Demand Theory: A Primer
Table of contents
Introduction
This primer aims to provide a concise and simple introduction to the theory of demand. As such it does not intend to be comprehensive but to provide enough background for the reader to progress rapidly to empirical work.
Notation
Bear in mind throughout that an agent need not be an individual but could be a household, or even when dealing with aggregation, the entire population. Also whenever a variable which is normally indexed (such as prices or quantities) is written without an index this indicates the vector of all of those variables.

Set of goods indexed usually by $i$

$p_{i} =$ Price of good $i$.

$q_{i} =$ Quantity of good $i$ (usually the amount demanded or actually consumed).

$w =$ total wealth of the agent

$y$ income of the agent (often indexed by i to indicate period)

$T$ will indicate time, often the total amount of hours available to an agent.

$g_{i}(w, p)$ is the (Marshallian) demand function for good $i$

$u(q)$ is the utility function. $u$ on its own will usually stand for a particular level of utility

$v(w,p) = max_{q} { u(q) : p \cdot q \leq w }$ is the indirect utility function

$h_{i}(u,p)$ is the Hicksian (compensated) demand function for good i. Known as compensated because they show how demand varies with utility held constant.

$e(u,p)$ is the expenditure/cost function.

The slutsky matrix S defined by $s_{ij} = \partial h_{i}(u,p) / \partial p_{j}$

$b_{i} = \frac{p_{i}q_{i}}{w}$ is the budget share for good i

$e_{i}$ is the wealth elasticity of demand for good k (at current prices and wealth) defined as: $ e_{i} = \partial \log( g_{i}(w,p)) / \partial \log w $

$e_{ij}$ are the crossprice elasticities of demand (if $i = j$ then this is the ownprice elasticity). Defined as: $ e_{ij} = \partial \log( g_{i}(w,p)) / \partial \log p_{j} $
The Basic Properties of Demand Functions
We shall assume a linear budget constraint.^{1} That is:^{2}
$ w = \sum_{k} p_{k}q_{k} $
Restrictions on the Demand Function: The Big Four

Adding Up: From the budget constraint we have: $ w = \sum_{k} p_{k}g_{k}(w,p) $.

Homogeneity The demand function is homogenous of degree 0 in prices and wealth (purely nominal changes should have no effect): $ g_{i}(\lambda w, \lambda p) = g_{i}(w,p) $.

Slutsky Matrix is Symmetric.

Negative SemiDefiniteness of the Slutsky Matrix.
Remarks
Taking derivatives, adding up implies:

Engel Aggregation: $ \sum_{k} p_{k} \frac{\partial g_{k}(w,p)}{\partial w} = 1 $

Cournot Aggregation: $ g_{i} + \sum_{k} p_{k} \frac{\partial g_{k}(w,p)}{\partial p_{i}} = 0, \forall i = 1, .... n $

Invertability (see Deaton and Muellbauer (1980) 48ff). Demand functions that satisfy the four properties above are integrable into a consistent preference ordering. That is we may construct a cost/utility function basis from which such demands can then be derived. Not only is this result important in itself but it demonstrates that these four properties are the only (necessary) consequences of utility maximization.
Other Useful Properties
Roy's Identity: This allows us to relate the Marshallian demand functions to the indirect utility function:
$ \frac{\partial v / \partial p_{i}} {\partial v / \partial w } = g_{i}(u,p) $
Slutsky equation:
$$
s_{ij} = \frac{\partial h_{i}} {\partial p_{j}} = \frac{\partial g_{i}} {\partial p_{j}}
+ \frac{\partial g_{i}} {\partial w} g_{j}
$$
Preference Foundations for Demand Functions
Omitted for time being. This is pretty dull. Refer people to MasCollel ea. .... Plan:

Axioms

Cardinal (value matter) and Ordinal (only order matters) preference theory

The existence of a realvalued utility function

WARP, GARP ... ?
Deriving Demand Functions from Utility
Utility Maximization
Given a cardinal utility function $u(q)$ the consumer has the following utility maximization problem (UMP):
$$ [ max_{q} u(q), \textrm{with } p.q \leq w, q \geq 0 ] $$
Let $ q^{s} $ be values that solve this problem (assuming they exist). Then the demand function is defined by $g(w,p) = q^{s}$
The Dual: Cost Minimization
There exists a problem dual to the original utility maximization problem, namely the expenditure minimization problem (EMP).
$$ [ min_{q} w = p . q, \textrm{subject to } u(q) = u ] $$
Properties of the Cost Function

The cost function is homogenous of degree 1 in prices, i.e. $ c(u,\lambda p) = \lambda c(u,p) $

The cost function is increasing in $u$, nondecreasing in $p$

$c(u,p)$ is concave in prices

The cost function is continuous in $p$ and its first and second derivatives with respect to prices exist everywhere (except possibly on a set of meausure 0)

(Shepherd’s Lemma) Partial derivatives of the cost function with respect to price yield the Hicksian demands: $ \frac{ \partial c(u,p) } { \partial p_{i} } = h{i}(u,p) $
Summary
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Aggregation
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Footnotes