<p>The <span class="math-container">$\beta$</span>-<span class="math-container">$\delta$</span> model, otherwise known as the <em>quasi-hyperbolic discounting</em> model, is introduced as an alternative to the traditional <em>exponential discounting</em> which suffers from the problem of being inconsistent with empirical evidence.</p>
<h2>Exponential Discounting</h2>
<p>Traditionally, economists use exponential discounting to capture the fact that things in the future have less value than things at present. Such a model presumes a <strong>constant discount factor</strong> <span class="math-container">$\delta$</span>, which is used to weigh utilities in different periods. For instance, the present value of one's utility from consumption over the next <span class="math-container">$T$</span> periods can be written as
<span class="math-container">\begin{equation}
u(x_0)+\delta u(x_1)+\delta^2u(x_2)+\cdots+\delta^Tu(x_T)=\sum_{t=0}^T\delta^tu(x_t),
\end{equation}</span>
where <span class="math-container">$x_t$</span> is the consumption/investment decision made at time <span class="math-container">$t$</span>. If we let <span class="math-container">$u(x_t)$</span> be the amount of money earned in period <span class="math-container">$t$</span>, then the above formula is also the usual way to calculate the present value of a stream of income. From a theoretical perspective, the exponential discounting model has the desirable feature that it produces dynamically consistent choices. That is, choices deemed as optimal today will still be considered optimal at arbitrary points in the future.</p>
<h2>Issue with Exponential Discounting</h2>
<p>Despite the theoretical appeals of exponential discounting, evidence suggests that this model does not measure well against actual human choices. The classic example involves two sets of choices:</p>
<ol>
<li>Getting <span class="math-container">$\\\$100$</span> today vs. Getting <span class="math-container">$\\\$101$</span> tomorrow</li>
<li>Getting <span class="math-container">$\\\$100$</span> 30 days from today vs. Getting <span class="math-container">$\\\$101$</span> 31 days from today</li>
</ol>
<p>When presented with these two choices, most people would choose to get <span class="math-container">$\\\$100$</span> today but would choose to wait one more day for the extra dollar. This is a clear violation of exponential discounting. If people are exponential discounters with a constant discount factor <span class="math-container">$\delta$</span>, then choosing <span class="math-container">$\\\$100$</span> today implies
<span class="math-container">\begin{equation}
 u($100)&gt;\delta u($101).\tag{1}
\end{equation}</span>
But choosing <span class="math-container">$\\\$101$</span> 31 days from today implies
<span class="math-container">\begin{equation}
\delta^{30}u($100)&lt;\delta^{31}u($101)\quad\Rightarrow\quad u($100)&lt;\delta u($101).\tag{2}
\end{equation}</span>
The incompatibility between <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span> suggests that exponential discounting is not a very good descriptive model of people's time preference.</p>
<h2>Quasi-Hyperbolic Discounting</h2>
<p>The above example illustrates that people have a strong &quot;bias&quot; toward current consumption, or receiving a reward <em>today</em> as opposed to any future date. In other words, the values of all future (<span class="math-container">$t\ge1$</span>) rewards/utility are downplayed relative to the reward/utility of the current period (<span class="math-container">$t=0$</span>). Therefore, based on the exponential discounting model, the quasi-hyperbolic discounting model introduces an extra parameter <span class="math-container">$\beta$</span> that captures this bias towards the present. The mathematical formulation is as follows:
<span class="math-container">\begin{equation}
u(x_0)+\beta\bigl(\delta u(x_1)+\delta^2u(x_2)+\cdots+\delta^Tu(x_T)\bigr)=u(x_0)+\beta\sum_{t=1}^T\delta^tu(x_t).
\end{equation}</span>
The parameter <span class="math-container">$\beta$</span> is the <strong>present bias parameter</strong>, as it measures the extent to which future outcomes are weighed relative to the current one. Usually <span class="math-container">$\beta$</span> is assumed to be less than <span class="math-container">$1$</span>, so that future utilities weigh less than the present one. But it is easy to see that if <span class="math-container">$\beta=1$</span>, then we get exponential discounting as a special case where there is no present bias. With an appropriately chosen <span class="math-container">$\beta$</span>, we can therefore reconcile the inconsistency in equations <span class="math-container">$(1)$</span> and <span class="math-container">$(2)$</span>.</p>

CC BY-SA 4.0
2019-03-28T19:38:16.060Z
This model, ( in a totally different context from that of Herr K's description ), is also referred to as the koyck distributed lag in econometrics. It can be written as: $y_t = \delta \times y_{t-1} + \beta \times u(x_{t})$ + an error term. error term might be gaussian but doesn't have to be.
mark leeds
CC BY-SA 4.0
2019-03-28T00:24:16.223Z
2021-11-05T14:44:12.507Z
2021-11-05T14:44:12.507Z
Herr K.
<p>What is the <span class="math-container">$\beta$</span>-<span class="math-container">$\delta$</span> model? How does it relate to present bias and the present value calculation?</p>

CC BY-SA 4.0
2019-03-28T04:54:11.493Z
[Hyperbolic discounting](https://en.wikipedia.org/wiki/Hyperbolic_discounting#Step-by-step_explanation)?
Giskard
<microeconomics><behavioral-economics>
The beta delta model
Brett
# The beta delta model

(Asked by: username_0 on 2019-03-27)

What is the $\beta$-$\delta$ model? How does it relate to present bias and the present value calculation?

**Tags:** microeconomics, behavioral-economics

> **Comments:**
> **username_1**: [Hyperbolic discounting](https://en.wikipedia.org/wiki/Hyperbolic_discounting#Step-by-step_explanation)?  

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## 1 answers

### Answer by username_2 on 2019-03-28. Score: 9

The $\beta$-$\delta$ model, otherwise known as the *quasi-hyperbolic discounting* model, is introduced as an alternative to the traditional *exponential discounting* which suffers from the problem of being inconsistent with empirical evidence.

## Exponential Discounting

Traditionally, economists use exponential discounting to capture the fact that things in the future have less value than things at present. Such a model presumes a **constant discount factor** $\delta$, which is used to weigh utilities in different periods. For instance, the present value of one's utility from consumption over the next $T$ periods can be written as
\begin{equation}
u(x\_0)+\delta u(x\_1)+\delta^2u(x\_2)+\cdots+\delta^Tu(x\_T)=\sum\_{t=0}^T\delta^tu(x\_t),
\end{equation}
where $x\_t$ is the consumption/investment decision made at time $t$. If we let $u(x\_t)$ be the amount of money earned in period $t$, then the above formula is also the usual way to calculate the present value of a stream of income. From a theoretical perspective, the exponential discounting model has the desirable feature that it produces dynamically consistent choices. That is, choices deemed as optimal today will still be considered optimal at arbitrary points in the future.

## Issue with Exponential Discounting

Despite the theoretical appeals of exponential discounting, evidence suggests that this model does not measure well against actual human choices. The classic example involves two sets of choices:

1. Getting $\\\$100$ today vs. Getting $\\\$101$ tomorrow
2. Getting $\\\$100$ 30 days from today vs. Getting $\\\$101$ 31 days from today

When presented with these two choices, most people would choose to get $\\\$100$ today but would choose to wait one more day for the extra dollar. This is a clear violation of exponential discounting. If people are exponential discounters with a constant discount factor $\delta$, then choosing $\\\$100$ today implies
\begin{equation}
u($100)>\delta u($101).\tag{1}
\end{equation}
But choosing $\\\$101$ 31 days from today implies
\begin{equation}
\delta^{30}u($100)<\delta^{31}u($101)\quad\Rightarrow\quad u($100)<\delta u($101).\tag{2}
\end{equation}
The incompatibility between $(1)$ and $(2)$ suggests that exponential discounting is not a very good descriptive model of people's time preference.

## Quasi-Hyperbolic Discounting

The above example illustrates that people have a strong "bias" toward current consumption, or receiving a reward *today* as opposed to any future date. In other words, the values of all future ($t\ge1$) rewards/utility are downplayed relative to the reward/utility of the current period ($t=0$). Therefore, based on the exponential discounting model, the quasi-hyperbolic discounting model introduces an extra parameter $\beta$ that captures this bias towards the present. The mathematical formulation is as follows:
\begin{equation}
u(x\_0)+\beta\bigl(\delta u(x\_1)+\delta^2u(x\_2)+\cdots+\delta^Tu(x\_T)\bigr)=u(x\_0)+\beta\sum\_{t=1}^T\delta^tu(x\_t).
\end{equation}
The parameter $\beta$ is the **present bias parameter**, as it measures the extent to which future outcomes are weighed relative to the current one. Usually $\beta$ is assumed to be less than $1$, so that future utilities weigh less than the present one. But it is easy to see that if $\beta=1$, then we get exponential discounting as a special case where there is no present bias. With an appropriately chosen $\beta$, we can therefore reconcile the inconsistency in equations $(1)$ and $(2)$.

> **Comments:**
> **username_3**: This model, ( in a totally different context from that of username_2's description ), is also referred to as the koyck distributed lag in econometrics. It can be written as: $y_t = \delta \times y_{t-1} + \beta \times u(x_{t})$ + an error term. error term might be gaussian but doesn't have to be.  

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