# Does consumer theory assume that utility functions are consistent overtime?

I'm interested in how we model a consumer with a changing utility function over time.

Example: an individual's demand for candy/sweets at 10 years of age will be different when he is 20,30 or 40 years old (based on casual observation).This would seem to indicate the utility derived from this "input" in our utility function has changed.

In basic intertemporal models the general assumption is that the objective utility function does not change. mathematically it makes the model simple to compute, however it does not reflect the changes in preferences over time.

However, if we do allow the individuals utility function to change over time in our model, does this not violate the requirements for a rational consumer ?

How do we model an individual with changing preferences over time and is it economically sound to do so?

• Relevant:researchgate.net/publication/…
– EconJohn
Dec 3 '17 at 21:57
• In The Economic Approach to Human Behavior, Gary Becker said: "The combined assumptions of maximizing behavior, market equilibrium, and stable preferences, used relentlessly and unflinchingly, form the heart of the economic approach as I see it." The emphasis is mine. Here "stable preferences" refer to preferences that are more or less the same across different periods. The reason for preference stability is obvious. If we allow preference or utility function to change arbitrarily, then we'd be able to explain pretty much everything by attributing the cause to such a change. Dec 4 '17 at 20:57
• @HerrK so allowing for a dynamic utility function would kill any formal analysis which can be done in consumer theory?
– EconJohn
Dec 4 '17 at 21:26

In The Economic Approach to Human Behavior, Gary Becker said:

The combined assumptions of maximizing behavior, market equilibrium, and stable preferences, used relentlessly and unflinchingly, form the heart of the economic approach as I see it.

The emphasis is mine. Here "stable preferences" refer to preferences (and by association, the utility functions representing them) that are more or less the same across different periods. The reason for preference stability is obvious. If we allow preference or utility function to change arbitrarily, then we'd be able to explain pretty much anything by attributing the cause to some appropriately chosen change in people's preference.

When it comes to the intertemporal preferences in particular, I agree with @MaartenPunt that, at least in principle, one can incorporate time-dependence into utility function. For instance, in the usual discounted utility framework, we have $$U(\mathbf x_t)=\sum_{t=0}^\infty D(t)u(\mathbf x_t)$$ where $\mathbf x_t$ is a vector of consumption goods at time $t$, $D(\cdot)$ is a discount function (e.g. $D(t)=\delta^t$ as in an exponential discounting model), and $u(\cdot)$ is the time-invariant period utility function. To incorporate time-dependence, we can simply allow $u(\cdot)$ to also be a function of time $$U(\mathbf x_t)=\sum_{t=0}^\infty D(t)u(\mathbf x_t,\color{red}t).$$ To make @MaartenPunt's third comment more explicit, suppose $$u(\mathbf x,t)=\alpha_t^1v(x^1)+\cdots+\alpha_t^iv(x^i)+\cdots+\alpha_t^nv(x^n)$$ where $x^i$ denotes the quantity consumed of good $i$ and $\alpha_t^i$s are the time-dependent weights on the utility derived from each good $i$. So the same consumption bundle will generate possibly different levels of utility in different time periods. For example, $\alpha_{10}^\text{candy}>\alpha_{40}^\text{candy}$ would capture the fact that a 10-year-old values a candy more than a 40-year-old does. On the other hand, a time-invariant preference would imply that $\alpha_t^i=\alpha^i$ for all $t=0,1,\dots$.

The above discussion is however distinctly different from the paper you linked to in the comment, which is about dynamic (in)consistency of choices. In the literature on intertemporal choices, the main focus is usually about whether some optimal consumption profile decided at time $t$ will remain optimal when reevaluated at some future time $t+k$. Usually papers in this literature maintain the assumption of time-invariant period utility function, i.e. $u(\mathbf x,t)=u(\mathbf x)$, but play with various forms of the discount function $D(\cdot)$ (e.g. hyperbolic or quasi-hyperbolic discounting) to generate predictions that match experimental data.

Not a real answer, but a few thoughts.

(1) In principle I think it can be done as long as the preferences changes relate to current consumption trade-offs e.g. sweets versus bread now, rather than sweets now versus sweets tomorrow. The latter is the problem of time inconsistency you link to in your comment. There's a large literature on time inconsistency, hyperbolic discounting etc., see for example also Laibson's classic. Incidentally, if the consumer can somehow commit to a certain path, that solves the problem of the inconsistency.

(2) If you're looking at the whole economy, a way around the problem is to work with a representative consumer that incorporates both the preferences of the young people and the elder people. As long as the relative proportions of the population do not change over time the representative consumer should make the same choices in the aggregate.

(3) How to do this in a practical way in the utility function: you could include time dependent weights on the different goods, where one increases and the other one decreases. It would definitely become more difficult to solve, but would be doable for small problems, and if the weights are simple functions of time perhaps even possible for larger problems.