Discounting according to output type.

Saving, Thomas R.

1. Introduction

Although the importance of discount rate choice for multiperiod public project evaluation is well understood, the theoretical literature on the appropriate discount rate is inconclusive. Specifically, what the discount rate should be when corporate taxes and personal income taxes create a wedge between gross (before tax) and net (after tax) rates of return is unresolved. (1) For example, Baumol (1968), Usher (1969), Sandmo and Dreze (1971), and Harberger (1973) suggested a "weighted average" approach, in which they envisioned that the so-called "social discount rate" was a weighted average of gross and net rates of return, with weights determined by the fractions of resources drawn from consumption and private investment, respectively. For the special case in which the gross rate of return (the marginal productivity of capital) is exogenous, public investment would come entirely from the displacement of private capital, and the social discount rate equals the gross rate of return. Diamond and Mirrlees (1971) also concluded that the public sector discount rate should equal the gross return. However, Diamond and Mirrlees made an important assumption that the tax system is optimized (in the sense of the second best) over a sufficiently rich set of tax instruments (say, a full range of commodity taxes), an assumption that is not satisfied by most studies on the public sector discount rate. On the other hand, Marglin (1963), Feldstein (1964), Bradford (1975), and Lind (1982) proposed a "shadow price of capital" approach, in which they insisted that future benefits be discounted at the net rate of return, but that costs be multiplied by a scale factor first, and then the resulting consumption equivalent costs also be discounted at the net rate. (2)

Reflecting this lack of theoretical consensus, U.S. federal government oversight agencies, in setting their respective discount rate policies, do not seem to follow any particular theory. Two alternative discount rates have emerged as policy recommendations by these agencies (Lyon 1990). A high rate of 7%, approximating the marginal productivity of capital, has been adopted by the Office of Management and Budget, while a low rate of 2-3%, approximating the federal borrowing rate, has been adopted by the General Accounting Office, the Congressional Budget Office, and the U.S. Water Resources Council. These differential discount rate policies could be inconsistent and biased in favor of a certain project on which the lower discount rate is applied.

This paper provides a justification for the coexistence of different discount rates by considering alternative types of project outputs. We focus on two special types of public project outputs: publicly provided goods that are perfect substitutes for private goods and publicly provided goods that enter individual utility functions in a separable fashion. Using a simple multiperiod project evaluation framework, we show that the appropriate discount rate for a public project depends on the nature of a project's outputs. If a project's outputs are perfect substitutes for private goods, future benefits should be discounted by the gross (before tax) rate of return. However, if a project produces utility-separable outputs, future benefits should be discounted by the net (after tax) rate of return.

2. A Simple Project Evaluation Framework with Two Types of Project Outputs

Since our focus is efficiency rather than inter- or intragenerational equity, we take the liberty of considering an economy consisting of identical, forever-living individuals. In each period, individuals consume a composite private good (treated as numeraire for each period) and two types of publicly provided goods. The first type is a perfect substitute for same-period private goods, and the second type enters individual utility functions in a weakly separable way. Therefore, individual utility functions can be written as

v(c, Q, G) [equivalent to] f [u(c + Q), G], (1)

where c = ([c.sub.0], [c.sub.1], ...), Q = ([Q.sub.0], [Q.sub.1], ...), and G = ([G.sub.0], [G.sub.1], ...) are, respectively, infinite dimensional vectors of private good consumption and type-one and type-two publicly provided goods.

For given Q and G, an individual's utility maximization problem can be stated as

max v(c, Q, G)

s.t. [a.sub.t] = [a.sub.t-1] (1 + [r.sub.n]) + [y.sub.t] - [L.sub.t] - [c.sub.t], t = 0,1, ..., (2)

where [a.sub.t] is the end-of-period t per capita wealth (including both productive capital and government bonds) after wage and interest income (net of taxes) have been earned and consumption has occurred, [r.sub.n] is net (after tax) rate of return on productive capital, and [y.sub.t] and [L.sub.t] are, respectively, period t wage earnings and lump-sum taxes. We assume a world of certainty, so that the after-tax bond interest rate is also [r.sub.n]. Let [a.sub.-1] be individual wealth one period before the decision making period (which is period 0), so that individual initial wealth is A = [a.sub.-1] (1 + [r.sub.n]).

The government collects lump-sum taxes and taxes on capital income to finance a preaccepted set of multiperiod public projects. These projects combined require a stream of (per capita) investments represented by the vector I = ([I.sub.0], [I.sub.1], ...) and generate a stream of two types of output, Q and G. Revenue per capita collected in period t is

[R.sub.t] = [b.sub.t-1]([r.sub.b] - [r.sub.n]) + [k.sub.t-1] ([r.sub.g] - [r.sub.n]) + [L.sub.t] (3)

where [b.sub.t-1] and [k.sub.t-1] are, respectively, end-of-period t - 1 per capita government debt and private capital, and [r.sub.b] and [r.sub.g] are, respectively, the before-tax interest rate on bonds and the marginal productivity of capital, that is, the gross (before tax) rate of return.

Since government can borrow, [R.sub.t] and [I.sub.t] need not be equal. Government debt per capita at the end of period t, [b.sub.t], evolves according to

[b.sub.t] = [b.sub.t-1] (1 + [r.sub.b]) + [I.sub.t] - [R.sub.t], t = 0, 1, ..., (4)

where [b.sub.-1] is inherited per capita debt. Individual wealth consists of government bonds and productive capital, so that

[a.sub.t-1] = [b.sub.t-1] + [k.sub.t-1]. (5)

Substituting Equations 3 and 5 into Equation 4, the government's intertemporal budget constraints can be rewritten as

[b.sub.t] = [b.sub.t-1] (1 + [r.sub.g]) + [I.sub.t] - [L.sub.t] - [a.sub.t-1] ([r.sub.g] - [r.sub.n]), t = 0,1, ..., (6)

which are equivalent to (3)

B + [[infinity].summation over t=0] [I.sub.t]/ [(1 + [r.sub.g]).sup.t] = [[infinity].summation over t=0] [L.sub.t] + [a.sub.t-1] ([r.sub.g] - [r.sub.n])/ [(1 + [r.sub.g]).sup.t] (7)

where B = [b.sub.-l] (1 + [r.sub.g]).

At the beginning of period 0, the government is committed to the preaccepted public sector production plan (I, Q, G). The concern of this paper is how a new, marginal project, denoted as ([DELTA]I, [DELTA]Q, [DELTA]G), is evaluated such that the representative individual experiences an increase in welfare. (4)

3. Differential Discount Rates Based on Project Output Type

According to the shadow price principle, the evaluation of a marginal project ([DELTA]I, [DELTA]Q, [DELTA]G) that is consistent with welfare maximization requires shadow prices for [I.sub.t], t [member of] {0, 1, ...}, [Q.sub.t], t [member of] {0, 1, ...}, and [G.sub.t], t [member of] {0, 1, ...}. Denoting these shadow prices, respectively, as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a project that is generally represented by ([DELTA]I, [DELTA]Q, [DELTA]G) enhances welfare if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

In the following paragraphs, we focus on the relationship within, as well as among, these three sets of shadow prices. Our approach to the discount rate issue is based on the relationship among shadow prices in different periods. Since the shadow price of a public sector input (output)--here, [I.sub.t] represents public sector inputs, whereas [Q.sub.t] and [G.sub.t] are public sector outputs--is defined as the welfare effect of a unit increase in the net demand (supply) of that input (output) by the public sector, we must first know how individual welfare is affected by various fiscal parameters representing taxation and public sector inputs and outputs.

Expressing the constraints in Equation 2 equivalently as (5)

[[infinity].summation over t=0] [c.sub.t]/ [(1 + [r.sub.n]).sup.t] = A + [[infinity].summation over t=0] [y.sub.t] - [L.sub.t]/[(1 + [r.sub.n])].sup.t], (9)

the Lagrange function of the individual's problem is

[psi] = v(c, Q, G) + [lambda] [A + [[infinity].summation over t=0] [y.sub.t] - [L.sub.t] - [c.sub.t]/ [(1 + [r.sub.n]).sup.t]].

Optimization requires

[differential]v/[differential[c.sub.t] = [lambda]/ [(1 + [r.sub.n]).sup.t], t = 0, 1, ..., (10)

where [lambda] is the marginal utility of initial wealth A. Noting the weak separability of G in v and letting L = ([L.sub.0], [L.sub.1], ...) be the infinite dimensional vector of lump-sum taxes, we can express the solution to the individual's problem as [c.sub.t](Q, L), t = 0, 1, ..., so that, from Equation 2, we have [a.sub.t](Q, L), t = 0, 1, .... (6) Denoting the value function (indirect utility) resulting from the individual's maximization problem as M(Q, G, L), we have, from the Envelope Theorem,

[differential]M/[differential] [Q.sub.t] = [differential]v/ [differential][Q.sub.t] = [differential]v/[differential][c.sub.t], [differential]M/[differential][G.sub.t] = [differential]v/ [differential][G.sub.t], [differential]M/[differential][L.sub.t] = -[lambda]/ [(1 + [r.sub.n]).sup.t]. (11)

For a preexisting public sector production plan (I, Q, G) to be fiscally feasible, the stream of lump-sum taxes L must satisfy the government intertemporal budget constraint (Eqn. 7). When consideration is given to adding a new marginal project ([DELTA]I, [DELTA]Q, [DELTA]G) to the public sector's production plan, one of the [L.sub.t]'s or some combination of them must be increased to satisfy the government budget constraint. The particular values of these shadow prices depend on the form of marginal financing and the measuring unit. (7) Assuming [L.sub.i], i [member of] {0, 1, ...} is increased to meet any additional project revenue requirement and using the numeraire good in period 0 as the measuring unit, we can calculate the three sets of shadow prices as follows:

First, the impact of a marginal increase in [I.sub.t] on [L.sub.i] is obtained by differentiating both sides of Equation 7 with respect to [I.sub.t], taking [L.sub.i] as a function of [I.sub.t]. The impact of a marginal increase in [Q.sub.t] and [G.sub.t] on [L.sub.i] can be obtained in a similar way. Therefore, we have, for any t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where

[OMEGA] = [(1 + [r.sub.g]).sup.-i] + [[infinity].summation over (j=0)] ([r.sub.g] - [r.sub.n]) [differential][a.sub.j-1]/[differential] [L.sub.i][(1 + [r.sub.g]).sup.j].

Note that [differential][L.sub.i]/[differential][G.sub.t] = 0, since utility-separability implies that G does not directly affect individual consumption decisions and therefore also leaves unaffected the government budget constraint (Eqn. 7).

The shadow prices of [I.sub.t], [Q.sub.t], and [G.sub.t] measured in terms of the numeraire good in period 0 when the marginal financing features an increase in [L.sub.i] are, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The last equality in each of the equations of Equation 13 is obtained by substituting relevant expressions from Equations 10, 11, and 12. In Appendix II, we prove the following lemma:

LEMMA. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The lemma indicates that the welfare effect of increasing government's investment in a period is equivalent to reducing the publicly provided private good in the period by the same amount. This result is intuitive for the following reason. Note first that the lemma is the same as saying that there is no net welfare effect by increasing both [I.sub.t] and [Q.sub.t] by one unit. With no loss of generality, suppose that the one-unit increase in [I.sub.t] is realized through an increase in period t lump-sum tax, [L.sub.t]. By how much should [L.sub.t] be increased? The answer is exactly one unit because, along with the one-unit increase in [Q.sub.t], the one-unit increase in [L.sub.t] will cover the one-unit increase in [I.sub.t] without generating revenue externalities in any other periods. Of course, increasing both [L.sub.t] and [Q.sub.t] by one unit does not have any net welfare effect since both are perfect substitutes for income.

Using the lemma and the shadow price expressions of Equation 13, we can see how future costs and benefits of public projects should be discounted, with an emphasis on the type of project outputs.

The [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] expression in Equation 13 indicates that the ratio of the shadow price of public investment in period t+1 to that in period t is [(1 + [r.sub.g]).sup.-1]. This implies that the appropriate discount rate for future project costs is the gross rate of return. The appropriate discount rate for future project benefits depends on the nature of project outputs. Specifically, if a project's outputs are perfect substitutes to private consumption goods, then project benefits in period t--directly measured by project outputs [DELTA][Q.sub.t] because they are perfect substitutes for period t private consumption [c.sub.t], which is the period's numeraire--should be discounted at the gross rate of return. This is the implication of the lemma and the above conclusion that the gross rate of return is right for the discounting of future costs. In addition, the lemma implies that the project benefits [DELTA][Q.sub.t] can be "netted out" against same-period investments [DELTA][I.sub.t]. On the other hand, if a project's outputs are separable in individual utility functions, then project benefits, conventionally measured by individual willingness to pay ([differential]v/ [differential][G.sub.t]/[differential]v/[differential][c.sub.t]) [DELTA][G.sub.t], should be discounted at the net rate of return. This is the implication of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] expression in Equation 13. Since costs and benefits are discounted at different rates of return for a project that produces utility-separable outputs, project benefits cannot be netted out against same-period investments. Moreover, for a project producing utility-separable outputs, the present value of its costs must be multiplied by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] before it is compared with the present value of the project's benefits.

Another interesting implication of our analysis is that the government borrowing rate [r.sub.b] is irrelevant for discount rate choice, regardless of the type of project outputs. In our model, the borrowing rate is determined by the tax policy given to bond returns. For example, [r.sub.b] = [r.sub.g] if bond returns are taxed as heavily as the returns on physical capital, but [r.sub.b] = [r.sub.n] if bond returns are tax exempted. However, how much the government decides to tax bond returns, and therefore the borrowing rate, has no real implication for the intertemporal trade-off in either the individual budget constraint 9 or the government budget constraint 7.

The above results can be summarized into the following proposition:

PROPOSITION. Given the assumptions in our simple multiperiod framework for project evaluation, (i) future costs should be discounted at the gross rate of return; (ii) for a project that produces outputs that are perfect substitutes to private goods, future benefits should be discounted at the gross rate of return, and benefits can be netted out against costs in the same period; (iii) for a project that produces outputs that are (weakly) separable in the individual utility function, future benefits should be discounted at the net rate of return, and benefits cannot be netted out against costs in the same period; and (iv) government borrowing rate per se has no role to play in discount rate choice.

Some intuitive explanations for our main result, which is that the appropriate discount rate for a project's future benefits depends on the type of project output, may be helpful. First, if a project produces outputs that are perfect substitutes for private consumption in various periods, then it can be generally represented by ([{[DELTA][I.sub.t]}.sup.[infinity].sub.t=0], [{[DELTA][Q.sub.t]}.sup.[infinity].sub.t=0]). Now, let us compare this target project with a hypothetical project that has the same output stream as the target project and an investment stream that is identical to the output stream, which can be represented by ([{[DELTA][Q.sub.t]}.sup.[infinity].sub.t=0], [{[DELTA][Q.sub.t]}.sup.[infinity].sub.t=0]). For the same intuitive argument we gave after the lemma, the hypothetical project is welfare-neutral. The only difference between the target project and the hypothetical project is that they have different investment streams -[{[DELTA][I.sub.t]}.sup.[infinity].sub.t=0] for the target project and [{[DELTA][Q.sub.t]}.sup.[infinity].sub.t=0] for the hypothetical project. Between the target and the hypothetical projects, the one with the lower present value of investment stream at the gross rate of return (the discount rate in the government budget constraint) is welfare-dominant. In other words, the target project is welfare enhancing if and only if [[infinity].summation over (t=0)] [DELTA][Q.sub.t][(1 + [r.sub.g]).sup.-t] > [[infinity].summation over (t=0)] [DELTA][I.sub.t] [(1 + [r.sub.g]).sup.-t]. Therefore, for a project that produces perfect substitutes for private consumption, its future benefits [{[DELTA][Q.sub.t]}.sup.[infinity].sub.t=0] should be discounted at the gross rate of return. Second, if a project produces outputs that are separable in the utility function, then it can be generally represented by ([{[DELTA][I.sub.t]}.sup.[infinity].sub.t=0], [{[DELTA][G.sub.t]}.sup.[infinity].sub.t=0]). Outputs [{[DELTA][G.sub.t]}.sup.[infinity].sub.t=0] do not have any effects on individual behaviors and, as a result, leave unaffected tax revenues in each and every period. Therefore, the only welfare effects from these project outputs are an individual's willingness to pay in various periods, ([differential]v/[differential][G.sub.t]/[differential]v/ [differential][c.sub.t]) [DELTA][G.sub.t] in period t in terms of units of [c.sub.t]. Optimization by the individuals means that the intertemporal trade-off of private good is the net-of-tax interest rate [r.sub.n]. Therefore, for a project that produces utility-separable outputs, its future benefits [{([differential]v/[differential][G.sub.t]/[differential]v/ [differential][c.sub.t]) [DELTA][G.sub.t]}.sup.[infinity].sub.t=0] should be discounted at the net rate of return.

The results in the proposition show that the discount rate for evaluating the benefits of a public project can vary systematically with the type of output it produces. Indeed, the results are so clear-cut that the appropriate discount rates are the gross and net rates of return, respectively, for the two output types considered in this paper. A query may arise as to which assumptions are responsible for the results being so clear-cut and how the results would be modified if restrictive assumptions were relaxed. (8)

The first major assumption we have made is that the outputs of any public project fall into one of two categories: outputs that are perfect substitutes for private goods and outputs that are separable in the utility function. While utility-separable publicly provided goods and publicly provided private goods are the two most often used specifications in literature, there also exist projects whose outputs cannot be classified into either of these two categories. What can be said about how future benefits from these more general project outputs should be discounted? Liu (2003) advocates an approach that proposes discounting all future benefits of a project--regardless of the type of the project's outputs--at the net rate of return and incorporating any effects on tax revenues from the project's outputs into analysis as negative costs in various periods. Therefore, the output type also matters in Liu (2003), but its effects are included in analysis through the revenue effects rather than manifested in differential discount rates. Since the approach in Liu (2003) can be applied to any type of project output, it is more general than the approach presented in our study. However, that more general approach would require project-specific information (on revenue effects) above and beyond the conventional cost and benefit measurements. By focusing on two specific types of project outputs, this paper shows how differential discount rates can coexist. Importantly, our approach requires only traditional cost-benefit information: project costs and consumer willingness to pay in various periods.

The second major assumption is that marginal projects are financed by the lump-sum tax along with government borrowing, whereas the capital income tax policy is taken as fixed. Although this assumption is typical in the social discount rate literature, Pestieau (1975) has shown that different tax structures may call on different discount rates. While the appropriate discount rates to use will not change with the tax structure or the marginal tax instrument in Liu (2003), the marginal cost of funds parameter in Liu's rule is determined by the tax structure and the marginal tax instrument. (9) Allowing a more complicated tax structure (that is, a distortionary tax system with both capital and labor income taxes, along with variable labor supply) alone would not alter our clear-cut quantitative results as long as marginal projects are financed by changes in lump-sum taxes along with government borrowing. However, marginal lump-sum tax financing is a critical assumption underlying our quantitative results. Nevertheless, our qualitative conclusion, which is that the type of public project outputs affects the discount rate for evaluating future benefits in a systematic way, remains the same.

The third major assumption is that the before-tax rate of return to capital is exogenous. This assumption is, of course, quite restrictive. However, it is very hard to deal with interest rate endogeneity in a multiperiod environment in which there is more than one interest rate. (10) Importantly, this simplifying assumption does not make the debate over the choice between the net and the gross rates disappear. For example, the weighted average approach would generate the gross rate of return as the correct discount rate when interest rates are exogenous, because public funds are obtained entirely from the displacement of private capital in this situation. On the other hand, the shadow price of capital approach would still recommend using the net rate of return for evaluating future benefits.

4. Conclusion

In this paper, we justify the use of different discount rates in evaluating government projects on the basis of the type of output produced. The marginal productivity of capital (the gross rate of return) is appropriate for discounting future benefits when project outputs are perfect substitutes to private consumption goods. In contrast, the net rate of return (which is also the government borrowing rate if bond yields are tax exempt) is appropriate for discounting future benefits if project outputs are utility-separable. The output type also has implications for whether project benefits can be netted out against same-period project costs.

Utility-separable publicly provided goods and publicly provided goods that are perfect substitutes for private goods are the two dominant specifications in the (static) literature of public good provision. A critical concept in that literature is the marginal cost of funds. It has been shown that the value of the marginal cost of funds for a given tax instrument depends on whether the public goods are separable or perfect substitutes for private goods. (11) Using a dynamic (multiperiod) framework, this paper shows that such a distinction between the two output types is also critical to the choice of the appropriate discount rate for the evaluation of a project's future benefits. Then, of course, there is a question of how to decide whether a particular project will generate a perfect substitute for private consumption or an output that is separable in the utility function. We suggest that if an output can also be bought by individuals from market, it can be regarded as a perfect substitute of private consumption. On the other hand, if an output does not seem to affect an individual's choices, such as savings, labor supply, and consumption (that is, outputs generated by space exploration projects), then it can be regarded as separable in the utility function.

Although this paper provides a justification for the coexistence of different discount rates by resorting to different types of project outputs, it does not provide a rationale for existing differences in government discount rates. (12) The two rates described as main alternatives in U.S. government practice are not the two rates identified in our theoretical analysis. Risk has a lot to do with the difference between the 7% high discount rate and the 2% low discount rate used by the federal oversight agencies. Only the differences due to taxes are relevant to the argument of this paper. Then, what are the risk-free gross and net rates of return, [r.sub.g] and [r.sub.n]? 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 inflation rate). To arrive at the risk-free gross rate of return, the 2.8% bond rate must be adjusted upward to incorporate the corporate income tax effect. The corporate income tax rate must reflect levies on corporations at the federal, state, and local levels. Feldstein (1998) suggests that the total corporate income tax rate is about 40%, implying a risk-free gross rate of return of [r.sub.g] = 4.7%. Using the 40% corporate income tax rate and the 20% personal income tax rate, the total levy on capital income is 52%. This, together with [r.sub.g] = 4.7%, implies that [r.sub.n] = 2.3%.

Appendix I: A Formal Demonstration of the Equivalence between the Government Budget Constraints 6 and 7

Consider the first t + 1 equations of those represented by Equation 6:

[b.sub.0] = [b.sub.-1](1 + [r.sub.g]) + [I.sub.0] - [L.sub.0] - [a.sub.-1] ([r.sub.g] - [r.sub.n]), [b.sub.1] = [b.sub.0](1 + [r.sub.g]) + [I.sub.1] - [L.sub.1] - [a.sub.0]([r.sub.g] - [r.sub.n]),

...

[b.sub.t-1] = [b.sub.t-2](1 + [r.sub.g]) + [I.sub.t-1] - [L.sub.t-1] - [a.sub.t-2] ([r.sub.g] - [r.sub.n]), [b.sub.t] = [b.sub.t-1](1 + [r.sub.g]) + [I.sub.t] - [L.sub.t] - [a.sub.t-1] ([r.sub.g] - [r.sub.n]).

First, divide the equation (both sides of it) beginning with [b.sub.0] by [(1 + [r.sub.g]).sup.0], the equation beginning with [b.sub.1] by [(1 + [r.sub.g]).sup.1].... Then, add the left sides and the right sides of the resulting equations, respectively. Noting that the intermediate b items ([b.sub.0], [b.sub.1], ..., [b.sub.t-1]) on the both sides of the final equation cancel out, we have

[b.sub.t]/[(1 + [r.sub.g]).sup.t] = [b.sub.-1] (1 + [r.sub.g]) + [t.summation over(k=0)] [[I.sub.k] - [L.sub.k] - [a.sub.k-1]([r.sub.g] - [r.sub.n])/ [(1 + [r.sub.g]).sup.k]].

Under the condition [lim.sub.t [right arrow] [infinity]] [b.sub.t] [(1 + [r.sub.g]).sup.-t] = 0 (no Ponzi Game), we have

[b.sub.-1](1 + [r.sub.g]) + [[infinity].summation over (k=0)] [[I.sub.k]/[(1 + [r.sub.g]).sup.k] = [[infinity].summation over (k=0)] [[L.sub.k] + [a.sub.k-1] ([r.sub.g] - [r.sub.n])/ [(1 + [r.sub.g]).sup.k],

which is Equation 7.

Appendix II: Proof of Lemma

From Equation 13 and using [differential] [a.sub.j-1]/[differential][Q.sub.t] = -[differential][a.sub.j-1]/ [differential][L.sub.t], we can see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A1)

To verify Equation A1, it is critical to express [differential][a.sub.j-1]/[differential][L.sub.i] and [differential][a.sub.j-1]/[differential][L.sub.t] in comparable terms.

Substituting the optimal levels of consumption and wealth into (individual) budget constraints 2 and differentiating both sides of each constraint with respect to [L.sub.i], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3)

Substituting Equations A2 and A3 into A1, the left-hand side of A1 becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A4)

From the integrated (individual) budget constraint 9, we have that

[differential][c.sub.h]/ [differential][L.sub.i] = [differential][c.sub.h]/[differential][L.sub.t] [(1 + [r.sub.n]).sup.t-i],

and A4 can be simplified to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A5)

which equals the right-hand side of Equation A1. QED.

We want to thank 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 February 2004; accepted November 2004.

(1) Intergenerational equity has also been used to justify the use of a social discount rate (SDR). For the SDR issue motivated by the consideration of equity across generations, see Intergovernmental Panel on Climate Change (1996) and Liu, Rettenmaier, and Saving (2004). However, the SDR issue that arises from the distortionary taxes on capital income is believed to be the most basic one and has drawn the most attention in the literature. The SDR issue in a tax-distorted closed economy is the concern of this paper.

(2) To obtain the scale factor in the shadow price of capital approach, a shadow price of capital must be calculated (hence, the name of the approach). Liu (2003) summarizes the practical difficulties of the existing views and proposes a new approach to the discount rate issue that is based on the (multiperiod) concept of the marginal cost of funds.

(3) The equivalence, which is formally established in Appendix I, holds under the "no Ponzi Game" condition that [lim.sub.t[infinity]] [b.sub.t] [(1 + [r.sub.g]).sup.-t] = 0. Note that this widely used condition does not require that government debt ever be fully paid off. It only requires that debt not permanently grow faster than the interest rate.

(4) This paper follows the tradition in the literature of cost-benefit analysis to focus on marginal project evaluation. See Dreze and Stern (1987) for an excellent demonstration of the basic theory and results in cost-benefit analysis.

(5) The equivalence between Equation 9 and the constraints in Equation 2 holds for the same reason and under the same (parallel) condition as the equivalence between Equations 7 and 6. See Footnote 3 and Appendix I.

(6) Strictly speaking, consumption in each period should also be a function of the vector of income y = ([y.sub.0], [y.sub.1], ...) and the interest rate [r.sub.n]. However, we do not include them as arguments since we treat them as exogenous in our simple framework.

(7) Shadow prices depend on both marginal financing and measuring unit. However, as long as the marginal financing is given, the evaluation result does not depend on the measuring unit, as the use of an alternative measuring unit would simply scale all shadow prices up or down by a common factor. We will use the numeraire good in period 0 as the measuring unit. For some interesting discussion of the implications of numeraire good choice for cost-benefit analysis in a heterogeneous environment, see Brekke (1997), Snow and Warren (1997), Dreze (1998), and Johansson (1998).

(8) We would like to point out that the result of the government borrowing rate being irrelevant to the discount rate choice does not hinge on any of the following assumptions. The reason is that how a government chooses to tax interest earnings on government bonds has no implications for the real economy.

(9) Another problem with the fixed capital market (tax) distortion in an economy with a representative individual is how the existence of the capital income taxes can be justified--see Wilson (1991).

(10) Note that previous studies allowing endogenous interest rates, such as those of Sandmo and Dreze (1971) and Harberger (1973), were all conducted in a two-period environment in which there was only one endogenous interest rate.

(11) For some well-known estimates of marginal costs of funds, see Stuart (1984), Ballard, Shoven, and Whalley (1985), and Browning (1987). For a reconciliation of divergent marginal cost of fund estimates that are based on the nature of public goods, see Wildasin (1984), Mayshar (1991), Ballard and Fullerton (1992), and Snow and Warren (1996).

(12) Also note that the discount rate required by each federal agency's discounting policy is supposed to be applied to all projects regardless of project type. Using one set of discount rates, rather than another, produces biases for some projects and biases against others.

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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 lliu@tamu.edu.

([dagger]) Private Enterprise Research Center, Texas A&M University, College Station, TX 77843-4231, USA; E-mail a-rettenmaier@tamu.edu.

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