Rationalization: 4

Intra-household rationality

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Introduction

This is a reading note for a series of work on the collective rationality of households. Compared to "individual rationality", this model explicitly recognizes that households consist of multiple individuals (or decision makers) with their own (rational) preferences, which cannot be taken as one "unitarily rational" decision maker. Compared to pure large-market regime, the model here does not assume the number of decision makers is large enough to make the aggregate demand "arbitrary" in the sense of Sonnenschein-Mantel-Debreu theorem. There is a comprehensive explanation for this phenomenon in Browning and Chiappori [1]. An earlier approach is to study the smooth demand functions to see whether the Slutsky matrix can be decomposed into the sum of the PND matrices and rank-one correctors [2]. In this note we focus on the non-parametric approach, which may only provide nonsmooth demand correspondences.

Historical remarks.

  • Basic: [3], which gave conditions for the collective rationality of households; the system is nonlinear so the authors proposed easier "testable" sufficient or necessary conditions. The collective consumption model there does not require a prior specification of the (public or private) nature of the goods; so one good may be impure and have private as well as public consumption.
  • With specification of the nature of the goods, the nonlinear testing system is sufficient and necessary, and it can later be translated to a mixed-integer linear system, which is also sufficient and necessary, then, the idea is applied to,
  • Welfare-related discussions [4] (earlier version [5]), and a later study [6]. In [4], the authors focused on collective consumption models that explicitly distinguish between public and private goods.
  • How good is the fit: "goodness-of-fit" test [7], like Varian's inconsistency index. The computational complexity of the index is studied in Smeulders et al. [12].
  • Then the authors had a paper on OR [8], which focused on computational complexity. This showed that, even with only two decision makers in the collective rationality model and there is no public goods, the problem is NP-complete.

In this note we revisit the nonparametric test in the same fashion of Afriat-Varian system [9], [10], which is an algorithmic approach to build a utility function from the observed consumption choices, and to declare when it is and is not possible to rationalize the observed choices. These tests have its roots in Samuelson's revealed preference theory [11]. For a finite number of rational players, the system needed to rationalize the collective consumption is a mixed-integer linear system [7], [4].

Household with private consumption

A preliminary setting for unitary rationality can be found in a previous post. We start with the simplest case where there are $n$ commodities or goods and $m$ decision makers in a household. We assume that each of the $n$ goods can only be consumed privately without externalities, as opposed to public goods. For demonstration purpose, $m=2$ is assumed.

Observed choices. Suppose we observe $K$ household consumption, $\mathbf{q}_k \in \mathbb{R}_+^n$ with price $\mathbf{p}_k \in \mathbb{R}_+^n$ at time $k \in [K] = \lbrace 1, \ldots, K\rbrace$. Sometimes we may also observe assignable information, which means we already know some individual $i$ has consumed at least $\mathbf{q}^i_k$. Hence, we obtain a dataset $\mathbf{\Xi} = \lbrace(\mathbf{p}_k, \mathbf{q}_k, \mathbf{q}^1_k, ..., \mathbf{q}^m_k)\rbrace_{k=1}^K$.

2-GARP and the MIL system. For a household of two person, a feasible split of personalized quantities is,

$$\mathbf{x} \in \mathcal{F} = \left\{ (\mathbf{x}^1_k, \mathbf{x}^2_k) \in \mathbb{R}_+^{2n} ~\big|~ \mathbf{x}^1_k + \mathbf{x}^2_k = \mathbf{q}_k, ~\forall k \in [K] \right\}.$$

We have the following definitions.

Collective rationality. We say a combination of utility function $(u^1, u^2)$ provides a collective rationalization of the dataset $\mathbf{\Xi}$ if for each $k$ there exists Pareto weights $(\mu_k^1, \mu_k^2) \in \mathbb{R}_+^2$, and a split $(\mathbf{x}^1_k, \mathbf{x}^2_k) \in \mathbb{R}_+^{2n}$ such thatNote that we do not require the weights to be equal for any $k$. $\mu^i_{\cdot}$ can be seen as the bargaining power of individual $i$ in the negotiation of the split.

$$\begin{align*} (\mathbf{x}_k^1, \mathbf{x}_k^2) \in \mathcal{F}, \quad\text{and}\quad \textstyle\sum_{i=1}^2 \mu_k^i u^i_k(\mathbf{x}^i_k) &\ge \textstyle\sum_{i=1}^2 \mu_k^i u^i_k(\mathbf{y}^i), \\ \forall (\mathbf{y}^1, \mathbf{y}^2) \in \mathbb{R}_+^n \times \mathbb{R}_+^n, \quad \left\langle \mathbf{p}_k, \textstyle\sum_{i=1}^2\mathbf{y}^i \right\rangle &\le \langle \mathbf{p}_k, \mathbf{q}_k \rangle. \end{align*}$$

2-GARP. A dataset $\mathbf{\Xi}$ is consistent with a 2-GARP household, if and only if, there exist a split $\lbrace(\mathbf{x}^1_k, \mathbf{x}^2_k)\rbrace_{k=1}^K$ in $\mathcal{F}$, and both each $\lbrace(\mathbf{p}_k, \mathbf{x}^i_k)\rbrace_{k=1}^K$ satisfies GARP (cf. [10]).

Let $\mathbf{\Xi}$ be a set of observations. The following conditions are equivalent:
(1) there exists a combination of concave and continuous utility functions $u^1, u^2$ that provide a collective rationalization;
(2) the split $\{(\mathbf{x}^1_k, \mathbf{x}^2_k)\}_{k=1}^K$ satisfies 2-GARP;
(3) the following system is feasible:

$$\begin{align*} U_k^i &\le U_t^i + \lambda_t^i \cdot \langle \mathbf{p}_t, \mathbf{x}_k^i-\mathbf{x}_t^i \rangle, && \forall k, t \in [K], \forall i \in [2], \\ (U_k^i, \lambda_k^i) &\in \mathbb{R}_+^2, && \forall k \in [K], \forall i \in [2], \\ (\mathbf{x}_k^1, \mathbf{x}_k^2) &\in \mathcal{F}, && \forall k \in [K]. \end{align*}$$

The proof to the above theorem is based on the Afriat-Varian system [9], [10]. Basically, GARP guarantees that the rationality is "observationally" equivalent to a concave utility function. Different from the original unitarian rationality, here because each of the individual $i$'s consumption is a variable, the system above is a bilinear system. On the other hand, because of alternative representation of GARP, the following MIL system is equivalent to the one in the theorem.

A dataset $\mathbf{\Xi}$ is consistent with a 2-GARP household if and only if the following mixed-integer linear system is feasible:

$$\begin{align*} \mathbf{x}_k^1 + \mathbf{x}_k^2 &= \mathbf{q}_k, && \forall k \in [K], \\ \mathbf{x}_k^i &\ge \mathbf{q}^i_k, && \forall i \in [2], \forall k \in [K], \\ y^i_{kt} + y^i_{ts} - y^i_{ks} &\le 1, && \forall k,t,s \in [K], \forall i \in [2], \\ \langle \mathbf{p}_k, \mathbf{x}_k^i - \mathbf{x}_{t}^i \rangle &~{\color{#BE5845}{<}}~ M y^i_{kt}, && \forall k, t \in [K], \forall i \in [2], \\ \langle \mathbf{p}_{t}, \mathbf{x}_{t}^i - \mathbf{x}_{k}^i \rangle &\le \langle \mathbf{p}_{t}, \mathbf{q}_{t} \rangle (1-y^i_{kt}), && \forall k, t \in [K], \forall i \in [2], \end{align*}$$

where $\mathbf{x}_k^i \in \mathbb{R}_+^n$ and $y^i_{kt} \in \{0,1\}$ for all $k,t \in [K]$ and $i \in [2]$.

Clearly, the system has a decentralized structure; apart from the split constraint, other constraints are the revealed preference constraints specifically for each individual $i$. If $k \succeq t$, then $y^i_{kt} = 1$, it enumerate all possible pairs of comparison $k,t$. The constraint on $y$ (the third one) means if $k\succeq t, t\succeq s \Longrightarrow k\succeq s.$ Similar structure may apply even if we limit $\mathbf{x}$ to be in some parametric family.One can show if we like $x$ to be demand from a linear utility, we arrive at a similar problem.

Note that, the "strict sign $<$" here means, if $y_{kt}^i = 0$, then we must have $\mathbf{x}_k^i \prec \mathbf{x}_t^i$. For implementation, we can write,

$$\langle \mathbf{p}_k, \mathbf{x}_k^i - \mathbf{x}_{t}^i \rangle + \varepsilon ~{\color{#BE5845}{\le}}~ M y^i_{kt}.$$

Using the system above, it is not hard to show that even 2-GARP is NP-complete, with or without assignable information.

References
  1. [1] Browning, M., Chiappori, P. A.: Efficient intra-household allocations: a general characterization and empirical tests. Econometrica. 66(6), 1241–1278 (1998).
  2. [2] Chiappori, P. A., Ekeland, I.: The micro economics of group behavior: general characterization. Journal of Economic Theory. 130(1), 1–26 (2006).
  3. [3] Cherchye, L., De Rock, B., Vermeulen, F.: The collective model of household consumption: a nonparametric characterization. Econometrica. 75(2), 553–574 (2007). https://doi.org/10.1111/j.1468-0262.2006.00757.x
  4. [4] Cherchye, L., De Rock, B., Vermeulen, F.: The revealed preference approach to collective consumption behaviour: testing and sharing rule recovery. The Review of Economic Studies. 78(1), 176–198 (2011).
  5. [5] Cherchye, L., De Rock, B., Vermeulen, F.: The revealed preference approach to collective consumption behavior: testing, recovery and welfare analysis. IZA Discussion Papers, Working Paper 3062 (2007). https://www.econstor.eu/handle/10419/34655
  6. [6] Cherchye, L., Cosaert, S., De Rock, B., Kerstens, P. J., Vermeulen, F.: Individual welfare analysis for collective households. Journal of Public Economics. 166, 98–114 (2018). https://doi.org/10.1016/j.jpubeco.2018.08.006
  7. [7] Cherchye, L., De Rock, B., Sabbe, J., Vermeulen, F.: Nonparametric tests of collectively rational consumption behavior: an integer programming procedure. Journal of Econometrics. 147(2), 258–265 (2008).
  8. [8] Nobibon, F. T., Cherchye, L., Crama, Y., Demuynck, T., De Rock, B., Spieksma, F. C. R.: Revealed preference tests of collectively rational consumption behavior: formulations and algorithms. Operations Research. 64(6), 1197–1216 (2016). https://doi.org/10.1287/opre.2016.1527
  9. [9] Afriat, S. N.: The construction of utility functions from expenditure data. International Economic Review. 8(1), 67–77 (1967).
  10. [10] Varian, H.: The nonparametric approach to demand analysis. Econometrica. 50(4), 945–973 (1982).
  11. [11] Samuelson, P. A.: Consumption theory in terms of revealed preference. Economica. 15(60), 243–253 (1948).
  12. [12] Smeulders, B., Spieksma, F. C. R., Cherchye, L., De Rock, B.: Goodness-of-fit measures for revealed preference tests: complexity results and algorithms. ACM Transactions on Economics and Computation. 2(1), 3:1–3:16 (2014).