Discounting according to output type.

• Author: Liu, Liqun

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...