# The beta delta model

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

• – Giskard Mar 28 '19 at 4:54

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 $$$$u(x_0)+\delta u(x_1)+\delta^2u(x_2)+\cdots+\delta^Tu(x_T)=\sum_{t=0}^T\delta^tu(x_t),$$$$ where $$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