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

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

Exponential Discounting

Traditionally, economists use exponential discounting to capture the fact that things in the future has less value compared to things at present. Such a model presumes a constant discount factor $\delta$, which is used to weight 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 measure 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 $(2)$.

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  • $\begingroup$ 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. $\endgroup$ – mark leeds Mar 28 at 19:38

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