Before we begin, it’s helpful to have this framework in mind: $$ \text{Utility Functions }u(\cdot) \Leftrightarrow\text{Preferences } a \succeq b \Leftrightarrow\text{Choices (aka Data) }(\beta, C) $$ Economists care about how to make the above concepts consistent (i.e. under what assumptions would the above $\Leftrightarrow$ hold mathematically).
Choices and Choice Structures
Given a finite set of mutually exclusive alternatives $X$:
Definition. A choice structure is a pair $(\beta, C)$ with
- $\beta \subseteq \mathcal{P}(X)$ a family of budget sets,
- $C : \beta \Rightarrow X$ a choice correspondence such that $C(B) \subseteq B$ for all $B \in \beta$.
Example.
$X=\lbrace a,b,c\rbrace $,
$\beta=\lbrace \lbrace a,b\rbrace ,\lbrace a,b,c\rbrace \rbrace $,
$C(\lbrace a,b\rbrace )=\lbrace a\rbrace , ; C(\lbrace a,b,c\rbrace )=\lbrace a,c\rbrace $.
Weak Axiom of Revealed Preference (WARP)
Definition. A choice structure satisfies WARP if:
If $x \in C(B)$ and $y \in B$ for some $B \in \beta$, then for any $B’ \in \beta$ with $x,y \in B’$, whenever $y \in C(B’)$ it must be that $x \in C(B’)$.
Observation. If $(\beta’,C’)$ satisfies WARP and $\beta \subseteq \beta’$ with $C(B)=C’(B)$ for all $B \in \beta$, then $(\beta,C)$ also satisfies WARP.
Example. If $\beta$ contains only sets of size $\leq 2$, then $(\beta,C)$ necessarily satisfies WARP.
Empirical Violations: for example Choice Overload.
$X=\lbrace \varnothing,a,b,\dots,z\rbrace $.
If $C(\lbrace \varnothing,a,b,c\rbrace )=\lbrace a\rbrace $ but $C(X)=\lbrace \varnothing\rbrace $, then WARP fails.
Preferences and Utility
Definition. A preference relation $\succeq$ on $X$ is rational if
- Complete: $\forall x,y$, $x \succeq y$ or $y \succeq x$,
- Transitive: if $x \succeq y, ; y \succeq z$ then $x \succeq z$.
From $\succeq$ we derive strict ($\succ$) and indifference ($\sim$).
Utility Functions. A function $u:X\to \mathbb{R}$ represents $\succeq$ if $x\succeq y \iff u(x)\geq u(y)$.
Proposition 1. If $u$ represents $\succeq$ and $f:\mathbb{R}\to\mathbb{R}$ is strictly increasing, then $v=f\circ u$ also represents $\succeq$.
Proposition 2. If some $u$ represents $\succeq$, then $\succeq$ is rational.
Rationalization
Definition. A rational preference $\succeq$ rationalizes $(\beta,C)$ if it generates a choice correspondence $C_\succeq$ on all subsets of $X$ such that $C(B)=C_\succeq(B)$ for all $B \in \beta$.
Proposition. If $(\beta,C)$ satisfies WARP and contains all subsets of $X$ of size $\leq 3$, then it is rationalizable.
Walrasian Demand
Environment: set of alternatives $X=\mathbb{R}^L_{\geq 0}$, price vector $p\in \mathbb{R}^L_{\geq 0}$, wealth $w\geq 0$.
Budget set. $B_{p,w}=\lbrace x\in X \mid p\cdot x\leq w\rbrace $.
Demand. $x(p,w)=C(B_{p,w})$, with assumptions:
- Homogeneity of degree 0,
- Single-valuedness,
- Walras’ Law: $p\cdot x(p,w)=w$.
WARP for demand.
$x(\cdot,\cdot)$ satisfies WARP iff:
if $p\cdot x(p’,w’) \leq w$ and $x(p,w)\neq x(p’,w’)$, then $p’\cdot x(p,w) > w’$.
Law of Compensated Demand.
Equivalent condition: with $w’ = p’\cdot x(p,w)$,
$$ (p’-p)\cdot (x(p’,w’)-x(p,w)) \leq 0, $$strict if $x(p’,w’)\neq x(p,w)$.
The Slutsky Matrix (I)
Assume differentiability. Define
$$ S_{ij}(p,w)=\frac{\partial x_i}{\partial p_j}(p,w) + \frac{\partial x_i}{\partial w}(p,w),x_j(p,w). $$
Collect to $S(p,w)=[S_{ij}]$.
Theorem. $x(\cdot,\cdot)$ satisfies WARP $\iff$ $S(p,w)$ is negative semidefinite $\forall(p,w)$.
Counter Example: Rational Preference that Cannot be represented by a utility function:
Definition. Lexicographic Preferences. On $\mathbb{R}^2$, $(x_1,x_2)\succeq_L(y_1,y_2)$ iff $x_1>y_1$ or ($x_1=y_1$ and $x_2\geq y_2$).
Claim. For Lexicographic Preferences, There exists no $$u : X \to \mathbb{R} ;; \text{s.t. } u(x) \geq u(y) \iff x \succeq_L y.$$
Proof Towards contradiction, suppose $u$ represents $\succeq_L$. Given any $x_1 \in \mathbb{R},$ there exists a rational number: (use density of rational number)
$$ r(x_1) \in \mathbb{Q}\quad \text{s.t. } u(x_1, 1) < r(x_1) < u(x_1, 2). $$ For any $x_1 < x_1’$, $u(x_1, 2) < u(x_1’, 1).$ So take $r(x_1’)$ similarly. $$ r(x_1) < u(x_1, 2) < u(x_1’, 1) < r(x_1’). $$ Thus $r(x_1) \neq r(x_1’)$. This implies $r : \mathbb{R} \to \mathbb{Q}$ is injective. This would imply $|\mathbb{Q}| \geq |\mathbb{R}|$, a contradiction.
Continuity and Representation
Contour sets: $\succeq(x)=\lbrace y\mid y\succeq x\rbrace $, $\preceq(x)=\lbrace y\mid x\succeq y\rbrace $.
Definition. $\succeq$ is continuous if these sets are closed for each $x$.
Theorem. If $\succeq$ is rational and continuous, there exists a continuous utility $u$ representing it.
Scaling and Canonical Representation
Fix $e\in \mathbb{R}^L_+$. For $x\in X$, define
$$ \Delta_B=\lbrace \lambda\geq 0\mid \lambda e\succeq x\rbrace ,\quad \Delta_W=\lbrace \lambda\geq 0\mid x\succeq \lambda e\rbrace . $$
By completeness, $\Delta_B\cup \Delta_W=\mathbb{R}_{\geq 0}$; by monotonicity and continuity, both are closed intervals. So $\Delta_B=[\bar\lambda,\infty)$, $\Delta_W=[0,\bar\lambda]$.
Thus each $x$ has a unique $\bar\lambda(x)$ such that $x\sim \bar\lambda e$.
This defines $\lambda:X\to \mathbb{R}$ with $x\mapsto\bar\lambda(x)$, a canonical (continuous, monotone) utility.
Remarks.
- Not every representing $u$ must be continuous.
- If $\succeq$ is monotone, every representing $u$ is monotone.
Convexity of Utility Function
Definition. $\succeq$ is convex if whenever $y\succeq x$ and $z\succeq x$, then all convex combinations $\theta y+(1-\theta)z\succeq x$.
Definition. $u$ is quasi-concave if $u(\theta x+(1-\theta)y)\geq \min\lbrace u(x),u(y)\rbrace $.
Claim. If $\succeq$ is convex, any $u$ representing it is quasi-concave.
Utility Maximization
$$ X(p,w) \equiv \arg\max_{x \in B_{p,w}} u(x), \qquad V(p,w) \equiv \max_{x \in B_{p,w}} u(x) $$
Here $V$ is the indirect utility function. By construction,
$$ V(p,w) = u(X(p,w)). $$
Propositions.
- $V$ is continuous,
- $V$ is homogeneous of degree $0$,
- $V$ is strictly increasing in $w$ and weakly decreasing in $p$,
- $V$ is quasi-convex in $p$.
The Expenditure Function and Hicksian demand:
$$ \begin{align*} & e(p,u_0) \equiv \min_{x} { p \cdot x \mid u(x) \geq u_0 }, \cr & h(p,u_0) \equiv \arg\min_{x} { p \cdot x \mid u(x) \geq u_0 }. \end{align*} $$
$h$ = Hicksian demand, $e$ = expenditure. Properties of $e$:
- Continuous,
- Homogeneous of degree $1$ in $p$,
- Strictly increasing in $u$, weakly increasing in $p$,
- Concave in $p$.
Duality:
The neat equalities tying everything we’ve defined before: (I think here we need to assume some convexity in the underlying utility function) $$ e(p, V(p,w)) = w, \qquad V(p, e(p,u_0)) = u_0 $$
$$ X(p,w) = h(p, V(p,w)), \qquad h(p,u) = X(p, e(p,u)). $$
Shephard’s Lemma
Use envelope theorem: (Shephard’s Lemma) $$ \frac{\partial e(p,u)}{\partial p_i} = h_i(p,u). $$
Thus, Shephard’s Lemma implies $$ \frac{\partial h_i(p,u)}{\partial p_j} = \frac{\partial^2 e(p,u)}{\partial p_i \partial p_j}. $$
Which says, Hicksian demands equal the gradient of $e$, and their derivatives are the Hessian of $e$.
The Slutsky Matrix from Hicksian Demand Function
By chain rule:
$$ \frac{\partial h_i(p,u)}{\partial p_j} = \frac{\partial x_i(p,w)}{\partial p_j} + \frac{\partial x_i(p,w)}{\partial w},x_j(p,w). $$
Thus the Slutsky Matrix is also equivalent to $$ S_{ij}(p,w) = \frac{\partial h_i(p,V(p,w))}{\partial p_j}. $$
Special Utility Structures to Keep in Mind:
Quasi-linear Utility
Definition. Preferences are quasi-linear if they can be represented by a utility function of the form
$$ U(x_0,x_1,\dots,x_L) = x_0 + \hat U(x_1,\dots,x_L). $$
Homothetic Preferences
Definition. Preferences are homothetic if they can be represented by a utility function that is homogeneous of degree 1.
Equivalently: $u(x)$ represents homothetic preferences $\iff \exists$ strictly increasing $f$ such that $f(u(x))$ is homogeneous of degree 1.
Property. If preferences are homothetic, then Marshallian demand satisfies
$$
X(p,w) = \hat X(p) w.
$$
Example.
$U(x) = \sum_{l=1}^L \beta_l \ln(x_l)$ with $\sum \beta_l = 1$.
Equivalent form: $\hat U(x)=x_1^{\beta_1}\cdots x_L^{\beta_L}$ (Cobb–Douglas).
Then $U(\lambda x)=U(x)+\ln \lambda$ and $\hat U(\lambda x)=\lambda \hat U(x)$.
Discounted Utility
$X =$ streams of consumption. Let $\vec c = (c_0,c_1,\dots,c_T) \in X$.
Definition. Discounted utility representation:
$$ U(\vec c) = \sum_{t=0}^T \delta^t u(c_t). $$
The $\delta$ represents trade-off between $c_t$ and $c_{t’}$: (MU := Marginal Rate of Substitution across time) $$ \frac{MU_t}{MU_{t’}} = \frac{\delta^t}{\delta^{t’}} \cdot \frac{u’(c_t)}{u’(c_{t’})}. $$