Benefit-Cost Analysis and the Role of Time

    The application of benefit-cost analysis is, of course, extremely old--it is at heart merely rational thinking!  This will seem odd to modern environmentalists who have often been led to believe that B-C analysis is an inappropriate way to make environmental decisions.  Consider the following letter from Ben Franklin to a friend written from London on September 19th, 1772:

"Dear Sir:
    In the affair of so much importance to you, wherein you ask my advice, I cannot, for want of sufficient premises, advise you what to determine, but if you please I will tell you how.  When those difficult cases occur, they are difficult chiefly because while we have them under consideration, all the reasons pro and con are not present to the mind at the same time; but sometimes one set present themselves, and at other times another, the first being out of sight.  Hence the various purposes or inclinations that alternatively prevail, and the uncertainty that perplexes us.  To get over this, my way is to divide half a sheet of paper by a line into two columns; writing over the one Pro, and over the other Con.  Then, during three or four days consideration, I put down under the different heads short hints of the different motives, that at different times occur to me, for or against the measure.  When I have thus got them all together in one view, I endeavor to estimate their respective weights; and where I find two, one on each side, that seem equal, I strike them both out.  If I find a reason pro equal to some two reasons con, I strike out the three.  If I judge some two reasons con, equal to some three reasons pro, I strike out the five; and thus proceeding I find at length where the balance lies; and if, after a day or two of further consideration, nothing new that is of importance occurs on either side, I come to a determination accordingly.  And, though the weight of reasons cannot be taken with the precision of algebraic quantities, yet when each is thus considered, separately and comparatively, and the whole lies before me, I think I can judge better, and am less liable to make a rash step, and in fact I have found great advantage from this kind of equaiton, in what may be called moral and prudential algebra.
    Wishing sincerely that you may determine for the best, I am ever, my dear friend, you most affectionately,
B. Franklin"

    Modern benefit-cost analysis uses fancy tools to do essentially what Ben Franklin's intuitive approach recommends above.  But, getting back to the primacy of human preferences (that might not be terribly "elevated"), consider the following quote:

"Is it progress if a cannibal uses a knife and fork?" --Stanislaw J. Lee

(Benefit-cost analysis can, and has, been (mis)used to justify many bad projects!)

    In appraising public sector investments bear in mind that 1) all projects are not equally important and hence don't merit the same resources for evaluation and 2) many projects are easily evaluated, clearly having benefits far greater or far less than costs.

The Role of Time in Environmental Economic Analysis

    First, we must recognize what an interest rate really is.  The interest rate is the "price" of consuming in this period rather than a year later--it is the foregone benefits of greater future consumption resulting from the decision to consume now.  That is, if you consume $1 worth of something in year 0, you will have $(1 + i) less of something to consume in year 1, where i is the real rate of interest (subtract the rate of inflation from the nominal interest rate and you have the so-called "real" interest rate).  [NOTE: just as with any price the interest rate "dollars" mean nothing themselves--what is being valued relatively is current consumption and future consumption.  This is exactly analogous to prices at a point in time that provide relative values of things consumed--you're not really giving up dollars, but rather the second-best "thing" when you by some"thing."]
    Second, we must recognize where interest rates come.  Some savers put funds send funds directly to borrowers (purchases of stocks and bonds do this).  Other savers put funds into accounts set up by "financial intermediaries" (banks, savings and loans, mutual funds, etc.) and the latter route those funds on to various borrowers.  The savers are saving because they value future consumption (the only reason people save...think about this).  Borrowers are borrowing because they value the profits that (they hope) can be obtained by using those funds productively (e.g. buying machines or building plants).  Savers would generally be expected to save more if they receive a higher interest rate (getting paid more to give up current consumption should encourage more saving), while borrowers would generally be expected to demand fewer funds at higher interest rates (fewer investments will be profitable if the costs of borrowing increase).  Thus, the interest rate in a society represents the interaction between productivity and thrift.  The equilibrium i-rate exists when the interest rate adjusts to equate S & D for these "loanable funds."  (GRAPH--show what happens as productivity increases and as thrift increases).  As with any price, buyers (in this case, borrowers) would like to pay less, while sellers (in this case, savers--those consuming less than their after-tax income) would like to receive a higher price.  But the equilibrium balances these desires, leaving everyone--while not necessarily "happy"--able to what they want to do (suppose "usury" laws--"preferences over preferences" again).
    Interest rates have something important to say about how one should (from an efficiency perspective) value benefits and costs that occur in the future.  If $1 "grows" through compounding to $1+i in one year, the natural conclusion is that $1 in one year must be worth less than $1 today (since $1 today will be worth $1+i next year).  For example, at 10% interest $1 next year is only worth about $.91 now since $.91 can be put in the bank and "become" $1 next year.  If you were to pay much more than $.91 you would be foolish in that you would be getting a below normal return on your investment ($.95 could earn, for example, $.095 in interest, hence you could have $1.045 at the end of the year but are getting only $1).  If you were to pay much less, that would be great (for you), but the borrower would be paying more in interest than they would need to (for example, if you could buy the future $1 for $.84 now that would result in a 20% interest payment; the borrower would never pay 20% if they could get funds for 10%).  So, a dollar next year is worth less now since something less than a dollar now (how much less depends on the rate of interest) will compound into a dollar then according to:
        X(1+i) = $1
where X = the present value of the $1 in one year.
Solving for X,
        X = $1/(1+i)
(in our example, X = $.91 when i = .10 or 10%)
    What if you get that $1 not one year from now but two years from now?  Well, you only have to figure out what amount will cumulate to $.91 at the end of the first year--because you already know that $.91 will become $1 in one more year.  So, you realize that $.83 will do that--after the first year you will have $.83(1+.10) = $.83 + $.083 = $.91+.  By the end of the next year, $.83 will, then, compound up to become $1 according to :
        X(1+i)(1+i) = $1
where X = the present value of the $1 in two years.
(in our example, X = $.83 when i= .10 or 10%)
Solving for X,
        X = $1/[(1+i)(1+i)] = $1/(1+i)2
    Similarly, a dollar received in any period, say period n, is worth today:
        X = $1/(1+i)n
    We are now ready to do Benefit-Cost analysis, since every "project investment" is merely a stream of dollar benefits or costs that occur in each period.  That is, if $C in costs occur in the second year, they have a present value of $C/(1+i)  If there are $B in benefits that occur in the sixth year, they have a present value of $B/(1+i)6   What we do, then, is to add up all the present values of all the benefits and costs that occur in each period for the life of the project and see if the resulting number is greater than or less than zero.  If it is greater than zero, the project has a positive "NPV" (net present value) and should--on efficiency grounds--be adopted; if the NPV is less than zero the project will lower social welfare and is inefficient (has costs greater than benefits when both are properly counted, by converting them to present values).