<p>Perhaps I misunderstand the question, because it seems trivial coming from such an established researcher.</p>
<p>As @HerrK. points out, utility functions that represent intertemporal discounting are generally of the form
<span class="math-container">$$
U\left((x_i)_{i=1}^T\right) = u(x_1) + \delta_1 u(x_2) + \delta_2^2 u(x_3) + \dots
$$</span>
where <span class="math-container">$\delta_i$</span> is the discount factor and <span class="math-container">$x_i$</span> is the consumption in period <span class="math-container">$i$</span>. This is covered in most micro textbooks, e.g., (for two periods) in Varian's <em>Intermediate</em> Microeconomics.</p>
<p>When <span class="math-container">$u() = \ln()$</span>, the function <span class="math-container">$U\left((x_i)_{i=1}^T\right)$</span> is Cobb-Douglas type. In case any <span class="math-container">$x_i$</span> converges to <span class="math-container">$0$</span>, utility converges to <span class="math-container">$-\infty$</span>, thus the optimal solution will be in the interior.</p>

CC BY-SA 4.0
2021-11-21T14:43:54.850Z
Thanks! I do not know why I got confused.
Erel Segal-Halevi
CC BY-SA 4.0
2021-11-21T07:41:07.403Z
2021-11-21T07:41:07.403Z
Giskard
<p>Consider an agent who has a fixed budget, and should decide how to split it between consumption today and consumption tomorrow. For simplicity, suppose there is no interest, and no borrowing/lending, and the price is the same in both days. In my studies, I encountered two ways to model the agent's decision problem:</p>
<ol>
<li><p>Consider each daily consumption as a different good, then model the agent's preferences using a utility function. For example, if the agent prefers consumption today to consumption tomorrow, but still wants to consume something tomorrow, then the utility function can be a Cobb-Douglas utility function, e.g. <span class="math-container">$x^{0.8} y^{0.2}$</span> where <span class="math-container">$x, y$</span> are the consumptions today and tomorrow respectively. Then, the agent maximizes his utility given the budget constraints, just like in a usual consumer's choice problem.</p>
</li>
<li><p>Assume that the agent has a certain discount factor <span class="math-container">$\delta$</span>, which determines how much the agent prefers to consume today over tomorrow. How can this be modeled using a utility function? Initially I thought that the corresponding utility function would be <span class="math-container">$x + \delta y$</span>. But then, the optimal solution (when <span class="math-container">$\delta&lt;1$</span>) is to consume everything today, and starve tomorrow. This does not make much sense.</p>
</li>
</ol>
<p>My question: what utility function corresponds to a discount factor of <span class="math-container">$\delta$</span>? Can it be represented, for example, by a Cobb-Douglas function?</p>

CC BY-SA 4.0
2021-11-20T21:08:11.553Z
How about $u(x)+\delta u(y)$ where $u$ satisfies $\lim_{z\to 0}u'(z)\to\infty$, so that a corner solution on either day is not optimal.
Herr K.
<utility><consumer-theory><discount-rate>
What utility function represents an agent with a time-discount factor?
Erel Segal-Halevi
# What utility function represents an agent with a time-discount factor?

(Asked by: username_0 on 2021-11-20)

Consider an agent who has a fixed budget, and should decide how to split it between consumption today and consumption tomorrow. For simplicity, suppose there is no interest, and no borrowing/lending, and the price is the same in both days. In my studies, I encountered two ways to model the agent's decision problem:

1. Consider each daily consumption as a different good, then model the agent's preferences using a utility function. For example, if the agent prefers consumption today to consumption tomorrow, but still wants to consume something tomorrow, then the utility function can be a Cobb-Douglas utility function, e.g. $x^{0.8} y^{0.2}$ where $x, y$ are the consumptions today and tomorrow respectively. Then, the agent maximizes his utility given the budget constraints, just like in a usual consumer's choice problem.
2. Assume that the agent has a certain discount factor $\delta$, which determines how much the agent prefers to consume today over tomorrow. How can this be modeled using a utility function? Initially I thought that the corresponding utility function would be $x + \delta y$. But then, the optimal solution (when $\delta<1$) is to consume everything today, and starve tomorrow. This does not make much sense.

My question: what utility function corresponds to a discount factor of $\delta$? Can it be represented, for example, by a Cobb-Douglas function?

**Tags:** utility, consumer-theory, discount-rate
> **Comments:**
> **username_1**: How about $u(x)+\delta u(y)$ where $u$ satisfies $\lim_{z\to 0}u'(z)\to\infty$, so that a corner solution on either day is not optimal.  

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## 1 answers

### Answer by username_2 on 2021-11-21. Score: 4 | Accepted answer

Perhaps I misunderstand the question, because it seems trivial coming from such an established researcher.

As @HerrK. points out, utility functions that represent intertemporal discounting are generally of the form
$$
U\left((x\_i)\_{i=1}^T\right) = u(x\_1) + \delta\_1 u(x\_2) + \delta\_2^2 u(x\_3) + \dots
$$
where $\delta\_i$ is the discount factor and $x\_i$ is the consumption in period $i$. This is covered in most micro textbooks, e.g., (for two periods) in Varian's *Intermediate* Microeconomics.

When $u() = \ln()$, the function $U\left((x\_i)\_{i=1}^T\right)$ is Cobb-Douglas type. In case any $x\_i$ converges to $0$, utility converges to $-\infty$, thus the optimal solution will be in the interior.

> **Comments:**
> **username_0**: Thanks! I do not know why I got confused.  

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