View source for 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 [[Finance:Preference (economics)|preferences]]) from a given walrasian demand function.<ref>https://core.ac.uk/download/pdf/14705907.pdf</ref> 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>X </math> and a known walrasian demand function <math> x: \mathbb{R}_{++}^{L} \times \mathbb{R}_{+} \rightarrow X </math>, solving the problem of integrability of demand consists in finding a utility function <math> u: X \rightarrow \mathbb{R} </math> such that

:<math> 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|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 [[Finance:Expenditure function|expenditure function]] <math>e(p, u)</math> for the consumer; then with it, one constructs an at-least-as-good set <math>V_u = \{x \in \mathbb{R}^L_+: u(x) \geq u\}</math>, which is equivalent to finding a utility function <math>u(x) </math>. If the demand function <math> x(p, w) </math> is [[Homogeneous function|homogenous]] of degree zero, satisfies Walras' Law, and has a [[Definite matrix#Definitions for real matrices|negative semi-definte]] [[Finance:Slutsky equation#Changes in Multiple Prices at Once: The Slutsky Matrix|substitution matrix]] <math> S(p, w)</math>, then it is possible to follow those steps to find a utility function <math> u(x) </math> that generates <math> x(p, w) </math>.<ref name = "mwg">{{cite book |  last1 = Mas-Colell| first1=Andreu | last2 = Whinston | first2 = Micheal D. | last3 = Green | first3 = Jerry R.| year=1995 |title=Microeconomic Theory | publisher=Oxford University Press |pages=75–80 |isbn=978-0195073409}}</ref>

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

:<math>x(p, w) = h(p, v(p, w))</math>

where <math>v(p, w) = u(x(p, w))</math> is the consumers' [[Finance:Indirect utility function|indirect utility function]] and <math>h(p, u)</math> is the consumers' hicksian demand function. Fix a utility level <math>u_0 = v(p, w)</math> {{r|g=nb|r=This arbirtrary choice is valid because utility levels are meaningless since preferences are preserved under monotone transformations of utility functions}}. From [[Finance:Shephard's lemma|Shephard's lemma]], and with the identity above we have

{{NumBlk|:|<math> \frac{\partial e(p)}{\partial p}  = x(p, e(p)) </math>|{{EquationRef|1}}}}

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

:<math>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>S(p, w)</math>, which we assumed to be symmetric firsthand. So ({{EquationNote|1}}) has a solution, and it is (at least theoretically) possible to find an expenditure function <math> e(p) </math> such that <math> p \cdot x(p, e(p)) = e(p)</math>.

For the second step, by definition,

:<math>e(p) = e(p, u_0) = \min \{p \cdot x : x \in V_{u_0}\}</math>

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

==Notes==
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==References==
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[[Category:Microeconomic theories]]

{{Sourceattribution|Integrability of demand}}