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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.

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  • $\begingroup$ 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. $\endgroup$ Feb 3 at 7:35

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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.

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    $\begingroup$ 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. $\endgroup$
    – AKdemy
    Feb 3 at 10:32
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    $\begingroup$ 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! $\endgroup$ Feb 5 at 10:14

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