# Utility Maximisation Subject to Income and Time Constraints

The consumption of economic goods often takes time. Consider, for example:

• Transport services, eg flights, rail journeys;
• Leisure goods, eg watching a film, visiting a park.

I would like to explore models of consumer or household behaviour in which utility in a period is maximised subject to both income and time constraints.

From a Google search, two important (but rather old) papers on this topic appear to be:

What are other (and more recent) key papers on this topic?

To give a simple example, suppose there are just two goods $X_1,X_2$ and utility $U$ is given by:

$$U = X_1^{0.5}X_2^{0.5}$$

Suppose further that the respective prices and time requirements are for $X_1 (1,2)$ and for $X_2 (2,1)$. Thus the income and time constraints are:

$$X_1 + 2X_2 \leq I$$

$$2X_1 + X_2 \leq T$$

where $I$ is income and $T$ is available time. If $T$ is much larger than $I$ (low-income person), then the income constraint will be binding, and the time constraint will be slack, and vice versa if $I$ is much larger than $T$ (high-income person). Over an intermediate range, including $T = I$, both constraints will be binding. These different scenarios have different implications for the effect of a marginal change in price. Perhaps time preference is relevant here but I can't immediately see how.

This book by Ian Steedman, reviewed here by Diane Coyle, looks relevant.

• – EconJohn May 30 '18 at 21:53