D01 | Macro Paper Warehouse

Choice and Opportunity Costs

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1 — Overview

This paper develops a unified choice-theoretic framework in which agents evaluate alternatives not in isolation but relative to their opportunity costs — the alternatives they forgo. The central departure from classical theory is the relaxation of additive separability between benefits and costs. In the standard additive model, accounting for opportunity costs is behaviourally equivalent to simple utility maximisation: a decision maker who correctly perceives the feasible set and maximises an additively separable utility will make identical choices whether or not opportunity costs are explicitly considered (the paper calls this the irrelevance of opportunity costs under additivity, formally establishing it as a general result). Once additive separability is relaxed, however, opportunity costs become non-trivial and generate a genuinely distinct theory of choice.

The primitive of the model is a net preference — an asymmetric binary relation on pairs (x, y) of distinct alternatives, where (x, y) ≻ (w, z) means the agent strictly prefers obtaining x while forgoing y over obtaining w while forgoing z. Because the opportunity cost of a chosen alternative depends on what else the agent would choose, and vice versa, choice emerges from an intrapersonal equilibrium rather than from direct maximisation.

The paper defines and axiomatically characterises two nested models. The Recursive Opportunity Model (ROM) adopts a behavioural definition of opportunity costs: the cost of the chosen alternative x in menu A is c(A \ x), the alternative that would actually be chosen were x unavailable; the cost of every unchosen alternative is x itself. This recursive structure is completely characterised by a single observable condition — Weak Path Independence (WPI): if x is chosen when added to a menu A, then x must also be chosen in a pairwise comparison against c(A). WPI is shown to imply Always Chosen (AC) — that a Condorcet winner is always selected — but it permits pairwise cycles of choice (failures of No Binary Cycles). Rationality within the ROM requires additionally that the net preference be a strict order satisfying Congruence, an acyclicity condition on the gross preference induced by the net preference. Even then, the utility function being maximised need not coincide with the gross preference naturally implied by the underlying psychological net preference, raising a welfare identification problem.

The Opportunity Model (OM) generalises the ROM by allowing the opportunity cost of the chosen alternative to be any unchosen alternative rather than the recursively determined one. This relaxation permits both pairwise cycles and menu effects (Condorcet violations). The OM is completely characterised by Never Chosen (NC): an alternative that loses every pairwise comparison within a menu (a Condorcet loser) cannot be chosen. Imposing a strict order and Congruence on the net preference of an OM rules out only pairwise cycles, leaving menu effects intact. Full rationality within the OM is restored only with the additional assumption that opportunity costs are non-decreasing in the induced gross preference as the feasible set expands (the Increasing Opportunity Model).

Extensions characterise multivalued versions of both models (M-ROM and M-OM) via adapted axioms on choice correspondences, and show that several known behavioural models in the literature — including list-rationalizable choice and game-tree rationalizable choice — satisfy WPI and thus are instances of ROM. Applications demonstrate that OMs can represent the attraction effect and the multiple decoy effect, providing a preference-maximisation account without appealing to bounded cognition, and that ROMs can represent intransitive pairwise choices via smooth parametric net preferences, avoiding the discontinuities of lexicographic semiorder models.

Q: What is the paper’s foundational definition of opportunity cost, and how does it differ from the standard textbook definition? A: The paper defines the opportunity cost of the chosen alternative x in menu A as the alternative that would actually be chosen from A \ {x} — that is, c(A \ {x}). The opportunity cost of any unchosen alternative y is the actual choice x. The standard textbook definition — “the next-best feasible alternative” — presupposes context-independent, additively separable preferences, precisely the assumption the paper relaxes. The behavioural definition is grounded directly in the agent’s own choice function, making it consistent with non-separable evaluations.

Q: Under what conditions do opportunity costs become irrelevant, and why? A: If preferences admit an additively separable utility representation u, then for any finite menu A and any two alternatives x and y, u(x) ≥ u(y) if and only if u(x) − max_{a ∈ A{x}} u(a) ≥ u(y) − max_{a ∈ A{y}} u(a). Net utility maximisation and gross utility maximisation rank alternatives identically. Opportunity costs become non-trivial only when additive separability is relaxed — at that point, the agent’s comparative evaluation of (alternative, cost) pairs can produce choices that no gross utility function rationalises.

Q: What is the Recursive Opportunity Model (ROM) and what single axiom characterises it? A: A choice function c is a ROM if there exists a net preference ≻ such that for every menu A and every unchosen alternative x, the chosen alternative evaluated at its opportunity cost is preferred to x evaluated at c(A). This is equivalent to the choice function satisfying Weak Path Independence (WPI): if x ∉ A and x = c(A ∪ {x}), then x = c({x, c(A)}). WPI is necessary and sufficient for a ROM (Theorem 1). It is not sufficient for full rationality, as it permits pairwise cycles while ruling out menu effects.

Q: What kinds of irrationality can a ROM exhibit, and what kinds does it preclude? A: The paper establishes (Corollary 1) that WPI implies Always Chosen — a ROM always selects the Condorcet winner when one exists. Therefore, the only admissible form of irrational behaviour in a ROM is pairwise cycles (failures of No Binary Cycles). Condorcet violations (menu effects) are precluded. A ROM becomes fully rational if and only if it additionally satisfies No Binary Cycles.

Q: What additional condition on the net preference guarantees that a ROM is rational? A: Theorem 2 establishes that a choice function is rational if and only if it is a ROM generated by a net preference that is a strict order (complete, asymmetric, transitive) satisfying Congruence. Congruence requires that the induced binary relation P≻ on alternatives — defined by xP≻y whenever there exists z such that (x, z) ≻ (y, z) or (z, y) ≻ (z, x) — is acyclic. For a (u, v)-additive net preference, Congruence holds if and only if u and v are ordinally equivalent.

Q: Can rational behaviour generated by a ROM be welfare-analysed using revealed preference in the standard sense? A: No — and this is a key warning in the paper. Even when a ROM with a strict order and Congruence produces fully rational behaviour, the utility function being maximised need not coincide with the gross preference P≻ naturally induced by the underlying net preference. The paper provides an explicit example (Remark 1, equation 10) in which the choice-rationalising order P is xPyPz while the induced preference is xP≻zP≻y. The utility “revealed” by choice may diverge from the psychological primitive driving that choice, undermining the normative authority of standard revealed preference welfare analysis.

Q: What is the Opportunity Model (OM) and how does it extend the ROM? A: The OM relaxes the recursive assumption by allowing the opportunity cost of the chosen alternative to be any unchosen element of the menu rather than specifically c(A \ c(A)). This breaks the recursive structure while preserving the intrapersonal equilibrium character (the choice still affects the net value of alternatives). The OM is completely characterised by Never Chosen (NC): no Condorcet loser can be chosen (Theorem 3). Unlike the ROM, an OM may fail to select the Condorcet winner, permitting both pairwise cycles and Condorcet violations.

Q: What is the Increasing Opportunity Model and when does it restore full rationality? A: An IOM is an OM in which the opportunity function o is monotone in the sense that if A ⊃ B and o(A) ≠ o(B), then o(A) is ranked higher than o(B) in the induced gross preference P≻. Intuitively, opportunity costs do not decrease as the feasible set expands. Theorem 5 establishes that a choice function is rational if and only if it is an IOM generated by a net preference that is a strict order satisfying Congruence. Full rationality within the OM thus requires both the internal consistency of the net preference (strict order, Congruence) and this monotonicity of opportunity costs.

Q: How does the paper explain the attraction effect using the OM? A: In the canonical formulation, c({x,y}) = x, c({y,d}) = y, c({x,d}) = x, and c({x,y,d}) = y, where d is a decoy. This pattern is incompatible with gross preference maximisation. The paper represents it as an OM with opportunity function o({x,y,d}) = d and a strict net preference order yd ≻ xy ≻ yx ≻ xd ≻ dx ≻ dy. The psychological interpretation is that the introduction of the decoy shifts the comparator for y from x to d; y looks more favourably comparable to d than x does, so the equilibrium where y is chosen is selected. No bounded cognition or imperfect attention is assumed.

Q: How does the framework account for multiple decoys? A: With decoys dx and dy specific to x and y respectively, the observed pattern c({x,y}) = x and c({x,y,dy}) = y and c({x,y,dx,dy}) = y can be represented as an OM with a transitive net preference satisfying xdx ≻ ydy ≻ xy ≻ yx ≻ dyy ≻ dxx and opportunity function o({x,y,dx,dy}) = dx, o({x,y,dy}) = dy. The paper notes this net preference can be extended to a strict order while preserving the choice pattern. This accommodates a phenomenon that poses a challenge to standard theoretical choice literature (per Masatlioglu, Nakajima and Ozbay [25]).

Q: How does the ROM explain intransitive choices more smoothly than lexicographic semiorder models? A: The paper shows that the Tversky (1969) cyclical pattern c({x,y}) = x, c({y,z}) = y, c({x,z}) = z with x=(115,7), y=(117,3), z=(120,0) can be generated by net preferences that admit smooth parametric representations. Specifically, for any two alternatives w=(a,b) and z=(c,d), the paper proposes (w,z) ≻ (z,w) iff (max{a−c, b−d})² > k(min{a−c, b−d})², where k is a relative sensitivity parameter. For k=1/2 this yields the required cycle. Lexicographic models require sharp discontinuities in preference and systematic avoidance of trade-offs, which are often viewed as implausible within the standard economic paradigm; the smooth parametric form avoids these features.

Q: What is the relationship between ROMs and previously studied choice models in the literature? A: Several known models satisfy WPI and are therefore, by Theorem 1, instances of ROMs: specifically, Rationalizability by Game Trees (Xu and Zhou) and List-Rationalizable Choice (Yildiz) are shown to satisfy WPI. The two-stage choice model of Bajraj and Ulku satisfies NC but not WPI, making it an OM but not a ROM. The net preference being maximised in each case can in principle be recovered using the explicit construction in the proof of Theorem 1.

Q: How does the ROM relate to Koszegi-Rabin personal equilibrium? A: Both models involve preferences that depend on a variable determined endogenously by choice, requiring an intrapersonal equilibrium concept in which the agent’s conjectures about their own behaviour must be internally consistent. The key difference is that in Koszegi-Rabin the psychological primitive is a set of reference-dependent preferences ≻r on alternatives in X (where r is the reference point), and equilibrium requires c(A) ≻{c(A)} y for all y ∈ A \ c(A). In the ROM, the primitive is a preference on pairs of distinct alternatives, and the opportunity cost differs for each alternative being compared (the chosen alternative has one opportunity cost, each unchosen alternative has a different one, namely c(A) itself).

Net preference: An asymmetric binary relation on pairs (x, y) of distinct alternatives, where (x, y) ≻ (w, z) means the agent strictly prefers to be in a situation where they choose x while forgoing y over a situation where they choose w while forgoing z. The primitive is defined on X = {(x, y) ∈ X × X : x ≠ y}, without imposing additive separability.

Recursive Opportunity Model (ROM): A choice function c is a ROM if there exists a net preference ≻ such that for every menu A and every unchosen x, the pair (c(A), c(A \ c(A))) ≻ (x, c(A)). The opportunity cost of the chosen alternative is defined recursively as c(A \ c(A)); choice results from intrapersonal equilibrium rather than simple maximisation.

Opportunity Model (OM): A generalisation of the ROM in which the opportunity cost of the chosen alternative can be any unchosen alternative in the menu (not necessarily the recursively determined one). Characterised by Never Chosen: no Condorcet loser can be chosen. Permits both pairwise cycles and Condorcet violations.

Weak Path Independence (WPI): The axiom characterising ROMs: if x ∉ A and x = c(A ∪ {x}), then x = c({x, c(A)}). Equivalently, if an alternative is chosen upon being added to a menu, it must also win in a pairwise comparison with what was previously chosen from the original menu.

Congruence: A consistency condition on net preferences requiring that the induced binary relation P≻ — defined by xP≻y whenever there exists z such that (x,z) ≻ (y,z) or (z,y) ≻ (z,x) — is acyclic. For a (u,v)-additive net preference, Congruence holds if and only if u and v are ordinally equivalent. Together with a strict net preference order, Congruence in a ROM is equivalent to rational choice.

Intrapersonal equilibrium: The concept underlying both models: an agent is in equilibrium when selecting x from A if they correctly anticipate their own contingent behaviour across hypothetical scenarios (i.e., they use the actual choice function c to evaluate what they would choose from A \ {x}), and the chosen alternative is net-preference-maximal given those consistent conjectures.

Never Chosen (NC): The axiom characterising OMs: an alternative that is a Condorcet loser — losing in every pairwise comparison within a menu — cannot be chosen from that menu. NC is weaker than WPI (which implies both Always Chosen and Never Chosen) and is the precise behavioural content of the OM.

Do The Effects of Nudges Persist? Theory and Evidence from 38 Natural Field Experiments

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks why the Home Energy Report (HER) — a widely deployed social-comparison nudge that shows households how their electricity consumption compares to their neighbors — produces behavioral changes that persist long after the nudge is discontinued, while analogous nudges in other domains (charitable giving, financial savings, voter turnout, tax compliance) fade almost entirely within a year or two. The authors formalize a research design to decompose the HER’s long-run effectiveness into two channels: technology adoption (a change in the stock of energy-efficient capital in the home) and habit formation (a change in the stock of habits or skills in the resident).

The identifying strategy exploits the administrative rule that when the initial resident in an HER experiment moves out, HER mailings stop immediately — but electricity consumption in the home continues to be observed as new residents occupy it. Under three assumptions — (1) treatment assignment did not influence the initial resident’s decision to move; (2) treatment assignment did not influence the type of resident who moved in; and (3) energy-efficient technology adopted in response to the HER remained in the home after the move — the post-move HER effect identifies the fraction of the long-run treatment effect attributable to technology adoption (ATK), and the remainder identifies the fraction attributable to habit formation (ATH).

Data come from 38 natural field experiments administered by Opower between 2008 and 2013 across 21 U.S. residential energy providers, comprising 61,310,166 electricity bills for 1,810,096 homes. The mover sample, restricted to homes where the initial resident deactivated service at or after the receipt of their fourth HER, contains 5,890,855 bills for 139,908 homes. Treatment and control homes enter the mover sample at statistically indistinguishable rates and have similar baseline electricity consumption.

The main findings: the HER reduced electricity consumption by 2.1 percent in the long run (the pre-move ATE). After the initial resident moved and the HER was discontinued, 1.1 percent of the reduction persisted in the home — attributable to technology. The habit channel accounts for the remaining 1.0 percent reduction. Normalizing by the ATE, 51.4 percent (s.e. = 13.1) of the long-run effectiveness is attributable to technology adoption and 48.6 percent to habit formation. The persistence of the post-move effect is robust across alternative specifications, different HER-receipt cutoffs, balanced panels, and exclusion of low-consumption move-period homes. A falsification test using rental homes — where tenants do not typically own appliances and the technology channel is therefore shut down — yields a null post-move effect, consistent with the balanced-habits assumption.

The authors use these results to explain a broader empirical pattern: one year after discontinuation, social comparison nudges targeting compliance, charitable giving, savings, and voter turnout retain on average only 4 percent of their initial effect, while nudges targeting energy and water conservation retain 65 percent. The paper argues this divergence reflects the relative abundance of enabling technologies in conservation contexts versus their absence in compliance or voting contexts. The findings also have cost-benefit implications: ignoring HER-induced technology adoption overstates net benefits by as much as 65 percent, depending on assumed technology cost per kWh saved (ranging from $0.03 per kWh saved per Gillingham et al. 2018 to $0.12 per kWh saved per Billingsley et al. 2014).

Scope conditions: results are specific to electricity-consumption nudges in the U.S. residential sector; the technology channel identification requires that adopted equipment stays in the home after a move; the decomposition rests on a linear production function for outcomes in habits and technology.

Q: What is the Home Energy Report and how was it administered in these experiments? A: The HER is a mailed social-comparison report that contrasts a household’s electricity consumption with that of similar neighbors. In each of the 38 waves, homes were observed for a 12-month baseline, then randomly assigned to treatment (receiving HERs) or control. HERs were mailed monthly, bimonthly, or quarterly; generation ceased when the initial resident deactivated electricity service.

Q: What is the paper’s central identification strategy? A: The authors exploit a discontinuity created when the initial treated resident moves out: HER mailings stop, but the home’s electricity consumption continues to be measured as new residents move in. Under three assumptions about non-interference of treatment with moving decisions, balanced habits of subsequent residents, and stability of adopted technology, the post-move HER effect point-identifies the technology-adoption component (ATK) of the long-run average treatment effect (ATE). The habit-formation component (ATH) is then inferred as ATE minus ATK.

Q: What are the three identifying assumptions and how are they tested? A: Assumption 1 (no effect of treatment on moving rates) and Assumption 2 (balanced habits of subsequent residents) are tested with the data; treatment and control homes enter the mover sample at statistically indistinguishable rates and have similar baseline consumption, supporting Assumption 1. The rental-home falsification test supports Assumption 2: rental homes show a null post-move effect, consistent with renters having balanced habits because the technology channel is inactive in rentals. Assumption 3 (stable technology after a move) is untestable from the data; the authors note that violation of this assumption would imply the post-move effect is a lower bound on ATK, making the technology-adoption estimate conservative.

Q: What are the main quantitative estimates of the decomposition? A: The pre-move (long-run) ATE is -2.1 percent of baseline electricity consumption. The post-move effect (ATK) is -1.1 percent, and the habit-formation component (ATH) is -1.0 percent. Normalizing by the ATE, 51.4 percent (s.e. = 13.1) is attributed to technology adoption and 48.6 percent to habits.

Q: How large is the HER effect in absolute terms during the comparison period? A: During the comparison period, the HER reduced average daily electricity consumption by approximately -1.8 to -2.3 percent in the first year and -1.5 to -2.0 percent in the second year, with 95 percent confidence intervals excluding zero. In levels, these correspond to roughly -0.6 to -0.9 kWh per day — equivalent to using 2 to 4 sixty-watt incandescent bulbs for 5 fewer hours per day.

Q: How persistent is the HER effect during the move period? A: In the first year of the move period the HER continues to produce reductions of -1.7 and -1.4 percent; more than a year after the initial resident’s departure the estimated effect is -1.2 percent. All move-period estimates are statistically significant at conventional levels.

Q: How does the paper explain variation in persistence across social-comparison nudge contexts? A: One year after discontinuation, nudges targeting compliance, charitable giving, savings, and voter turnout retain on average only 4 percent of their initial effect, while nudges targeting energy or water conservation retain 65 percent on average. The paper argues the divergence reflects the relative availability of enabling technologies: households can adopt long-lived, input-efficient technologies (appliances, fixtures) to reduce energy and water use, but analogous technologies to facilitate compliance, donations, or voting are largely unavailable or absent.

Q: How does this paper’s finding about technology adoption compare to Allcott and Rogers (2014)? A: Allcott and Rogers (2014) used participation in utility-sponsored energy-efficiency programs as a proxy for technology adoption and found it explained no more than 2 percent of the HER’s long-run effectiveness. The authors reject this conclusion: their decomposition attributes 51.4 percent to technology, which is estimated precisely enough to statistically reject the 2 percent figure from Allcott and Rogers (2014). They attribute the discrepancy to the imperfect proxy used by Allcott and Rogers and low statistical power in analogous analyses.

Q: What are the cost-benefit implications of accounting for HER-induced technology adoption? A: Assuming monthly HERs for one year, a household electricity price of $0.10/kWh, and benefits accruing over two years, the baseline net benefit (ignoring technology costs) is $32.38 per household (electricity savings of $44.38 minus $12 administration cost). Using a technology cost of $0.03/kWh saved (Gillingham et al. 2018), net benefits fall to $27.14. Using $0.12/kWh saved (Billingsley et al. 2014), net benefits drop to $11.43 — a reduction of up to 65 percent from the baseline estimate. The HER still passes cost-benefit analysis but prior evaluations that ignore technology costs overstate net benefits substantially.

Q: How robust are the decomposition results to alternative sample definitions and specifications? A: The qualitative findings are stable across: alternative sets of control variables (Table A1); mover samples defined by receiving as few as 1 or as many as 5 HERs before moving (Table A2, with pre-move effects of -2.08 and post-move effects of -0.93 to -1.04 across cutoffs); balanced panels requiring fixed observation windows in each period (Table A3); and exclusion of homes showing unusually low consumption in the move period (Table A4, post-move effects of -1.19 to -1.48).

Q: What policy implications does the paper draw for nudge design? A: Policymakers seeking persistent nudge effects should target behaviors that can be augmented by readily available technologies, or pair social-comparison nudges with opportunities to adopt new technologies. In voting contexts, combining social-comparison nudges with opt-in mail-in or online ballot defaults could produce more persistent effects. In savings and charitable giving, pairing social comparisons with automatic contribution-rate defaults (as in Madrian and Shea 2001; Thaler and Benartzi 2004) is predicted to produce longer-lived effects than the nudge alone.

Q: What methodological contribution does the paper offer beyond the HER application? A: The mover-based decomposition is a generalizable research design for separating human capital (habits, skills) from physical capital (technology, infrastructure) as channels of policy effectiveness. The authors suggest it can be applied using other natural separation events — such as student graduation or employee departure — to assess the extent to which nudges build human capital in both recipients and the organizations in which they are embedded.

Technology adoption channel (ATK): The component of the HER’s long-run average treatment effect attributable to increases in the stock of energy-efficient technologies in the home — identified empirically as the post-move HER effect that persists after the treated resident departs and the HER is discontinued.

Habit formation channel (ATH): The component of the HER’s long-run treatment effect attributable to changes in the habits or skills of the resident — inferred as the residual after netting the technology component (ATK) from the total long-run effect (ATE).

Post-move effect: The estimated difference in electricity consumption between treatment and control homes after the initial resident has moved out, the HER has been discontinued, and a new resident has taken occupancy; under the paper’s identifying assumptions this equals ATK.

Balanced-habits assumption: The identifying assumption that treatment assignment did not influence the characteristics or habits of residents who subsequently moved into homes in the experimental sample, so that the habits of incoming residents are comparable across treated and control homes.

Stable-technology assumption: The identifying assumption that energy-efficient technologies adopted in response to the HER remain in the home after the initial resident moves; relaxing this assumption implies the post-move effect is a lower bound on ATK.

Home Energy Report (HER): A mailed social-comparison report that contrasts a recipient household’s electricity consumption with that of similar neighboring households; the treatment studied across all 38 experiments in this paper.

Enabling technologies: Long-lived, input-efficient capital goods (appliances, lighting, insulation) that reduce the marginal cost of conservation and thereby lock in behavioral changes induced by a nudge; their relative abundance in energy and water conservation contexts — versus their absence in voting, giving, or compliance contexts — is the paper’s proposed explanation for cross-context variation in nudge persistence.