A generalized approach to multigeneration project evaluation.

1. Introduction

It is well-known that the choice of discount rate plays a significant role in the ranking of projects with varying time paths of costs and benefits. For public projects, the choice of the appropriate social discount rate (SDR) is not clear-cut even in the context of short-lived projects in which costs and benefits occur within a single generation. In particular, when capital taxes produce a wedge between gross and net rates of return, should one discount at the economy's rate of return (the before tax rate) or the individual's rate of return (the after tax rate)? In the single-generation context, the literature falls in two camps. (1) One camp argues that the appropriate SDR is a weighted average of the gross and net rates of return with weights typically determined by the fractions of resources drawn from consumption and private investment, respectively. (2) A second camp argues that future benefits should be discounted at the net rate of return, but that the initial investment should be multiplied by a scale factor so that the direct costs of the project can be expressed in terms of consumption units. (3)

Both approaches suffer from severe implementation problems. For the first approach, there is no general formula for the weights needed to calculate the SDR as a weighted average of two market rates of return (Dreze 1974). Indeed, the appropriate SDR to use is project-specific (Stiglitz 1982). Without a systematic methodology to derive project specific SDRs, the weighted average approach is hard to implement. For the second approach, which equalizes the SDR to the net rate of return, the scale factor--the opportunity cost of public investment--must be quantified. The opportunity cost of public investment depends on whether resources come from private investment or consumption and on the calculation of the present value of the future consumption yielded by a unit of capital discounted at the net rate of return; that is, the shadow price of capital. As Lind (1997) points out, however, there is no general agreement on a specific procedure for calculating the shadow price of capital. Indeed, the concept of the opportunity cost of public investment suffers the same project dependence problem as the first view of the SDR (Diamond 1968).

Multigeneration projects such as nuclear waste disposal, natural resource conservation, or even Social Security reform further complicate discount rate choice. Specifically, intergenerational equity, in addition to intertemporal efficiency, plays a role in determining the appropriate social discount rate for multigeneration public project evaluation. (4) As in the single-generation context, there are two competing views on the appropriate discount rate when evaluating multigeneration projects: the descriptive and the prescriptive approaches. (5)

The descriptive approach to discounting is a market-based approach. It does not rely on an explicit social welfare function (SWF) or the pure time preference rate embedded in the SWF. However, the descriptive approach works under the assumption that intergenerational compensation through changes in the tax/debt policy is available. (6) It is argued that, under appropriate intergenerational compensation, the market rates of return are still relevant for calculating the SDR for multigeneration project evaluation, and, in general, the SDR should be a weighted average of the gross and net rates of return. This descriptive view of setting the SDR equal to the market return appears to underlie the Office of Management and Budget's (OMB) adoption of 7% as the appropriate discount rate for federal programs. (7)

The intent of the prescriptive approach is to use the social discount rate as an expression of ethical judgment on how consumption of future generations should be compared to consumption of current generations. The starting point of the prescriptive approach is a social welfare function or the pure time preference rate embedded in the SWF. (8) The prescriptive approach is based on the widely accepted principle that if the full impacts of a project's costs and benefits are expressed as net additions to each generation's consumption, then the appropriate discount rate should be the social rate of time preference (SRTP), expressed as the sum of the pure rate of time preference and the product of the elasticity of marginal utility with respect to consumption and the growth rate of per capita consumption. If one believes that future generations' utility should weigh as much as the current generations', the pure rate of time preference is zero. The remaining growth component yields an SRTP in the range of 1.5% to 3.0% if one assumes the growth rate of per capita consumption to be 1% to 2% and the elasticity of marginal utility with respect to consumption to be 1.5. The prescriptive approach is adopted by the Congressional Budget Office (CBO), which requires converting all costs and benefits to consumption units and then applying an SRTP of 2%. (9)

Emphasizing the difference between the respective discount rates suggested by the two approaches misses the point, as what is discounted is different in the two approaches. (10) The descriptive approach discounts the direct costs and benefits of a project. In contrast, the prescriptive approach discounts the project's net incremental consumption to each and every generation. The two approaches are not necessarily inconsistent, but to date, there are problems with both approaches. First, for the descriptive approach, when taxes falling on capital income drive a wedge between these two returns, it is not clear whether the gross or the net rate of return is the appropriate discount rate. Though it may be argued that the appropriate discount rote should be a weighted average of gross and net rates of return, it is still a point of debate how these weights can be practically determined. Second, although the prescriptive approach using the SRTP to discount converted consumption flows of a project is theoretically sound, it is difficult to apply because there is no general procedure to translate the direct cost--benefit stream of a project into net additions to each generation's consumption. Moreover, there is always the question of what exactly the pure time preference rate is.

In this article, we generalize the existing descriptive approach to multigeneration public project evaluation, taking into account distortionary taxes on capital income. We show that such a generalization does not lead to a project-specific SDR, which is the weighted average of gross and net rates of return. What emerges is the concept of the marginal cost of public funds (MCF), which has the convenient property of project independence. In a sense, the approach to multigeneration project evaluation presented here can also be regarded as a generalization of the MCF approach to public project evaluation developed in the static setting. Thus, this article connects the two separate literatures on public project evaluation: MCF and SDR. (11) In addition to the key parameter (the MCF) being project-independent, another desirable feature of the MCF-based approach is that it identifies projects that, along with appropriate intergenerational transfers through time-varying head taxes, are Pareto improving, and is, therefore, independent of any utilitarian social welfare function being used. Thus, our approach allows a clean separation between two related, but distinct, issues that arise in literature concerned with the SDR in a multigeneration context. On one hand, it is argued that the SDR should reflect the value placed on the welfare of future generations. On the other hand, it is argued that the SDR should reflect tax distortions, notably distortions of intertemporal decision-making arising from capital income taxes. Availability of time-varying head taxes essentially eliminates the first of these considerations and focuses attention on the second.

Our analysis yields two primary conclusions that qualify how the social discount rate is used in evaluating multigeneration projects. First, the central role played by the SDR in long-term project evaluation should be replaced with parameters of the marginal cost of funds. Second. for the typical cases where the MCF is larger than one and projects incur costs before they generate benefits, the value of the traditional social discount rate will be larger than the gross return. This higher discount rate exceeds the value currently in use, implying fewer long-term projects would be accepted using our criterion.

We begin with a simple two-period overlapping generations model with multigeneration public projects financed by an intertemporal tax policy including a distortionary tax on capital income and time-varying head taxes. In section 3, we derive a criterion for evaluating new, small multigeneration projects that is based on the marginal cost of funds. The MCF role, which also holds for a life cycle with an arbitrary number of periods, is a generalization of the descriptive approach in the second-best environment. Two features of the generalized descriptive approach--project-independence and SWF-independence--are highlighted. Then, in section 4, we discuss the implications of the MCF-based criterion for the (traditional) concept of social discount rate. Finally, in section 5 we offer an estimate of the values of involved parameters and discuss extensions to alternative tax structures, followed by a concluding summary.

2. A Formal Framework for Multigeneration Project Evaluation

Consider a simple overlapping generations (OLG) model in which individuals live two periods, earning wage income in period 1 by supplying a fixed amount of labor and retiring on savings in period 2. Neither the two-period life nor the fixed labor supply is crucial for obtaining our results, but working with an OLG model with few complications significantly simplifies exposition. Assume that individuals are identical within generations but may differ across generations, and, for simplicity, let generation size be constant over time. We identify each generation by the period in which its members are young and assume that generation t individuals have the following general utility function

(1) [u.sup.t]([c.sup.t.sub.t+1], [G.sub.t], [G.sub.t+1]), t [greater than or equal to] 0

where [c.sup.t.sub.t] ([c.sup.t.sub.t+1]) is consumption and young (old) by a generation t individual and [G.sub.t], [G.sub.t+1] are levels of government-provided goods and services in periods t and t + 1, respectively. Note that generations born in periods earlier than t =-1 are irrelevant for the government's decision making in period zero. On the other hand, the generation born in period t = -1 is alive and in its second period of life at the decision making time. At the first pass of our analysis, however, we assume that all impacts of a project are on the newborn and yet-to-be-born generations. Generalizations accounting for generations that are in their mid-life at the decision-making time, as well as individuals living more than two periods and variable generation sizes, are given later in the article.

Let wages and the marginal productivity of capital (the gross rate of return) be exogenous and. for notational simplicity, assume that the gross rate of return is constant over time. (12) For any given [G.sub.t] and [G.sub.t+1], generation t(t [greater than or equal to] 0) individuals maximize Equation 1 subject to a lifetime budget constraint 

(2) [c.sup.t.sub.t] + [[beta].sub.N][c.sup.t.sub.t+1] = [w.sub.t] - [h.sub.t] - [[beta].sub.N][h.sub.t+1] = [w.sub.t] - [T.sup.t],

where [[beta].sub.N] = [[1 + r(1 - [tau])].sup.-1] is the net discount factor, r is the gross (before tax) rate of return on private investment, [tau] is the tax rate on capital earnings [hence r(l - [tau]) is the net rate of return], (13) [w.sub.t] is the individual's wage earnings in the first period of life, and [h.sub.t] is a general head tax in period t. As an alternative to head taxes {[h.sub.0], [h.sub.1], ... }, we define [T.sup.t] (t [greater than or equal to] 0) as a generation t- specific lump-sum tax paid in period t such that for generation t lifetime lump-sum taxes are ([T.sup.t],0).

For expositional convenience we proceed as if generation-specific lump-sum taxes {[T.sup.0],[T.sup.1], ...} are imposed instead of the head taxes. Whereas nonindividualized lump-sum taxes (head taxes) are regarded as feasible, individualized lump-sum taxes are usually ruled out as impractical. (14) However, in our intergenerational context, generation-specific lump-sum taxes are equivalent to time-varying head taxes and are therefore implementable. In Appendix A, we show that for a fairly general environment, for any set of time-varying head taxes that satisfies the intertemporal government budget constraint, there exists a set of generation-specific lump-sum taxes that also satisfies the government budget constraint and generates the same consumption pattern and welfare for each generation, and vice versa. Using a similar procedure to that in Appendix A, one can also show that collecting generation-specific lump-sum taxes in the second period of life or in both periods of life is equivalent to collecting generation-specific lump-sum taxes only in the first period. Hence, our assumption that the generation-specific lump-sum taxes are only collected in the first period of life is nonessential but imposed to economize on the notation and to simplify the exposition.

The first-order condition for the generation t [greater than or equal to] 0 individual's problem is

(3) ([partial derivative][u.sup.t]/[partial derivative][c.sup.t.sub.t+1])[([partial derivative][u.sup.t]/[partial derivative][c.sup.t.sub.t]).sup.-1] = [[beta].sub.N].

Individual (private) utility maximization yields private consumption in both periods, [c.sup.t.sub.t]([T.sup.t], [G.sub.t], [G.sup.t+1]) and [c.sup.t.sub.t+1]([T.sup.t], [G.sub.t], [G.sub.t+1]), saving in period t equal to (15)

[s.sup.t]([T.sup.t], [G.sub.t], [G.sub.t+1]) [equivalent to] [w.sub.t] - [T.sup.t] - [c.sup.t.sub.t]([T.sup.t], [G.sub.t], [G.sub.t+1])

and indirect utility

[v.sup.t]([T.sub.t], [G.sub.t], [G.sub.t+1]) [equivalent to] [u.sup.t][[c.sup.t.sub.t],([T.sup.t], [G.sub.t], [G.sub.t+1]), [c.sup.t.sub.t+1]([T.sup.t], [G.sub.t], [G.sub.t+1]), [G.sub.t], [G.sub.t+1]].

From the Envelope Theorem,

(4) [partial derivative][v.sup.t]/[partial derivative][T.sup.t] = - [partial derivative][u.sup.t]/[partial derivative][c.sup.t.sub.t] = - [[gamma].sup.t], [partial derivative][v.sup.t]/[partial derivative][G.sub.t] = [partial derivative][v.sup.t]/[partial derivative][G.sub.t], [partial derivative][v.sup.t]/[partial derivative][G.sub.t+1] = [partial derivative][u.sup.t]/[partial derivative][G.sub.t+1],

where [[lambda].sup.t] is generation t [greater than or equal to] 0 individuals' marginal utility of income.

As a mechanism to account for all generations, both present and future, assume an external benevolent dictator who spans all current and future generations and can neither take nor give anything to the economy. The dictator chooses intertemporal tax policy to finance public projects and make intergenerational transfers to maximize a SWF. We assume two types of taxes: a fixed level of distortionary taxes (in terms of rotes), which include taxes falling on capital income, and time-varying head taxes or subsidies (which are equivalent to generation-specific lump-sum taxes or subsidies) that are chosen to optimize the dictator's intergenerational objective function. To simplify the notation and focus on the issue of project evaluation when gross and net rates of return differ, the distortionary portion of the tax system initially consists solely of taxes on capital earnings. (16) A discussion of more complicated distortionary taxes is presented in section 5.

We do not allow the capital income tax rate to be subject to optimization because it would imply a zero tax rate on capital income in our model given the availability of time-varying head taxes, and therefore, rendering the framework unsuitable for analyzing the SDR. Like all other studies on SDR, the preexisting distortionary capital income tax is imposed here, which is subject to the question of how to justify its existence. For example, Wilson (1991), Dahlby (1998), and Sandmo (1998) criticized early studies on the MCF in the framework of a representative individual for working with a distortionary tax system without being able to justify its existence. However, the assertion of preexisting distortionary taxes in this article seems to be less critical than in those early studies on MCF because the distortionary taxes can be regarded as being imposed to handle intragenerational redistribution, an issue we abstract from to focus on intergenerational resource allocations.

The assumed existence of a benevolent dictator maximizing an SWF raises an important issue. Many researchers find the interpersonal comparisons implied by an SWF unacceptable because of questions concerning the precise form of the SWF. Fortunately, in the end, the project evaluation rule derived in this article is independent of the form of SWF. Indeed, projects selected using our criterion are Pareto improving, given appropriate intergenerational compensation through head taxes, or, equivalently, generation-specific lump-sum taxes. The property of Pareto improvement of our criterion may seem to be trivial with the availability of generation-specific lump-sum taxes. However, what is new is that these generation-specific lump-sum taxes can actually be implemented through time-varying head taxes.

The dictator oversees financing a set of accepted public projects through debt management and tax collections. Government outlays required of and government services provided by these accepted projects can cumulatively be represented by (E, G), where E [{[E.sub.t]}.sup.[infinity].sub.t=0] is the outlay stream of accepted projects and G = [{[G.sub.t]}.sup.[infinity].sub.t=0] is the service stream of these projects. (17) Intertemporal government budget balance requires

(5) [infinity. summation over t=0][[beta].sup.t.sub.G][E.sub.t] = A + [infinity. summation over t=0][[beta].sup.t.sub.G][R.sub.t] = A + H [[s.sup.-1]r[tau] + [infinity. summation over t=0] [[beta].sup.t.sub.G]([[beta].sub.G][s.sup.t]r[tau] + [T.sup.t])],

where [[beta].sub.G] = [(1 + r).sup.-1] is the gross discount factor, A is initial dictator assets, [R.sub.t] is tax revenue collected in period t, H is the constant generation size, and [s.sup.-1] is savings made by a generation - 1 individual one period before the decision-making time. For any given set of preaccepted public projects represented by (E, G), the dictator chooses {[T.sup.0], [T.sup.t], ... } to maximize some SWF subject to Equation 5.

At this point, it is appropriate to emphasize the role of investment in the government budget constraint represented in Equation 5. If, as the consequence of deficits/surpluses in previous periods, the dictator's assets have a positive balance at the end of a period, they will be invested and earn the gross rate of return. On the other hand, if the dictator's assets have a negative balance at the end of a period (the absolute value of which is outstanding public debt), the dictator will pay the gross rate of return on this debt the next period. At the same time, the dictator taxes bond and capital returns the same way and individuals treat bonds and capital as perfect substitutes in their investment decisions.

3. A Marginal Cost of Public Funds--Based Criterion of Project Evaluation

A Marginal Cost of Public Funds-Based Criterion for the Two-Period Life Cycle

Following convention in the social discount rate literature, the problem at hand is how to evaluate a new small mulligeneration project, given the preaccepted set of projects (E, G). (18) Represent a marginal project by ([DELTA]E, [DELTA]G), where [DELTA]E = [{[DELTA][E.sub.t]}.sup.[infinity].sub.t=0] is the stream of additional government outlays, measured in units of contemporary private goods, that are required by the project, and [DELTA]G = [{[DELTA][G.sub.t]}.sup.[infinity].sub.t=0] is the stream of government services, measured in units of the service produced, that are provided by the project. (19) To conduct cost-benefit analysis, benefits of public projects are measured by consumers' willingness to pay and the project's indirect revenue effects. (20) Denote [B.sup.t.sub.k], t [greater than or equal to] 0, t [less than or equal to] k [less than or equal to] t + 1 as a generation t individual's willingness to pay for [DELTA][G.sub.k] in terms of contemporaneous private consumption. (21) In addition, denote [DELTA]R = [{[DELTA][R.sub.t]}.sup.[infinity].sub.t=0] as the indirect revenue stream generated by the project. (22) Both [B.sup.t.sub.k] and [DELTA][R.sub.t] will precisely be defined below. The raw materials for cost-benefit analysis of a public project ([DELTA]E, [DELTA]G) can also be expressed in terms of [{[DELTA][E.sub.t]}.sup.[infinity].sub.t=0], [{B.sup.t.sub.k}.sup.t[greater than or equal to].sub.t[less than or equal to]k[less than or equal to]t+1, and [{[DELTA][R.sub.t]}.sup.[infinity].sub.t=0].

To make generation t (t [greater than or equal to] 0) individuals better-off with project ([DELTA]E, [DELTA]G), the additional lump-sum tax collected from a generation t individual, [DELTA][T.sup.t], must satisfy

[partial derivative][v.sup.t]/[partial derivative][T.sup.t][DELTA][T.sup.t] + [partial derivative][v.sup.t]/[partial derivative][G.sub.t][DELTA][G.sub.t] + [partial derivative][v.sup.t]/[partial derivative][G.sub.t+1][DELTA][G.sub.t+1] > 0,

or

(6) [DELTA][T.sup.t] < [B.sup.t.sub.t] + [[beta].sub.N][B.sup.t.sub.t+1], t [greater than or equal to] 0,

where

[B.sup.t.sub.t] [equivalent to] [partial derivative][u.sup.t]/[partial derivative][G.sub.t][DELTA][G.sub.t]/[partial derivative][u.sup.t]/[partial derivative][c.sup.t.sub.t]

[B.sup.t.sub.t+1] [equivalent to] [partial derivative][u.sup.t]/[partial derivative] [G.sub.t+1] [DELTA][G.sub.t+1]/[partial derivative][u.sub.t]/[partial derivative][c.sup.t.sub.t+1].

Inequality 6 simply says that to make an individual better-off with the combination of a new project and an increase in his generation-specific lump-sum tax payment, the additional lump-sum tax payment must be smaller than the individual's willingness to pay for the project's outputs in terms of the first-period consumption.

On the other hand, [{[DELTA][T.sup.t]}.sup.[infinity].sub.t=0] must be sufficiently large so that the government budget in Equation 5 remains satisfied (extra revenues are allowed) after the acceptance of the project ([DELTA]E, [DELTA]G). That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Condition 7 implies two points that we want to emphasize. First, the project's outputs, [DELTA]G, have impacts on various generations' savings, which, in turn, generate revenue effects through the capital income tax. The revenue effect in period t is denoted as [DELTA][R.sub.t], as shown above. As a result, although the project's investment requirement in period t is [DELTA][E.sub.t], its net revenue requirement is [DELTA][E.sub.t] - [DELTA][R.sub.t]. Second, the present value (discounted at the gross rate of return) of the additional lump-sum tax payments (H[summation of.sup.[infinity].sub.t=0][[beta].sup.t.sub.G][DELTA][T.sup.t]) in various periods must be strictly larger than the present value of the net revenue requirements (the left side of Condition 7) as long as the marginal propensity to save for each generation is positive; namely, [partial derivative][s.sup.t]/[partial derivative][T.sup.t] < 0. As we show below, the second point here leads to the marginal cost of funds concept.

From Inequality 6 and Condition 7, acceptance of project ([DELTA]E, [DELTA]G), along with changes in generation-specific lump-sum taxes, can make everyone better-off, if and only if,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[gamma].sub.t], = -([partial derivative][s.sup.t]/[partial derivative][T.sub.t]) is generation t individuals' marginal propensity to save. As a consequence of the tax wedge in the capital market, collecting a generation-specific lump-sum tax entails an efficiency cost in the sense that a dollar increase in a generation's lump-sum tax--which is the loss felt by the generation--would generate a present value increase of less than a dollar in the generation's total tax payments due to reduced savings, and therefore, reduced tax payments on the capital income tax. Expression 9 measures the required lump-sum tax increase per dollar increase in the present value of the generation's total tax payments and is the marginal cost of funds for generation t individuals when the marginal tax instrument is the generation-specific lump-sum tax. The welfare interpretation of MC[F.sub.t] is the real cost to a generation t individual (in terms of the first-period consumption) of an additional unit of total taxes collected from him (in terms of present value). Note that MC[F.sub.t] > I in this model because [[gamma].sub.t] > 0. (24)

PROPOSITION 1. In a two-period overlapping generations model, assume that a dictator can make intergenerational transfers through time-varying head taxes given preexisting taxes on capital income. Then a marginal project represented by

[{[DELTA][E.sub.t]}.sup.[infinity].sub.t=0], [{[B.sup.t.sub.k]}.sup.t[greater than or equal to]0.sub.t[less than or equal to]k[less than or equal to] t+1], [{[DELTA][R.sup.t]}.sup.[infinity].sub.t=0]

can be Pareto improving if and only if Inequality 8 is satisfied.

Some elaboration of the criterion in Equality 8 may be helpful at this point. The left-hand side of Inequality 8 brings together net and gross discounting and the marginal cost of funds. First, because individuals can trade between periods at only the net rate, each generation's benefits are discounted to its first period using the net rate of return. Second, before each generation's present value benefit can be compared to government costs, it must be converted to revenue-equivalent values using the generation's MCF. Third, because the gross rate of return represents the opportunity cost of government spending, the adjusted present value benefits, as well as the costs and indirect revenue benefits on the right-hand side of Inequality 8, are discounted at the gross rate of return. Criterion 8 has a simple intuitive explanation. Given the welfare interpretation of MC[F.sub.t] as the real cost to a generation t individual of an additional unit of lifetime taxes (in present value terms including both generation-specific lump-sum taxes and the capital income tax), the maximum additional lifetime taxes the dictator can collect from a generation t (t [greater than or equal to] 0) individual after the provision of AG without making that individual worse off is, in terms of present value, ([B.sup.t.sub.t] + [[beta].sub.N][B.sup.t.sub.t+1)/MC[F.sub.t]. Essentially, Condition 8 requires that the present value of additional tax payments collected must exceed the present value of the net project costs, [DELTA]E - [DELTA]R.

The General Marginal Cost of Public Funds-Based Criterion

Criterion 8 is formally derived, and the MCF Expression 9 is formally defined, for an overlapping generations model in which each generation has a two-period life. We now generalize the criterion to a life of an arbitrary number of periods. At the same time, we shall take care of the generations that have already lived one or more periods at the decision-making time and allow for time-varying generation sizes. We leave the detailed derivation of the general criterion to Appendix B because it is similar to the one we have just given for the two-period life case.

Assume that a generation lives D + 1 periods (D = 1 corresponds to the two-period life cycle). At decision-making time (period zero), generation -(D + 1) and earlier generations are already gone. Generations that must be considered in the cost-benefit analysis are those born in period -D and later. Suppose Ht is the number of generation t (t [greater than or equal to] -D) individuals and [B.sup.t.sub.k](t [greater than or equal to] -D, max[0, t] [less than or equal to] k [less than or equal to] t + D) is a generation t individual's willingness to pay, in terms of contemporaneous private consumption, for [DELTA][G.sub.k]. Then, Proposition 1 can be generalized to the following Proposition 1'.

PROPOSITION 1'. In an overlapping generations model with D + 1 generations overlapping in every period, assume that a dictator can make intergenerational transfers through time-varying head taxes given preexisting taxes on capital income. Then, a marginal project represented by

[{[DELTA][E.sub.t]}.sub.[infinity].sub.t=0], [{[B.sup.t.sub.k]}.sup.t[greater than or equal to]-D.sub.max[0,t][less than or equal to]k[less than or equal to]t+D], [{[DELTA][R.sub.t]}.sup.[infinity].sub.t=0] can be Pareto improving if and only if

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where all variables and parameters have the same interpretation as before. (25)

The Optimal Criterion 10 contains two sets of benefits: benefits for those generations already in existence before the decision-making period (period zero), generations dated -D to 1, and benefits for generations born in or after the decision-making period, generations dated t [greater than or equal to] 0. The criterion requires the following discount rates for a multigeneration public project. First, future project benefits of any given generation are discounted to the first period of the generation's life or the decision-making period, whichever comes later, at the return available to that generation when it makes its consumption decisions, the net rate of return. Thus, this present value represents the total willingness to pay for this project by individuals of any specific generation. Second, the dictator adjusts the present value of each generation's benefits by a generation-specific marginal cost of funds that reflects any efficiency distortion due to taxes borne by the generation. Third, the dictator discounts the adjusted present value of each generation's benefits to the decision-making period at the alternative cost of public investment, in our case, the gross rate of return. Fourth, the future project costs net of indirect revenue effects are discounted at the economy's alternative cost of public investment, that is, the gross rate of return.

The above MCF-based criterion generalizes the existing descriptive approach for the situation of first best taxation. The tax-induced wedge between gross and net rates of return plays two roles in the MCF-based criterion. First, within-generation benefits should be discounted at the net rate of return because that is the rate of return governing individual intertemporal consumption choice at the margin. On the other hand, the publicly borne project costs (net of the indirect revenue effects from a project) should be discounted by the gross rate of return because the gross rate represents the opportunity cost of government spending. Second, the tax-induced wedge indicates that the tax system is distortionary. Thus, an individual's monetary valuation of a project--the generational present value benefit--is not directly comparable to public money. That is exactly why generational benefits must be divided by their respective generation-specific MCF before they can be treated as public money to be discounted by the gross rate and then be directly compared with the present value of costs. Note that when [tau] = 0, gross and net rates of return are identical and the MCF is one for all generations so that the MCF-based approach becomes the traditional descriptive approach where only one market rate is relevant. In this case, no MCF concept is needed, and project benefits in a given period do not have to be disaggregated among various coexisting generations.

Our MCF-based generalized descriptive approach to multigeneration project evaluation has a salient feature. That is, it separates "rule parameters" (gross and net rates of return and generation-specific MCF) and "project parameters" (consumers' willingness to pay, direct project costs, and indirect revenue effects) in such a way that the rule parameters are project-independent.

4. Implications for the Social Discount Rate

The Concept of Social Discount Rate

Accepting our criterion, the emphasis on the SDR in the literature is misplaced because it presumes that a single rate exists that should be used to discount all project costs and benefits. A direct consequence of this misplaced emphasis is that the SDR must be project-specific, causing an implementation problem. In contrast, our approach treats costs and benefits differently and within- and between-generation benefits differently as well. As a result, only the marginal cost of public funds is required to evaluate public projects, which has the attractive feature of project independence.

Thus, the first major implication of our MCF-based approach for the SDR is that determining the size of the SDR in the descriptive approach is no longer necessary and the more meaningful concept is the MCF. Although gross and net market rates of return still carry with them important information for project evaluation, they are directly incorporated into the project evaluation rule rather than being used to calculate a weighted average rate of return. Nonetheless, our criterion can be used to calculate the appropriate SDR for any given project, and therefore, to derive the project-specific weights to be applied to the gross and net rates of return. (26) The following example illustrates this point. Consider a simplified world in which each generation lives for two periods and a two-period project in which the cost incurred in period zero is [DELTA][E.sub.0], the benefits (measured as willingness to pay) received in period one are [B.sup.0.sub.1] for each member of generation zero and [B.sup.1.sub.1] for each member of generation one, and there are no indirect revenue effects. (27) Assuming the two generations have the same number of people H and the same marginal cost of funds, by our criterion, the project should be accepted if and only if

(11) [[beta].sub.N]H[B.sup.0.sub.1]/MCF + [[beta].sub.G]H[B.sup.1.sub.1]/MCF - [DELTA][E.sub.0] > 0

which, assuming that [B.sup.0.sub.1] = [B.sup.1.sub.1] = B, can be written as

(12) 2HB/2MCF ([[beta].sub.G] + [[beta].sub.N] - [DELTA][E.sub.0] > 0.

We can relate our results to the SDR by noting that the usual criterion for project acceptance that uses the concept of the SDR can be expressed as

(13) 2HB/1 + SDR - [DELTA][E.sub.0] > 0.

To aid in the discussion, we define the discount factor associated with the social discount rate as

[[beta].sub.S] = [(1 + SDR).sup.-1].

From Inequalities 12 and 13, we have that

(14) [[beta].sub.S] = ([[beta.sub.G] + [[beta].sub.N]/2MCF).

Thus, for the special case of MCF = 1, the SDR discount factor is the simple average of the gross and net discount factors. More generally, the SDR discount factor depends on the value of MCF associated with the marginal financing.

The conventional wisdom is that the gross return forms an upper bound for the SDR and the net return a lower bound. From Equation 14 it is apparent that the appropriate SDR discount factor is not bounded from above by the net discount factor or from below by the gross discount factor. Specifically,

(15) SDR > r or [[beta].sub.S] < [[beta].sub.G] if MCF > [[beta].sub.G + [[beta].sub.N]/2[[beta].sub.G] > 1,

SDR < r(1 - [tau]) or [[beta].sub.S] > [[beta].sub.N] if MCF < [[beta].sub.G] + [[beta].sub.N]/2[[beta].sub.N < 1.

Thus, even for the simple project considered here, the SDR could be larger than the gross rate of return or smaller than the net rate of return, depending on the MCF associated with the marginal tax instrument. (28)

The Typical Magnitude of Social Discount Rate

As has been explained above, the magnitude of SDR--for the old method that does not distinguish between costs and benefits in terms of discounting and the use of the MCF--depends on both the cost-benefit structure of a project and the tax instrument used at the margin. Then, what can we say about the likely magnitude of a "typical SDR"? Because the MCF of a typical tax instrument tends to be larger than one (see the next section for a discussion of the value of MCF) and multigeneration public projects usually produce costs in earlier stages and capture consumption benefits later on, we can obtain the second major implication of our results for the SDR: The SDR as traditionally understood is typically larger than the gross rate of return. This point can be most clearly understood with an example. Suppose a generational project, if undertaken, produces a (present value) benefit of B for each future generation (t = 1, 2, ...) but no benefit for currently living generations (t = -D, -D + 1, ..., 0). The project requires an immediate one-time investment [DELTA][E.sub.0] and has no indirect revenue effects. Assume all generations are the same size H and all future generations have the same MCF. Then, according to our criterion, the project should be accepted if and only if

HB/MCF [[infinity].summation over (t = 1)] [[beta].sup.t.sub.G] > [DELTA][E.sub.0].

Comparing this with the rule using the concept of SDR,

HB [[infinity].summation over (t = 1)] [[beta].sup.t.sub.S] > [DELTA][E.sub.0],

it is clear that MCF > 1 implies [[beta].sub.S] < [[beta.sub.G] or SDR > r. Thus, even though most discussions of the social discount rate seem to suggest that the gross rate of return would be too high for intergenerational discounting, our criterion suggests that, in most cases, the appropriate project specific SDR may be even larger than the gross rate of return. (29)

Many hold the view that applying any positive discount rate ignores the long run and that a change in value judgment concerning intergenerational equity embedded in an SWF would naturally alter the criterion for multigeneration project evaluation. To the contrary, our criterion suggests that the discount rate for both future costs and generational benefits should always be the gross rate of return, a number significantly larger than zero. (30) Indeed, the discount rate as traditionally understood (i.e., when a difference between costs and benefits with respect to the use of MCF is not considered) may well be larger than the gross rate if the MCF is larger than unity and if costs come earlier than benefits. Moreover, value judgments concerning intergenerational equity seem to have no role to play in our criterion.

The above observations do not mean that the long run is ignored under our criterion or that the criterion represents an intergenerational policy that is insensitive to ethical considerations. The high discount rate suggested by our criterion emphasizes the opportunity cost of future public-good provision (such as environmental protection) and that a better way to help future generations might be to leave them less debt. (31) Similarly, though the SWF value judgment does not show up in our criterion, it may still (and must) play a role in intergenerational equity. Our analysis merely separates project evaluation from debt management, with the latter dealing with the equity issue across generations.

5. Parameter Estimation and Extensions

Discussion of Parameter Values

The application of our project evaluation criterion requires estimates of the "rule parameters" including the marginal productivity of capital (gross rate of return), the consumer's net rate of return and generation-specific MCF, in addition to the "project parameters," namely, project-specific cost and benefit estimates. (32) In the following, we discuss the appropriate values for [tau], r, and MC[F.sub.t]. The capital income tax rate must reflect levies on corporations at the federal, state, and local levels as well as taxes on personal income. Feldstein (1998) suggests that the total corporate income tax rate is about 40% and that an appropriate estimate of the personal income tax rate is 20%, which together implies a tax rate on capital income [tau] = 52%.

The risk premium cum gross rate of return has variously been estimated to be between 8.5% and 9.3%. For example, Poterba (1999) estimates the real rate of return on nonfinancial corporate capital as 8.5% between 1959 and 1996. Consistent with the theory in this article, however, the relevant gross rate of return should be risk-free. Investment assets that am closest to being risk-free are government bonds. For the purpose of this article, long-term government bonds are most appropriate. The Trustees of the Social Security system currently use 2.8% for their long-term estimate of real returns (6.3% of special-issue U.S. Treasury obligations minus 3.5% long-term rate of inflation). We adjust it upward to incorporate the corporate income taxes (at a rate of 40%) to arrive at the risk-free gross rate of return r = 4.7%. Hence, the risk-free net rate of return is r(1 - [tau]) = 2.3%.

Using Expression 9 to estimate MC[F.sub.t] requires converting the gross return (both r and [[beta].sub.G]) and the propensity to save ([[gamma].sub.t]) to their generational equivalent. As an approximation, we convert the annual rate of return to a generational return by calculating the rate of return in effect from midpoint to midpoint of a generation's span. With a span of 30 years and a 4.7% annual rate of return, the generational equivalent rate of return is (1.047) (30) - 1 = 2.97. In a two-period Cobb-Douglas utility function with first- and second-period consumption shares of two-thirds and one-third, the propensity to save is 0.333 (Atkinson and Stiglitz 1980). Substituting these parameter values and [tau] = 0.52 into Expression 9 yields a marginal cost of funds MCF = 1.15. To see how sensitive the marginal cost of funds is to the values of the annual gross rate of return and the generational propensity to save, we present Table 1 with the first row showing the alternative values for the risk-free annual gross rate of return and the first column showing the alternative values for the generational propensity to save.

Extensions to More Complicated Tax Structures

We have so far considered a simple yet meaningful tax structure to finance multigeneration projects where the dictator makes intergenerational transfers through time-varying head taxes given a preexisting tax on capital income. However, what if the preexisting distortionary taxes take a more complicated structure and include consumption taxes or labor income taxes? (33) Moreover, what if head taxes are not available, or, even if available, the dictator also uses other distortionary taxes to make intergenerational transfers? The extension to allowing preexisting distortionary taxes to include (in addition to capital income taxes) labor income taxes and/or consumption taxes is straightforward. The only thing that must be changed is the expression for generation-specific MCFs. In addition to the savings effects, labor supply effects and consumption effects would also come to play in determining the magnitude of the MCF in this more complicated tax structure.

The extension to the situation in which the dictator also optimizes on some distortionary taxes, on the other hand, is nontrivial. As proved in Appendix A and used in the last section, the dictator optimizing on time-varying head taxes is equivalent to optimizing on generation-specific lump-sum taxes. Unfortunately, this nice property of head taxes is not shared by other taxes. However, if we further assume that the distortionary taxes to be optimized can be set on a generation-specific basis, then all the results derived in the last section go through with the only difference being the expression of generation-specific MCFs. (34)

6. Conclusions

By postulating a benevolent dictator who neither takes from nor gives anything to the economy but merely exists to maximize an arbitrary intergenerational social welfare function, we have derived a criterion for the evaluation of multigeneration projects. The fact that many multigeneration projects are evaluated and some implemented suggests governments apply some criterion in their evaluation. We derive a criterion that, if imposed on a government, will make its behavior consistent, at least on the margin, with the notion of our benevolent dictator. The criterion we have developed is intended to provide real decision-makers with an appropriate tool for evaluating multigeneration projects. It is important to keep in mind that governments are not, and cannot be, the benevolent dictator on whose behavior this work is based. First, governments, even benevolent ones, use resources. Second, governments make decisions based on the constraints and preferences of those in power. In this world of "public choice" economics, our criterion plays the role of a constitution that prevents an opportunistic government from accepting inefficient projects--projects that penalize one generation at the expense of another.

The beauty of our generalized descriptive approach is twofold. First, the parameters required for implementing our MCF-based criterion are project-independent compared with the project-specific SDR. Second, our MCF-based criterion is welfare function independent in that the pure rate of time preference inherent in a social welfare function plays no role in our criterion. We show that a criterion that discounts each generation's future benefits (to the generation's starting period) at the consumer's rate of interest and adjusts each generation's present value benefit by that generation's marginal cost of public funds, and then discounts both these adjusted generational benefits and project costs at the opportunity cost of public investment, the marginal productivity of capital, will only select projects that have the potential to be Pareto improving.

Because the gross rate of return plays an important role in our evaluation procedure, a comparison of our article to Diamond and Mirrlees (1971) (hereafter D&M), which concluded that the public sector discount rate should equal the producer rate of return, seems appropriate. D&M derived their conclusion under one important condition: optimal (in the sense of the second-best) taxation that optimizes over a sufficiently rich set of tax instruments (say, a full range of commodity taxes). This assumption is regarded as quite restrictive, and most studies on the SDR, including this article, do not require or allow for optimal commodity taxation. On the conclusion side, the capital market tax-distortion, although present, plays no role in D&M's evaluation procedure: one does not need the consumer rate of return and concepts like the MCF and the indirect revenue effects, which are associated with the capital market distortion. On the other hand, the capital market distortion presents itself prominently in our evaluation procedure through the inclusion of both gross and net rates of return, the MCF, and the indirect revenue effects.

We conclude with some final comments concerning the debate on descriptive versus prescriptive approaches and how this article may inform the debate. The descriptive approach tries to simplify the task of project evaluation by separating it from equity concerns, leaving those concerns to be handled by redistributive tax policy. To achieve such separation, however, individualized lump-sum taxation must be available, which is unrealistic. (35) In contrast, head taxes (uniform lump-sum taxation) are often regarded as feasible (Wilson 1991, Sandmo 1998). This article takes advantage of a property of head taxes, proven in Appendix A, that time-varying head taxes are equivalent to generation-specific lump-sum taxes, even in the presence of other, distortionary taxes. As shown in this article, an MCF-based approach makes the separation of multigeneration project evaluation from concerns of intergenerational equity feasible. Nonetheless, one must keep in mind that using the descriptive approach to evaluate a project is only the first half of the job; the second half is to reoptimize debt policy with a focus on intergenerational equity concerns in response to project acceptance.

In comparison, the prescriptive approach, as it is currently applied, rejects the notion that government intertemporal tax/debt policy would be reoptimized in response to acceptance of a project. Thus, the assumption underlying this approach is that, with or without a multigeneration project, government tax policies remain the same. Therefore, a long-term project's merit should be judged by how it improves the existing status of intergenerational equity, in addition to its efficiency implications. However, in turning project costs and benefits into incremental consumption of each generation in a tax-distorted economy, the distinction between public and private money must still be made, and the concept of MCF still has a role to play.

To summarize, in resolving the "prescriptive versus descriptive" debate in the future, the focus should not be the discount rate or value judgment concerning intergenerational equity. Different approaches obviously require different discount rate(s), and the same across-generation value judgment can be embedded in both approaches. Instead, the focus should be to what extent a government's debt policy and its project policy can be coordinated and how to improve such coordination. We expect that the concept of MCF will prove very useful in such a reoriented debate.

Appendix A: Equivalence Between Head Taxes and Generation-Specific Lump-Sum Taxes

In this appendix we prove, in a fairly general framework, that time-varying heed taxes and generation-specific lump-sum taxes are equivalent instruments in the dictator's problem of choosing how to finance multigeneration projects (E, G) in the presence of taxes on capital income. Specifically, we show that for any set of head taxes {[h.sub.0], [h.sub.1], ...} that satisfies the intertemporal government budget constraint, there exists a set of generation-specific lump-sum taxes {[T.sup.-D], ...[T.sup.- 1],[[T.sup.0],[T.sup.1], ...} that also satisfies the government budget constraint and generates the same consumption pattern and welfare for each generation, and vice versa.

Suppose that a generation lives D + 1 periods (D [greater than or equal to] 1 and D = 1 corresponds to the two-period life cycle). At decision-making time (period zero), the only generations that are relevant to be included in cost-benefit analysis are those born in period -D and later. Suppose [H.sup.t] is the number of generation t (t [greater than or equal to] -D) individuals.

The dictator oversees financing a set of accepted public projects through debt management and tax collections. Government outlays required of and government services provided by these accepted projects can be cumulatively represented by (E, G). where E = [{[E.sub.t]}.sup.[infinity].sub.t=0] is the outlay stream of accepted projects and G = [{[G.sub.t]}.sup.[infinity].sub.t=0] is the service stream of these projects. The government's tax instruments are a capital income tax (at the rate [tau] and time-varying head taxes [{[h.sub.t]}.sup.[infinity].sub.t=0], or alternatively, the capital income tax and generation-specific lump-sum taxes [{[T.sup.t]}.sup.[infinity].sub.t=-D], where [T.sup.t] is collected from a generation t individual in period t or zero, whichever comes later.

Under time-varying head taxes {[h.sub.0], [h.sub.1], ...}, a generation t (t [greater than or equal to] -D) individual's lifetime (from period max[t, 0] on) budget constraint is

(A1) [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N] [C.sup.t.sub.k] = [a.sup.t.sub.max[t,0] + [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N] [w.sup.t.sub.k] - [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N][h.sub.k],

where [C.sup.t.sub.k] and [w.sup.t.sub.k] are a generation t individual's consumption and wage earnings in period k (max[t,0] [less than or equal to] k [less than or equal to] t + D), respectively, and [a.sup.t.sub.max[t,0]] is the individual's starting asset in period max[t,0]. Under generation-specific lump-sum taxes {[T.sup.-D], ...[T.sup.-1], [T.sup.0], [T.sup.1], ...}. a generation t (t [greater than or equal to] -D) individual's lifetime (from period max[t, 0] on) budget constraint is

(A2) [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N] [C.sup.t.sub.k] = [a.sup.t.sub.max[t,0]] + [t+D.summation over (k=max[t,0])] + [[beta].sup.k-max[t,0].sub.N] [w.sup.t.sub.k] - [T.sup.t].

For t [greater than or equal to] -D, let

(A3) [T.sup.t] = [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N] [h.sub.k].

Equation A3 defines a conversion mechanism from tax regime {[h.sub.0], [h.sub.1], ...} to tax regime {[T.sup.-D], ... [T.sup.-1], [T.sup.0], [T.sup.1], ...}. and vice versa. Obviously, when Equation A3 is satisfied, Equations A1 and A2 represent the same budget constraint to a generation t individual. Therefore, the consumption pattern and welfare is the same under both tax regimes for each generation.

To prove our equivalence result, what remained to be proven is that when the government budget constraint is satisfied under one tax regime, it is also satisfied under the other. For this purpose, it is sufficient to show that, when Equation A3 is satisfied, the present value of a generation t (t [greater than or equal to] -D) individual's lifetime tax payments--discounted to period max[t,0] at the gross rate of return--is the same under both tax regimes. We first need expressions for savings under both tax regimes.

Under tax regime {[h.sub.0], [h.sub.1], ...}, a generation t individual's savings in period k (where max[t, -] [less than or equal to] k [less than or equal to] t + D) is

(A4) [S.sup.t.sub.k](h) = [a.sup.t.sub.max[t,0][ (1 + [r.sub.N]).sup.k-max[t,0]] + [k.summation over (j=max[t,0])] [(1 + [r.sub.N]).sup.k-j] ([w.sup.t.sub.j] - [c.sup.t.sub.j] - [h.sub.j]).

Under tax regime {[T.sup.-D], ..., [T.sup.-1], [T.sup.0], [T.sup.1], ...}, a generation t individual's savings in period k (where max[t, 0] [less than or equal to] k [less than or equal to] t + D) is

(A5) [s.sup.t.sub.k] (T) = [a.sup.t.sub.max[t,0]] [(1 + [r.sub.N]).sup.k-max[t,0]] + [k.summation over (j=max[t,0])] [(1 + [r.sub.N]).sup.k-j] ([w.sup.t.sub.j] - [c.sup.t.sub.j]) - [T.sup.t] [(1 + [r.sub.N]).sup.k-max[t,0]].

As mentioned earlier, [c.sup.t.sub.j] in Equation A4 and [c.sup.t.sub.j] in Equation A5 are the same when Equation A3 is satisfied. Therefore,

(A6) [s.sup.t.sub.k](T) = [s.sup.t.sub.k](h) - [T.sup.t] [(1 + [r.sub.N]).sup.k-max[t,0]] + [k.summation over (j=max[t,0])] [(1 + [r.sub.N]).sup.k-j] [h.sub.j].

Under tax regime {[h.sub.0], [h.sub.1], ...}, the present value of a generation t (t [greater than or equal to] -D) individual's lifetime tax payments-discounted to period max[t, 0] at the gross rate of return--is

(A7) [t+D.summation over (k-max[t,0])] [[beta].sup.k-max[t,0].sub.G] [h.sub.k] + [t+D.summation over (k-max[t,0].sub.G] [[beta].sup.k+1-max[t,0].sub.G] [s.sup.t.sub.k](h)(r - [r.sub.N]).

On the other hand, under tax regime {[T.sup.-D], ..., [T.sup.-1], [T.sup.0], [T.sup.1], ...}, the present value of a generation t (t [greater than or equal] -D) individual's lifetime tax payment--discounted to period max[t, 0] at the gross rate of return--is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where the equality in Equation A8 is obtained using Equations A3 and A6.

To show that Equations A7 and A8 are equal, we simply the last term of the right side of A8 by changing the order of summations. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Replacing index j with k in the last expression of Equation A9 and substituting it into Equation A8, the right side of A8 becomes exactly A7. Therefore. for any generation t (t [greater or equal to] -D), the present value of a individual's lifetime tax payments--discounted to period max[t, 0] at the gross rate of return--is the same under both tax regimes. This concludes the proof of the equivalence of two tax regimes.

Appendix B: Derivation of Criterion 10

To make generation t (t [greater than or equal to] -D) individuals better-off with project ([DELTA]E, [DELTA]G), the additional lump-sum tax collected from a generation t individual, [DELTA][T.sup.t], must satisfy

(A10) [DELTA][T.sup.t] < [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N] [B.sup.t.sub.k],

where [B.sup.t.sub.k] (t [greater than or equal to] -D; t + D [greater than or equal to] k [greater than or equal to] max[t, 0]) is a generation t individual's willingness to pay, in units of contemporary numeraire goods, for [DELTA][G.sub.k].

On the other hand, [{[DELTA][T.sup.t]}.sup.[infinity][.sub.t = -D] must be sufficiently large so that the government budget constraint,

(A11) [[infinity].summation over (t=0)] [[beta][.sup.t.sub.G][E.sub.t] = [[infinity].summation over (t=0)] [[beta].sup.t.sub.G][R.sub.t] = A + [[infinity].summation over (t=-D)] [[beta].sup.max[t,0].sub.G] [H.sub.t] ([T.sup.t] + [t+D.summation over (k=max[t,0])] [[beta].sup.k+1-max[t,0].sub.G] [s.sup.t.sub.k]r[tau]),

remains satisfied (extra revenues are allowed) after the acceptance of the project ([DELTA]E, [DELTA]G), along with the additional financing [{[DELTA][T.sup.t]}.sup.[infinity].sub.t=-D]. That is,

(A12) [[infinity].summation over (t=0)] [[beta].sup.t.sub.G] ([DELTA][E.sub.t] - [DELTA][R.sub.t]) [less than or equal to] [[infinity].summation over (t=-D)] [[beta].sup.max[t,0].sub.G] [H.sub.t] (1 + [t+D.summation over (k=max[t,0])] [[beta].sup.k+1-max[t,0].sub.G]r[tau] [partial derivative] [s.sup.t.sub.k]/[partial derivative] [T.sup.t])[DELTA][T.sup.t],

where [DELTA][R.sub.t] is the change in revenue in period t caused by [DELTA]G. [DELTA]G has impacts on various generations' savings in various periods, which, in turn, generates revenue effects through the capital income tax. Therefore, although the project's investment requirement in period t is [DELTA][E.sub.t], its net revenue requirement is [DELTA][E.sub.t] - [DELTA][R.sub.t]. Inequality A12 says that the present value (discounted at the gross rate of return) of the additional tax payments caused by [{[DELTA][T.sup.t]}.sup.[infinity]][.sub.t=-D] must be no less than the present value of the net revenue requirements.

From Inequalities A10 and A12, acceptance of project ([DELTA]E, [DELTA]G), along with changes on generation-specific lump-sum taxes, can make everyone better-off if and only if

(A13) [[infinity].summation over (t=0)] [[beta].sup.t.sub.G]([DELTA][E.sub.t] - [DELTA][R.sub.t]) < [[infinity].summation over (t=-D)] [[beta].sup.max[t,0].sub.G][H.sub.t]/MC[F.sub.t] [t+D.summation over (k=max[t,0])] [[beta].sup.k-max[t,0].sub.N][B.sup.t.sub.k],

where

(A14) MC[F.sub.t] = [(1 + [t+D.summation over (k=max[t,0])] [[beta].sup.k+1-max[t,0].sub.G] r[tau] [partial derivative][s.sup.t.sub.k]/[partial derivative] [T.sup.t]).sup.-1], t [greater than or equal to] -D.

Obviously, Criterion A13 is Criterion 10 in Section 3.

Table 1. The Margin Cost of Funds by the Propensity to Save and the
Rate of Return

                      Risk-Free Annual Gross Rate of Return

Generational
Propensity to Save         3.0%      4.7%      6.0%

0.20                       1.07      1.08      1.09
0.33                       1.11      1.15      1.17
0.50                       1.18      1.24      1.27

(1) Rigorously speaking, there are no "single-generation" multiperiod projects because even projects extending over a short time present issues of distribution across generations that overlap in a continuous way. However, the earlier literature on the SDR discussed the discounting issue as if the costs and benefits of a project accrue to a single generation. On the other hand, more recent discussions of the SDR tend to explicitly address the intergenerational distribution of the costs and benefits of a project.

(2) This camp is represented by Ramsey (1969), Usher (1969), Sandmo and Dreze (1971), Harberger (1973), Pestieau (1975). Sjaastad and Wisecarver (1977), and Burgess (1988).

(3) This camp is represented by Marglin (1963a. b), Feldstein (1964), Bradford (1975), Mendelsohn (1981), and Lind (1982). The scale factor in this approach is often referred to as the opportunity cost of public investment. Because a key parameter in calculating the opportunity cost of public investment is the shadow price of capital, this approach to the SDR is olden referred to as the shadow price of capital approach.

(4) The distinction between discounting due to ethical considerations and discounting due to patience is emphasized in Schelling (1995).

(5) The terminology is due to the Intergovernmental Panel on Climate Change (IPCC). Discussions of these two approaches can be found in IPCC (1996) and Lind (1997).

(6) Advocates of the descriptive approach include Nordhaus (1994), Manne (1995). and Horowitz (2002), among many others.

(7) According to the Office of Management and Budget (OMB) (1992), 7% "approximates the marginal pretax rate of return on an average investment in the private sector in recent years" (Section 8: Discount Rate Policy).

(8) Studies applying the prescriptive approach include, among many others, Arrow and Kurz (1970) and Cline (1992). An important development in the prescriptive approach is Howarth (1996), which explicitly works with overlapping generations.

(9) In practice, however, the CBO equalizes the SRTP to the consumer rate of interest, as the 2% SRTP comes from a study of rates of return on government Treasury bills (Hartman 1990). (10) For example, one confusion due to not fully understanding the logic of the two approaches is believing that using the prescriptive

approach tends to generate relatively low discount rates and thus favor more spending on environmental protection. (11) How the standard Samuelson Rule should be modified to incorporate the "distortionary effects" of the second-best taxation has been studied extensively in the considerable literature on the marginal cost of public funds (MCF). For example, beginning with Stiglitz and Dasgupta (1971) and Atkinson and Stern (1974), Wildasin (1984), Browning (1987), Mayshar (1991), Ballard and Fullerton (1992), Ahmed and Croushore (1996), and Snow and Warren (1996) studied the role of public-good feedback effects in determining the MCF of a distortionary tax; King (1986), Batina (1990), Kaplow (1996), Allgood and Snow (1998), Dahlby (1998), Sandmo (1998), Ng (2000), and Slemrod and Yitzhaki (2001) investigated the cost--benefit rule in economies with heterogenous agents. Noticeably, however, how MCF should be defined and used in an intertemporal setting, with one exception, has not been addressed in the literature. Liu (2003) established a multiperiod project evaluation rule that is based on the concept of MCF and shed some light on the weighted average approach and the shadow price of capital approach to the SDR. However, the model in Liu (2003) is the one of a representative forever-living agent, which cannot be used to study the discounting issues that arise in a multigeneration context.

(12) This world can be thought of as an economy with linear production technology where the only technological progress is of the labor augmenting type. We sacrifice completeness of analysis by adopting a partial equilibrium framework to locus on the discounting aspect of project evaluation. Most analytical studies on the marginal cost of funds use a partial equilibrium framework (e.g., see Atkinson and Stern 1974. Wildasin 1984, Browning 1987, Dahlby 1998, and Sandmo 1998). A noticeable exception is Mayshar's (1991) formula for MCF in a general equilibrium framework. See Wildasin (1988) for implications of general equilibrium analysis for project evaluation.

(13) Note that in the actual economy there are likely to be many different capital income taxes confronting different individuals. This analysis sidesteps that issue by making everyone in a generation the same. On the other hand, this problem was prominent in some early discussions on the issue of discounting public investments, for example, in Krutilla and Eckstein (1958) and Bradford (1970).

(14) General lump-sum taxes as opposed to individual lump-sum taxes have been studied in static optimal taxation models (e.g., Wilson 1991, Sandmo 1998).

(15) Saving can take two forms: government bonds and productive capital, which, in this model, are perfect substitutes from the consumer's point of view.

(16) The existence of a wedge between gross and net rates of return due to taxes on capital incomes gives rise to the so-called Baumol Paradox (Baumol 1968), which has been at the center of the social discount rate debate.

(17) We do not introduce the complication of uncertain future costs and benefits. One can imagine that explicit consideration of uncertainty can be eliminated by using certainty equivalents.

(18) The marginal (or small) project approach to cost--benefit analysis is emphasized by Dreze and Stem (1987) and followed by all studies in the marginal cost of funds literature and most studies in the SDR literature. One notable exception in the SDR literature seems to be Quirk and Terasawa (1991), in which project evaluation is viewed as a problem of choosing the projects with highest internal rates of return given the government budget.

(19) One can think of AE and AG as the cost and benefit stream of the project, respectively. On the other hand, note that our concept of a "project" is general enough to include a proposal to reduce certain public services and recover some cash as a project. In this case. both [DELTA]E and [DELTA]G would be negative vectors, and we would call -[DELTA]E the benefit stream and -[DELTA]G the cost stream.

(20) We recognize the difficulty of evaluating the benefits of public-good provision in terms of willingness to pay, especially in a multigenaration context, because future benefits may be evaluated by future generations with different value systems from those we have known. However, discussing implications of value system variation is beyond the scope of this article. Including indirect revenue effects in the measure of a project's benefits may not naturally occur to many economists because they do not seem to be associated with any particular individual. However, in the presence of distortionary taxation, individuals' behavioral response to changes in government service will cause revenue changes even under the unchanged parameter values of the tax system. These indirect revenue effects have welfare implications because their existence implies that the net revenue requirements of a project may be higher or lower than what are estimated from a purely physical point of view.

(21) Generation t = -1 is still alive in period zero and should receive some benefits from [DELTA][G.sub.0]. But as we mentioned earlier, to simplify notation, we have initially assumed that all impacts of a project are on the newborn and yet-to-be-born generations. Generalizations will be given in the next subsection.

(22) In the static literature of the marginal cost of funds, two alternative approaches have been proposed to address the "feedback revenue effects" of public projects. The first links the MCF itself to the feedback revenue effects of public projects. As a result, the MCF varies, even for the same tax instrument, with the nature of public projects. Correspondingly, the benefits of public projects consist only of consumers" willingness to pay. Wildasin (1984) was among the first to emphasize the revenue effect from public-good provision oft the measure of the MCF. Ballard and Fullerton (1992) and Snow and Warren (1996) adopt this version of MCF. The second school of thought, as represented in Mayshar (1991) and adopted by Dahlby (1998), Sandmo (1998) and Liu (2003), suggests that any project-specific revenue effects should be measured and treated as project benefits. Thus, there is a unique MCF, calculated holding public-good provision constant, for any given tax instrument. We adopt the second approach to nonseparable public-good provision, as the main concern here is to establish a project-independent evaluation criterion.

(23) [DELTA]G affects [R.sub.t], (tax revenue collected in period t) only through its effect on [s.sup.t-1, the latter being a function of [DELTA][G.sub.t-1] and [DELTA][G.sub.t]. [DELTA][R.sub.0] = 0 because [s.sup.-1] has already been decided by the time ([DELTA]E, [DELTA]G) is evaluated, and therefore is not affected by [DELTA]G. Note that in the definition of [DELTA][R.sub.t]. the indirect revenue effect caused by the project, the thought experiment here is to change the government outputs without changing tax parameters.

(24) It may appear that the value of the MCF of the generation-specific lump-sum taxation would depend on its timing. The emphasis here is on the fact that an increase in the first-period lump-sum tax induces a reduction in first-period saving and therefore a decrease in second-period capital income tax. On the other hand, if the second-period, rather than the first-period, lump-sum tax is increased, there would be an increase in first-period saving (to help finance the increased tax in the second period). Would this suggest that the MCF for the second-period lump sum tax be less than one? The answer is "no" because the increase in the second-period lump sum tax, plus the increase in the second-period capital income tax due to increased saving, is discounted to the first period at the gross rate of return, whereas the present value loss to the individual from the increase in the second-period lump sum tax is calculated by discounting it at the net rate of return, Indeed, using a similar procedure to that in Appendix A, one can show that the MCF for the first-period lump sum tax is exactly the same as the MCF for the second-period lump sum tax.

(25) The exact expressions of [B.sup.t.sub.k], [DELTA][R.sub.t], and MC[F.sub.t] will be different from their counterparts in a two-period life cycle.

(26) This by itself is important because although it has long been recognized that the SDR should be project-specific, there has been no systematic way to calculate the SDR for each and every project at hand.

(27) Imagine this simple two-period project to be a national flu-shot program. The current investment is [DELTA][E.sub.0], in terms of vaccine and administration costs, which will generate health benefits next year. In the next year, the then old generation receives direct immunization benefits ([B.sup.0.sub.1] for each member) and the then newborn generation receives indirect benefits from others' immunization ([B.sup.1.sub.1] for each member).

(28) Though MCF > 1 is probably what most people would expect of a distortionary tax instrument, MCF < 1 is also possible. See Ballard and Fullerton (1992) and Sandmo (1998) for a discussion of situations where the MCF goes below unity.

(29) As we emphasized in the last subsection, the appropriate SDR to use is project-specific. As a result, there could be no definite general conclusion concerning the likely magnitude of the SDR. Our claim that the SDR for a typical project (which generates benefits after incurring costs) under typical financing (which has a MCF larger than one) is larger than the gross rate of return is only suggestive. In particular, we have assumed in our illustrative example that the revenue impact of a project's outputs is zero. Allowing nonzero revenue effects may significantly complicate the matter. In addition, the SDR discussed here is intended to be applied only to project costs and generational benefits, not to within-generation benefits. The future benefits accrued to any given generation should still be discounted to the first period of the generation's life at the net rate of return.

(30) Note that the differential treatment of costs and generational benefits is made through the use of MCF, not through differential discounting.

(31) It would be inefficient to evaluate environmental protection projects with a discount rate smaller than what is implied by the criterion of this article and at the same time leave future generations with tax burdens in the form of both explicit government debts and implicit Social Security and Medicare unfunded obligations. See Auerbach, Kotlikoff, and Leibfritz (1999) and Kotlikoff (2002) for detailed discussions of the intergenerational inequality in tax burden distribution.

(32) One of the main implications of our project evaluation rule is that the revenue consequences of increments to government service should be incorporated into a project's evaluation as terms offsetting project costs. This would require project-specific information above and beyond the conventional cost and benefit measurements and may impose significant challenge in the application of our method.

(33) In the latter case, our basic model must be extended to allow variable labor supply.

(34) For a demonstration of this point, consult an early version of this article, see Liu, Rettenmaier, and Saving (2002).

(35) Christiansen (1981), among others, studied conditions under which the conventional cost-benefit criterion using no distributional weights continues to be valid even in the absence of lump-sum taxes. However. those conditions, usually about individuals' preferences, tend to be ad hoc and very restrictive.

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Liqun Liu, * Andrew J. Rettenmaier, ([dagger]) and Thomas R. Saving ([double dagger])

* Private Enterprise Research Center. Texas A&M University. College Station, TX 77843-4231, USA; E-mail: [[e-mail].

([dagger]) Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231. USA: E-mail: [e-mail].

([double dagger]) Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231, USA: E-mail: [e-mail]; corresponding author.

We want to thank David Bradford, Jonathan Hamilton, and several anonymous referees for very helpful comments and suggestions. We also thank The Lynde and Harry Bradley Foundation and the National Center for Policy Analysis (NCPA) for financial support.

Received August 2000; accepted January 2004.