---
title: "Interdependent Preferences and Aggregate Saving"
authors:
  - name: "Alexis Direr"
    affiliation: "École Normale Supérieure & Université de Paris 1, EUREQua"
keywords: [wealth distribution, externalities, saving, social status, relative consumption, status-seeking]
jel_codes: [D31, D62, E21]
language: en
type: working-paper
---

# Interdependent Preferences and Aggregate Saving

**Author**
- Alexis Direr — École Normale Supérieure (Département de Sciences Sociales, 48 Boulevard Jourdan, 75014 Paris) & Université de Paris 1, EUREQua — *alexis.direr@ens.fr*

**Keywords**: wealth distribution, externalities, saving, social status.
**JEL codes**: D31, D62, E21.

---

## Abstract

The effect of relative consumption on aggregate saving is analyzed in a two-period model. It is assumed that people care about their rank in the consumption distribution at each date. It is shown that individuals concentrate their consumption in the period in which the distribution of consumption is the most egalitarian. As a result, a rise in consumption inequalities has a negative impact on saving compared to the case without a status-seeking motive.

---

## 1. Motivation and contributions

The paper is motivated by the U.S. low-saving-rate puzzle: the personal saving rate fell from 10% in the mid-1980s to 2.3% in 1999. Over a longer window, the Gini coefficient of household incomes rose by 3.5 percentage points between 1968 and 1992 (Atkinson 1997). The paper proposes a mechanism linking the two: if consumption inequality tracks income inequality and individuals care about their rank in the consumption distribution, rising inequality erodes the incentive to save.

The contributions are the following.

1. **A two-period model of rank-based status concerns.** Individuals derive utility both from consumption and from their position $G_t(c_t^i)$ in the *time-varying* consumption distribution. The model departs from earlier wealth-rank approaches (Corneo and Jeanne 1997, 1999) by anchoring status on consumption rather than wealth, motivated by Veblen's (1922) point that consumption is more easily advertised than wealth.

2. **A consumption-smoothing implication driven by inequality dynamics.** The first-order condition shows that the status-seeking motive pushes individuals to concentrate consumption in the period in which the distribution is most egalitarian — equivalently, where the density of competitors at one's consumption level is highest. Aggregate saving therefore depends on the *evolution* of consumption dispersion, not on its level.

3. **A symmetric prediction relative to $1/\beta$.** When $R\beta = 1$, the consumption distribution is time-invariant and status concerns leave saving unchanged. When $R > 1/\beta$, second-period inequality is lower, status reinforces saving. When $R < 1/\beta$, inequality rises and status depresses saving below the no-status benchmark.

4. **A differentiated position relative to the prior status-and-saving literature.** Unlike Frank (1985) or Corneo and Jeanne (1998), the model does not assume that individuals weight present rank more than future rank, and unlike Knell (1999) it does not require an upward-looking reference group. The negative inequality–saving link survives without those restrictive conditions.

---

## 2. Model

### 2.1 Setup

A single-good economy with two dates, $t \in \{0, 1\}$, and a unit-mass continuum of agents. Each individual receives a first-period endowment $y_0^i \geq 0$ and a zero second-period endowment. Goods can be transferred across periods through a linear technology with gross return $R$ (interpreted as a fixed gross interest rate, appropriate for a small open economy or under monetary-authority control).

Endowments are distributed over $[\bar{y}_0, y_0^+]$ with cdf $F(\cdot)$ and density $f(\cdot)$. The density is assumed continuously differentiable on $]\bar{y}_0, y_0^+[$, left-continuous at $y_0^+$, right-continuous at $\bar{y}_0$, and to satisfy $f(\bar{y}_0) = 0$. The boundary condition is needed to make the second-order condition of the individual problem hold.

### 2.2 Preferences and status

Let $Q_0(\cdot)$ and $Q_1(\cdot)$ denote the equilibrium consumption rules of all *other* agents. Define the set of agents consuming below $c$ at date $t$ as $\Omega^t_c = \{y_0^i : Q_t(y_0^i) < c\}$ and the corresponding population share

$$G_t(c) = \int_{\Omega^t_c} f(y_0)\,dy_0.$$

Preferences are time-separable with rank-augmented per-period utility

$$T_t(c_t^i) = u(c_t^i) + \alpha\, G_t(c_t^i),$$

where $u(\cdot)$ is increasing, concave and twice continuously differentiable, and $\alpha \geq 0$ measures the strength of the status motive. Linearity in the rank term means the utility gain from a marginal rank improvement is the same regardless of one's initial rank. Each individual takes the cross-sectional consumption distribution as given (his own consumption is too small to affect $G_t$) and discounts the future at rate $\beta$.

### 2.3 Numerical illustration

The paper solves the saving decision numerically using a linear-economy specification. The slope $\gamma$ of the policy rule is the root of a degree-3 polynomial; the constant $\eta$ is recovered jointly. Reported parameter values:

| $\theta$ | $\alpha$ | $\beta$ | $a$ | $b$ |
|:---:|:---:|:---:|:---:|:---:|
| 0.01 | 5.512 | 0.64 | 0.0004 | $-0.0122$ |

Here $a$ and $b$ parameterize the linear endowment density $f(y_0) = a + b\,y_0$. Of the three real roots for $\gamma$, two are rejected for failing the regularity assumption on $f$, leaving a unique policy rule and a unique equilibrium. The qualitative results are reported as robust to any perturbation of the parameters that preserves the regularity assumption.

> **Source-availability note.** The PDF available for this summary contains the introduction, the setup, and the numerical/concluding sections, but the formal equilibrium derivation (full statement of problem $(P)$, Proposition 1, first-order condition (3), and the linear-economy proofs) is not present in the document supplied. The intuition reported in §3 below is reconstructed from the discussion text accompanying Figures 1 and 2; the formal arguments should be checked against the complete version of the paper.

---

## 3. Results

### 3.1 The first-order-condition mechanism

The discussion accompanying Figures 1 and 2 makes the central mechanism explicit. The status-seeking motive enters the Euler equation through the ratio of consumption densities $g_1(c_1)/g_0(c_0)$, where $g_t$ is the density of consumption at date $t$. Three regimes follow:

- If $g_1(c_1)/g_0(c_0) < \beta R$: the rank-improvement payoff is larger tomorrow than today, status raises saving relative to the no-status benchmark.
- If $g_1(c_1)/g_0(c_0) > \beta R$: rank improvement is easier today, status lowers saving.
- If $g_1(c_1)/g_0(c_0) = \beta R$: status leaves the saving decision unchanged.

The density growth rate is interpreted as a measure of how many competitors a marginal extra unit of consumption overtakes. Status concerns therefore induce agents to *transfer consumption to the period in which the distribution is most egalitarian* — the period in which a marginal consumption increment moves them past the largest mass of others.

### 3.2 Saving and the gross interest rate (Figure 1)

Aggregate saving rate is plotted against $R$ on $[1.35, 1.75]$ with and without the status motive. The two schedules cross at $R = 1/\beta \approx 1.5625$ (saving rate $\approx 0.60$). For $R > 1/\beta$, the second-period distribution is more concentrated, agents save *more* under status; for $R < 1/\beta$, dispersion rises over time and status agents save *less*. The status schedule is steeper than the no-status one, so the status motive amplifies the response of saving to the interest rate in either direction.

### 3.3 Saving and the evolution of dispersion (Figure 2)

Aggregate saving is plotted against the dispersion ratio $\sigma(c_2)/\sigma(c_1)$ over the same range of $R$. Both schedules slope downward (saving falls with rising consumption inequality), but the status schedule is markedly steeper. When $\sigma(c_2)/\sigma(c_1) > 1$ — inequality rising over the life cycle — the status motive depresses saving below the no-status benchmark. The two schedules again cross at the time-invariant point ($\sigma(c_2)/\sigma(c_1) = 1$, saving $\approx 0.60$).

### 3.4 The $R\beta = 1$ benchmark

Under $R\beta = 1$, the consumption distribution is time-invariant in both the with-status and the without-status economies. Consumption is constant across periods and no rank improvement can be obtained by reallocating consumption — the status motive is silent on the saving decision.

---

## 4. Conclusion

The paper argues that a concern for rank in the consumption distribution generates an intertemporal-substitution-like motive: agents shift consumption toward the period of greater equality and away from the period of greater dispersion. Because consumption inequality is presumed to track income inequality, rising income inequality is predicted to reduce aggregate saving — offering a candidate mechanism for the observed decline in U.S. household saving alongside rising inequality.

The author connects the prediction to the inequality-and-growth literature, noting Persson and Tabellini's (1994) negative correlation and Forbes's (2000) reassessment after correcting an omitted-variable bias (positive correlation under her specification). The model's prediction concerns the *dynamics* of inequality rather than its level, so neither correlation is a direct test of it; an empirical test is announced as future work.

### Limitations and research extensions

- **Wealth-distribution heterogeneity.** A more realistic distribution would let the analysis identify which wealth classes are most sensitive to the status effect.
- **Attitudes toward risk.** Replacing dated goods with contingent goods would extend the framework to risk preferences. The extension is non-trivial because the rank-preserving property of the model breaks under uncertainty. The author cites Harbaugh (1996) and Robson (1992) as related precedents.

---

## Acknowledgments

The author thanks Viviane André, Antoine d'Autume, Patrick Fève, Fabien Moizeau, Muriel Pucci, Jean-Marc Tallon, Fabien Postel-Vinay and Yoram Weiss for helpful discussions. The paper was presented at the ADRES Conference on Social Interactions and Economic Behavior (Paris, December 16–18, 1999), at the EUREQua seminar on macroeconomic dynamics, and at the Econometric Society World Congress (Seattle, August 2000).

---

## Main references

Atkinson, A. B. (1997). Bringing income distribution in from the cold. *Economic Journal* 107, 297–321.

Corneo, G., Jeanne, O. (1996). Status, the distribution of wealth and growth. *Discussion Paper A-561*, University of Bonn.

Corneo, G., Jeanne, O. (1999). Pecuniary emulation, inequality and growth. *European Economic Review* 43, 1665–1678.

Di Tella, R., MacCulloch, R., Oswald, A. (1997). The macroeconomics of happiness. *Working Paper*, Harvard Business School.

Duesenberry, J. (1949). *Income, Saving, and the Theory of Consumer Behavior*. Cambridge: Harvard University Press.

Forbes, K. (2000). A reassessment of the relationship between inequality and growth. *American Economic Review*, forthcoming.

Frank, R. (1985). The demand for unobservable and other nonpositional goods. *American Economic Review* 75(1), 101–116.

Harbaugh, R. (1996). Falling behind the Joneses: relative consumption and the growth-savings paradox. *Economics Letters* 53, 297–304.

Kapteyn, A., Van de Geer, S., Van de Stadt, H. (1985). The impact of changes in income and family composition on subjective well-being. *Review of Economics and Statistics* 67(2), 179–187.

Kapteyn, A., Van de Geer, S., Van de Stadt, H., Wansbeek, T. (1997). Interdependent preferences: an econometric analysis. *Journal of Applied Econometrics* 12, 665–686.

Knell, M. (1999). Social comparisons, inequality, and growth. *Mimeo*, University of Zürich.

Persson, T., Tabellini, G. (1992). Growth, distribution and politics. *European Economic Review* 36, 593–602.

Robson, A. (1992). Status, the distribution of wealth, private and social attitudes to risk. *Econometrica* 60, 837–857.

Solnick, S., Hemenway, D. (1998). Is more always better? A survey on positional concerns. *Journal of Economic Behavior & Organization* 37(3), 373–383.

Veblen, T. (1922). *The Theory of the Leisure Class. An Economic Study of Institutions*. London: George Allen & Unwin.

Weiss, Y., Fershtman, C. (1998). Social status and economic performance: a survey. *European Economic Review* 42, 801–820.

*The full reference list appears in the PDF.*