<p>Despite its funny title and although it is quite old you may find this paper interesting:</p>

<p><a href="https://www.google.nl/url?sa=t&amp;source=web&amp;rct=j&amp;url=https://www.journals.uchicago.edu/doi/abs/10.1086/260860&amp;ved=2ahUKEwjS_9TAoa7bAhUOzqQKHbmGAogQFjAAegQICRAB&amp;usg=AOvVaw23LctmgSAzihTpKed4mu1l" rel="nofollow noreferrer" title="The economics of the afterlife">The economics of the afterlife</a> by Scott Gordon in the JPE. His argument is basically that in the presence of an infinite amount of time there would still be a rate of time preference.</p>

<p>A follow-up paper disproving Gordon's claim was written by Amegashie, but I haven't checked that one in detail. You can find that <a href="https://www.google.nl/url?sa=t&amp;source=web&amp;rct=j&amp;url=http://www.uoguelph.ca/~jamegash/Economics_000.pdf&amp;ved=2ahUKEwjS_9TAoa7bAhUOzqQKHbmGAogQFjADegQIAxAB&amp;usg=AOvVaw33oDInjjTKsC6-RXRXNW_o" rel="nofollow noreferrer"> here</a>.</p>

<p>Nb credit where credit is due: I would probably never have read the first paper without Yoram Bauman's presentation "Hyperinflation in hell"</p>

CC BY-SA 4.0
2018-06-02T11:12:16.900Z
Interesting (and unusually concise) papers.  But I can't see a direct relevance to my question, except perhaps to underline that time constraints are most relevant to utility maximisation _within a defined time period_, and more problematic over a lifetime of unknown duration or hypothetically over infinite time.
Adam Bailey
CC BY-SA 4.0
2018-06-02T12:11:07.010Z
@AdamBailey I thought you might be interested to see what happens if one relaxes the time constraint. Moreover Gordon's paper partly hinges on the fact that one cannot do things *at the same time*  that is, order may matter, which seemed to be  part of your motivation as well. But I'll admit that the relevance is not as direct as you might like
Maarten Punt
CC BY-SA 4.0
2018-05-30T20:37:26.147Z
2018-05-30T20:37:26.147Z
Maarten Punt
<p>The consumption of economic goods often takes time. Consider, for example:</p>

<ul>
<li>Transport services, eg flights, rail journeys;</li>
<li>Leisure goods, eg watching a film, visiting a park.</li>
</ul>

<p>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. </p>

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

<ul>
<li><a href="https://www.researchgate.net/publication/215665892_A_Theory_of_Allocation_of_Time" rel="nofollow noreferrer">Becker, G S (1965) <em>A Theory of the Allocation of Time</em></a></li>
<li><a href="https://econpapers.repec.org/article/ecjeconjl/v_3a81_3ay_3a1971_3ai_3a324_3ap_3a828-46.htm" rel="nofollow noreferrer">De Serpa, A C (1971) <em>A Theory of the Economics of Time</em></a></li>
</ul>

<p>What are other (and more recent) key papers on this topic?</p>

<p><strong>Addendum 31 May 2018</strong></p>

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

<p>$$U = X_1^{0.5}X_2^{0.5}$$</p>

<p>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:</p>

<p>$$X_1 + 2X_2 \leq I$$</p>

<p>$$2X_1 + X_2 \leq T$$</p>

<p>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.</p>

<p><strong>Addendum 23 July 2018</strong></p>

<p><a href="https://www.amazon.co.uk/Consumption-Takes-Time-Implications-Schumpeter-ebook/dp/B000OI0R3I/ref=sr_1_10?ie=UTF8&amp;qid=1532263760&amp;sr=8-10&amp;keywords=ian%20steedman&amp;tag=enlighteconom-21" rel="nofollow noreferrer">This book</a> by Ian Steedman, reviewed <a href="http://www.enlightenmenteconomics.com/blog/index.php/2018/07/taking-time-seriously-in-economics/" rel="nofollow noreferrer">here</a> by Diane Coyle, looks relevant.</p>

CC BY-SA 4.0
2018-05-30T21:53:38.050Z
Relevant: https://economics.stackexchange.com/questions/19569/does-consumer-theory-assume-that-utility-functions-are-consistent-overtime
EconJohn
<reference-request><utility><consumer-theory>
Utility Maximisation Subject to Income and Time Constraints
Adam Bailey
# Utility Maximisation Subject to Income and Time Constraints

(Asked by: username_0 on 2018-05-29)

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:

* [Becker, G S (1965) *A Theory of the Allocation of Time*](https://www.researchgate.net/publication/215665892_A_Theory_of_Allocation_of_Time)
* [De Serpa, A C (1971) *A Theory of the Economics of Time*](https://econpapers.repec.org/article/ecjeconjl/v_3a81_3ay_3a1971_3ai_3a324_3ap_3a828-46.htm)

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

**Addendum 31 May 2018**

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.

**Addendum 23 July 2018**

[This book](https://www.amazon.co.uk/Consumption-Takes-Time-Implications-Schumpeter-ebook/dp/B000OI0R3I/ref=sr_1_10?ie=UTF8&qid=1532263760&sr=8-10&keywords=ian%20steedman&tag=enlighteconom-21) by Ian Steedman, reviewed [here](http://www.enlightenmenteconomics.com/blog/index.php/2018/07/taking-time-seriously-in-economics/) by Diane Coyle, looks relevant.

**Tags:** reference-request, utility, consumer-theory

> **Comments:**
> **username_1**: Relevant: https://economics.stackexchange.com/questions/19569/does-consumer-theory-assume-that-utility-functions-are-consistent-overtime  

---

## 1 answers

### Answer by username_2 on 2018-05-30. Score: 1

Despite its funny title and although it is quite old you may find this paper interesting:

[The economics of the afterlife](https://www.google.nl/url?sa=t&source=web&rct=j&url=https://www.journals.uchicago.edu/doi/abs/10.1086/260860&ved=2ahUKEwjS_9TAoa7bAhUOzqQKHbmGAogQFjAAegQICRAB&usg=AOvVaw23LctmgSAzihTpKed4mu1l "The economics of the afterlife") by Scott Gordon in the JPE. His argument is basically that in the presence of an infinite amount of time there would still be a rate of time preference.

A follow-up paper disproving Gordon's claim was written by Amegashie, but I haven't checked that one in detail. You can find that  [here](https://www.google.nl/url?sa=t&source=web&rct=j&url=http://www.uoguelph.ca/~jamegash/Economics_000.pdf&ved=2ahUKEwjS_9TAoa7bAhUOzqQKHbmGAogQFjADegQIAxAB&usg=AOvVaw33oDInjjTKsC6-RXRXNW_o).

Nb credit where credit is due: I would probably never have read the first paper without Yoram Bauman's presentation "Hyperinflation in hell"

> **Comments:**
> **username_0**: Interesting (and unusually concise) papers.  But I can't see a direct relevance to my question, except perhaps to underline that time constraints are most relevant to utility maximisation _within a defined time period_, and more problematic over a lifetime of unknown duration or hypothetically over infinite time.  
> **username_2**: @AdamBailey I thought you might be interested to see what happens if one relaxes the time constraint. Moreover Gordon's paper partly hinges on the fact that one cannot do things *at the same time*  that is, order may matter, which seemed to be  part of your motivation as well. But I'll admit that the relevance is not as direct as you might like  

---

