Rationalization

Limitation of using individual data.

Rationalizability

Consider a historical dataset of $K$ samples of revealed preferences of one agent,

$$\boldsymbol{\Xi} = \{(\mathbf{p}_k, \mathbf{x}_k)\}_{k\in[K]},$$

$$\mathbf{x}(\mathbf{p}) = \arg\max_{\mathbf{x} \in \mathbb{R}_+^n} u(\mathbf{x}) \quad \mathrm{s.t.}~ \langle \mathbf{p}, \mathbf{x} \rangle \le w(\mathbf{p}),$$

$$\mathbf{x}(\mathbf{p}_k) = \mathbf{x}_{k}.$$

Varian ^[2] showed that if Generalized Axiom of Revealed Preference (GARP) holds, that is, an agent does not have cyclic revealed preferences, then the data can be rationalized by a monotone concave utility function. They give the following construction of Afriat pairs $(u_k, \lambda_k) \in \mathbb{R}\times\mathbb{R}_+$. Consider the following system of inequalities:

$$u_{k'}\le u_k + \lambda_k \langle \mathbf{p}_k, \mathbf{x}_{k'}-\mathbf{x}_k \rangle \quad \forall k,k' \in [K].$$

$$u_i(\mathbf{x})=\min_{k \in [K]}\big\{u_{ik} + \lambda_{ik} \langle \mathbf{p}_k, \mathbf{x}-\mathbf{x}_{ik} \rangle\big\}.$$

• GARP is a stronger condition and Samuelson's Weak Axiom of Revealed Preference (WARP) and weaker than SARP:

  $$\text{WARP} \succeq \text{GARP} \succeq \text{SARP} \succeq \text{SSARP}.$$

• GARP is weaker than the Strong Axiom of Revealed Preference (SARP), which requires the demand function is not set-valued.※There are several versions of SARP. An excellent explanation is given in [2].
• A stronger version, SSARP (Strong SARP), further requires the demand to be injective --- i.e., the same bundle cannot be induced by two distinct (normalized) prices. This is the Chiappori-Rochet ^[3] condition under which the data can be rationalized by a smooth, strictly concave utility.

A proposal to use parametric models can be found in ^[4]. Because they limit to parametric subspace of utility functions, part of the "error" is due to model misspecification, referred to as the inconsistency index. ^[5][6] extends the classical RP approach to study if the budget is also chosen by the agent, accommodating the effect of income changes. This provides rational intervals on the demand function. An important application of the approach is to detect the failure of rationalization, and more importantly, to fix it by locally perturbing the Engel curves in the experiment; see also, ^[7].

Limitation

Rationalizability is a consistency property of the observed data; it says nothing directly about the predictive power of the recovered utility on unseen prices. Two distinct obstacles arise.

Statistical: learnability. For a general utility class $\mathscr{U}$, predictive recovery is unlikely. Beigman and Vohra ^[9] showed that revealed-preference learning is, in general, not PAC-learnable: the relevant function class can possess infinite fat-shattering dimension. They identify a restricted regime (single-valued, sufficiently smooth demand) with finite dimension, but PLC utilities fall outside it.※PLC utilities can have set-valued demand functions, violating the single-valuedness assumption in [9].

Computational: equilibrium hardness. Even with full knowledge of every agent's utility, computing the market outcome is intractable. In the Fisher model※The Fisher model is a special case of the Arrow--Debreu model where the budgets $w_i(\mathbf{p}) \equiv w_i \in \mathbb{R}_+$ are fixed and exogenous to the prices. with PLC utilities $u_i(\cdot)$, finding a Walrasian equilibrium is PPAD-hard ^[10], and checking the existence of an exact or approximate equilibrium is NP-complete ^[11].

1. [1] Afriat, S. N.: The construction of utility functions from expenditure data. International Economic Review. 8(1), 67–77 (1967).
2. [2] Varian, H.: The nonparametric approach to demand analysis. Econometrica. 50(4), 945–973 (1982).
3. [3] Chiappori, P.-A., Rochet, J.-C.: Revealed preferences and differentiable demand. Econometrica. 55(3), 687–691 (1987).
4. [4] Halevy, Y., Persitz, D., Zrill, L.: Parametric recoverability of preferences. Journal of Political Economy. 126(4), 1558–1593 (2018).
5. [5] Blundell, R. W., Browning, M., Crawford, I. A.: Nonparametric Engel curves and revealed preference. Econometrica. 71(1), 205–240 (2003).
6. [6] Blundell, R., Browning, M., Crawford, I.: Best nonparametric bounds on demand responses. Econometrica. 76(6), 1227–1262 (2008).
7. [7] Blundell, R., Browning, M., Cherchye, L., Crawford, I., De Rock, B., Vermeulen, F.: Sharp for SARP: nonparametric bounds on counterfactual demands. American Economic Journal: Microeconomics. 7(1), 43–60 (2015).
8. [8] Mas-Colell, A.: On revealed preference analysis. The Review of Economic Studies. 45(1), 121–131 (1978).
9. [9] Beigman, E., Vohra, R.: Learning from revealed preference. Proceedings of the 7th ACM Conference on Electronic Commerce (EC '06), 36–42 (2006). https://dl.acm.org/doi/10.1145/1134707.1134712
10. [10] Chen, X., Teng, S.-H.: Spending is not easier than trading: on the computational equivalence of Fisher and Arrow-Debreu equilibria. Algorithms and Computation (ISAAC 2009), Springer (2009).
11. [11] Vazirani, V. V., Yannakakis, M.: Market equilibrium under separable, piecewise-linear, concave utilities. J. ACM. 58(3), 10:1–10:25 (2011). https://dl.acm.org/doi/10.1145/1970392.1970394
12. [12] Debreu, G.: Representation of a preference ordering by a numerical function. In: Mathematical Economics: Twenty Papers of Gerard Debreu. Cambridge University Press (1983).
13. [13] Samuelson, P. A.: Foundations of Economic Analysis. Harvard University Press, Cambridge, Mass. (1983).
14. [14] Houthakker, H. S.: Revealed preference and the utility function. Economica. 17(66), 159–174 (1950).
15. [15] Gale, D.: A note on revealed preference. Economica. 27(108), 348–354 (1960).

Concepts in Microeconomics

Preferences. Consider alternative set $\mathscr{X}$, the preference $\succeq$ is a binary relation on $\mathscr{X}$. For any $\mathbf{x}, \mathbf{y} \in \mathscr{X}$, we say $\mathbf{x} \succeq \mathbf{y}$ if $\mathbf{x}$ is preferred to $\mathbf{y}$. We say $\mathbf{x} \sim \mathbf{y}$ if $\mathbf{x}$ is indifferent to $\mathbf{y}$. We say $\mathbf{x} \prec \mathbf{y}$ if $\mathbf{y} \succeq \mathbf{x}$.

• The preference $\succeq$ is rational if it is complete and transitive.
• Convex if for any $\mathbf{x}, \mathbf{y} \in \mathscr{X}$, we have $\mathbf{x} \succeq \mathbf{z}$ and $\mathbf{y} \succeq \mathbf{z}$, then $\alpha \mathbf{x} + (1-\alpha) \mathbf{y} \succeq \mathbf{z}$ for all $\alpha \in [0, 1]$.
• Homothetic if, for any $\mathbf{x}, \mathbf{y} \in \mathscr{X}$, we have $\mathbf{x} \succeq \mathbf{y}$ if and only if $\alpha \mathbf{x} \succeq \alpha \mathbf{y}$ for all $\alpha > 0$.
• Continuous if $\succeq$ is preserved under limits: $\mathbf{x}_n \to \mathbf{x}$ and $\mathbf{y}_n \to \mathbf{y}$ such that $\mathbf{x}_n \succeq \mathbf{y}_n$ for all $n$.
• (Locally) Non-satiated: there exists $\mathbf{x} \succeq \mathbf{y}$ for any $\mathbf{y} \in \mathscr{X}$.
• Quasilinear w.r.t. good 1 (the numéraire) if, typically $\mathscr{X} = \mathbb{R}_+^n$:

  $$\begin{align*} & \mathbf{x} \sim \mathbf{y} \Rightarrow \mathbf{x} + \mathbf{e}_1 \sim \mathbf{y} + \mathbf{e}_1 \\ & \mathbf{x} + \mathbf{e}_1 \succeq \mathbf{x}, \quad \forall \mathbf{x} \in \mathbb{R}_+^n \end{align*}$$

Choices. A choice is a more subtle concept than a preference. Let $\mathscr{B} \subseteq 2^{\mathscr{X}}$ be a family of "budget sets". A choice correspondence is a map $c: \mathscr{B} \rightrightarrows \mathscr{X}$ with $c(B) \subseteq B$ for all $B \in \mathscr{B}$.

• In other words, choices are like "revealed" preferences. I give subset of alternatives, and you reveal your preferences.
• Note $B$ could be a continuous set. For example, $B(\mathbf{p}, w) = \{\mathbf{x} \in \mathbb{R}_+^n : \langle \mathbf{p}, \mathbf{x} \rangle \le w\}$ is a Walrasian budget set; also called competitive budget set.
• The family $\mathscr{B} = \{B(\mathbf{p}, w) : \mathbf{p} \in \mathbb{R}_{++}^n,\ w \in \mathbb{R}_+\}$ gives the Walrasian choice $c(B(\mathbf{p}, w)) = \mathbf{x}(\mathbf{p}, w)$.

For preferences $\succeq$, there is a "most preferred rule" for $\succeq$: For any $B \in \mathscr{B}$,

$$c_{\succeq}(B) = \{\mathbf{x}\in B \mid \mathbf{x} \succeq \mathbf{y}, \forall \mathbf{y} \in B \}.$$

Utility. A utility function $u: \mathscr{X} \to \mathbb{R}$ represents the preference $\succeq$ if

$$\mathbf{x} \succeq \mathbf{y} \iff u(\mathbf{x}) \ge u(\mathbf{y}), \quad \forall \mathbf{x}, \mathbf{y} \in \mathscr{X}.$$

• Necessity. If $\succeq$ admits a utility representation, then $\succeq$ is rational (complete and transitive); these properties transfer from $\ge$ on $\mathbb{R}$.
• Failure of the converse. A rational $\succeq$ on $\mathscr{X}$ need not admit a utility. The canonical counterexample is the lexicographic order on $\mathbb{R}^2$, which is rational but has no real-valued representation (it embeds an uncountable chain).
• Debreu's theorem ^[12]. If $\succeq$ is rational and continuous on a separable (or, more generally, second-countable) topological space $\mathscr{X}$, then it admits a continuous utility representation $u: \mathscr{X} \to \mathbb{R}$.
• Ordinal invariance. A utility representation is unique only up to strictly increasing transformation: if $u$ represents $\succeq$, so does $f \circ u$ for any strictly increasing $f: \mathbb{R} \to \mathbb{R}$.※In particular, continuity of the representation is not pinned down by $\succeq$: a step-function representation can coexist with a continuous one. The logarithmic transform is a familiar case, convenient for homothetic preferences.
• Finite/countable $\mathscr{X}$. If $\mathscr{X}$ is finite (or countable), every rational $\succeq$ on $\mathscr{X}$ admits a utility representation.

Demand and Axioms of Revealed Preference

Rational preferences $\succeq$ (complete and transitive) imply we know every possible comparison between alternatives. So essentially, it is not trivial to "test" from what is revealed from the observations, the choices.

Alternative $\mathbf{x}$ is revealed preferred to $\mathbf{y}$ by budget $B$ if $\mathbf{y} \in c(B)$ implies $\mathbf{x} \in c(B)$. If $\succeq$ is rational, then $c_{\succeq}(B)$ satisfies WARP.

For longer chains, write $\mathbf{x} \mathscr{R} \mathbf{y}$ (directly revealed preferred) if $\mathbf{x} \in c(B)$ and $\mathbf{y} \in B$ for some $B \in \mathscr{B}$, and $\mathbf{x} \mathscr{P} \mathbf{y}$ (strictly revealed preferred) if additionally $\mathbf{y} \notin c(B)$.

That is, no strict revealed-preference cycle among distinct bundles is allowed, a chain $\mathbf{x}_1 \mathscr{R} \mathbf{x}_2 \mathscr{R} \cdots \mathscr{R} \mathbf{x}_K \mathscr{R} \mathbf{x}_1$ forces all $\mathbf{x}_k$ to coincide. SARP is the natural axiom when $c(\cdot)$ is single-valued.

That is, a chain of weak revealed preferences cannot be closed by a strict revealed preference. GARP allows ties (set-valued $c$), where SARP would force equality of all bundles in the cycle. The hierarchy is then transparent:

• WARP is the $K = 2$ case (no 2-cycles).
• GARP is no strict cycles of any length (allows weak ties / set-valued $c$).
• SARP is no nontrivial cycles of any length among distinct bundles (forces single-valued $c$).
• SSARP is SARP + injectivity in prices ^[3].

This looks very similar to subtour elimination problem in traveling salesman problem in combinatorial optimization. So the cycle cannot exist for $2^{\mathscr{X}}$, the power set. For continuous spaces, this characterization can be hard. A historical remark for this can be found in ^[15].

Specified to Walrasian demand. Walrasian demand $\mathbf{x}(\mathbf{p}, w)$ is a special choice correspondence in which the budget $B(\mathbf{p}, w) \subset \mathbb{R}_+^n$ is the convex polytope cut out by a single price hyperplane. Two structural restrictions are standard:

• Homogeneity of degree zero in $(\mathbf{p}, w)$:

  $$\mathbf{x}(\mathbf{p}, w) = \mathbf{x}(\alpha \mathbf{p}, \alpha w) \quad \forall \alpha > 0.$$

  Only relative prices and real wealth matter; nominal scaling is irrelevant.
• Walras' law:

  $$\langle \mathbf{p}, \mathbf{x}(\mathbf{p}, w) \rangle = w,$$

  i.e., the budget is exhausted (a consequence of local non-satiation of the underlying preference).

On $\mathscr{X} = \mathbb{R}_+^n$ the natural background preference is the componentwise order: $\mathbf{x} \succeq \mathbf{y}$ whenever $\mathbf{x} \ge \mathbf{y}$. Demand consistent with a monotone $\succeq$ then prefers larger bundles whenever the budget allows.

WARP on Walrasian demand. For Walrasian demand, WARP admits a clean compensated-price characterization:

• Sufficient condition. If, whenever $\mathbf{x}(\mathbf{p}',w')$ is affordable at $(\mathbf{p}, w)$, the bundle $\mathbf{x}(\mathbf{p}, w)$ becomes unaffordable at $(\mathbf{p}', w')$,

  $$\langle \mathbf{p}, \mathbf{x}(\mathbf{p}',w') \rangle \le w \Rightarrow \langle \mathbf{p}', \mathbf{x}(\mathbf{p},w) \rangle \ge w',$$

  then $\mathbf{x}(\mathbf{p}, w)$ satisfies WARP.
• Equivalent Slutsky form. Under Walras' law, $\mathbf{x}(\mathbf{p}, w)$ satisfies WARP if and only if, for every Slutsky-compensated price change --- i.e., $(\mathbf{p}', w')$ such that $\mathbf{x} = \mathbf{x}(\mathbf{p}, w)$ remains just affordable, $\langle \mathbf{p}', \mathbf{x} \rangle = w'$ --- we have

  $$\langle \mathbf{p}' - \mathbf{p}, \mathbf{x}(\mathbf{p}',w') - \mathbf{x}(\mathbf{p},w) \rangle \le 0.$$

  Geometrically: when prices change but $\mathbf{x}$ is still affordable, demand moves in the direction opposite to the price change. For differentiable $\mathbf{x}$, this is exactly the negative semidefiniteness of the Slutsky matrix.

What WARP does not give. The Slutsky condition above alone does not imply rationalizability by a utility function --- it only rules out 2-cycles (cf. the WARP/SARP gap on Walrasian budgets). This is the magic of the Afriat-Varian approach: by upgrading WARP to GARP (no cycles of any length), rationalizability, convexity of preferences, and concavity of the rationalizing utility collapse into a single finite system of linear inequalities.