# Are there two different meanings of the discount rate?

Today I suddenly became aware of how the term "discount rate" are frequently used in two different contexts:

1. When people prefer to have something now rather than in the future. The reason is supposedly human psychology. The discount rate in this case is inherent to the individual.

2. The same amount of money now is more valuable than in the future. The reason is that if one has that money now, he can invest and earn interest. The discount rate in this case is the real interest rate, determined by the market.

Am I correct in thinking that these are essentially two distinct phenomena subsumed under the same term? Or there is actually a deeper connection between them?

It is standard approach by now to acknowledge the existence of a "rate of pure time preference", denoted usually by $\rho$, that characterizes individuals. This is a fundamental aspect of preferences - a "primitive" parameter. It is not a proxy for the existence of uncertainty (this is why it is found also in deterministic models), neither does it reflect "economic opportunity costs" (this is why it discounts future utility also).

In continuous-time model this discount factor takes the form $e^{-\rho t}$, while in discrete-time models, this discount factor takes the form $\beta = 1/(1+\rho)$.

In the individual's intertemporal maximization problem, the interest rate does not enter in the utility/objective function, but in the law of motion of personal wealth. So a negative real interest rate does not reflect on the personal discount rate.

Traditionally also, firms are considered to discount the future only with respect to the opportunity cost which in turn is assumed to be reflected in the (real) interest rate $r$. So when modelling the intertemporal problem of a firm we usually see $e^{-r t}$ and $\beta = 1/(1+ r)$.

• So how would one explain the following phenomenon: A young person saves for retirement even if the real interest rate is negative (i.e. he's losing money in savings to inflation). Is the personal discount rate part of the explanation? Jul 1 '15 at 18:33
• @Heisenberg "Saving for retirement" has also a strong subsistence/biological survival motive. "Losing money to inflation" is better than "having no money at all" when old and not working anymore. This can be modeled using a utility function like Stone-Geary, where utility is generated if consumption exceeds a certain level. The amount of consumption below this level is the amount required for survival -and to guarantee this level when old, you need to save when young, even at a financial loss. Jul 2 '15 at 7:17

You're correct to think of them as different phenomena.

Often there is what's called the personal discount rate $\rho$. Typically it gets rolled up into a personal discount factor $\beta\equiv\frac{1}{1+\rho}$. You refer to this object in #1. We often see it in lifetime problems where a return function (like utility) is maximized over time. Here are a few perspectives: Yao et al., Frederick et al., or Samuelson.

There's also a distinct discount rate (often $r$ or potentially $r_t$ if you allow it to vary with time) on assets. You seize on this in #2.

• So the personal discount rate is straightforward in case of consumption. What about saving? If I save for retirement despite the real interest is negative, do I have a negative personal discount rate? Jun 30 '15 at 20:50
• @Heisenberg, in the traditional literature you're discounting preferences with $\rho$. As Herr K. mentions, there's a neat way to generalize the problem, and Alecos Papadopoulous gives examples in continuous time. Jun 30 '15 at 23:50

If we think of a discount factor, $\delta$, as the probability of survival to the next period and the utility of death is normalized to $0$, then a simple intertemporal utility function can be written as $$u(x_t)+\delta u(x_{t+1})+(1-\delta)u(\text{death})=u(x_t)+\delta u(x_{t+1}),$$ where the argument in the utility function can either be consumption goods (your case 1) or financial assets (your case 2).

Arthur Robson also proposes that time preferences have, aside from possible mortality, deeper evolutionary underpinnings, which has to do with the growth rate of offspring. See Robson and Samuelson (2007), Robson and Samuelson (2009), and Robson and Szentes (2014).