A general revealed preference test for quasilinear preferences: theory and experiments

Abstract

We provide a generalized revealed preference test for quasilinear preferences. The test applies to nonlinear budget sets and non-convex preferences as those found in taxation and nonlinear pricing contexts. We study the prevalence of quasilinear preferences in a laboratory real-effort task experiment with nonlinear wages. The experiment demonstrates the empirical relevance of our test. We find support for either convex (non-separable) preferences or quasilinear preferences but weak support for the hypothesis of both quasilinear and convex preferences.

Appendices

Appendix 1: Proof of Theorem 1

Proof

($\Rightarrow$) Consider a function $v(x,m) = u(x) +m$ that rationalizes the consumption experiment. For clarity, we maintain the notation $m^t(x^s)$, $m^t(x^t)$ and $m^t(x)$ for the rest of the proof. Then, for every sequence of observations $k_1,k_2,\ldots , k_n \in \{1,\ldots ,T\}$, the following is true:

$$\begin{aligned} u(x^{k_{j+1}})+m^{k_{j+1}}(x^{k_{j+1}})\ge u(x^{k_{j}}) + m^{k_{j+1}}(x^{k_{j}}), \end{aligned}$$

where $k_{n+1}=k_1$. We can simplify this and obtain the following inequalities:

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} m^{k_1}(x^{k_1}) - m^{k_1}(x^{k_2}) \ge u(x^{k_2}) - u(x^{k_1}), \\ &{} m^{k_2}(x^{k_2}) - m^{k_2}(x^{k_3}) \ge u(x^{k_3}) - u(x^{k_2}), \\ &{}\ldots \ \ldots \ \ldots \ \ldots \ \ldots \ \ldots \ \ldots \ \ldots \ \ldots \ \ldots \\ &{} m^{k_n}(x^{k_n}) - m^{k_n}(x^{k_1}) \ge u(x^{k_1})-u(x^{k_n}). \end{array}\right. } \end{aligned}$$

Summing up these inequalities, we obtain the following:

$$\begin{aligned} m^{k_1}(x^{k_1}) - m^{k_1}(x^{k_2})+ m^{k_2}(x^{k_2}) - m^{k_2}(x^{k_3}) +\ldots + m^{k_n}(x^{k_n}) - m^{k_n}(x^{k_1})\ge 0. \end{aligned}$$

This is exactly the cyclical monotonicity condition.

($\Leftarrow$) This part of the proof is split in two parts. First, we construct the subutility numbers corresponding to the observed choices and show that the observed chosen point $(x^t,m^t)$ is at least as good as any other point $(x^s,m^t(x^s))$. Next, we extend the subutility function to the entire $\mathbb {R}^n_+$ and show that corresponding QLU rationalizes the data.

Let

$$\begin{aligned} u^t = \min \{m^{k_1}(x^{k_1})-m^{k_1}(x^{t})+\ldots +m^{k_{n}}(x^{k_n})-m^{k_n}(x^{k_{n-1}})\} \end{aligned}$$

over all sequences in the data, including those with repeating elements.¹⁴ Note that all the sequences are starting with $k_1$, however, the real “anchoring” element of the sequence is $k_2=t$ for the observation $t\in T$. We will show that the minimum is well-defined. It will be enough to show that there will be no cycles in the minimal sequence, since the rest will follow from the fact that the minimum is taken over finite sums of finite numbers. So assume, to the contrary, that the minimum sequence contains a cycle; then we have

$$\begin{aligned} \ldots + \left( m^{k_n}(x^{k_n})-m^{k_n}(x^{k_{n-1}}) + \ldots + m^{k_1}(x^{k_1}) - m^{k_1}(x^{k_n})\right) + \ldots . \end{aligned}$$

However, cyclical monotonicity implies that this term is greater or equal than zero, and hence, excluding it would make the sequence even smaller. That contradicts the original assumption that the sequence was the smallest.

Further, we show that such a construction of $u^t$ will guarantee that the following system of inequalities is satisfied. For any $t,s\in \{1,\ldots , T\}$ we want to show that

$$\begin{aligned} u^t - u^s \ge m^t(x^s)-m^t(x^t). \end{aligned}$$

By the construction of u(x), we can guarantee that

$$\begin{aligned} u^s \le m^t(x^t)-m^t(x^s)+u^t, \end{aligned}$$

since

$$\begin{aligned} u^t = m^{k_1}(x^{k_1})-m^{k_1}(x^t)+\ldots +m^{k_{n}}(x^{k_n})-m^{k_n}(x^{k_{n-1}}) \end{aligned}$$

for some (minimal) sequence, and we construct $u(x^s)$ using that minimal sequence. Recall that we allowed taking every sequence, including those with repeating elements, so we can add any element to the existing sequence and the utility level $u(x^s)$ will not exceed the value of the new extended sequence. Therefore,

$$\begin{aligned} u^t-u^s\ge u^t - \left( m^t(x^t)-m^t(x^s)+u(x^t)\right) =m^t(x^s)-m^t(x^t). \end{aligned}$$

This concludes the first part of the proof. Next, we use the numbers constructed above to recover the entire utility function.

For every $x\in X$, let

$$\begin{aligned} u(x) = \min \limits _{t\in \{1,\ldots , T\}} \{u^t+m^t(x^t)-m^t(x)\}. \end{aligned}$$

Note that since $m^t$ is continuous and monotone, so is u(x).

First, we will show that for every $t\in \{1,\ldots , T\}$, $u(x^t)=u^t$. As we have shown above for every $s\in \{1,\ldots , T\}$ for which $m^s(x^t)$ is defined,

$$\begin{aligned} u^t\le u^s + m^s(x^s)-m^s(x^t). \end{aligned}$$

Therefore, $u(x^t)=u^t$. Next, we show that the constructed utility rationalizes the data, that is $v(x,m)\le v(x^t,m^t(x^t))$ for every $(x,m)\in B^t$, where $v(x,m) = u(x)+m$. By the construction of u(x), we know that $u(x) = \min \limits _{t\in \{1,\ldots , T\}} \{u^t+m^t(x^t)-m^t(x)\} \le u^t+m^t(x^t)-m^t(x)$. Therefore, $u(x)+m \le u(x)+m^t(x) \le u(x^t)+m^t(x^t)$. $\square$

Given the proof of the Theorem 1 one can immediately infer the following remark.

Remark 2

A consumption experiment $E=((x^t,m^t),B^t)_{t=1}^T$ can be rationalized by a QLU function if and only if there is a monotone function $u: C\rightarrow \mathbb {R}$ such that

$$\begin{aligned} u(x^t)-u(x^s) \ge m^t(x^s)-m^t(x^t) \text { for every } t,s\in \{1,\ldots ,T\}. \end{aligned}$$

where $C = \bigcup \limits _{t\in \{1,\ldots ,T\} } \{x^t\}$ is the set of chosen points.

Remark 2 crucially important for the following reason. Cyclical monotonicity is an elegant condition, it may be less computationally tractable especially once mixed with measures of distance to rationality. However, the linear programming approach provides the definitely computationally tractable way of testing QLU rationalizability.

Appendix 2: Additional empirical results

This appendix presents additional empirical analysis. We first present additional results using the HMI index. We then present analysis using the Critical Cost Efficiency Index (CCEI) introduced by Afriat (1973). The results are robust to alternative measures of distance to rationality.

1.1 Additional results for HMI

We present two additional tables. Table 3 presents unconditional pass rates. Table 4 presents an alternative construction of confidence intervals using normal approximations instead of the exact procedure.

Table 3 Left panel: Pass rates for each theory considered. Unlike Table 2 in the main text, this table reports unconditional pass rates. The first number is the percentage pass rate; the fraction in parentheses is the numbers of subjects used to compute the pass rates; the numbers brackets are the 95% confidence intervals. Right panel: HMI cutoffs for different p-levels

Table 3 replicates the results of Table 2 but uses unconditional pass rates instead. These numbers serve for purely illustrative purposes, as it is not totally correct to analyze nested theories using unconditional pass rates. Recall that the nested structure can be summarized as follows.

$$\begin{aligned} \text {C-QLU(C)} \subseteq \text {C-LNU} \subseteq \text { LNU } \text { and } \text {C-QLU(Q)} \subseteq \text {QLU} \subseteq \text { LNU } \end{aligned}$$

That is, C-QLU is nested within both C-LNU and QLU, while both C-LNU and QLU are nested within LNU. Finally, C-LNU and QLU are independent theories.

Table 4 replicates the results of Table 2 using the CLT approximations for the confidence intervals of pass rates. Note that unlike the confidence intervals generated by Clopper-Pearson procedure (exact confidence intervals), the CLT confidence intervals are symmetric. Thus, they might run out of the domain: returning values above one or below zero. Even though there are no bound violations in our data we prefer to report the exact confidence intervals. Using the CLT confidence intervals does not alter the results presented in Table 2.

Table 4 Left panel: Pass rates for each theory considered. Unlike Table 2 in the main text, this table reports confidence intervals for pass rates using CLT approximations. The first number is the percentage pass rate; the fraction in parentheses is the numbers of subjects used to compute the pass rates; the numbers brackets are the 95% confidence intervals. Right panel: HMI cutoffs for different p-levels

1.2 Analysis using CCEI

Another measure of distance to rationality is the Critical Cost Efficiency Index (CCEI) introduced by Afriat (1973). The definition of the CCEI in the augmented consumption space we consider is, however, ambiguous. An interpretation of the CCEI is the share of wealth to be left on the table needed to make an agent rational. This can be done in linear budget sets straightforwardly. Assume all prices (p) are set up such that for each budget set income is normalized to be equal to 1. Then $p^t y \le 1$ defines the “revealed preference relation.” The CCEI redefines this constraint by introducing $e\in [0,1]$ such that $p^t y \le e$ defines a new revealed preference relation. In this case $1-e$ provides the relative measure of the wealth one needs to sacrifice in order to rationalize observed choices. Further, the CCEI finds the maximal e (minimizing potential wealth losses) such that observed choices are rationalizable.

However, in the augmented consumption space, “wealth” (wage in our context) is already one of the dimensions of the consumption space. Thus one option is to consider only changes in wages. This corresponds to the case described on the Fig. 9b. An alternative explanation of the CCEI is based on the idea of “thick indifference curves” or “just noticeable differences” as axiomatized by Dziewulski (2020). This interpretation allows us to disregard the fact that the consumption space includes money already. This interpretation allows shifting the budget downwards as presented in Fig. 9a. The nonlinearity of the budget set, however, produces another complication. Since we are using piece-wise linear budgets, it matters if the value of x at which price regime changes Varies with the level of the CCEI. The just noticeable difference interpretation of the CCEI (see Fig. 9a) implies that the “break-point” varies with e. The wealth interpretation of the CCEI (see Fig. 9b) does not allow the break-point to vary with e. Next we present the results for the pass rates for both interpretations of CCEI.

Fig. 9

Introducing CCEI in augmented space

1.2.1 CCEI based on wealth

Table 5 presents the results as in Table 2 using the CCEI based on changes in wealth only (version from Fig. 9b) as the measure of distance to rationality. The pass rates for LNU and QLU are comparable to those using HMI. However, comparing C-LNU to QLU we observe that the pass rates for C-LNU are lower. This is evidence that the assumption of concavity is more restrictive than quasilinearity. Overall we see that the pass rates for C-QLU(Q) are lower than those for QLU. We observe that concavity is a restrictive assumption. However, pass rates for C-QLU(C) and C-LNU are comparable. Thus, adding the quasilinearity assumption to C-LNU does not seem to be restrictive.These results are consistent with the results presented in Table 2.

Table 5 Left panel: pass rates with absolute numbers in parentheses and confidence intervals for the pass rates in brackets. Tests for QLU and C-LNU are conditional on passing LNU; tests for CQLU are conditional on either QLU or C-LNU. Right panel: CCEI cutoffs for the given “significance” level

1.2.2 CCEI based on just noticeable differences

Table 6 presents results using the just noticeable differences interpretation of the CCEI (version from Fig. 9a). We see that the main results still hold. First, we see that pass rates for QLU condition on LNU are high. Moreover, using this version of the CCEI we obtain higher pass rates for QLU than C-LNU. We also see that pass rates for C-LNU are weakly lower than those of C-QLU(C) for all significance levels. This means that under the assumption of concavity adding quasilinearity is costless. Finally, we see that pass rates for C-QLU(Q) are lower than those of QLU. This agrees, accounting for noise, with the results obtained with alternative measures of rationality. We conclude that our main finding of quasilinearity is not more restrictive than concavity (see Table 2) does not depend on the measure of distance to rationality utilized.

Table 6 Left panel: pass rates with absolute numbers in parentheses and corresponding confidence intervals in brackets. Tests for QLU and C-LNU are conditional on passing LNU; tests for CQLU are conditional on either QLU or C-LNU. Right panel: CCEI cutoffs for the given “significance” level