Abstract
We present a rational model of consumer choice, which can also serve as a behavioral model. The central construct is \(\lambda \), the marginal utility of money, derived from the consumer’s rest-of-life problem. It provides a simple criterion for choosing a consumption bundle in a separable consumption problem. We derive a robust approximation of \(\lambda \) and show how to incorporate liquidity constraints, indivisibilities, and adaptation to a changing environment. We find connections with numerous historical and recent constructs, both behavioral and neoclassical, and draw contrasts with standard partial equilibrium analysis. The result is a better grounded, more flexible, and more intuitive description of consumer choice.
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Notes
Indeed, \(N\) could even be infinite and not affect our analysis.
Prices that apply to future goods are discounted appropriately. For simplicity, we treat \(L\) as a constant, but later note how it can be endogenized.
In the special case that the same separable subproblem recurs every period, (6) closely resembles the Bellman equation familiar to macroeconomists (where time discounting is built into \(V(.)\)). Note, however, that in general, \( \widetilde{V}\) differs from \(V\) not only in the value of the state variable (remaining wealth) but also in the set of (dated) goods available to purchase.
Of course, expenditure \(L-\mathbf {p}\cdot \mathbf {x}\) available for the continuation problem falls as we move out along the IEP so, by concavity, \( V^{\prime }\) increases. Thus, a more complete informal description is that the falling proportionality factor meets the rising \(V^{\prime }\) at a unique point, \(\mathbf {x}^{*}\), on the IEP; more formally, under present assumptions, the intermediate value theorem and implicit function theorem guarantee a unique smooth interior solution described by (7).
The maximand in (9) slightly generalizes quasi-linear utility. Textbook treatments (e.g., Varian 1992, p. 154, 164–167) often assume that \(n=1\) so the variable of interest \(x\) is scalar, and that \(\widehat{\lambda }=1\). In our notation, the textbook quasi-linear utility function would be written as \(u(x)+m\) with budget constraint \(m=L-px.\) Our approach shows that textbook quasi-linear preferences can be justified for single separable goods, and can be generalized directly for separable bundles of related goods. Given a constant exogenous \(\lambda \), there is no loss of generality in using a VNM utility function for \(u\) (and implicitly, \( U\)) that normalizes its value to 1, but our analysis sheds light on the conditions for which the constancy assumption is justified.
For given constant \(\lambda ,\) the Frisch demand function is formally equivalent to \(\mathbf {x}^{\lambda }\), notwithstanding important differences in interpretation (see the second segment of Appendix ).
This is intuitive, as a constant \(\lambda \) means that there are no income effects. As an empirical matter, it is not clear that Giffen goods exist at all (c.f. Dwyer and Lindsay 1984; Nachbar 1998), with the possible exception of extreme poverty (c.f. Jensen and Miller 2008) when indeed we would not expect consumers to take the future into account and to exhaust whatever purchasing power they have.
Only approximately linear because \(\beta \) is a second derivative evaluated at a point that can depend on \(y\).
The logic is reminiscent of quasi-hyperbolic discounting: there is a big difference between today and tomorrow, but tomorrow and the day after look similar from the vantage point of today.
Here we assume, for simplicity, that the consumer does not try to extrapolate from individual observed prices to changes in the price level. See Deaton (1977) for an exploration of that idea.
Textbooks often refer to \(\mathbf {x}^{B}\) as the Marshallian demand function, in distinction to the Hicksian demand function which holds constant the utility level rather than \(B\) or \(\lambda \). The literature survey in Appendix will note that \(\mathbf {x}^{B}\) actually owes more to Hicks than to Marshall, whose preferred demand function was a special case of \(\mathbf {x}^{\lambda }.\)
For the single good case, we have the following simplified expressions
$$\begin{aligned} \frac{\partial x^{*}}{\partial p}(\widehat{p})&= \frac{ \lambda - BV^{\prime \prime }(L-B)}{u^{\prime \prime }(x^{*})-p^2V^{\prime \prime }(L-B)}, \\ \frac{\partial x^{\lambda }}{\partial p}(\widehat{p})&= \frac{\lambda }{ u^{\prime \prime }(x^{*})}, \text{ and } \\ \frac{\partial x^{B}}{\partial p}(\widehat{p})&= \frac{ B}{p^2}. \end{aligned}$$If such a consumer is faced with several purchasing decisions (subproblems) subject to a unified liquidity constraint, then Eq. (27) and Fig. 1, with \(q=0\) and \(r=\infty \), apply only to the final decision. For the prior decisions, she should still use the unconstrained \( \lambda \) rule, where the constraint will be built into her continuation (indirect) utility. When the borrowing constraint binds, the consumer’s time horizon effectively shrinks to a single pay period. This would decrease the precision of the estimate of \(\lambda ,\) leading to a normative and a positive prediction. First, the final decision should incorporate more goods than usual to decrease the bias in \(\lambda .\) Second, as the estimate is an underestimate—as \(V^{\prime }(L-px)>V^{\prime }(L)\)—paycheck-to-paycheck consumers using the moneysworth demand should be bad at consumption smoothing and get to the end of the month short of money.
Hauser and Urban (1986) pose as alternative hypotheses that consumers use “value for money,” \(u/p,\) or “net value,” \(u-\lambda p,\) to prioritize purchases of indivisibles. Our analysis shows that the two rankings are equivalent for yes/no decisions, but we shall now show that “net value” is the appropriate criterion for mutually exclusive alternatives.
Note that this crucial detail is overlooked in the melioration theory of Herrnstein and Prelec (1991), which posits that the option with higher utility per $ is the one that is chosen (in a distributed choice problem).
Note that for yes/no decisions the quality-price ratio is also incremental but the benchmark is normalized to zero.
We suspect that it is also quite descriptive of actual consumer behavior, but we are not aware of any empirical studies so far that compare the predictive power of \(\mathbf {x}^{\lambda }\) and \(\mathbf {x}^{B}\).
In public finance, for example, it is well known that a uniform income tax is more efficient for raising a given amount of revenue than a collection of specific taxes on individual items.
Arguably, revealed preference theory is the real contribution of ordinality. A companion paper (Sákovics 2013) shows that revealed preference theory has a cardinal counterpart, with the Generalized Axiom of Revealed Preference replaced by the stronger Axiom of Revealed Valuation (ARV). Formally, assume that we have \(T\) observations of a consumer’s chosen bundles, \(x^{t}\in \mathfrak {R}_{+}^{n},\) at price vectors, \( p^{t}\in \mathfrak {R}_{++}^{n}\). Let \(a_{tj}=p^{t}\cdot (x^{j}-x^{t})\) denote the pecuniary advantage of the chosen bundle, \(x^{t}\), relative to an arbitrary bundle, \(x^{j}.\) Then ARV states For every ordered subset \( \{i,j,k,...,s\}\subseteq \{1,2,...,T\}, \ \ a_{ij}+a_{jk}+...+a_{si}\ge 0.\) The main theorem states If and only if the observed choices satisfy ARV, they are rationalizable by a cardinal utility function of the form \( u(x)-\lambda p\cdot x\).
See also Bordalo et al. 2013, for a related model based on the salience of past observations.
We have chosen the \(i\)th coordinates as we will be checking sensitivity to the \(i\)th price.
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Acknowledgments
This paper supersedes the previously circulated “The marginal utility of money: a new Marshallian approach to consumer choice.” We thank Olga Rud Rabanal for research assistance. For helpful suggestions, we thank two anonymous referees of this Journal, as well as Luciano Andreozzi, Buz Brock, Roberto Burguet, Andrew Clausen, Joan-Maria Esteban, Steffen Huck, Axel Leijonhufvud, Youn Kim, Michael Mandler, Albert Marcet, Carmen Matutes, David de Meza, Ryan Oprea, Martin Shubik, Peter Simmons, Nirvikar Singh, Hal Varian, Donald Wittman, and seminar audiences at Barcelona GSE, Cambridge, CEU (Budapest), Edinburgh, Glasgow, Roy Seminar (Paris), UBC-UHK Summer Micro Seminars (Hong Kong), UCIII (Madrid), UC at Santa Cruz, and York (England).
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Friedman, D., Sákovics, J. Tractable consumer choice. Theory Decis 79, 333–358 (2015). https://doi.org/10.1007/s11238-014-9461-0
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DOI: https://doi.org/10.1007/s11238-014-9461-0