2 The Classical Theory of Demand and Preferences

2.1 Introduction

This section introduces the classical theory of individual preference and the connection between this and the observable choices that individuals make. To do this, we derive the demand that is implied by the preferences of an individual and the budget set available to them.

Reading material: MWG Ch. 3, Kreps Ch. 2
Proved results: A Representation Theorem

2.2 Preferences

Definition 2.1 Let \(X\) be a set. The Cartesian product of the set \(X\) is the set of all pairs \((x_{1},x_{2})\) where \(x_{i} \in X\) for \(i \in \{1,2\}\). This cartesian product is denoted \(X \times X\).

Definition 2.2 Let \(X\) be a set. A preference (or binary) relation \(\succsim\) is a subset of \(X \times X\). We say that for \(x,y \in X\), \(x \succsim y\) if \((x,y) \in \succsim\).

The preferences of an individual are represented by a weak preference relation over bundles in a commodity set. Let \(X\) denote a set of elements that we will call the consumption set.

From the definition of a weak preference relation we can define strong preference.

Definition 2.3 For the weak preference relation \(\succsim\), we define the strong preference relation \(\succ\) by saying that \(x \succ y\) if \(x \succsim y\) and not \(y \succsim x\). Likewise we define the indifference relation as \(x \sim y\) if \(x \succsim y\) and \(y \succsim x\).

The following two definitions will be useful for the preference relations used in this class.

Definition 2.4 A preference relation \(\succsim\) is complete if for any two \(x,y \in X\), either \(x \succsim y\), \(y \succsim x\) or both.

Definition 2.5 A preference relation \(\succsim\) is transitive if for any three \(x,y,z \in X\), if \(x \succsim y\) and \(y \succsim z\) then \(x \succsim z\).

These two properties are assumptions. That is, an individual’s preferences need not satisfy them. However, if completeness does not hold for a particular preference relation then there are choice situations where the model assumed about individuals does not say what the person will do. Such situations are interpreted as a problem with the model, as opposed to a problem with the individual. That is, a model’s utility comes from it’s ability to make predictions. If the model doesn’t make predictions in some situations, then it is not useful.

If a preference relation doesn’t satisfy transitivity, then there where will be sets of choice situations where an individual displays cyclic behavior. If \(x \succsim y\) and \(y \succsim z\) but \(z \succsim x\) then the individual would be unable to choose a weakly preferred element from the set \(\{x,y,z\}\) because for every element of the set there is another element that is preferred to it. As we will see, this idea is mathematically analogous to the existance of a maximum of a continuous function over a compact set.

It will often be useful to make additional assumptions about a preference relation \(\succsim\). Consider a consumption set \(X\) where every element \(x \in X\) can be characterized by a set of elements \(\{x_{i}\}\) for \(i\) in some index set \(I\).

Definition 2.6 A preference relation \(\succsim\) is monotone if for every \(x,y \in X\) for which \(y_{i} \ge x_{i}\) for all \(i \in I\), \(y \succsim x\). A relation is strictly monotone if \(y_{i} > x_{i}\) for all \(i \in I\) implies that \(y \succ x\).

Intuitively, a preference relation is monotonic if having more of each of the elements of consumption results in a choice maker being better off.

Definition 2.7 (This requires a vector-space like consumption set.) A preference relation \(\succsim\) is locally nonsatiated if for every \(x \in X\) and every \(\epsilon > 0\), there exists a \(y \in X\) such that \(||x - y|| < \epsilon\) and \(y \succ x\).

To understand the next potential assumption, we must recall the definition of convexity of a set.

Definition 2.8 A set \(X\) is convex if for any two elements \(x,y \in X\) and real number \(\alpha \in [0,1]\), the object \(\alpha x + (1-\alpha) y \in X\).

Note.: Notice that the definition of convexity requires that addition and scalar multiplication be defined over the set \(X\). Much (but not all) of the analysis of preference relations occurs in vector spaces (discussed elsewhere) which have these two operations defined.

Given this definition, we can define the convexity of a preference relation.

Definition 2.9 A preference relation \(\succsim\) is convex if for every \(x \in X\), the set \(\{y \in X: y \succsim x\}\) is convex (in the set sense).

This set \(\{y \in X: y \succsim x\}\) is known as the upper contour set of the point \(x\).

Definition 2.10 A preference relation \(\succsim\) is strictly convex if for every \(x \in X\), the set \(\{y \in X: y \succsim x\}\) is convex (in the set sense).

Definition 2.11 The relation \(\succsim\) is continuous if for any sequence of pairs \(\{(x^n,y^n)\}_{n=1}^{\infty}\) with limit \((x,y)\), \(x^n \succsim y^n\) for all \(n\) implies that \(x \succsim y\).

Definition 2.12 A subset \(Z\) of \(X\) is called \(\succ\) -order dense if for all \(x,y \in X\) such that \(x \succ y\), there exists some \(z \in Z\) such that either \(x \succ z \succsim y\) or \(x \succsim z \succ y\).

Proposition.

Let \(X\) be a choice set. A preference relation \(\succsim\) over \(X\) is complete, transitive and has a countable \(\succsim\)-order dense subset \(Z\) if and only if there exists a continuous utility function \(u:X \rightarrow \mathbb{R}\) for which \(u(x) \ge u(y)\) if and only if \(x \succsim y\).

Proof.

First, let’s prove that if there exists a \(u(\cdot)\) satisfying the above property then, \(\succsim\) is complete, transitive and continuous. Assume that such a \(u\) exists. Completeness of \(\succsim\) follows from the fact that \(>\) is complete. That is, for any two real numbers \(u(x)\) and \(u(y)\), either \(u(x) \ge u(y)\), \(u(y) \ge u(x)\) or both. Likewise for transitivity. For any three of \(u(x), u(y)\) and \(u(z)\), if \(u(x) \ge u(y)\) and \(u(y) \ge u(z)\), then \(u(x) \ge u(z)\).

To go the other way, let \(Z = \{z_{1},z_{2}, \ldots\}\). For each \(z_n\), define the number \(r(z_n) = \left(\frac{1}{2}\right)^{n}\). Now, for each \(x \in X\), let \(\overline{Z}(x) = \{z \in Z: z \succsim x\}\) and let \(\underline{Z}(x) = \{z \in Z: x \succsim z\}\). Notice that if \(x \succsim y\) then \(\overline{Z}(x) \subseteq \overline{Z}(y)\) and \(\underline{Z}(x) \supseteq \underline{Z}(y)\). Moreover, if \(x \succ y\) then at least one of these subset relations is strict (by \(\succ\)-order density) which implies that there is some \(z \in Z\) for which either \(x \succ z \succsim y\) or \(x \succsim z \succ y\) or both. Define

\[u(x) = \sum_{z \in \underline{Z}(x)} r(z) - \sum_{z \in \overline{Z}(x)} r(z).\]

Notice that both of these sums are well defined and therefore \(u(x)\) is well defined. Since \(\underline{Z}(x) \supseteq \underline{Z}(y)\), \(\sum_{z \in \underline{Z}(x)} r(z) > \sum_{z \in \underline{Z}(y)} r(z)\) and since \(\overline{Z}(x) \subseteq \overline{Z}(y)\), \(\sum_{z \in \overline{Z}(x)} r(z) < \sum_{z \in \overline{Z}(y)} r(z)\), where again, at least one of these relationships is strict. Thus, \(u(x) \ge u(y)\) and \(u(x) > u(y)\) if \(x \succ y\). You are left to prove that this function will be continuous if the preference relation is continuous.

2.3 Utility Maximization

For the remainder of this section, we will consider preferences that can be represented by a utility function \(u(x)\) over a choice set \(X\).

2.3.1 Budget sets

Definition 2.13 A linear functional is a function \(f: X \rightarrow \mathbb{R}\) that has the property that \(f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)\) for all \(x,y \in X\) and \(\alpha,\beta \in \mathbb{R}\). The space of all linear functionals of the set \(X\) is denote \(X^{*}\) and is known as the dual space of \(X\) or sometimes just the "dual of \(X\)".

In the arena of consumer demand, linear functionals play the role of producing the cost of any particular consumption bundle. That is, if I want to consume the bundle \(x \in X\), then the cost of consuming such a bundle is \(f(x)\).

Example 2.1 In the space \(\mathbb{R}^n\), the set of linear functionals is the set of inner products formed with elements of \(\mathbb{R}^n\). Specifically, if my consumption space is the set of all possible bundles of apples and oranges then (assuming we allow for fractional consumption of fruit) the choice set is \(X = \mathbb{R}^{2}_{+}\). The set of linear functionals over this set is the space of functions \(\{f: f(x) = a'x for a \in \mathbb{R}^2\}\).

Example 2.2 Let \(X = \ell\) (the space of all countably infinite sequences of real numbers). A natural first place to look for the dual \(X^{*}\) is the set of all functionals of the form \(\\{f: f(x) = \sum_{i=0}^{\infty}a_{i}x_{i} \mbox{ for } a_{i} \in \mathbb{R}}\). This however, is not the set of linear functionals over this set. Can you see why?

This is a first example of a recurrent theme in economic (and mathematical) analysis: infinite dimensional spaces require extra thought. However, as we will see, it is both possible and quite useful to conduct economic analysis over these infinite dimensional spaces.

Definition 2.14 Prices over a commodity space \(X\) is an element of the set of linear functionals

Proposition 2.1 (Heine-Borel) A subset \(S\) of \(\mathbb{R}^{n}\) is compact if and only if it is closed and bounded.

2.4 Optimal Demand

Definition 2.15 A function \(f:X \rightarrow \mathbb{R}\) is quasiconcave if its upper contour sets are convex.

Theorem 2.1 (Weierstrass) A continuous function over a compact set attains a maximum.

Proposition 2.2 If \(p \in \mathbb{R}^{n}_{++}\), \(x \ge 0\) and \(u(\cdot)\) is continuous then the utility maximization problem has a solution.

Proof. If \(p \in \mathbb{R}^{n}_{++}\) then \(B(p,w)\) is compact. [Can you provide a counter example from \(\mathbb{R}^{n}_{+}\)?]. Since \(u(\cdot)\) is continuous, it has a solution.

Proposition 2.3 If \(p \in \mathbb{R}^{n}_{++}\) and \(u(\cdot)\) is continuous and locally non-satiated, then the demand correspondence \(x(p,w)\) has the following properties:

\(x(\alpha p, \alpha w) = x(p,w)\) (homogeneity of degree 0)

\(px = w\) for all \(x \in x(p,w)\).

If \(\succsim\) is convex, so that \(u(\cdot)\) is quasiconcave, then \(x(p,w)\) is a convex set. Moreover, if \(\succsim\) is strictly convex, so that \(u(\cdot)\) is strictly quasiconcave, then \(x(p,w)\) is a single element.

Proof.

2.4.1 Comparative statics of the UMP

Consider the first order conditions from the UMP.

\[\begin{aligned} \begin{aligned} \bigtriangledown u(x) \le \lambda p \\ x'[\bigtriangledown u(x) - \lambda p] = 0 \end{aligned} \end{aligned}\]

Theorem 2.2 (Implicit Function Theorem) Consider the system of equations \(f(x;q) = 0\). Let \(D_{x}f(x_{0};q_{0})\) be non-singular. Then there exists a function \(\hat x(q)\) defined at a neighborhood of solution \((x_{0},q_{0})\) such that \(f(\hat x(q),q) = 0\) and further more

\[D_{q}\hat x(q) = -D_{x}f(x;q)^{-1}D_{q}f(x;q)\]

Proof (sketch). Consider \(D_{x}f(x;q)D_{q}\hat{x}(q) - D_{q}f(x;q) = 0\).

2.4.2 Indirect utility

The indirect utility function \(v(p,w)\) is

Homogenous of degree zero

Strictly increasing in \(w\) and nonincreasing in \(p_{l}\) for an \(l\).

Quasiconvex (the set \(\{(p,w): v(p,w) \le \overline{v}\}\) is convex for any \(\overline{v}\).

Continuous in \(p\) and \(w\).

Proof. We prove number 4. Let \(v(p,w) = u(x(p,w))\). If \(x(p,w)\) is continuous, then \(v(p,w)\) is continuous, but we can’t guarantee that \(x(p,w)\) is single-valued. However, we can prove that for some sequence of \((p^{n},w^{n})\) that converges to \((p,w)\) and \(x^{n} \in x(p^{n},w^{n})\) where \(x^{n} \rightarrow \hat{x}\), that \(\hat{x} \in x(p,w)\). To see this, consider the sequence \(x^{n} \in x(p^{n},w^{n})\) with \(x^{n} \rightarrow \hat{x}\), but \(\hat{x} \notin x(p,w)\). We know that \(p^{n}\cdot x^{n} \le w^{n}\) for all \(n\) and that both sides of this inequality are continuous, so \(p\cdot \hat{x} \le w\). So \(\hat{x}\) is feasible under \(p,w\). Since it is not optimal there must be \(\overline{x} \in x(p,w)\) for which \(u(\overline{x}) > u(\hat{x})\). Now, since \(u(\cdot)\) is continuous, there is another bundle \(y\), arbitrarily close to \(\overline{x}\) such that \(p\cdot y < w\) and \(u(y) > u(\hat{x})\). If \(n\) is large enough, by continuity of the linear functional, it must be that \(p^{n}\cdot y < w^{n}\). Since \(y\) is feasible we must have that \(u(x^{n}) > u(y)\). By the contintuity of \(u(\cdot)\) and taking limits we have that \(u(\hat{x}) \ge u(y)\) which is a contradiction.

We can use this result to show that \(v(p,w)\) is continuous in \(p\) and \(w\).

2.5 Expenditure Minization

The problem \(\min_{x} p\cdot x\) s.t. \(u(x) \ge \overline{u}\) is called the expenditure minimization problem (EMP). The minimized expenditure is denoted \(e(p,u)\) and called the expenditure function and the solution to this problem is \(h(p,u)\) and is called Hicksian demand.

Proposition 2.4 The expenditure function \(e(p,u)\) is

Homogenous of degree one in p

Strictly increasing in u and non-decreasing in \(p_l\)

Concave in \(p\)

Continuous in \(p\) and \(u\)

Proof. For concavity, let \(\overline{u}\) be the required utility level. Define \(p'' = \alpha p + (1-\alpha)p'\). If \(x''\) is optimal under \(p''\), then

\[\begin{aligned} \begin{aligned} e(p'',\overline{u}) = p'' \cdot x'' \\ = \alpha p \cdot x'' + (1-\alpha) p' \cdot x'' \\ \ge \alpha e(p,\overline{u}) + (1-\alpha)e(p',\overline{u}) \end{aligned} \end{aligned}\]

Proposition 2.5 Let \(h(p,u)\) be a solution to the EMP for a continuous, locally non-satiated preference relation. We call this Hicksian demand. Hicksian demand has the following properties:

Homogeneity of degree zero in \(p\). That is, \(h(\alpha p, u) = h(p,u)\).

No excess utility. For any \(x \in h(p,u), u(x) = u\).

Convexity/uniqueness. If \(\succsim\) is convex, then \(h(p,u)\) is a convex set. If \(\succsim\) is strictly convex, so \(u(\cdot)\) is strictly quasiconcave, then there is a unique element in \(h(p,u)\).

Proof. See book. These methods are similar to those of marshallian demand.

2.6 Analytical Properties of Demand

Proposition 2.6 (Compensated Law of demand) Suppose that \(u(\cdot)\) is continuous and represents a locally non-satiated preference relation. Assume that \(h(p,u)\) is single valued for all \(p \in \mathbb{R}^{n}_{++}\). Then Hicksian Demand satisfies

\[(p' - p)\cdot[h(p',u) - h(p,u)] \le 0\]

Proof. Note that

\[\begin{aligned} \begin{aligned} p'\cdot h(p',u) & \le p'\cdot h(p,u) \\ p\cdot h(p,u) & \le p\cdot h(p',u) \end{aligned} \end{aligned}\]

Now subtract right from left and then add the top and bottom (or vice versa).

Proposition 2.7 Suppose that \(u(\cdot)\) is a continuous utility function representing a locally nonsatiated and strictly convex preference relation \(\succsim\) (note that this implies uniqueness). For all \(p\) and \(u\), \(h(p,u)\) is the derivative vector of the expenditure function with respect to prices:

\[h(p,u) = D_{p}e(p,u)\]

That is, \(h_{l}(p,u) = \partial e(p,u)/\partial p_{l}\) for all \(l\).

Proof (First order condition.). Assume that \(h(p,u)\) is strictly positive for all goods \(l\) and that \(h(p,u)\) is differentiable. Then,

\[\begin{aligned} \begin{aligned} D_{p}e(p,u) & = D_{p}[p\cdot h(p,u)] \\ & = h(p,u) + [p\cdot D_{p}h(p,u)]' \end{aligned} \end{aligned}\]

From the first order conditions of the EMP, \(p = \lambda D_{x} u(h(p,u))\), so plugging this into the second term gives

\[D_{p}e(p,u) = h(p,u) + \lambda[ D_{x}u(h(p,u)) \cdot D_{p}h(p,u)]'\]

Since \(u(h(p,u)) = u\) for all \(p\), \(D_{x}u(h(p,u)) = 0\).

Proposition 2.8 Suppose \(u(\cdot)\) is a continuous utility function representing a locally nonsatiated and strictly convex preference relation \(\succsim\). Suppose that \(h(\cdot, u)\) is continuously differentiable at \((p,u)\). Then

\(D_{p}h(p,u) = D^{2}_{p}e(p,u)\)
\(D_{p}h(p,u)\) is a negative semidefinite matrix (\(x'Mx \le 0\), all eigenvalues are weakly negative, follows from the concavity of \(h\))
\(D_{p}h(p,u)\) is a symmetric matrix
\(D_{p}h(p,u)p = 0\) (follows from homogeneity of degree zero of \(h(p,u)\) in \(p\). Since \(h(\alpha p,u) - h(p,u) = 0\) one can differentiate with respect to \(\alpha\) to get the result. )

Definition 2.16 Two goods \(i\) and \(j\) are substitutes if \(\partial h_{i}(p,u)/\partial p_{j} \ge 0\) and complements otherwise.

Note.: Property 4 above implies that each good must have at least 1 substitute.

Proposition 2.9 (Relationship between x and h) Let \(u(\cdot)\) be continuous and represent a LNS preference relation \(\succsim\). Let \(p \in \mathbb{R}^{n}_{++}\). Let \(w > 0\).

If \(x\) solves the UMP then \(x\) solves the EMP with \(u = u(x)\). Furthermove, \(p \cdot x\) from the EMP is exactly \(w\).
If \(x\) solves the EMP with utility level \(u\), then \(x\) solves the UMP when \(w = p\cdot x\). Furthermore, u(x) = u.

Beyond the law of demand, what do we know about the matrix \(D_{p}h\)?

Proposition 2.10 (Slutsky) Suppose \(u(\cdot)\) is a continuous utility function representing a locally nonsatiated preference relation \(\succsim\). Then for all \((p,w)\), and \(u = v(p,w)\),

\[D_{p}h(p,u) = D_{p}x(p,w) + D_{w}x(p,w)x(p,w)'\]

Proof. \(h(p,u) = x(p,e(p,u))\). Now differentiate this with respect to \(p\) and substitute the fact that \(h(p,u) = x(p,e(p,u)) = x(p,w)\) to get the result.

\(So D_{p}h = S(p,w)\) is negative semidefinite and relates to the marshallian demand function \(x\).

Proposition 2.11 (Roy’s identity) Suppose that \(u(\cdot)\) is continuous and represents a lns pref. \(\succsim\). Assume that \(v(p,w)\) is differentiable everywhere. Then

\[x(p,w) = -\frac{1}{D_{w}v(p,w)} D_{p}v(p,w)\]

Proof. Differentiate \(v(p,e(p,u))\) with respect to \(p\). Then subsitute \(x\) for \(h\) and rearrange.

2.7 Integrability

2.8 Choice, Demand and the Weak Axiom

Definition 2.17 Demand satisfies the WA of revealed preference if for any two price-wealth combo \((p,w)\) and \((p',w')\), if \(px(p',w') \le w\) and \(x(p',w') \ne x(p,w)\) then \(p'x(p,w) > w'\).

Proposition 2.12 Demand \(x(p,w)\) satisfies WARP if for any compensated price change we have \((p'-p)[x(p',w')-x(p,w)] \le 0\).

2.9 Individual Welfare (Partial equilibrium)

Indirect utility measures the change in utility that arises from changes in the economic environment. However, since any monotonic function of utility also represents preferences, think about the following monotonic transformation. Recall that the expenditure function for continuous, LNS preferences is increasing in \(u\). So for the indirect utility function \(v(p,w)\), the function \(e(p,v(p,w))\) is a monotonic transformation of utility and so also represents preferences. It does so in a way that is measured in dollars. Two ways to use this to measure welfare changes in individuals are

Definition 2.18 Compensating variation is

\[CV(p^{0},p^{1},w) = e(p^{1},u^{1}) - e(p^{1},u^{0}) = w - e(p^{1},u^{0})\]

This can be interpreted as the net revenue of a planner who is required to compensate a person for the change in prices after they have occurred.

Definition 2.19 Equivalent variation is

\[EV(p^{0},p^{1},w) = e(p^{0},u^{1}) - e(p^{0},u^{0}) = e(p^{0},u^{1}) - w\]

This can be interpreted as the amount that a person has to be compensated in order to accept a price change.

Lemma 2.1 With normal goods, EV > CV.

Lemma.

\[EV = \int_{p_{1}^{1}}^{p^{0}_{1}}h_{1}(p_{1},\overline{p}_{-1},u^{1})dp_{1}\]

Lemma.

\[CV = \int_{p_{1}^{1}}^{p^{0}_{1}}h_{1}(p_{1},\overline{p}_{-1},u^{0})dp_{1}\]

What if we can’t observe the expenditure function? Can we say anything about welfare changes?

Lemma 2.2 (Revealed Preference) If a consumer has a locally non-satiated rational preference relation and \((p^{1} - p^{0})x^{0} < 0\) then the price change is strictly welfare improving.

Proof. Revealed preference. \(x^{0}\) is still affordable under \(p^{1}\) and is in the interior of the budget set. Therefore, local non-satiation implies that the consumer can do strictly better.

2.10 Aggregate Demand

Aggregate demand is the aggregation of individual demands. However, aggregate demand does not analytically necessarily behave like individual demands. Three questions one could ask about aggregate demand are: 1) In what circumstances do the wealth effects in aggregate demand look like the wealth effects of individual demand?, 2) does aggregate demand satisfy the weak axiom of revealed preference? and more specifically 3) Can aggregate demand be represented by the preferences of a single consumer?

2.11 AD and Aggregate Wealth

Aggregate demand is

\[x(p,w_{1}, w_{1}, \ldots, w_{I}) = \sum_{i=1}^{I}x_{i}(p,w_{i})\]

If \(\overline{w} = \sum_{i=1}^{I}\), when is it okay to write \(x(p,w_{1},w_{2}, \ldots, w_{I}) = x(p,\overline{w})\)? If this is to be true, then for any two wealth allocations \(w = (w_{1}, \ldots, w_{n} )\) and \(w' = (w_{1}', \ldots, w_{I}')\) for which \(\sum w_i = \sum w_{i}'\) demand must be equal.

To understand what needs to be true in order for this to hold, consider a small change \(\Delta w\) in the wealth distribution that leaves aggregate wealth unchanged. The first order change in aggregate demand from this change in wealth is

\[\sum_{i}\frac{\partial x_{li}(p,w_i)}{\partial w_{i}}\Delta w_{i}\]

and this must be equal to zero if demand is not going to change. This needs to be zero for all redistributions \(\Delta w\) and initial wealth levels \(w\). This will only be true if

\[\frac{\partial x_{li}(p,w_i)}{\partial w_i} = \frac{\partial x_{lj}(p,w_j)}{\partial w_j}\]

for all \(i\) and \(j\). This means that wealth expansion paths must be parallel straight lines.

Proposition 2.13 Consumers will exhibit parallel and straight wealth expansion paths if and only if their preferences all admit indirect utility of the formed

\[v_{i}(p,w_{i}) = a_{i}(p) + b(p)w_{i}.\]

Proof. To see that this is sufficient, assume that all consumers have indirect utility of the form above. Then by Roy’s identity

\[x(p,w) = -\frac{D_{p}a_{i}(p)}{b(p)} - \frac{D_{p}b(p)}{b(p)}w_{i}\]

Notice now that the coefficient on \(w_i\) does not vary with individuals, so \(\partial x_{il}(p,w)/\partial w = \partial x_{jl}(p,w)/\partial w\) for all \(i,j\) and any \(l\)

2.12 AD and the Weak Axiom

The weak axiom of revealed preference says the following. For any set \(B\) of potential choices (think a budget set), let \(C(B)\) be the choice that is made from \(B\). Let \((\mathcal{B},C(\cdot))\) be a choice structure.

Definition 2.20 A choice \((\mathcal{B},C)\), satisfies the weak axiom of revealed preference if for some \(B \in \mathcal{B}\) with \(x,y \in B\), we have \(x \in C(B)\), then for any \(B' \in \mathcal{B}\) with \(x,y \in B'\) and \(y \in C(B')\), we must also have \(x \in C(B')\).

Now that we understand the properties of demand itself, (continuity, homogeneity of degree zero and Walras’ law). We note that a desirable property of demand would be that it satisfies the weak axiom of revealed preference. This is because revealed preference is observationally very useful.

Counterexample.: Do example 4.C.1 from the book.

Therefore, aggregate preferences may not satisfy WARP even if individual preferences do.

Example 2.3 Consider a question of whether an ad valorem tax should be levied on consumption of good 2 and returned in a lump sum to all consumers. Will consumption of good 2 be reduced? Without further assumptions this cannot be assurred.

What conditions are necessary to prove that a reduction of aggregate consumption of good 1 occurs? Basically, assuming the result.

Lemma 2.3 If every consumer’s demand function satisfies the uncompensated law of demand \((p' - p)(x(p',w) - x(p,w)) \le 0\), then the aggregate demand function satisfies the uncompensted law of demand and hence WARP.

2.13 Existence of a Representative Consumer

Gorman form stuff.