The composition of public expenditure in a dynamic macro model of monopolistic competition.

Balvers, Ronald J.

I. Introduction

It is well known that market power causes firms to underproduce from a social welfare perspective. Widespread existence of monopolistic elements, as documented for example by Hall [14; 15] and Lebow [16], may provide an argument for government policy designed to increase production at the aggregate level. The literature on macro models of monopolistic competition, e.g., Rotemberg [26], Blanchard and Kiyotaki [10], and others finds that changes in the money stock in principle do not affect aggregate production; a further source of market imperfection in the form of a nominal rigidity is necessary to allow changes in the money stock to generate production effects.

Another branch of literature considers public expenditure policy rather than monetary policy. Aiyagari, Christiano, and Eichenbaum [1] most recently, and others, such as Barro [6], examine a perfectly competitive economy where public expenditure stimulates output, possibly with a multiplier effect, via its effect on the marginal utility of wealth and labor supply. Starting from Pareto optimality such policy, however, will lower rather than increase welfare. Rotemberg and Woodford [27; 28] study the effect of public expenditure in an imperfectly competitive economy. An increase in current expenditure entices individual firms to deviate from a collusive equilibrium so that mark-ups fall and aggregate production and welfare are enhanced.

In these papers public expenditure provides a pure stabilization good that has neither a direct utility benefit nor a productivity benefit. Recent work by Aschauer [2; 3], Morrison and Schwartz [20], and others, see the survey by Munnell [21], reveals however that public expenditure apart from providing a spending stimulus may be important in providing a productivity stimulus. In this vein, Barro [7] considers a publicly provided good that serves as an input in the private production processes of a perfectly competitive economy. He concludes that such a good should be provided in the same quantity as would be produced in a perfectly competitive industry. Glomm and Ravikumar [13] similarly consider a publicly provided good that enhances private sector productivity in a perfectly competitive economy. They, however, allow for different degrees of non-rivalry (congestion) in this good and demonstrate that the optimal proportional tax is independent of the degree of nonrivalry.

My paper intends to examine the consumptive and productive benefits of a publicly provided good within a dynamic macro model of monopolistic competition. To this end, I introduce monopolistic competition in the neoclassical growth model of Long and Plosser [18]. This model is chosen for several reasons. First, it is generally known and presents a simple closed-form solution. Second, it incorporates a diversity of sectors that allows, for instance, a distinction between investment and consumption-oriented production. Differences in the degree of monopoly power between sectors with a different investment content may affect how optimal expenditure policy should be conducted. Third, the different sectors enable examination of different types of government expenditure programs, including those in which the government provides an input to the production process. It thus becomes possible to examine in addition to the effect of the level of government expenditure, also the effect of the composition of government expenditure on aggregate activity. The composition issue was first addressed in the context of a traditional macro model by Barth and Cordes [9] and Ramirez [22] and is here re-examined in the context of a dynamic general equilibrium model.(1)

Section II generalizes the Long and Plosser model to allow for monopoly power, which requires not only the introduction of monopolistic competition in some subset of the sectors, but also requires a disaggregative approach different from Long and Plosser's, who may rely on a social planner due to the equivalence of the perfectly competitive outcome and the social planning outcome. The solution in section III demonstrates that overconsumption (relative to the Pareto optimal outcome) is inherent, even if investment-oriented sectors exhibit the lower degree of monopoly power. Section IV first points out that a micro-management approach to government policy in the sense of an industrial policy may lead to a social welfare optimum. Such a policy imposes tremendous information requirements on the government and would seem to be impracticable.

Section IV thus continues by discussing a smaller class of policies - macro-based policies - and, in particular, those based on government expenditure. It is found that expenditure on infrastructure and other types of government investment can improve welfare and should be provided in higher quantity than would be produced under privatization in a perfectly competitive industry. In contrast, government consumptive expenditure should be less than would be provided in a perfectly competitive industry. Such policies would help to offset part of the aggregate consequences of market power. Section V concludes.

II. The Long and Plosser Model with Monopolistically Competitive Firms

The purpose of this section is to present a version of a neoclassical growth model that can be amended to incorporate elements of monopolistic competition. The model of Long and Plosser [18] is chosen here because it is tractable, known to most macroeconomists, and general enough to allow incorporation of a variety of different sectors. Note that the intent is not to develop a real business cycle model and obtain positive implications; rather the emphasis is on generating normative results concerning the optimal size and composition of government expenditure, for which the Long and Plosser model seems an appropriate vehicle.

In adapting the Long and Plosser Model (henceforth LP) two basic changes must be implemented. First, since in monopolistic competition the outcome is not generally Pareto optimal, the social planner approach cannot be used and a decentralized approach is employed in formulating and solving the model. Second, monopolistically competitive firms are embedded by assuming that some or all sectors consist of firms with monopoly power derived from the production of imperfect substitutes.

The layout of this section is to first describe the model from the representative household's perspective and subsequently consider the decision problem for the different firms and the ensuing outcome at the sectoral level and lastly identify the equations relevant for solving the model. In this pursuit the original LP notation is maintained wherever possible.

The Representative Household

Given the information available at time t the household maximizes a standard time-additive utility function:

[E.sub.t] [summation of] [[Beta].sup.s] u([C.sub.s], [Z.sub.s]) where s = t to [infinity], (1)

with one-period utility a function of the consumption index [C.sub.t] and leisure [Z.sub.t]. Employing the same example as LP, further specify:

u([C.sub.t], [Z.sub.t]) = [[Theta].sub.0] ln [Z.sub.t] + [summation of] [[Theta].sub.i] ln [C.sub.it] where i = 1 to n, (2)

where [[Sigma].sub.i][[Theta].sub.i] = 1 (sum from 1 to n) by normalization. All [[Theta].sub.i] [greater than or equal to] 0 and n represents the number of different sectors (or industries) in the economy.

As in Lucas [19], the representative household holds shares that represent claims to the dividends of the firms. Additionally the household receives labor income, implying the following budget constraint,

[C.sub.t] + [summation of] [Q.sub.it][S.sub.it + 1] where i = 1 to n = [W.sub.t] [summation of] [L.sub.it] where i = 1 to n + [summation of]([Q.sub.it] + [D.sub.it])[S.sub.it] where i = 1 to n. (3)

The [S.sub.it] indicates the proportion of the shares in sector i held by the household and [L.sub.it] denotes the household's labor supply to sector i. All prices (here the wage [W.sub.t] and share prices [Q.sub.it], as well as dividends [D.sub.it], are presented in real terms, that is, units of the consumption basket [C.sub.t]). The same applies to the output prices in the various industries [P.sub.it] so that

[C.sub.t] = [summation of][P.sub.it][C.sub.it] where i = 1 to n. (4)

An additional constraint sets leisure plus the sum of all labor inputs equal to H the number of hours available to the household,(2)

[Z.sub.t] + [summation of][L.sub.it] where i = 1 to n = H. (5)

The household chooses the [S.sub.it + 1], [C.sub.it], and [Z.sub.t] to maximize (1) subject to (3) and (5), yielding, respectively, for all i:

[Beta][E.sub.t]([R.sub.it + 1][C.sub.t]/[C.sub.t + 1]) = 1, (6)

with [R.sub.it + 1] [approximately equal to]([Q.sub.it + 1] + [D.sub.it + 1])/[Q.sub.it] denoting the realized shareholder return for holdings in sector i, and:

[P.sub.it][C.sub.it]/[[Theta].sub.i] = [C.sub.t], (7)

[W.sub.t] = [[Theta].sub.0][C.sub.t]/[Z.sub.t]. (8)

In solving the model, equations (5) through (8) will represent the consumer side of the economy. First, however, the production side will be considered.

The Monopolistic Firms and Sectoral Equilibrium

In each sector a given fixed number of firms produces to maximize the expected present value of profits (dividends).(3) Firms are identical within each sector except that they produce different varieties of the sectoral good. These firms derive monopoly power from the fact that consumers and other firms value variety in their use of the sectoral output, as in the static models of Ethier [12], Weitzman [31], and Woglom [32]. Appendix A derives the market outcomes under the assumption of Cobb-Douglas production functions and CES sub-production and sub-utility functions (the latter two producing constant-elasticity demand functions). Given that in equilibrium all firms in a sector produce the same quantities, using the same inputs, output in sector i at time t + 1, [Y.sub.it + 1], is just the sum of the output of all firms in this sector, yielding:

[Mathematical Expression Omitted]. (9)

Here [[Lambda].sub.it + 1] represents a stochastic parameter with the Markov property. In contrast to LP, production need not exhibit constant returns to scale, instead returns to scale vary between sectors and are given by [s.sub.i] = [[Sigma].sub.j][a.sub.ij] (sum from 0 to n), where all [a.sub.ij] [greater than or equal to] 0. The capital inputs [X.sub.ijt] denote the inputs of sector j used by the firms in sector i at time t.

Appendix A also derives the first-order conditions for the input decisions in sector i as

[Mathematical Expression Omitted]. (10)

[Mathematical Expression Omitted], (11)

where [[Eta].sub.i] is a parameter of the CES subutility function such that 0 [less than] [[Eta].sub.i] [less than or equal to] 1. This parameter characterizes the degree of monopoly power in each sector, with lower [[Eta].sub.i] revealing more monopoly power. In equations (10) and (11) the expected marginal revenue product is set equal to the marginal resource cost for each input and in each sector.

The household budget equation (3), given that in equilibrium all shares are held so that [S.sub.it + 1] = [S.sub.it] = 1 together with the fact that sectoral dividends equal [D.sub.it] = [P.sub.it] [Y.sub.it] - [W.sub.t][L.sub.it] - [[Sigma].sub.j]([P.sub.jt][X.sub.ijt]) yields the accounting condition for each sector j:

[Y.sub.jt] = [C.sub.jt] + [summation of][X.sub.ijt] where i = 1 to n. (12)

Equation numbers (5) through (8) summarizing the consumer side and equation numbers (9) through (12) summarizing the production side provide [n.sup.2] + 5n + 2 equations that can be used to obtain the quantities [X.sub.ijt], [C.sub.it], [L.sub.it], and [Z.sub.t] as will be presented in the next section (the remaining 3n + 1 equations solve for quantities [Y.sub.it + 1] and prices [P.sub.it + 1], [R.sub.it + 1] and [W.sub.t] as presented in Appendix B).

III. Solution and Interpretation

The Solution

The procedure for obtaining the solution is similar to LP and provided in Appendix B. The results for all sectors are as follows:

[X.sub.ijt] = ([Beta][a.sub.ij][[Eta].sub.i][[Gamma].sub.i]/[[Gamma].sub.j])[Y.sub.jt], [for every]i, j [element of] {1, n}, (13)

[C.sub.it] = ([[Theta].sub.i]/[[Gamma].sub.i])[Y.sub.it], [for every]i [element of] {1, n}, (14)

Z = ([[Theta].sub.0]/[[Gamma].sub.0])H, (15)

[L.sub.i] = ([Beta][[Eta].sub.i][a.sub.i0][[Gamma].sub.i]/[[Gamma].sub.0])H, [for every]i [element of] {0, n}. (16)

with:

[[Gamma].sub.j] = [[Theta].sub.j] + [Beta][summation of][a.sub.ij][[Eta].sub.i][[Gamma].sub.i] where i = 1 to n, [for every]i [element of] {0, n}. (17)

The [[Gamma].sub.j] may be viewed as representing an index of the value of good j.

Interpretation of the Results: Overconsumption

The crucial distinction in the solution compared to LP is related to the [[Eta].sub.i] parameters. These parameters are related to the demand elasticities which, themselves, relate to the taste for diversity in equations (A4) and (A5). Further, the [[Eta].sub.i] equal the inverse of the mark-up so that as [[Eta].sub.i] falls the monopoly power in sector i increases. In the perfect competition case one may set [[Eta].sub.i] = 1 so that for the firms in sector i the expected value of the marginal product equals marginal input cost in equations (10) and (11). If all industries are perfectly competitive ([[Eta].sub.i] = 1 for all i) the results are equivalent to those in LP. Since the [[Eta].sub.i] can take any value between zero and one this model generalizes LP.

From the first welfare theorem it is clear that the perfectly competitive solution is Pareto optimal. Further, due to the assumption of a representative household, this Pareto optimal outcome is unique. The outcome for any [[Eta].sub.i] [less than] 1 thus is Pareto inefficient as is typical in the presence of monopolistic elements.

Equation (15) together with the definition of [[Gamma].sub.0] in equation (17) shows immediately that aggregate employment decreases with the monopoly power in any industry ([[Eta].sub.i] [less than] 1 for any i). This is consistent with the static literature on macro models of monopolistic competition, for instance Blanchard and Kiyotaki [10]. For [[Theta].sub.0] [greater than] 0, the incentive to work decreases as products are sold at higher relative prices; or, equivalently, as real wages decrease. When [[Theta].sub.0] = 0 labor supply is perfectly inelastic and employment remains constant at H. Interestingly, employment may still increase in some sectors. In the case of [[Theta].sub.0] = 0, for instance, the decrease of employment in one sector must mean that another sector employs more.

A more surprising result emerges for consumption. Increased monopoly power in any sector unambiguously increases consumption of all goods. That is, monopoly power leads to overconsumption compared to the perfectly competitive or Pareto optimal outcome. More precisely,

PROPOSITION 1 (Overconsumption). Given the current state of the economy, in particular the set [Mathematical Expression Omitted], current consumption does not decrease (and increases for at least one good) in the monopoly power of any particular sector.

Proof. Immediate from equations (14) and (17) since [C.sub.j] falls in [[Gamma].sub.j] and [[Gamma].sub.j] rises in [[Eta].sub.i].

The result is akin to the underinvestment result in Shleifer and Vishny [29] arising from demand externalities. The intuition here, however, is of a different nature. Consider a firm's incentive to invest. To raise value to the shareholders, firms will attempt to keep price high by keeping production low; the latter calls for low investment. For a given level of production in any sector, low demand from the firms means that more is left to the consumers. One might expect that high product prices discourage consumption but, this is equally true for investment. On the whole, monopoly power thus distorts economic decisions to favor current consumption over investment and future consumption. This is most clearly demonstrated setting, by means of illustration, [[Eta].sub.i] = [Eta] for all i - all sectors have identical monopoly power. The solution here is then equivalent to the LP solution with [Beta] replaced by [Beta][Eta]. In other words, monopoly power has the effect of raising the economy's effective rate of time preference. One crucial distinction, however, is that the increased time preference is undesirable from the perspective of the representative household and that government policy may be desirable to reduce overconsumption.

It follows directly from equations (13) and (17) that monopoly power in sector i decreases capital investment in that sector itself. The same proviso applies as for employment, however, that the introduction of differing degrees of monopoly power may in particular sectors raise investment (even though, from equation (14), aggregate investment decreases), when the value of the sector's good (measured by [[Gamma].sub.i]) rises relative to that of the sectors from which it buys its capital goods.

A final interpretation relates monopoly power to decreased productivity. Equations (13)-(17) reveal that [[Eta].sub.i] is always coupled with [a.sub.ij]. The production function, equation (9) and the comparison of the solution here to LP, implies therefore that monopolistic competition with [Mathematical Expression Omitted] is equivalent to perfect competition with [a.sub.ij][[Eta].sub.i] as productivity parameters instead of [a.sub.ij], implying returns to scale with scale parameter [s.sub.i][[Eta].sub.i] in each sector. This interpretation may explain a decline in productivity as caused by increased monopoly power over time.

IV. Government Policy

Industrial Policy and Pareto Optimality

In examining the types of government policies that may improve welfare assume first that the government's information about an individual firm is as good as that of the firm itself. The Pareto optimal outcome can then be obtained straightforwardly:

PROPOSITION 2 (Pareto Optimal Policy). An industrial policy, subsidizing all inputs purchased in sector i at rate [v.sub.i] = 1 - [[Eta].sub.i], and financing these subsidies through lump-sum taxes, is Pareto optimal in the monopolistic LP model.

Proof. Multiply the input prices on the right-hand sides of equations (10) and (11) by their after-subsidy cost of 1 - [v.sub.i] = [[Eta].sub.i] so that the [[Eta].sub.i] drop from the model and the solution becomes equivalent to the Pareto optimal outcome in LP. (More detailed proof available from the author).

Interestingly, the optimal rate of subsidy, 1 - [[Eta].sub.i], equals exactly Lerner's [17] measure of monopoly power, so that the optimal policy subsidizes firms proportionately to their degree of monopoly power. The reason, of course, is that firms with higher monopoly power underproduce more and need more enticement to hire the competitive quantity of inputs.

One major problem (not modeled) with this type of policy is the micro management it implies - for the policy to be effective the government needs precise information about individual firms and industries to avoid firms imitating high-monopoly-power firms to increase subsidies. I.e., the policy may not be implementable in the absence of symmetric information.

Government Expenditure Policy

Optimal policies rely on micro management as was demonstrated in the previous section. Such policies are bound to become less desirable, however, when informational considerations enter the model explicitly. A practically more interesting class of policies concerns macro-based policies - those policies that do not require sector-specific allowances. Examples of such "second-best" policies may be uniform price controls or a general investment tax credit, but I will focus on government expenditure policy. In particular, I will consider whether the existence of monopoly power provides a motivation for the government to spend "excessively" and to what extent optimal spending depends on the investment aspect of the publicly provided goods. For instance, should the government spend more on infrastructure than a competitive firm would under a privatization arrangement? How does the answer depend on what fraction of infrastructure spending must be considered consumptive?

To focus on the issue at hand - the underproduction resulting from monopoly power - it is convenient to abstract from the traditional arguments for public expenditure, nonexclusion and nonrivalry. See Barro and Sala i Martin [8], for a macroeconomic discussion of these aspects of public expenditure.

Instead I will assume that the government provides a regular good that could as well be provided by private firms. The question becomes however how much to produce (or, equivalently, procure) of this particular good. I will presume that the cost of the publicly provided good can be financed through a nondistortionary lump-sum tax.(4) I aim to compare private production in sector g as presented in equations (13)-(17) to the quantity of good g optimally provided by the government. To this end consider the social planner problem in setting production for this sector while taking private sector decisions as given.

The most convenient way to derive the socially optimal quantity produced of good g is to act as if [[Eta].sub.g] is the choice variable (it is monotonically related to the inputs needed to produce good g but further implies that all inputs are used in their most efficient proportions), allowing it to take values higher than 1 when the optimal quantity is above the level supplied in a perfectly competitive sector.(5)

In the private sectors decisions are made in accordance with equations (13)-(17), which the government takes as given. The dynamic programming formulation of the government's decision problem - choosing production of the publicly provided good (by choice of [[Eta].sub.g]) to maximize the expected utility of the representative consumer subject to private sector decisions - is presented and solved in Appendix C and produces

[Mathematical Expression Omitted], (18)

where [[Gamma].sub.j] represents the market value of sector j's good (the set of j includes g) as given in equation (17); [[Gamma][prime].sub.j] [equivalent] [Delta][[Gamma].sub.j]/[Delta][[Eta].sub.g] so that [[Gamma][prime].sub.j] = [Beta][a.sub.nj][[Gamma].sub.n] + [Beta] [summation of] [a.sub.ij][[Eta].sub.i][[Gamma][prime].sub.i] where i=1 to n from (17); and [Mathematical Expression Omitted], with [Mathematical Expression Omitted] representing the value of good j to society. Equation (18) presents the level of public provision as related to the ratio of the supply stimulus due to the publicly provided good and the net cost of the input absorption by the public sector.

It is easy to check that the [[Gamma][prime].sub.j] must be positive. One result then follows directly from equation (18): if the "average" of the returns to scale in each sector, [s.sub.i], rises towards 1/[Beta], the optimal size of the government sector g becomes unbounded. The reason lies in the strategic complementarities inherent in economies of scale; see Romer [23; 25] for other policy implications in the presence of economies of scale, arising there from knowledge externalities.

A further result, that [[Eta].sub.g] = 1 for the economy without monopoly power, i.e., [[Eta].sub.i] = 1 for all i, is easily verified since it follows from equation (17) and the definition of [Mathematical Expression Omitted] that in this case [Mathematical Expression Omitted] for all i. Summing over all j in equation (18) using the expression for [[Gamma][prime].sub.j] and the fact that [[Sigma].sub.j][a.sub.ij] = [s.sub.i] then implies [[Eta].sub.g] = 1, reproducing the result in Barro [7] that, in a perfectly competitive economy, the publicly provided good should be produced in the same quantity as would occur if the good were produced in the private sector.

It is difficult to ascertain in general under which conditions [[Eta].sub.g] will exceed 1 implying that the government produces more than a perfectly competitive stand-in. The difficulties lie in the fact that the public sector could potentially be hiring some inputs away from sectors that have better scale economies and a low degree of monopoly power. Results then readily become difficult to sign when outliers are possible. To avoid such ambiguity two simplifying assumptions will be introduced at this point.

The first assumption, of "proportionality", is that inputs are optimally used in the same proportions by all sectors. Technically this implies that the [a.sub.ij], the productivity of the input from sector j in the production of sector i, can be factored as [a.sub.ij] = [s.sub.i][a.sub.j] with the [s.sub.i] representing the general productivity specific to sector i and the [a.sub.j] interpreted as the productivity enhancement specific to sector j. To normalize set [[Sigma].sub.j][a.sub.j] = 1 so that the [s.sub.i] equal the returns to scale in sector i as previously defined.

The second simplifying assumption posits "polarity": each sector i produces either a pure capital good, [[Theta].sub.i] = 0, or a pure consumption good, [a.sub.i] = 0. For labor, sector 0, polarity implies either a vertical labor supply curve ([[Theta].sub.0] = 0) or that labor is unproductive ([a.sub.0] = 0). For publicly provided goods it is possible to think of two different government sectors, [g.sub.c] and [g.sub.k], each producing one of the polar goods, a consumption good and a capital good, respectively.

The optimal production level for both polar types of publicly provided goods is characterized in the following proposition:

PROPOSITION 3 (Public Expenditure). Given the assumption of proportionality that [a.sub.ij] = [s.sub.i][a.sub.j] for all i and j, with [[Sigma].sub.j][a.sub.j] = 1, and the assumption of polarity that [[Theta].sub.i] = 0 or [a.sub.i] = 0 for all i:

(a) optimal provision of the public consumption good is determined by

[[Eta].sub.gc] = ([summation of] [s.sub.i][[Theta].sub.i][[Eta].sub.i] where i=0 to n) / [summation of] [s.sub.i][[Theta].sub.i] where i=0 to n. (19)

(b) optimal provision of the public investment good is determined by

[[Eta].sub.gk] = (1 - [Beta] [summation of] [s.sub.i][a.sub.i][[Eta].sub.i] where i=0 to n) / (1 - [Beta] [summation of] [s.sub.i][a.sub.i] where i=0 to n). (20)

Proof. Polarity implies that [[Gamma].sub.i] = [[Theta].sub.i] for all consumption goods and together with proportionality implies that [[Gamma].sub.i]/[a.sub.i] = [[Gamma].sub.j]/[a.sub.j] for all capital goods. Substituting these relations into equation (18) and employing equation (17) for [[Gamma].sub.i] and the definitions of [[Gamma][prime].sub.i] and [Mathematical Expression Omitted] below equation (18) produces equations (19) and (20) for the cases of the publicly provided consumption good and the publicly provided capital good, respectively.

The intuition for Proposition 3 is the following. In the pure consumption good case the optimal spending amount is related to the weighted average degree of monopoly power in all sectors, where the weights are given by [[Theta].sub.i][s.sub.i] - the usefulness of each sector's good as a consumption good multiplied by that sector's returns to scale. Since [[Theta].sub.i] = 0 for all capital goods this implies that the government should produce the good as though it had the average degree of monopoly power in the consumption goods sector. One might have naively thought that the optimal policy would be to simply correct for monopoly power in this sector itself and supply the competitive quantity, [[Eta].sub.gc] = 1. Proposition 3a shows that this intuition is false. The reason is that, for this good, the true value, [Mathematical Expression Omitted], equals the market value, [[Gamma].sub.gc]; even though the good is underproduced at this level, all goods are (on average) and to take more than the proportionate quantity of inputs would lower welfare.

In the pure capital good case the true value exceeds the market value so that additional production helps decrease the economy-wide investment shortfall. Accordingly, [[Eta].sub.gk] [greater than] 1 as follows from equation (20). Straightforward differentiation implies that an increase in monopoly power in any sector i ([[Eta].sub.i] falls) raises the optimal quantity produced of the public capital good. Similarly, an increase in [s.sub.i] (or [Beta]) in any sector raises the optimal quantity produced of the public capital good. Note, however, that increasing or constant returns to scale in no way is necessary for public investment to exceed the competitive level; all that is necessary is monopoly power in at least one capital goods sector. The degree of monopoly power is in equation (20) weighted by the [s.sub.i][a.sub.i] which account for both the productivity (returns to scale) in sector i itself as well as the degree to which sector i enhances productivity in the other sectors. An increase in [a.sub.i] (and equivalent decrease in some [a.sub.j], as the [a.sub.i] must add to one), implying that sector i contributes more to productivity and sector j less, raises public investment depending on the sign of (1 - [Eta].sub.i])[s.sub.i] - (1 - [[Eta].sub.j])[s.sub.j] as follows from equation (20): if the extent to which marginal output value exceeds marginal cost in sector i, 1 - [Eta].sub.i], weighted by productivity, outweighs that of sector j then an increase in the productivity enhancement value of sector i relative to sector j raises optimal public investment.

The previous analysis has relied on some bold simplifications concerning the nature of the public good provision that will now briefly be discussed. First, the absence of elements of nonrivalry and nonexclusion that form the traditional arguments for public goods provision. The reason for ignoring these issues is that the here developed argument for public goods provision is completely independent of these public good characteristics; the presence of nonexclusion or nonrivalry will only add a complementary reason for increasing the size of the public sector. Second, the allocation of the government good occurs at market prices. This assumption can be viewed in two ways; either the government sells the good in the market like any firm, or allocation is free, based on the projected value to each firm. The latter case comes again close to micro management and may be undesirable for that reason. Third, the input mix employed in the public sector was chosen at market prices; selecting inputs based on their "true" values would likely result in an even larger optimal size of the public sector. Fourth, taxes were assumed to be nondistortionary. Explicit consideration of the first three issues may be interesting but is not likely to alter the qualitative results of the model. The incorporation of nondistortionary taxes likely will reduce optimal government expenditure, possibly below the competitive level in the case of government investment.

Quantitative Significance

Quantitatively, the optimal level of government investment implied by Proposition 3b may be substantially above the level otherwise produced in a perfectly competitive sector. Assume for the sake of obtaining a rough numerical impression that the [s.sub.i], [a.sub.i], and [[Eta].sub.i] are not systematically correlated. Then the variables in equation (20) can be replaced by their averages, producing

[[Eta].sub.gk] = (1 - [Beta]s[Eta])/(1 - [Beta]s), (20[prime])

with s representing the average returns to scale and [Eta] the average degree of monopoly power.(6)

Consider some typical estimates for [Eta], s, and [Beta]: for U.S. data the work of Hall [14; 15], and Morrison and Schwartz [20] implies a value for [Eta] of around 0.6, which is also consistent with the numbers used in Rotemberg and Woodford [27]. Morrison and Schwartz also finds average returns to scale of around 1.1, which is consistent with Romer [24]. In this paper the average returns to scale include the production of labor which, of course, has zero returns to scale. Thus, a better measure of average returns to scale here should weigh in the zero scale returns of labor. Romer [24] finds that the exponent on labor (excluding human capital) in a Cobb-Douglas production function, measuring labor's contribution to productivity, should be around 0.2. Thus our average returns to scale measure is approximately 0.9. Taking [Beta] conventionally as 0.95 the value of [[Eta].sub.gk] in equation (20[prime]) equals about 3.3.

Hence, from equations (13) and (16), the governments input demand would be more than three times as high as under perfect competition. Given Munnell's [21] estimate that in the U.S. the stock of public capital is approximately half the size of the private capital stock, the additional input demand implied by optimal - as compared to perfectly competitive - production of the publicly provided capital good appears to be sizeable.

V. Conclusion

It is tempting to advocate government investment on the premise that individuals invest too little and that the public sector must compensate. The danger is that even when growth is stimulated individuals may be forced into an intertemporal consumption pattern that puts too much weight on the future and does not respect individual tastes. The above model provides a very different perspective. Individuals save and invest too little, not out of their own volition, but as a result of socially suboptimal behavior by monopolistic firms. Government investment is now warranted as a way of improving not only growth but also social welfare.

The intuition is that, given the standard production function, an increase in government output (infrastructure) raises productivity in the private sector, making investment goods bought in the private sector cheaper and providing further incentives for government output. This strategic complementarity issue is correctly ignored by private firms within a sector who do not internalize the benefits to the sector and society as a whole.

In a perfectly competitive economy such complementarity does not occur. A marginal increase in productivity may stimulate input demands but this indirect effect has no benefit to profit and welfare as follows from application of the envelope theorem - the value of the marginal product of the inputs equals the marginal cost of providing these inputs. In a monopolistically competitive economy, on the other hand, the value of the marginal product of the inputs always exceeds the marginal cost so that the indirect input stimulus does raise welfare and provides the government with a motive for providing such stimulus.

The positive impact of government expenditure thus derives from a pecuniary externality; as such this paper is consistent with modern macroeconomic theory which views some form of externality as an essential condition for expenditure policy. A very fundamental issue arises, however, which is broached in the paper. When externalities are present it becomes important to explain why certain price correcting policies are not necessarily superior to expenditure policy. In principle, this paper provides a framework for doing so by allowing inefficiencies in a large number of heterogeneous sectors. In the presence of informational acquisition costs or asymmetries in information between firms and the government, sector-specific price correcting policies may become unworkable.

Appendix A: Derivation of Sectoral Input Demands and Production

Consider a firm k in sector i. Its production [Y.sub.ikt] is given by the standard Cobb-Douglas technology as

[Mathematical Expression Omitted]. (A1)

[[Lambda].sub.it+1] represents a stochastic parameter with the Markov property. Returns to scale vary between sectors and are given by [s.sub.i] = [[Sigma].sub.j][a.sub.ij] (sum from 0 to n). The capital inputs [X.sub.ijkt] denote the inputs of sector j used by firm k in sector i at time t; the number of firms in sector i is given by a constant [k.sub.i].

A firm is assumed to choose its inputs [L.sub.ikt] and [X.sub.ijkt] to maximize the expected present value of dividends, i.e.,(7)

[Mathematical Expression Omitted], (A2)

with

[D.sub.ikt] = [P.sub.ikt][Y.sub.ikt] - [W.sub.t][L.sub.ikt] - [summation of] [P.sub.jt][X.sub.ijkt] where j=1 to n. (A3)

In its decisions the firm is constrained by the demand curve which will be derived subsequently. Firms can derive monopoly power from a variety of sources. Here one such source will be assumed. Consumers are considered to derive utility from a composite of the imperfect substitutes produced in a particular sector, i.e., consumers like diversity. The consumption index (or subutility function) for the good produced in sector i is given by

[Mathematical Expression Omitted]. (A4)

The symbol [k.sub.i] denotes the number of firms in sector i and [[Eta].sub.i] is a parameter of the CES subutility function such that 0 [less than] [[Eta].sub.i] [less than or equal to] 1.

Firms, similarly, use a composite of the inputs from a particular sector, as in the formulation of Ethier [12]:

[Mathematical Expression Omitted] (A5)

for all i and j. For simplicity, set [[Eta].sub.ij] = [[Eta].sub.j] [less than] 1 for all i and j. Notice that all firms within a particular sector are treated symmetrically and face equivalent constraints. The firm's decision problem can now easily be solved for the symmetric equilibrium.

Consider the firm's demand coming from the households and the other sectors. The household in the two-stage budgeting process appropriate for these preferences maximizes [C.sub.jt] in equation (12) for a particular budget, yielding [C.sub.jkt] = (1/[k.sub.j])[([P.sub.jkt]/[P.sub.jt]).sup.1/([[Eta].sub.j]-1)][C.sub.jt]. A similar relation holds for the demand coming from firms so that total demand to firm k in industry j is given by:

[Y.sub.jkt] = (1/[k.sub.j])[([P.sub.jkt]/[P.sub.jt]).sup.1/([[Eta].sub.j]-1)]([C.sub.jt] + [summation of] [X.sub.ijt] where i=1 to n). (A6)

As is common in models of monopolistic competition it will be assumed that the individual firm takes variables at the sector level as given. Given the accounting condition for sectoral output it follows then that the individual firm in sector i faces a constant elasticity demand curve,

[Y.sub.ikt] = (1/[k.sub.i])[([P.sub.ikt]/[P.sub.it]).sup.1/([[Eta].sub.i]-1)][Y.sub.it]. (A7)

Similar demand curves where obtained in the static monopolistic macro models of Weitzman [31] and Woglom [32]. Now choose inputs to maximize equation (A2) given equation (A3) and subject to equation (A7). This produces for the choice of labor inputs:

[Mathematical Expression Omitted].

Use symmetry to set [P.sub.ikt] = [P.sub.it] then equations (A1) and (A7) and updating can be employed in the above equation to obtain for all i equations (10) and (11) in the text.

Equation (9) in the text follows from (A1) due to symmetry since [X.sub.ijt] = [k.sub.i][X.sub.ijkt], [L.sub.it] = [k.sub.i][L.sub.ikt] in that case as follows from (A5) and the homogeneity of labor, and since [Y.sub.it] = [k.sub.i][Y.sub.ikt] from (A7).

Appendix B: Solution of the Model

As in LP the "guess and verify" method is used to find a solution. Here, instead of guessing a value function, it is checked that the candidate solution satisfies all relevant equations. Suppose that for all i and j:

[X.sub.ijt] = [F.sub.ij][Y.sub.jt], (B1)

[C.sub.it] = [V.sub.i][Y.sub.it], (B2)

[L.sub.it] = [L.sub.i], (B3)

[Z.sub.t] = Z, (B4)

[R.sub.it+1] = (1/[Beta])[P.sub.it+1][Y.sub.it+1]/[P.sub.it][Y.sub.it], (B5)

where [F.sub.ij], [V.sub.i], [L.sub.i] and Z are constants to be determined.

First verify that the guess for [R.sub.it+1] is correct. Substitute equations (7) and (B2) into equation (6) to see that it holds as an identity. (Note that it can furthermore be checked, using equation (A3) for dividends that, from (10), (11), and (B5), [D.sub.it] = (1 - [Beta][s.sub.i][[Eta].sub.i])[P.sub.it][Y.sub.it]. This implies from the definition of return and equations (6), (7) and (B2) that [Q.sub.it] = [[Beta](1 - [Beta][s.sub.i][[Eta].sub.i])/(1 - [Beta][s.sub.i])][P.sub.it][Y.sub.it].)

Eliminate the real wage [W.sub.t] by substituting equation (8) into equation (10). Given equation (B5) for return, equations (10) and (11) now become for all i and j (j not equal to 0):

[Beta][[Eta].sub.i][a.sub.i0]Z[[Theta].sub.i] = [[Theta].sub.0][V.sub.i][L.sub.i], (B6)

[Beta][[Eta].sub.i][a.sub.ij]([[Gamma].sub.i]/[[Gamma].sub.j]) = ([V.sub.i]/[V.sub.j])[F.sub.ij], (B7)

where equation (7) was used in the derivation of equation (B7).

Analogously to LP define:

[[Gamma].sub.j] = [[Theta].sub.j] + [Beta] [summation of] [a.sub.ij][[Eta].sub.i][[Gamma].sub.i] where i=1 to n. (B8)

Equation (12) implies that [V.sub.j] = 1 - [[Sigma].sub.i][F.sub.ij] (sum from 1 to n) given the assumed solution. It then follows from equations (B7) and (B8) that

[V.sub.i] = [[Theta].sub.i]/[[Gamma].sub.i]. (B9)

Hence, equation (B7) implies that for all i and j:

[F.sub.ij] = [Beta][a.sub.ij][[Eta].sub.i][[Gamma].sub.i]/[[Gamma].sub.j]. (B10)

From equations (B6) and (B9),

[L.sub.i] = [Beta][[Eta].sub.i][a.sub.i0][[Gamma].sub.i]Z/[[Theta].sub.0].

Using equation (5) together with the above equation produces

Z = ([[Theta].sub.0]/[[Gamma].sub.0])H. (B11)

Here [[Gamma].sub.0] is defined in equation (B8) for j = 0. From the above two equations,

[L.sub.i] = ([Beta][[Eta].sub.i][b.sub.i][[Gamma].sub.i]/[[Gamma].sub.0])H. (B12)

Equations (B8) through (B12) together with the assumed solution, equations (B1) through (B5), provide the solution to the model as presented in the text. It is easily checked that the solution satisfies all relevant equations.

Appendix C: Derivation of Equation (18).

[Mathematical Expression Omitted], (C1)

where ln [Y.sub.t] (and [[Lambda].sub.t]) are n-element vectors with,

ln [Y.sub.it+1] = ln [[Lambda].sub.it+1] + [a.sub.i0] ln([Beta][[Eta].sub.i][a.sub.i0][[Gamma].sub.i]/[[Gamma].sub.0])H + [summation of] [a.sub.ij] ln([Beta][[Eta].sub.i][a.sub.ij][[Gamma].sub.i]/[[Gamma].sub.j])[Y.sub.jt] where j=1 to n, [for every]i [element of] {1, n} (C2)

As before,

[[Gamma].sub.j] = [[Theta].sub.j] + [Beta] [summation of] [a.sub.ij][[Eta].sub.i][[Gamma].sub.i] where i=1 to n, [for every]j [element of] {0, n}. (C3)

Now consider the envelope conditions for all In [Y.sub.jt] and define [Mathematical Expression Omitted], then

[Mathematical Expression Omitted]. (C4)

Further define [Delta][[Gamma].sub.j]/[Delta][[Eta].sub.n] [equivalent to] [[Gamma][prime].sub.j], so that the first-order condition can be presented as:

[Mathematical Expression Omitted]. (C5)

In words, the left-hand side of (C5) represents the decrease in consumption caused by the price increases from the increased input demand by the public sector; the right-hand side represents the next-period benefit from increased public good provision, plus the next-period benefit from increased production (stimulated by higher prices) of the goods used as inputs in the public sector, minus the cost of the overall decreased production incentive effected by higher prices for the inputs used in the public sector.

Rewriting (C5) with the help of (C4) yields equation (18) in the text.

The author thanks Rafael Tenorio, an anonymous referee, and seminar participants at West Virginia University for helpful comments.

1. Barth and Cordes [9] allow government investment to complement (or substitute for) private investment in a traditional Keynesian macro model so that government expenditure can affect aggregate activity directly rather than solely through changes in interest rates. Ramirez [22] subsequently has explained in the same context the significance of considering government expenditure composition as an additional policy instrument.

2. All labor inputs are perfect substitutes from the household's perspective. Accordingly, it will follow that there is no monopsony power in the labor market. It is possible to reformulate preferences and allow monopsony power, but this is avoided for simplicity.

3. The number of firms in each sector is taken as given. This assumption is common in most macro models of imperfect competition; see for instance Blanchard and Kiyotaki [10], Rotemberg and Woodford [28], and Balvers and Cosimano [5]. An exception is Balvers [4] who shows that results are not substantially affected when the number of firms is made endogenous. Essentially, expected profits are pushed to zero by free entry but this does not change the fact that in equilibrium marginal revenue must exceed marginal cost for each firm, as in Dixit and Stiglitz [11]. In Startz [30] entry and exit by firms fundamentally alters the outcome. This occurs because aggregate demand externalities derived from the effect of demand increases on profit income (which fuels further demand increases) disappear as profits are competed to zero. In the case of aggregate supply externalities, however, the flow of profit income is irrelevant.

4. The nondistortionary tax policy allows me to isolate the effects of the distortion due to monopolistic elements. Note that the basic solution of the model in equations (13)-(17) is therefore only affected by the inputs in and output of the publicly provided good and not by its financing. For the effects of a distortionary proportional tax in the context of a publicly provided good see Glomm and Ravikumar [13].

5. [[Eta].sub.g] also serves as an actual utility and productivity parameter (see equations A4 and A5) which of course cannot be altered by the government. However, as all subsectoral quantities are equal in equilibrium, the value of [[Eta].sub.g] has no direct effect on equilibrium utility other than via the indirect effect of the quantities produced, which can be altered by the government.

6. The "uncorrelatedness" assumption used to derive (20[prime]) can also be employed without using the polarity and proportionality assumptions needed to derive (20). In this case the expression for [[Eta].sub.gk] in (20[prime]) can be derived as a lower bound. Employing polarity together with uncorrelatedness produces the expression for [[Eta].sub.gk] in (20[prime]) with equality. Derivations of these results are available from the author upon request.

7. This objective is appropriate as long as the firm may ignore its impact on aggregate variables. The typical assumption in models of monopolistic competition is that there are many firms so that the firm may be considered small. The same assumption is made here; however an additional assumption must be made in the context of the share economy in this model. If all households were represented as shareholders in the firm it might be rational for the firm to internalize its impact on the overall economy. This issue is avoided by assuming an [Epsilon] cost to diversification. As firms within a sector are subject to the same random shocks, consumers may diversify across sectors but will not diversify within a sector, thus forsaking positive externalities and causing individual firms to ignore the greater good.

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