The discount rate is often approximated as the real rate of return on capital, R, along an optimal path. This is given by the Ramsey equation, which in turn is made up of three parts.
The first is the growth rate of consumption per person in the economy, Ġ.
The second is the elasticity of the marginal utility of consumption, S. The marginal utility of consumption is how much good you get from consuming something, and its elasticity here is taken as how this good changes as you consume more. Wealthy people get less good from the next dollar than poor people do, but how much less? If we think that the good decreases rapidly as people get richer, and we don't want to help rich people in the future who won't appreciate the help, then we have a large discount rate and tend to help ourselves rather than them by spending on us rather than investing for them.
The third part of the discount rate is the pure rate of time preference, E. This is related to our observed tendency to choose to have something, such as an apple, or an Apple, now rather than in the future.
The Ramsey equation puts these together to give the real rate of return on capital as R=E+ĠS. And, economists often set this as being equal to the discount rate.
In the example in the text, 1000 dollars this year becomes 1040 dollars next year after the bank adds the interest. You can say that the present value is P, the future value F, the interest i=F-P, and the interest rate r=i/P=(F-P)/P. The discount rate is d=(F-P)/F. A little algebra will show you that d=r/(1+r) and r=d/(1-d). Here r and d are in the decimal forms (0.04, not 4%). In the main text, we will use D=100d for the discount rate in percent. The economic models eventually assume a discount rate, such as 4%, but often you will see calculations made with 3% and 5%, and sometimes 1% and 7%, because the discount rate is quite uncertain. This uncertainty in the discount rate is much larger than the difference between the interest and discount rates, so using either one will get you close.
The example in the text used an interest rate of 0.04, which gives d=0.04/(1+0.04)=0.03846. . . , which looks messy. But, we could have taken d=0.04, which would have given r=0.04/(1-0.04)=0.041666. . ., which makes the interest rate look messy. Your bank may indeed advertise interest rates such as '4.17%!!!'
Inflation is the tendency for all prices and wages to rise in an economy. Measuring inflation is not a trivial task, but useful estimates are available for the inflation rate at different times in different places. Mathematically, it is not difficult to remove the effects of inflation - if everything goes up together, we can correct everything together, reducing the values back to what they would have been at some chosen time (or inflating them to what they will be at some other chosen time). When effects of inflation have been removed from a calculation, you may see costs and benefits referred to in 'constant dollars' or '2005 dollars' or '(some other specific year) dollars' in a government report on decision-making about energy.
This module has covered the techniques for calculating the present value of future events. But, how uncertainty shows up in this is possibly surprising.
Suppose you estimate that damages of $1000 will occur in 100 years, and you want to estimate their present value as properly as possible. But, you aren’t sure whether the discount rate is D=1% or D=7%—you think that 1% and 7% are equally likely to be correct. You decide to split the difference, by assuming that the average of the present values of these cases is your best estimate of the present value.
You recall that earlier in this module we found that P=F/(1+d)t, allowing you to calculate the present value P from the future value F=1000 for time t=100 using the discount rate d=D/100.
You might be tempted to assume that the answer can be obtained by averaging 1% and 7% to get 4% and then calculating P for this discount rate of 4%. You would be wrong.
If you got out your calculator, you would find
For D=7%, P=1.15
For D=4%, P=19.80
For D=1%, P=369.71
You want the average of the present values for the 1% and 7% cases, which is (369.71+1.15)/2=185.43. That is almost 10 times larger than the value P=19.80 you get for a 4% discount rate.
What happened? For sufficiently high discount rate and sufficiently long time, the present value of any future event is very small, and you can’t lower a small value very much more by using a higher discount rate or longer time. Under those conditions, the average of the present values with an uncertain discount rate is not too far from being half of the present value, with the lower discount rate.
You could put the average present value, 185.43, into the equation above and solve for the discount rate that would give it. The result is D=1.7%, far below the 4% you would get by averaging 1% and 7%.
If you then decided to look at the present value of damages of 1000 occurring 200 years in the future rather than 100 years, you would find
D=7%, P=0.001 (that’s one-tenth of a cent)
D=4%, P=0.39 (that’s 39 cents)
D=1%, P=137.69
The average of 137.69 and 0.001 is 68.85, almost exactly half of the low-discount case. And if you calculate the one discount rate that would give this, it is 1.3%, even lower than the 1.7% for 100 years.
So, if you want to use a single number for the discount rate, but you know that there really is uncertainty about what that number should be, the single number is lower than the average of the possible discount rates, and the single number gets smaller as you look further into the future. In turn, our lack of knowledge about the future means that an economically efficient response involves more actions now to prevent global warming than you would calculate by simply taking a discount rate in the middle of the possible values.
For additional insights on why the discount rate should be made smaller when looking further into the future, motivating more action now to reduce global warming, see K. Arrow et al., 2013, Determining benefits and costs for future generations, Science 341, 349-350. (Note that in the Newell and Pizer paper referenced in the main text, and in this Arrow paper, time is made continuous and discounting is exponential; their equation looks different from ours, but the numerical difference from what we did here with annual values is very small—for example, Newell and Pizer get 34 cents rather than 39 cents for the present value of 4% discounting of 1000 in damages 200 years in the future.)