Money Pumps and Bounded Rationality

A dataset $D=(\mathbf{p}^{t},\mathbf{x}^{t})_{t\leqslant T}$ contains a money pump if there are observations $t_{1},\,t_{2},\ldots,\,t_{K}$ (drawn from $D$) such that $t_{1}=t_{K}$ and $\text{MP}_{t_{1},t_{2},\ldots,t_{K}}$ as defined by (1) is strictly positive. A dataset $D$ that is free of money pumps is said to satisfy cyclical monotonicity. Another way of saying the same thing is that a money pump exists if there is a permutation of $\{1,2,\ldots,T\}$, denoted $\sigma$, where, in period $t$, the arbitrageur sells $\mathbf{x}^{t}$ and buys $\mathbf{x}^{\sigma(t)}$; and the arbitrageur pumps a strictly positive amount of money,

from the consumer. We define the total money pump (TMP) as the amount of money which would be extracted from the consumer by an arbitrageur following an optimal trading strategy. That is,

where the supremum is taken over all permutations $\sigma$.

It is not hard to show that if $D$ is quasilinear rationalized by $U$ then $D$ is also additively rationalized by $U$; on the other hand, there is no guarantee that if $D$ is additively rationalized by $U$, then the same $U$ provides a quasilinear rationalization of $D$. The following result shows, among other things, that in fact these two types of rationalizations are empirically equivalent, and holds whenever $\text{TMP}=0$.

The claims in Theorem 1 are not new in the sense that they can easily be pieced together using existing results. In particular, the equivalence between statements 1 and 2 is obvious. The equivalence between statements 2, 3, and 4 is found in Browning (1989) and the equivalence between statements 2, 5, and 6 is found in Brown and Calsamiglia (2007). To keep this article reasonably self-contained, we provide a proof of Theorem 1 in the Appendix.

The equivalence between statements 3 and 4 and the equivalence between items 5 and 6 shows that there are no additional restrictions placed on the data by assuming that $U$ is well-behaved in either the additive rationalization or the quasilinear rationalization.

The equivalence between 1, 4, and 6 shows that there is a tight relationship between the existence of a money pump and additive and quasilinear rationalizations. This suggests that the value of the total money pump may be useful as a measure of the degree to which the consumer fails to act as an additive or quasilinear utility maximizer. We address this issue in the next subsection.

By defnition, a dataset $D=(\mathbf{p}^{t},\mathbf{x}^{t})_{t\leqslant T}$ can be additively rationalized if the observations, when taken as a whole, maximize an overall utility function $V:\mathbb{R}^{LT}_{+}\to\mathbb{R}$ that is additive across bundles at each observations, i.e., $V(\tilde{\mathbf{x}}^{1},\tilde{\mathbf{x}}^{2},\ldots,\tilde{\mathbf{x}}^{T})=\sum_{t=1}^{T}U(\tilde{\mathbf{x}}^{t})$, subject to the overall expenditure not exceeding the consumer’s total expenditure, which is $\sum_{t=1}^{T}\mathbf{p}^{t}\cdot\mathbf{x}^{t}$. When $D$ cannot be additively rationalized, how should we measure the extent of the violation? The cost-efficiency approach proposed by Afriat (1973) measures the amount of money which could have been saved by the consumer were they to have acted perfectly in line with the utility maximization hypothesis under investigation (the additive utility model in our case).⁷⁷7Appendix A discusses in greater detail the relationship between $A$ and Afriat’s critical cost efficiency index. For a utility function $U$ let $e^{U}$ denote the smallest amount of money for which a consumer with additive (across periods) utility function $U$ could obtain utility $\sum_{t=1}^{T}U(\mathbf{x}^{t})$. That is,

The additive cost inefficiency displayed by the consumer is the difference between the amount that the consumer actually spent and the smallest amount of money they could have spent to achieve the same utility. More formally, the additive cost inefficiency is the number

where the infimum is taken over all utility functions.

Next, let us suppose that we would like to measure the extent to which the consumer has failed to act as a quasilinear utility maximizer. One approach, which might be termed the utility efficiency approach, is to measure the additional utility which the consumer could have derived had they acted perfectly in line with the model of utility maximization under investigation. Of course, this approach only makes sense when utility is cardinal (in particular, the utility function must be identified up to translation by the consumer’s behavior). If the utility function is only identified up to monotonic transformation then it makes no sense to talk about differences in utility. As the quasilinear utility function is indeed cardinal in the requisite sense, it is reasonable to define the quasilinear utility inefficiency displayed by the consumer as

The term in the supremum represents the largest amount of quasilinear utility which could have been attained by the consumer in period $t$ whereas the rightmost term represents the amount of quasilinear utility actually achieved. The object $Q$ was introduced in Allen and Rehbeck (2021) as a measure of deviation from quasilinear utility maximization.⁸⁸8What we call $Q$ corresponds to the “minimum deviations” considered in Appendix B of Allen and Rehbeck (2021) when using (in their language) the aggregator $f(e_{1},e_{2},\ldots,e_{T})=\sum_{t=1}^{T}e_{t}$.

Importantly, Allen and Rehbeck (2021) show that $Q$ is easy to calculate. In particular, $Q$ is equal to $\bar{\varepsilon}$, the solution value for the linear programming problem of finding $(u_{1},u_{2},\ldots,u_{T})\in\mathbb{R}^{T}$ and $(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{T})\in\mathbb{R}_{+}^{T}$ to solve {IEEEeqnarray}rCl min&   ∑_t=1^T ε_t
s.t.   u_s ⩽u_t + p^t ⋅( x^s - x^t ) + ε_t,    for all s,t

To recap, we have introduced three distinct ways of quantifying deviations from cyclical monotonicity. It turns out that they are all the same.

As noted above, the result $Q=\bar{\varepsilon}$ is shown in Allen and Rehbeck (2021). We include it here and provide a proof for the sake of completeness.

To give some insight into the proof of Theorem 2 let us focus on how we show that $\text{TMP}=Q$ (the idea behind showing that $\text{TMP}=A$ is similar). The “easy direction” is showing that $Q\geqslant\text{TMP}$. Indeed, for any $U$ and permutation $\sigma$ we have {IEEEeqnarray*}rCl ∑_t=1^T sup_x∈R_+^L [ U(x) - p^t ⋅x] - [ U(x^t) - p^t ⋅x^t ] & ⩾ ∑_t=1^T [ U( x^σ(t) ) - p^t ⋅x^σ(t) ] - [ U(x^t) - p^t ⋅x^t ]
= ∑_t=1^T p^t ⋅( x^t - x^σ(t) ) and so the amount of quasilinear utility wasted for any $U$ is always greater than the amount of money which can be pumped for any $\sigma$. Thus, $Q\geqslant\text{TMP}$.

Showing that $\text{TMP}\geqslant Q$ is more delicate. The key insight we utilize is that for any permutation $\sigma$ which achieves the supremum in the definition of TMP it happens that the permuted dataset $D_{\sigma}=(\mathbf{p}^{t},\mathbf{x}^{\sigma(t)})_{t\leqslant T}$ satisfies cyclical monotonicity. In other words, the purchasing behavior of the arbitrageur, which is given by $D_{\sigma}$, must satisfy cyclical monotonicity. Once this fact is established Theorem 1 can be applied to show that there exists a well-behaved utility function $U$ which rationalizes the permuted data. It can then be shown that the amount of money pumped via $\sigma$ is weakly greater than the amount of quasilinear utility wasted according to $U$ which establishes that $\text{TMP}\geqslant Q$.

Recall that from items 3-6 of Theorem 1 we learned that when dealing with additive or quasilinear rationalizations there are no additional restrictions imposed on the data by requiring that the rationalizing utility function is well-behaved. This insight is preserved in Theorem 2 in the sense that the infima in the definitions of $A$ and $Q$ can always be achieved by well-behaved utility functions and thus we could have defined $A$ and $Q$ using infima over the collection of well-behaved utility functions without changing the content of these objects.

A dataset $D=(\mathbf{p}^{t},\mathbf{x}^{t})_{t\leqslant T}$ satisfies the generalized axiom of revealed preferences (GARP) if, for all $t_{1},t_{2},\ldots,t_{K}$ with $t_{1}=t_{K}$ satisfying

it is not the case that any of the inequalities in (6) hold strictly. We know from Afriat (1967) that GARP is a necessary and sufficient condition for a dataset to be rationalized by a well-behaved utility function.

A permutation of $\{1,2,\ldots,T\}$, denoted $\sigma$, is constrained for $D$ if $\mathbf{p}^{t}\cdot\mathbf{x}^{t}\geqslant\mathbf{p}^{t}\cdot\mathbf{x}^{\sigma(t)}$ for all $t$. A constrained permutation $\sigma$ represents a trading strategy (sell $\mathbf{x}^{t}$ and buy $\mathbf{x}^{\sigma(t)}$ in each period $t$) which nets the arbitrageur a weakly positive sum in each period. The constrained total money pump index, denoted $\text{TMP}_{c}$, is defined as in (2), but with the supremum taken over all constrained permutations. In other words, $\text{TMP}_{c}$ is the amount of money which would be extracted from the consumer by an arbitrageur who was following his optimal trading strategy under the provision that he cannot lose money in any round of trading.

From Theorem 1 we know that a consumer who satisfies GARP can be turned into a money pump provided they fail to satisfy the stronger property of cyclical monotonicity. It is however plain from the definitions that the consumer who satisfies GARP cannot be turned into a constrained money pump. We next present two models of behavior which turn out to be characterized by GARP.

The dataset $D$ is constrained additively rationalized by $U:\mathbb{R}_{+}^{L}\rightarrow\mathbb{R}$ if

where the leftmost inequality is required to be strict if any of the inequalities on the right hand side hold strictly. The constrained additive rationalization requires the consumer to maximize additive utility subject to period specific budget sets. In particular, the consumer is not allowed to reduce spending in one period in order to increasing spending in a different period. Due to this credit constraint it is easy to see that a dataset is rationalized by some utility function $U$ if and only if the dataset is constrained additively rationalized.

The dataset $D$ is constrained quasilinear rationalized by $U:\mathbb{R}_{+}^{L}\rightarrow\mathbb{R}$ if

In a constrained quasilinear rationalization, the observed choice $\mathbf{x}^{t}$ need only be superior (net of expenditure) to bundles which are cheaper than itself. Unlike an (unconstrained) quasilinear rationalization, it is not required that the net utility of $\mathbf{x}^{t}$ is higher than that of all alternative bundles.

The following result relates the concepts just introduced.

It is easy to check that items 1 and 2 are equivalent. It is also easy to see that a utility function $U$ rationalizes $D$ if and only if $U$ constrained additively rationalizes $D$ and thus items 4 and 5 are equivalent. The equivalence between items 2, 3, and 4 is well-known and is part of Afriat’s Theorem. Thus, the only novel part of Theorem 3 is the equivalences involving item 6. Note that we do not guarantee that the constrained quasilinear utility function is well-behaved (however, it is increasing and continuous).

Theorem 3 suggests that $\text{TMP}_{c}$ can be used to measure the extent to which any of the equivalent conditions in the theorem are violated. This is the theme of the next subsection.

Here we introduce constrained versions of the additive cost inefficiency and quasilinear utility inefficiency measures introduced in Section 2. Because Theorem 3 establishes that constrained additive and constrained quasilinear utility maximization are equivalent to GARP these measures can be thought of as reporting the extent to which GARP is violated.

To proceed, let $D=(\mathbf{p}^{t},\mathbf{x}^{t})_{t\leqslant T}$ be a dataset. For a utility function $U$ let $e_{c}^{U}$ denote the smallest amount of money which a consumer with additive (across periods) utility function $U$ would need in order to obtain the utility level $\sum_{t=1}^{T}U(\mathbf{x}^{t})$, while not spending more than $\mathbf{p}^{t}\cdot\mathbf{x}^{t}$ in each period $t$, i.e.,

The constrained additive cost inefficiency displayed by the consumer is the amount of money the consumer could save while still obtaining the same additive utility provided that in each period spending is kept within the original budget set. More precisely, the constrained additive cost inefficiency is

where the infimum is taken over all utility functions. The interpretation of $A_{c}$ is the same as with $A$ and, in particular, it could be thought of a measure of deviation from the model of constrained additive utility maximization via the cost efficiency approach.

For a price vector $\mathbf{p}\in\mathbb{R}_{++}^{L}$ and a number $m>0$ let $B(\mathbf{p},m)$ denote the linear budget set $B(\mathbf{p},m)=\{\mathbf{x}\in\mathbb{R}_{+}^{L}:m\geqslant\mathbf{p}\cdot\mathbf{x}\}$. The constrained quasilinear utility inefficiency displayed by the consumer is the amount of extra quasilinear utility the consumer could have acquired by deviating from $\mathbf{x}^{t}$, provided that in each period the consumer’s expenditure does not exceed $\mathbf{p}^{t}\cdot\mathbf{x}^{t}$. Formally, the constrained quasilinear utility inefficiency is

where the infimum is taken over all utility functions. The interpretation of $Q_{c}$ is the same as with $Q$. We know that $Q$ can be calculated by solving a linear programming problem and it is natural to conjecture that $Q_{c}$ can also be calculated in this fashion. Indeed, this happens to be the case. $Q_{c}$ coincides with $\bar{\varepsilon}_{c}$, the value of the linear programming problem of finding $(u_{1},u_{2},\ldots,u_{T})\in\mathbb{R}^{T}$ and $(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{T})\in\mathbb{R}_{+}^{T}$ to solve {IEEEeqnarray}rCl min&   ∑_t=1^T ε_t
s.t.   u_s ⩽u_t + p^t ⋅( x^s - x^t ) + ε_t,   ∀s,t   such that  p^t ⋅x^t ⩾p^t ⋅x^s Note that the linear programs of (2.2) and (8) are almost identical. The difference is that the constraints in (8) only apply for pairs of observations $t,s$ where $\mathbf{p}^{t}\cdot\mathbf{x}^{t}\geqslant\mathbf{p}^{t}\cdot\mathbf{x}^{s}$ whereas the constraints in (2.2) apply regardless of whether this condition holds or not.

The following result shows that all the measures of deviation from GARP just introduced are equivalent.

The proof of Theorem 4 shares much in common with the proof of Theorem 2 (with a couple of added nuances). To explain further we focus on the proof that $\text{TMP}_{c}=Q_{c}$ (the proof that $\text{TMP}_{c}=A_{c}$ is similar). The “easy direction” is showing that $Q_{c}\geqslant\text{TMP}_{c}$ and the proof approach is essentially the same as the one we used to show $Q\geqslant\text{TMP}$ in Theorem 2.

To show that $\text{TMP}_{c}\geqslant Q_{c}$ we proceed as follows. Let $\sigma$ be a constrained permutation which achieves the supremum in the definition of $\text{TMP}_{c}$. The first step of our proof is to show that the permuted dataset $D_{\sigma}=(\mathbf{p}^{t},\mathbf{x}^{\sigma(t)})$ cannot be pumped by any permutation $\sigma^{\prime}$ which is constrained for $D$ (note that we consider $\sigma^{\prime}$ which are constrained for $D$ and not permutations constrained for $D_{\sigma}$). Once we establish that $D_{\sigma}$ cannot be pumped in this fashion we employ Lemma 1 in the Appendix which guarantees that there exists a continuous and increasing utility function $U$ that satisfies $U(\mathbf{x}^{\sigma(t)})-\mathbf{p}^{t}\cdot\mathbf{x}^{\sigma(t)}\geqslant U(\mathbf{x})-\mathbf{p}^{t}\cdot\mathbf{x}$ for all $\mathbf{x}\in B(\mathbf{p}^{t},\mathbf{p}^{t}\cdot\mathbf{x}^{t})$. We then show that the amount of money which can be pumped with $\sigma$ is weakly greater than the amount of quasilinear utility wasted according to $U$; in other words, the extra utility the consumer could have gained by deviating from $\mathbf{x}^{t}$, subject to any deviation costing weakly less than $\mathbf{p}^{t}\cdot\mathbf{x}^{t}$. This establishes $\text{TMP}_{c}\geqslant Q_{c}$.

Here we compare our $\text{TMP}_{c}$ with related measures introduced in Echenique, Lee, and Shum (2011) (henceforth ELS) and Smeulders, Cherchye, Spieksma, and De Rock (2013) (henceforth SCSD). The starting point for ELS is the observation that any violation of GARP (i.e. any sequence $t_{1},t_{2},\ldots,t_{K}$ with $t_{1}=t_{K}$ so that (6) holds with at least one strict inequality) can be exploited by an arbitrageur to pump money from the consumer. This observation led ELS to quantify the degree of the violation of GARP $t_{1},t_{2},\ldots,t_{K}$ in accordance with the amount of money which could be extracted by the arbitrageur. As a single dataset $D=(\mathbf{p}^{t},\mathbf{x}^{t})_{t\leqslant T}$ can contain multiple violations of GARP it was proposed by ELS to take the average (either the mean or the median) of the amount of money which can be pumped (where the average is taken over each violation of GARP) in order to measure the degree of irrationality exhibited by the consumer. SCSD proved that calculating ELS’ average money pump (using either the mean or median) is NP-hard suggesting that this measure can be difficult to calculate in practice. SCSD proposed taking the maximum amount of money which can be pumped from the consumer through any single violation of GARP (i.e. any sequence $t_{1},t_{2},\ldots,t_{K}$ with $t_{1}=t_{K}$ so that (6) holds with at least one strict inequality). They show that this maximum money pump, in contrast to the average money pump, can be calculated in polynomial time. Recall that from Theorem 4 we know that our measure $\text{TMP}_{c}$ can be calculated by solving a linear programming problem and thus our measure too can be calculated in polynomial time.

The main difference between our measure, $\text{TMP}_{c}$, and the measures introduced by ELS and SCSD is that our index is the amount of money which can be extracted using the optimal trading strategy of the arbitrageur. On the other hand, the indices of ELS and SCSD aggregate (using either an average or a maximum) the amount of money which can be extracted over each violation of GARP whether or not the optimal strategy of the arbitrageur would actually exploit this violation (and, as we shall see in the examples below, he may not). We also note that there are no theoretical results connecting either the measure of ELS or the measure of SCSD to quasilinear or additive rationalizations (or any other rationalization concept) in the sense of our Theorem 4.

To further elucidate the differences between our $\text{TMP}_{c}$ and the measures of ELS and SCSD we present two examples.

The two preceding examples help make clear the distinction between our measure, which corresponds to the arbitrageur’s optimal strategy, with the measures of ELS and SCSD. In Example 1 we see that ELS’ average money pump incorporates several small violations of GARP which the arbitrageur’s optimal strategy ignores. Similarly, in Example 2 we that SCSD’s maximum money pump exploits the most severe single violation of GARP however the arbitrageur’s optimal strategy ignores this severe violation because he can make even more money by pumping two smaller violations.