. pp. 114–148.
1 2 Mas-Colell, Andreu; Whinston, Micheal D.; Green, Jerry R. (1995). Microeconomic Theory. Oxford University Press. pp. 75–80. ISBN 978-0195073409.

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[5] This arbirtrary choice is valid because utility levels are meaningless since preferences are preserved under monotone transformations of utility functions

[1] ttps://core.ac.uk/download/pdf/14705907.pdf

[2] Samuelson, Paul (1950). "The Problem of Integrability in Utility Theory". Economia. 17 (68): 355–385. doi:10.2307/2549499.

[3] Hurwicz, Leonid; Uzawa, Hirofumi (1971). "Chapter 6: On the integrability of demand functions". In Chipman, John S.; Richter, Marcel K.; Sonnenschein, Hugo F. (eds.). Preferences, utility, and demand: A Minnesota symposium. New York: Harcourt, Brace, Jovanovich. pp. 114–148.

[mwg-4] 1 2 Mas-Colell, Andreu; Whinston, Micheal D.; Green, Jerry R. (1995). Microeconomic Theory. Oxford University Press. pp. 75–80. ISBN 978-0195073409.

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