Income and Price Effects in Intertemporal Consumer Problems

1 Introduction

Consumer behavior over time is often influenced by both monetary and non-monetary constraints that affect the way in which demand responds to changes in prices and income. While the drivers behind intertemporal price and income responses have been identified in specific contexts, a general characterization is still missing.

To advance in this direction, we consider a general class of intertemporal consumer problems with monetary and non-monetary state variables, and we characterize the price and income response on impact and in the long-run. Our framework encompasses models of endogenous preferences and endogenous income, it does not rely on specific functional forms and, in line with many empirical investigations, it is based on exogenous changes to income or price. Hence, it can be applied to market-driven or policy-driven changes in income or price in a variety of dynamic contexts, including addiction, habits, taste formation, and physical and mental health (see, e.g. Atkin 2013; Bronnenberg, Dubé, and Gentzkow 2012; Dalgaard and Strulik 2014; Darden, Gilleskie, and Strumpf 2018; Della Vigna and Malmendier 2006; Galama and Van Kippersluis 2019).

We derive general expressions for income and price responses on impact and in steady state. For the purposes of this paper, these responses will be denoted as the short- and the long-run responses, respectively. We show that both the short- and the long-run price effect can be decomposed into income and substitution effects. This provides a dynamic counterpart to the Slutsky equation. We then focus on common cases in the economic literature, and show the crucial role of the law of motion of the non-monetary state variable, together with time preferences.

After presenting the general model, we first focus on dynamic income responses. A common assumption is that the non-monetary state variable is affected by only one choice variable. In this case, the relation between short and long-run income effects is particularly clear. When the non-monetary state variable is self-depleting, meaning that an increase in its stock reduces its speed of accumulation, dynamic income effects have the same sign in the short and in the long run. In the opposite case, that is when the state variable is self-productive, short and long-run income effects have opposite signs. For example, the short and long-run income elasticity of demand for education or health would have opposite sign if, as suggested by Cunha and Heckman (2007) and by Dalgaard and Strulik (2014), skill formation and health deficit accumulation are self-productive processes.

We also investigate the relation between static and dynamic income responses. Under some conditions, if the non-monetary state variable is self-depleting, or if the consumer is sufficiently impatient, the short-run income response has the same sign as in a static consumer problem. By contrast, if the non-monetary state variable is self-productive and the consumer is sufficiently patient, the long-run income response is coherent in sign with that of a static problem, whereas the short-run response has the opposite sign.

We next turn to dynamic price responses. Intuitively, one may expect them to be larger in the long run because agents have more time to adjust their behavior, but we show that this is not a general result. We identify conditions under which the elasticity of demand is smaller in the long than in the short run.^[1] Moreover, we show that short and long-run price elasticities can have opposite sign. This case occurs when the non-monetary state variable is self-productive. An example is provided by dental health, which is self-productive because early health issues tend to trigger subsequent ones. After a permanent increase in the price of dental care, a patient consumer will react by raising demand immediately in order to prevent higher expenditures in the long run. An impatient consumer will instead reduce dental care in the short run, generating worse dental health and, consequently, higher expenditures in the long run. This illustrates how in a self-productive scenario patient consumers react to price changes in opposite ways with respect to impatient ones.

Our results imply that heterogeneity in the consumers’ discount factor is key, as policies designed for more patient consumers may have opposite consequences for less patient ones. In addition, our findings show that, even for a given consumer type, evaluating the efficacy of a policy by its short-run impact may be misleading: policies that seem ‘right’ in the short run may prove ‘wrong’ in the long run, or the other way around. Consider for instance taxing an addictive good such as cigarettes. Most of the empirical literature finds that the demand for smoking is inelastic, which would imply that taxes are not a powerful deterrent (Cawley and Ruhm 2012; Hansen, Sabia, and Rees 2017). This conclusion is typically based on short-run responses. When considering a longer time interval, Dragone and Raggi (2018) find that demand is instead elastic. In this case, small effects of the policy in the short run are not a sign of failure, but rather a lower bound for long-run effects.

Our findings are related to Caputo (1997), who derives steady state comparative statics and local comparative dynamics of a general class of optimal control problems with one state variable.^[2] Our approach considers two state variables (monetary and non-monetary), is explicitly framed in terms of consumer problem, and highlights the role of self-productivity and time discounting.

In a companion paper (Dragone and Vanin 2022), we focus on dynamic substitution effects under the restriction that the marginal utility of assets is constant (see, e.g. Becker and Murphy 1988; Heckman 1974). This restriction, which assumes a Frisch compensation of income, ignores income effects under the implicit assumption that they are negligible. While this may be reasonable in case of transitory shocks, or under specific circumstances, in many applications income effects are relevant and cannot be safely ignored (Haushofer and Shapiro 2016, 2018). In the current paper we move beyond the analysis of substitution effects in two ways. First, we study dynamic income effects, which are sometimes the direct target of policy interventions and are more generally relevant whenever the consumer problem is a building block of a general equilibrium model. Second, we show that the key drivers of dynamic income and substitution effects are common. This allows to compactly decompose uncompensated dynamic price effects into substitution and income effects, both in the short and in the long run, and to draw a parallel with the familiar (static) Slustky equation.

More generally, we contribute to the understanding of intertemporal consumer problems beyond models considering only dynamic budget constraints (see, e.g. Hall 2010; Jappelli and Pistaferri 2010). As non-monetary constraints shape consumer behavior in several contexts, and their neglect may bias empirical estimates (see, e.g. Wallenius 2011), our results may be useful for empirical analysis and policy evaluation.

2 Dynamic Income and Price Effects

Consider the following intertemporal maximization problem,

where t denotes time, ρ is the intertemporal discount rate, U x , y , Z is the instantaneous utility function, x and y are consumption goods, and Z is a state variable with law of motion (3) and initial condition Z(0) = Z ₀. We assume that U x , y , Z and f x , y , Z are twice differentiable and concave functions, with U _x , U _y > 0 and U _xx < 0. No restriction on the sign of the cross-derivatives of these functions is imposed.

In the dynamic budget constraint (2), A represents assets, with initial condition A(0) = A ₀ ≥ 0, and function r A is returns on assets, with r _A > 0 and r _AA < 0. Term p describes the market price of good x, and M and w(Z) are the exogenous and endogenous components of income, respectively. To avoid an exploding debt, we also impose the No-Ponzi condition lim t → ∞ A t e − ∫ 0 t r A A s d s ≥ 0 .^[3]

As in Dragone and Vanin (2022), we call Z self-productive when f _Z > 0, and self-depleting when f _Z < 0. Simple and common examples of the two cases are Z ̇ = f ( Z ) − x (with f Z > 0 ) and Z ̇ = x − δ Z , respectively.

In the Appendix we solve the optimization problem and derive conditions for a steady state with saddle-point stability. Starting from such steady state, we consider an unexpected permanent change in income or price. We investigate the dynamic response of the demand for x at two different points in time: in the short run, i.e. on impact at the time of the change, and in the long run, i.e. when a new steady state is reached. We also study the long-run response of Z (the response of y is reported in the Appendix).^[4] We will use superscripts S and L to denote the short and the long run.

We first consider income responses.

Expressions for a, b, c, d, ɛ are complicated and little insightful in the general case. Notice that term a is common to all three expressions. This allows to considerably simplify the short and long-run income responses in the special case f _y = 0 considered in Section 3, and to highlight their connection.

Short and long-run income responses derived in Proposition 1 can be used to decompose dynamic price responses.

In analogy to the Slutsky equation of static consumer problems, the above Proposition states that dynamic price effects can also be decomposed into two components. The first one is the dynamic counterpart of the substitution effect, represented by the first term in equations (8)–(10). The second one is the dynamic counterpart of the income effect: − x L ⋅ x M S , − x L ⋅ x M L and − x L ⋅ Z M L . This decomposition holds both in the short and in the long run, and for both consumption goods and the non-monetary state variable. Notably, it does not depend on the functional form of the utility function or the law of motion of Z.

Differently from a static consumer problem, here a permanent price change triggers a dynamic response of consumption that need not be monotonic over time and that can feature responses of opposite sign in the short and in the long run. These results do not necessarily depend on income effects, because in a dynamic consumer problem substitution effects can be positive and can therefore be themselves the source of apparent violations of the (static) law of demand. To see this in the clearest way, in the following Section we focus attention on common simple cases.

3 Insights from Simple Cases

In this section we shed light on the relation between short and long run responses to income and price changes under common simplifying assumptions. As a benchmark, it is useful to recall that the solution x ̃ to the standard static consumer problem, max x , y U ( x , y ) s.t. px + y = M, satisfies

where ν > 0 by concavity.

Let us start with dynamic income responses. Consider the common case in which the dynamics of state variable Z only depends on x and Z, and not on y, i.e. f _y = 0. Income responses from Proposition 1 simplify to

where

and q < 0 (see eq. (67)). Expressions (12) immediately yield

In the following discussion we assume ɛ < 0.^[5] Expressions (12) show that in the short run good x is a normal good when a < 0. The long-run response has the same sign as the response on impact if f _Z < 0. If instead f _Z > 0, good x is an inferior good in the long run whenever it responds as a normal good on impact, and the other way around.^[6] Accordingly, the above Corollary shows that the demand for x responds to a permanent change in income with the same sign in the short and in the long run if Z is self-depleting, and with opposite sign if Z is self-productive.

Observe that expression U _xy − pU _yy in (13) is the one that determines the sign of the static income response in (11). Clearly, dynamic responses also depend on the law of motion of the state variable and on time discounting. To appreciate their role in the clearest way, it is useful to introduce some additional assumptions, as in the following Corollary.

We now turn to dynamic price effects. Further assuming that utility is separable in x and y, i.e. U _xy = 0, and using Proposition 2 and expression (13), price responses simplify to

where g < 0.^[7] This yields

The results of the above Corollary emphasize the role of time discounting and of the marginal effect of Z on its speed of change for dynamic price effects. These findings are in line with those obtained in Dragone and Vanin (2022) in the context of a dynamic consumer problem where demand is compensated to maintain the marginal utility of wealth constant. Here, instead, there is no income compensation and the expressions derived for price effects include both substitution and income effects.

When Z is self-depleting, the future consequences of current behavior are dampened, and just as for ordinary goods in a static set-up, demand falls after a price increase, both in the short and in the long run. This response may or may not be gradual and monotonic over time. An example of non-monotonic response is given by the fall in the price of fast-food and exotic fruit in the former East Germany after reunification, which induced an initial spike in demand that faded away as the novelty vanished. Dragone and Ziebarth (2017) rationalize it through a model of habit formation, in which f _Z < 0.

When Z is self-productive the future consequences of current behavior are amplified over time. This creates a tension with time discounting, which instead dampens the importance of the future. The dynamic response depends on which force prevails. If consumers are sufficiently impatient, their focus on the present induces them to reduce consumption in the short run (but they must increase it in the long run). If they are sufficiently patient, the opposite holds. Hence a price-based policy can deter consumption of x in the short run, but stimulate it in the long run, or the other way around.

4 Conclusions

We study the dynamic responses to unanticipated permanent changes in income or price in a consumer problem featuring monetary and non-monetary state variables. We show that both in the short and in the long run price effects can be decomposed into the dynamic counterparts of income and substitution effect. For both income and price effects, short and long-run responses may have opposite sign. We highlight the crucial role of time discounting, which dampens the importance of the future, and of the law of motion of non-monetary dynamic constraints.

When the non-monetary state variable is self-productive, the consequences of current behavior are amplified over time. This creates an intertemporal trade-off that produces an increase in demand in the short run followed by a long-run decrease, or the other way around, depending on the degree of impatience. When instead the state variable is self-depleting, a condition that is easily satisfied in models with capital depreciation, the consequences of current behavior are dampened, in line with time discounting. Accordingly, we show that the short and long-run responses have the same sign, and we identify the conditions under which the short-run elasticity of demand is larger than the long-run elasticity, or, as commonly expected, the latter is larger.

Dynamic consumer problems with monetary and non-monetary state variables are a building block of many economic models. Given the increasing availability of panel microdata and the empirical relevance of exogenous shocks, we hope our results can be useful to guide and interpret empirical analysis, policy design and policy evaluation.