1 We must now reconsider our supposition of a single rate of time preference for society, say r, and a single yield, say ρ, on private investment.
The latter task is simple inprinciple. Onthe assumptionthat risk-aversion among investors predominates, received doctrine has it that there is a tendency for private investment with a greater expected risk to carry a higher actuarial rate of return– that is, a higher average rate of returnover a long-term period.1
Provided that classification of private investment according to actuarial rates of returnis feasible, it might be thought that the highest actuarial rate of return (corresponding to the riskiest private investment) is the appropriate opportunity rate of returnfor public projects – onthe argument that the outlay K raised to finance the public project could always have been invested in this type of private investment. Even if the argument were accepted, however, there may be relatively little of this riskiest private investment about, and it may therefore take too long to discover its actuarial rate of return. The economist might then choose for the appropriate opportunity rate ρ the rate of returnona less risky private investment, but one that is more common and more likely to be maintained over the future.
2 The reader is reminded, however, that, where the necessary funds for the public project are to be raised entirely by borrowing from the public, the relevant oppor-tunity rate of return, which is to be used as the discount rate, has to be calculated, in general, not by reference simply to the rate paid by the government on the nominal value of the bonds issued. In so far as private investment is displaced –‘crowded out’, inthe jargon– the higher actuarial rate of returnonprivate investment ρ is the appropriate rate.
Only in the limiting case in which the full amount of the initial outlay K that is borrowed by the government does not in fact displace any private investment – so
1 Under a progressive income tax, the total tax paid over, say, a 20-year period is greater for a riskier investment with the same total gross return than for a less risky investment in as much as the gross returnonthe former is more unevenly spread thanthe same gross returnonthe latter. Thus, evenin the absence of risk-aversion, the gross rate of return – which is what ρ measures – has to be higher for the riskier private investment simply in order to yield the same net returnover time as for the safer private investment.
QUAH: “CHAP27” — 2007/1/25 — 14:02 — PAGE 150 — #2 that the aggregate volume of investment that would have taken place during the year is increased by the amount K – does the relevant rate of discount (which has to have reference to the opportunities forgone when the amount K is diverted from the use to which it would otherwise be put) equal society’s rate of time preference, inasmuch as those who buy the K value of government bonds are reducing some part of their current consumption for some additional consumption over the future.
In the more general case where the amount borrowed by the government has the effect of displacing some part only of this amount of private investment, say
$4 millionof anoutlay K of $10 million, the opportunity rate of this $4 million is equal to ρ, with society’s rate of time preference r on the remaining $6 million.
The appropriate discount rate in this general case is, therefore, a weighted average of r and ρ.
3 Anestimate of the social rate of time preference is more elusive. We have already discussed anideal capital market inwhich everybody’s rate of time pref-erence, whether or not he is a borrower or lender, is the same and exactly equal to the rate of interest prevailing on the market, and equal also to the rate of return on existing capital and new investment.
Although we canmove a little inthe directionof realism by envisaging a large number of loan markets, each differing from the other according to the terms of the loan, it is not possible to suppose that a person can borrow all he wishes at the going rate of interest. For if he could, he would also be able to renew the loan when it expired, so postponing repayment indefinitely. Yet, even if borrowers were all equally honest, unless we want also to suppose them equally wealthy, prudent and shrewd, they would not be equally creditworthy. For example, for an initial
$100,000 loanto runfor five years, the more creditworthy the borrower the lower, in general, will be the rate of interest charged.
In order to estimate a community’s rate of time preference, however, it is not enough to take account of all the different loan markets, and within each such market the different categories of borrowers, for, as a result of rationing the amount of money lent to each borrower, the rate of interest he pays on the marginal dollar borrowed may be well below his rate of time preference. For more reliable estimates of people’s time preferences, then, we must go beyond market data. We must use questionnaire surveys.
4 Following the basic maxim, it would seem that, if a person says he will defer consumption of 100 this year for no less than 105 next year, the implied rate of time preference of 5 per cent has to be accepted.2And if this 5 per cent holds over
2 From the fact that a person is indifferent as between consuming an additional 100 this year and consuming an additional 105 next year, it is not to be inferred that he is ‘myopic’ or ‘impatient’.
As demonstrated in Appendix 10, in the complete absence of a loans market, a person may regard a givensum as being of equal worth whether he receives it today or some time inthe future. Yet, once a loans market is introduced, this same person may adjust his pattern of consumption so that, indeed, he then becomes indifferent as between, say, 100 today and 105 in a year’s time.
QUAH: “CHAP27” — 2007/1/25 — 14:02 — PAGE 151 — #3 Rates of return and of time preference 151 the entire time span, he will be indifferent as between consuming 100 this year, year 0, and consuming 100 (1.05)tinyear t.
Canthe rate of time preference r be higher than ρ, the average rate of returnon private investment? Although virtually impossible for society as a whole, we must recognize the possibility that some individuals who perforce must, via taxation, reduce their consumption have very high rates of time preference. Let us take an extreme example of a manof 90 years of age, Mr A, who has to put a value onthe amount of consumptionhe would require in10 years’ time inorder to compen-sate him for sacrificing the consumption today of an additional 200. His average rate of time preference over the next 10 years may be inordinately high, and not unreasonably so. If he believes that his chances of surviving the next 10 years are very low, he may truly claim to be indifferent as between consuming an additional 100 this year and an additional 20,000 in 10 years’ time. This average rate of time preference of about 100 cent per annum is clearly expressive of his impatience to consume while he is still alive: it is the minimum incentive needed to persuade him to forgo present consumptioninfavour of consumptioninthe tenth year.
It would seem to follow that the age distribution of the beneficiaries and the losers in the different projects being compared would significantly affect the weighted average rate of time preference and, therefore might be a critical variable in determining their ranking. For example, a public investment whose benefici-aries were largely elderly people would certainly have a higher average rate of time preference and, in so far as it enters the discount rate, would reduce the present value of that project below that of a project whose beneficiaries were mainly young people.
5 It would seem reasonable to calculate society’s rate of time preference as the weighted average of the several groups in the community that are affected, making society’s rate of time preference R equal to nwiri, where there are n different groups, ribeing the rate of time preference of group i, an d wibeing the weight of group i, with nwiequal to unity.
It transpires, however, that the R so calculated is generally slightly smaller than anexact measure of society’s rate of time preference as, if a sum x is compounded for a number of years at this rate R, it will compound to a sum that is slightly smaller than the sum compounded for each group separately and then added. Over the years, of course, the absolute difference between R and the true measure will grow.
But save in exceptional circumstances the difference will remain relatively slight.3
3 To illustrate with only two groups (group 1 with a weight of 0.7 and a rate of time preference of 10 per cent, the other group with a weight of 0.3 with a rate of time preference of 0.05): R would thenbe equal to (0.7 × 0.1) + (0.3 × 0.05) equal, therefore, to 8.5 per cent. If x is $1,000, then compounded at R for two years it becomes $1000(1.085)2or $1,177.25. For five years, it becomes equal to $1,503.65. If now we compound each group separately, after two years we have $700(1.1)2 plus $300(1.05)2, a total of $1,177.75 – a difference from compounding R of only 50 cents. After five years, the compounded sum of the two separate groups becomes $700(1, 1) plus $300(1.05)5, a total of $1,510 – a difference now of less than $7.
QUAH: “CHAP27” — 2007/1/25 — 14:02 — PAGE 152 — #4 Where the range of the different rates of time preference for the community affected is not great, at least for the larger groups, the use of this weighted average R as society’s rate of time preference is unlikely to make a significant difference to the calculationas compared with the use of the rate of time preference of each of the separate n groups.
QUAH: “CHAP28” — 2007/1/25 — 08:03 — PAGE 153 — #1