# Differences of two discount factors

In game theory at the undergraduate level, the discount factor is denoted as $$\delta \in [0,1]$$.
On the other hand, in macroeconomics at the graduate level, the notation $$e^{-\rho t}$$ is used($$t$$ is time).
I am unclear as to why the second notation is introduced.

Additionally, I would like to confirm if the range of $$\rho$$ is $$\rho \in [0, \infty]$$.

Thank you for any assistance you can provide.

• I heard that when I consider $p' = e^\rho -1$, I can derive $e^{-\rho t} = (e^\rho)^{-t} = (\rho'+1)^{-t} = \frac{1}{1+\rho'}^t \equiv \beta^t$. But it's still unclear what makes this attractive. Feb 3 at 7:35

The value $$e^{-\rho t}$$ is the discount rate that you need to apply if the per period interest is $$\rho$$, but interest is accumulated continuously. The value of $$\rho$$ can be interpreted as the per period interest rate, so it usually takes values in the interval $$(0,1)$$.

To see how to get to this value, assume that you invest an amount of $$a$$ at an interest rate of $$\rho$$ per time period. After $$t$$ periods, you will have an amount equal to: $$a\underbrace{(1+\rho)(1+ \rho)\ldots (1+\rho)}_{t \text{ periods }} = a (1+\rho)^t.$$ Now assume that instead you accumulate interest not every period but ever (1/2) period. In particular, the bank gives you an interest rate of $$\rho/2$$ every (1/2) period. Then after 1 period, the amount of money will be: $$a \left(1+\frac{\rho}{2}\right)\left(1+ \frac{\rho}{2}\right) = a \left(1+\frac{\rho}{2}\right)^2.$$
So after $$t$$ periods, you will have: $$a \left(1 + \frac{\rho}{2}\right)^{2t}.$$ We can generalize this. Assume that that every period is split into $$n$$ subperiods and you receive an interest of $$\rho/n$$ every subperiod, i.e. every $$(1/n)$$ period. Then after $$t$$ periods, you'll have: $$a \left(1 + \frac{\rho}{n}\right)^{n t}.$$ If interest is gathered continuously, this means that we take this value for $$n$$ going to infinity. This gives: $$\lim_{n \to \infty} a \left(1 + \frac{\rho}{n}\right)^{n t} = a e^{\rho t}.$$ So if the per period interest rate is $$\rho$$ but interest is accumulated continuously, then you will receive an amount of $$a e^{\rho t}$$ after $$t$$ periods.

In order to have accumulated an amount of, say $$A$$ after $$t$$ periods (i.e. $$A = a e^{\rho t}$$), you need to invest an amount equal to: $$a = A e^{-\rho t},$$ This means that $$e^{-\rho t}$$ is the period $$t$$ discount rate that you need to apply if interest is accumulated continuously.

• Just to add that in real world calculations, the internet rate can be negative as was the case for a long time in Europe and Japan. In this case, the discount factor is actually >1. Feb 3 at 10:32
• Thank you for your clear explanation! I understand that the discount rate is the inverse of the interest rate, and $e^{\rho t}$ represents the interest accumulated over t periods in continuous time. @AKdemy also thank you for your insightful comment! Feb 5 at 10:14