Dynamic Afriat Theorem

• Dynamic Afriat Theorem is a nonparametric framework that tests dynamic stochastic choices with period-specific utility draws under the DRUM model.
• It employs a mixture over deterministic dynamic preference profiles to capture individual heterogeneity and time correlations in sequential choices.
• The theorem’s four equivalent characterizations, including dynamic Afriat inequalities, bridge deterministic and random utility models in a dynamic panel-data setting.

The Dynamic Afriat Theorem provides a finite-data, nonparametric revealed preference framework for testing consistency of dynamic stochastic choice with the Dynamic Random Utility Model (DRUM). The DRUM extends Afriat’s classic utility maximization and the static Random Utility Model (RUM) of McFadden–Richter by permitting individual-specific, period-by-period random utility draws, with arbitrary time correlation and cross-sectional heterogeneity. DRUM rationalizability amounts to the observed dynamic stochastic choice function being representable as a mixture over deterministic dynamic preference profiles, paralleling Afriat's finite mixture result but in a fully dynamic, panel-data setting. The Dynamic Afriat Theorem delivers necessary and sufficient conditions—via four equivalent characterizations—that generalize static revealed preference tests to the dynamic, stochastic context (Kashaev et al., 2023).

1. Primitives and Definition of DRUM

Consider a finite time-horizon $\mathcal T=\{1,\dots,T\}$. At each period $t$, the agent faces a finite choice set $X^t$ and a finite collection of menus $B^t_j \subseteq X^t$, indexed by $j\in\mathcal J^t$. A budget-path (or menu-path) is specified by $\bm j=(j_1,\dots,j_T)$, with each $j_t\in\mathcal J^t$. Given this menu-path, a choice-path is $$\bm x=(x^1_{i_1|j_1},\dots,x^T_{i_T|j_T}) \in \prod_t B^t_{j_t}$$. The central object of study is the dynamic stochastic choice function

$$\rho = \bigl(\rho_{\bm j}(\bm x)\bigr), \quad \rho_{\bm j}(\bm x) = \Pr\{\text{choice-path} = \bm x \mid \text{menu-path} = \bm j\}.$$

A utility-process is $u=(u^1,\dots,u^T)$, with each $u^t: X^t\to\mathbb{R}$, injective and monotone for some acyclic order $>^t$ on $2^{X^t}\setminus\{\emptyset\}$. Under DRUM, the agent draws, for each period, a possibly time-correlated sequence of utility functions and chooses a period-optimal element from each menu.

DRUM-rationalizability: There exists a probability measure $\mu$ on the set of utility-processes $\mathcal U$ such that for every menu-path $\bm j$ and choice-path $\bm x \in \prod_t B^t_{j_t}$,

$$\rho_{\bm j}(\bm x) = \int_{\mathcal U} \prod_{t=1}^T \mathbf{1}\Bigl\{ x^t_{i_t|j_t} \in \arg\max_{y\in B^t_{j_t}} u^t(y) \Bigr\} \;d\mu(u).$$

2. The Four Equivalent Characterizations (Dynamic Afriat Theorem)

Let $\mathcal R^t$ denote the set of all strict linear orders $r^t$ on $X^t$ extending $>^t$. A dynamic preference profile is $$\bm r = (r^1, \dots, r^T) \in \mathcal R^1 \times \cdots \times \mathcal R^T =: \mathcal R$$. For each $\bm r$ and $(\bm i,\bm j)$, define

$$a_{\bm r,\,\bm i,\,\bm j} = \begin{cases} 1, & \text{if for every } t,\; x^t_{i_t|j_t} \text{ is the best in } B^t_{j_t} \text{ according to } r^t \ 0, & \text{otherwise}. \end{cases}$$

Stacking these indicator-columns, form $A_T$, whose rows index choice-path/menu-path pairs, and whose columns index dynamic preference profiles $\bm r$.

Theorem (Dynamic Afriat): For an observed dynamic stochastic choice $\rho$, the following are equivalent:

1. $\rho$ is DRUM-rationalizable.
2. There exists a nonnegative vector $\nu \in \Delta^{|\mathcal R|-1}$ (a mixture over $\mathcal R$) such that $\rho = A_T\nu$.
3. There exists a $\nu\geq 0$ with no sum-to-one constraint such that $\rho = A_T\nu$.
4. (Dynamic Afriat inequalities/Axiom of Dynamic Stochastic Revealed Preference): For every finite sequence $\{(\bm i_k,\bm j_k)\}_{k=1}^K$,

$$\sum_{k=1}^K \rho_{\bm j_k}(\bm x_{\bm i_k|\bm j_k}) \leq \max_{\bm r \in \mathcal R} \sum_{k=1}^K a_{\bm r,\,\bm i_k,\,\bm j_k}.$$

The right-hand side is the maximal number of those choice-paths that a single deterministic profile $\bm r$ could generate.

3. Connection to Static Afriat Theorem and Static RUM

When $T=1$, the DRUM conditions reduce precisely to the McFadden–Richter characterization of static RUMs: $\rho = A_1\nu$ holds if and only if $\rho$ lies in the cone generated by the deterministic demand types. Equivalently, for all finite collections $\{(i_k,j_k)\}$,

$$\sum_k\rho_{j_k}(i_k) \leq \max_{r\in\mathcal R^1}\sum_k a_{r,i_k,j_k}.$$

In the deterministic, time-series Afriat theorem, observed budget–choice pairs $(p_t, x_t)$ must satisfy $$u(x_s) - u(x_t) \geq \lambda_t p_t \cdot (x_s - x_t)$$ for some $u$ and scalars $\lambda_t > 0$. This is the special case of the DRUM inequalities when $\rho \in \{0,1\}$ and all draws coincide with one fixed $\bm r$.

The DRUM constraints thus generalize both static random-utility mixture cones and the deterministic Afriat inequalities, providing a unified revealed-preference test for dynamic, stochastic, and heterogeneous panel data.

4. Necessity and Sufficiency of the Characterizations

The implication (ii) $\Rightarrow$ (iv) follows since, for any sequence of choice-paths, $\rho$ as a mixture cannot have more mass on a sequence than any single deterministic profile can generate: $$\sum_k\rho(\dots) = \sum_k\sum_{\bm r}\nu_{\bm r} a_{\bm r,\dots} = \sum_{\bm r}\nu_{\bm r} \sum_k a_{\bm r,\dots} \leq \max_{\bm r}\sum_k a_{\bm r,\dots}.$$

For sufficiency, (iv) $\Rightarrow$ (ii), linear programming duality (Farkas' lemma) guarantees that any $\rho$ not in the cone generated by $A_T$ can be separated by a hyperplane, which would yield a finite sequence violating (iv). Thus, satisfaction of the dynamic Afriat inequalities implies existence of a nonnegative solution $\nu$ with $A_T\nu = \rho$.

Equivalence between (ii) and (iii) is immediate up to normalization. The equivalence (iii) $\Leftrightarrow$ (i) holds via direct construction of the mixture measure $\mu$ from the (possibly unnormalized) finite vector $\nu$.

5. Illustrative Example: Two-Period Binary Menus

Let $X^t = \{x, y\}$ for $t=1,2$ with $B^t = \{x, y\}$ as the only menu. Then $\mathcal R^t = \{x\succ y, y\succ x\}$, resulting in $|\mathcal R| = 4$ dynamic profiles:

• $r^{11}: (x\succ y, x\succ y)$,
• $r^{12}: (x\succ y, y\succ x)$,
• $r^{21}: (y\succ x, x\succ y)$,
• $r^{22}: (y\succ x, y\succ x)$.

Suppose empirical $\rho$ is: $$\rho(x,x) = 0.4,\, \rho(x,y) = 0.2,\, \rho(y,x) = 0.3,\, \rho(y,y) = 0.1$$ (with $\sum = 1$). The $4\times4$ matrix $A_2$ encodes which deterministic profiles could generate each path. Solving $\rho=A_2\,\nu$, with $\nu\geq 0$ and $\sum\nu=1$, yields a nonnegative solution, confirming DRUM rationalizability. When testing the dynamic Afriat inequalities, e.g., for the sequence $((x,x), (y,x), (x,y))$,

$$\rho(x,x) + \rho(y,x) + \rho(x,y) = 0.4 + 0.3 + 0.2 = 0.9 \leq 2,$$

where $2$ is the maximum over profiles. All finite sequences satisfy the bound, so the observed behavior is DRUM-rationalizable.

6. Economic and Empirical Significance

Time correlation in tastes is captured by DRUM—utility draws may follow any joint law over $T$ periods, allowing taste shocks, learning, or fatigue in panel data. Cross-sectional heterogeneity is also modeled: different agents may possess different dynamic preference profiles, mixed in the observed population.

Empirically, requiring each marginal $\rho^t$ to be consistent with a static RUM, along with the joint $\rho$ satisfying dynamic Afriat inequalities, provides a strictly stronger test than separate static RUMs, but a weaker restriction than enforcing constant utility across periods. The DRUM model thus bridges the deterministic Afriat (panel, fixed tastes) and McFadden–Richter (static cross-section, varied tastes) settings, unifying and extending both approaches within a dynamic, stochastic, nonparametric revealed preference paradigm (Kashaev et al., 2023).