ECON/ENVS 231 – Prof. Gaudin p.1 of 3
Tools for benefit cost analysis
1. Present Value Calculations/Discounting: Adding real dollar values to be paid or received at
different points in time
The opportunity cost of consuming something worth one dollar today instead of saving it for future
consumption is not $1. Remember that the opportunity cost of something is the value of the best alternative.
The best alternative to lending me $100 could be to put the $10 in a safe interest bearing account. If you
save $X, you can buy something worth X(1+r) next period if the interest rate between the two periods is r.
(So if you leave $100 dollar your savings account that pays a 5% interest (r = 0.05), you would have 100 x
(1+0.05) = $105 in the account a year from now. Therefore, the opportunity cost of lending me $100 for a
year is (100) (1.05) =$105).
Equivalently if you get $X only next year, it is not the same as getting $X today, since saving $X today
would give you $X(1+r) next year and $X(1+r) is greater than X. If you are given only X/(1+r) today and
save it, you end up with X tomorrow. So receiving X next period is equivalent to receiving X/(1+r) today.
We say that the present value of X is X/(1+r).
• Discount rate: r. For benefit cost analysis of government policy, the discount rate should be the
relevant social rate of discount (rs,) that represents the best alternative use of the public funds. The rate
of return should include all external benefits that would go with the best alternative use of the funds.
• Discount factor = β = 1/(1+r)
• In general, X dollars t year from now is equal in present value to X / (1+r) t dollars if r is the yearly
net real interest rate so, using FV for “future value” and PV for “present value”
PV(X)=FV(X)/(1+r)t= βt FV
• Also, an X dollar benefit/cost every period to “infinity” (or for a significant number of years)
starting from next period or the end of the first period (a steady stream of $X), knowing that the rate of
interest will be equal to r in all periods, is worth X/r dollars in present value.
Example 1: We estimate that a cancer operation 10 years from now will cost $100,000 in today’s dollars.
How much money do we need to save today to have the $100,000 to do the operation 10 years from now.
We need to calculate the PV of $100,000
If the normal safe REAL (net of inflation) rate of return on funds is 5%
PV = $100,000/(1.05)10 = $61,392
Will the PV increase or decrease if r=0.1 instead?
2. Expected Value Calculation: Taking Account of Uncertainty/Risk
We have assumed that everything about the present and the future is known with certainty. This would be
nice but it doesn’t happen that way! Information is not free and if you want to inform yourself you will
have to spend time and money, and even then there will always be some information that you cannot
obtain. This imperfect information makes a big difference on economic outcomes. When it comes to
knowing FUTURE outcomes, perfect knowledge is an even more unrealistic assumption. Future events are
always subject to unknown fluctuations (shocks). We will focus here on the notion of uncertainty.
To make rational decisions when faced with uncertainty we use our expectations of the likelihood of every
possible outcome. In particular, we use the expected value of our possible actions.
Expected value = the sum of every possible outcome weighted by the probability of each outcome
