Productivity growth and public sector employment.

Ho, Kong Weng ; Hoon, Hian Teck

1. Introduction

There is broad recognition that, with reference to U.S. data, there has been a rise in the share of public sector employment over the past several decades. [See, for instance, the standard labor economics textbook in its fourth edition by McConnell and Brue (1995, p. 347).] In fig. 1, we show that the U.S. share of public sector employment in total civilian labor force increased from 9.7 percent in 1950 to a peak of 15.7 percent in 1975, and, thereafter, gradually declined to 14.6 percent in 1995.(1)

Popular explanations for the rise in public sector employment share typically appeal to demand factors. In particular, it is commonly argued that in a growing economy, there is a rise in relative demand for high income-elastic government services such as higher education, health services, parks and a clean environment. By implication, the derived demand for public sector workers increases. In this paper, we provide an alternative explanation for the rising share of public sector employment up to the early 1970s before gradually declining, using a general-equilibrium model whose predictions accord well with the stylized facts of U.S. productivity growth.

More generally, economists' interest in the growth of public sector expenditure goes back at least to the German economist Adolph Wagner who wrote in the 1880s. [See his paper in Musgrave and Peacock (1958).] In Musgrave (1969), it is argued that because infrastructure is particularly important at the early stage of development, public capital expenditures occupy a larger share in the earlier development phase, and then show a relative decline as a higher level of income is reached. The classic study of Peacock and Wiseman (1961), on the other hand, puts forward an alternative hypothesis regarding the rising share of public expenditure to GNP. The authors argue that during national emergencies, particularly war, voters are willing to cross the old "tax threshold" and to accept a higher level of taxation, which they would otherwise resist. After the emergency has passed, they are willing to retain the higher level of taxation, making possible higher levels of civilian public expenditures. The volume edited by Forte and Peacock (1985) contains a useful collection of papers applying economic analysis to look at the underlying causes of growth of public expenditure.

The theory developed in this paper focuses on only one component of public expenditure - the payroll of public sector employees. It augments the neoclassical growth model to incorporate Harrod-neutral technical progress and a public sector input that enters productively into private sector production. Making the empirically relevant assumptions that the production of the public sector good is relatively labor-intensive, and that the elasticity of substitution between capital and labor in private production is less than unity, we show that in a market economy where the public and private sectors hire labor competitively, the observed rise and fall in the share of public sector employment can be explained basically by the role of technical progress. In particular, we demonstrate that a temporary rise in the rate of Harrod-neutral technical progress (corresponding to the 50s and 60s in [ILLUSTRATION FOR FIGURE 1 OMITTED]) raises the share of public sector employment before declining (in the 70s and 80s). For empirical studies on the productivity slowdown, we refer the reader to the following literature. Denison (1985, p. 34) computes that the average growth rate of potential national income in the whole economy per person potentially employed dropped from 2.26 percent (1948-73) to 0.23 percent (1973-82), in which residual productivity decreased by 1.13 percent. Baily and Gordon (1988) note that there might be a common impetus to productivity advance in the early postwar years across all sectors (p. 420) and there was a widespread productivity slowdown after 1973 (p. 362). Table 1 in their paper shows that the average annual aggregate productivity growth for business was 2.94 percent in terms of output per hour and 2.00 percent in terms of multi-factor productivity for the period 1948-73. In 1973-87, the figures dropped to 1.02 and 0.39 percent, respectively. Their extensive studies claim that measurement errors cannot account completely for the productivity slowdown. Jorgenson (1988, Table 3) calculates that the average growth in aggregate productivity for 1948-79 was 0.81 percent and it was 0.34 percent for 1973-1979, considering the contributions of productivity from different industrial sectors. Hence, the stylized fact is that the productivity growth rate in the 50s and 60s was higher than the later period and the productivity slowdown began in the early to middle of the 70s.

The paper is organized as follows. In section 2, we develop the basic model. Section 3 studies the effects of a shock to productivity growth and reports a simulation of the model. Section 4 provides the conclusion.

2. The basic model

The economy consists of the public and private sectors. Population grows at the rate [Mu] and Harrod-neutral technical progress, indexed by [[Lambda].sub.t], grows at the rate [Lambda] such that effective labor force at time t is [[Lambda].sub.0][L.sub.0][e.sup.([Lambda] +[Mu])t] where [L.sub.0] is the initial labor force. The reason we have worked with Harrod-neutral technical progress is that we want to have a model which has a steady state that is consistent with the broad stylized facts of growth. [See Solow (1970) for a list of the stylized facts of growth.] In particular, we want a model with the steady-state properties that along a balanced-growth path the capital-output ratio is a constant, and net saving and investment are a constant fraction of output. As argued by Solow (1970, p. 34-37), in a one-sector constant-returns-to-scale model, the only form of disembodied technical progress consistent with these facts is Harrod-neutral technical progress. [See also Burmeister and Dobell (1970, p. 77-78).]

We let [k.sub.t] = [K.sub.t]/([[Lambda].sub.t][L.sub.t]) be the capital per unit of effective Worker and [n.sub.pt] = [[Lambda].sub.t][N.sub.pt]/([[Lambda].sub.t][L.sub.t]) be the share of private sector employment where [N.sub.pt] is the raw number of private sector workers. The real effective wage is [v.sub.t]/[[Lambda].sub.t] where [v.sub.t] is the real wage. The public sector produces an intermediate product, which contributes to the productivity of the private sector. Taking the intermediate input as given using capital and labor, private firms competitively produce output per effective worker [Mathematical Expression Omitted] where f ([center dot]) is the private production function, g, is the productive contribution of the public intermediate input, which affects private production in a Hicks-neutral manner, [Gamma] is the elasticity of private sector output with respect to [g.sub.t] and is between zero and unity. With suitable normalization, we define [g.sub.t] = [n.sub.gt], the share of public sector employment, assuming that the public sector uses only labor to produce the intermediate product.

2.1 Agents' optimization problems

Each private competitive firm solves the problem: Maximize [Mathematical Expression Omitted] where r, is the real rental rate. The first order conditions, also sufficient under our assumptions, are:

[Mathematical Expression Omitted]; (1)

[Mathematical Expression Omitted]. (2)

Equation (1) equates the marginal revenue product of a private sector worker to the private sector's real demand wage [([v.sub.t]/[[Lambda].sub.t]).sup.p] while (2) equates marginal revenue product of capital to the real rental rate.

With the public sector hiring workers competitively from a homogeneous labor pool, we derive the following condition:(2)

[Mathematical Expression Omitted]. (3)

Equation (3) equates the marginal revenue product of a public sector worker to the public sector's real demand wage [([v.sub.t]/[[Lambda].sub.t]).sup.g]. Making a simplifying assumption that the public sector wage bill is financed by an ad valorem wage income tax, the government budget constraint pins down the marginal rate of wage income tax: [[Tau].sub.t] = [n.sub.gt].(3)

We adopt Weil's (1989) set-up of infinitely-lived households extended by Harrod-neutral technical progress. Assuming each agent has a logarithmic utility function, we can derive the law of motion of aggregate per capita consumption given by

[Mathematical Expression Omitted] (4)

where [c.sub.t] is per effective capita aggregate consumption, p is the rate of time preference, and [w.sub.t] is the non-human wealth per unit of effective labor.

2.2 Wage equalization and goods market equilibrium

We assume that labor is mobile across sectors so wage equalization obtains in equilibrium. Noting that [n.sub.gt] + [n.sub.pt] = 1, we can depict (1) and (3) in the employment-wage diagram, which is familiar from the international trade literature and shown in fig. 2 for a given k. Equating [([v.sub.t]/[[Lambda].sub.t]).sup.p] = [([v.sub.t]/[[Lambda].sub.t]).sup.g], we obtain

[n.sub.gt] = [Gamma] f/[f.sub.2]. (5)

To see how a rise in k affects [n.sub.g], we ask how the two loci in fig. 2 shift in response to a rise in k. At any given employment share, using (5), we obtain

[Mathematical Expression Omitted]

where [Sigma] is the elasticity of substitution between capital and labor. Gaude (1985), in a critical survey of estimates of the elasticity of substitution between capital and labor, reports that most of the time-series estimates of elasticity are lower than unity, while the cross-section estimates are generally higher than the time-series estimates and close to unity. Lucas (1969), after contrasting the two approaches, concludes that the time-series estimates are preferred [see Summers (1981)]. In the analysis that follows, we use the maintained hypothesis that [Sigma] [less than] 1. As fig. 2 demonstrates, we then have the result that [n.sub.g] = [Phi](k) with [Phi][prime](k) [less than] 0. The intuition is simple: an increase in k raises the marginal productivity of labor and therefore raises the demand wage of the capital-intensive private sector more than the wage of the labor-intensive public sector such that the private sector employment expands while the public sector employment shrinks when the wage is equalized across the two sectors. Capital deepening (per effective worker) is hence associated with a declining share of public sector employment.

The goods market clearing condition is given by

[Mathematical Expression Omitted]. (6)

Using (2) in (4), and noting that the only nonhuman wealth held is capital, we also obtain the general equilibrium evolution of consumption:

[Mathematical Expression Omitted]. (7)

The evolution of the economy is summarized by (6) and (7), with saddle path stability, given the initial [k.sub.0]. [ILLUSTRATION FOR FIGURE 3 OMITTED].

Before proceeding to use the model to analyze the effects of a temporary increase in the rate of Harrod-neutral technical progress, it will be useful to understand the steady-state properties of the model. Note that along a balanced-growth path, with capital growing at the rate of effective labor, the public sector share of employment is constant. The share of labor in national income, [Mathematical Expression Omitted], being equal to [Gamma]/[n.sub.g] is also constant, since as just noted, public sector share of employment is a constant in the steady state. Finally, under our simplifying assumption that the public sector wage bill is financed by an ad valorem wage income tax, what is the share of taxes in GNP? By definition, the share of taxes in GNP is given by [Mathematical Expression Omitted], and hence is equal to [Tau][Gamma]/[n.sub.g]. However, as we have noted earlier, the government budget constraint pins down the marginal rate of wage income tax: [[Tau].sub.t] = [n.sub.gt]. Hence, the share of taxes in GNP is a constant. (Indeed, this property is true also outside the steady state. As the share of labor in national income declines during the period of unusually high productivity growth, the tax rate increases in tandem with the rise of the public sector share of employment to keep total taxes a constant share of national income.)

3. Effects of productivity shock

Suppose that the economy is initially at a steady state at time [t.sub.0]. It experiences an unanticipated increase in productivity growth rate from [[Lambda].sub.0] to [[Lambda].sub.1] ([[Lambda].sub.1] [greater than] [[Lambda].sub.0]) known to last until [t.sub.1] ([t.sub.1] [greater than] [t.sub.0]). Human wealth rises on account of higher productivity growth but declines on account of the upward shift of the term structure of real interest rates. If the former (latter) effect dominates, current consumption rises (falls). In either case, capital per effective worker gradually falls as the required investment ([Lambda][k.sub.t]) rises with higher [Lambda]. Fig. 3 shows that.along the transition path BC, [k.sub.t] gradually falls as capital accumulation fails to catch up with higher productivity growth. Since [n.sub.gt] = [Phi]([k.sub.t]) with [Phi][prime]([k.sub.t]) [less than] 0, it follows that during the period of high productivity growth, the public sector share of employment rises. However, at [t.sub.1], with [Lambda] reduced, [k.sub.t] begins to rise. During this phase of slower productivity growth, [n.sub.gt] gradually declines as the rising [k.sub.t] leads the private sector to offer higher real demand wages than the public sector, and so attracting more workers.

Our computer simulation also verifies the phase diagram analysis from fig. 3. Fig. 4 simulates the time paths of [n.sub.g], k, and c. The dotted lines denote the steady-state levels. At [t.sub.0] = 0, the productivity growth rate increases unexpectedly and temporarily till [t.sub.1] = 9.4089. (The appendix describes the computer simulation in more detail.) Consumption per effective worker c shoots up upon impact and decreases below the steady-state level till [t.sub.1] while ns increases, reaching a peak before [t.sub.1], and k decreases, touching a valley before [t.sub.1]. (The peak and valley occur at [t.sub.1] when a smaller deviation from the fixed point is used in the simulation program.) From [t.sub.1] onward, [n.sub.g] decreases, k increases, and c increases, converging to the steady-state levels. The shape of the simulated time trend of [n.sub.g] is similar to the actual time trend in fig. 1, which is still in the process of convergence.

4. Conclusion

We have developed a simple model to explain the trend of the share of public sector employment in the U.S. from 1950 to 1995. It may seem counter-intuitive that from 1950 to 1975 when the economy was enjoying a higher productivity growth, the private sector employment contracted while the public sector employment expanded. However, this is not surprising since the nature of productivity growth is labor-augmenting and the capital per effective worker becomes smaller, implying that the labor-intensive public sector would hire a greater share of the labor force. Beginning 1975, the U.S. economy experienced a productivity slowdown and our model easily predicts a declining share of public sector employment. Hence, our simple theory provides a neat account of the rising and then falling trend of the share of public sector employment of the U.S. economy from 1950 to 1995.

Notes

1. Since we will be interested in endogenizing the public sector share of employment, we have used a definition of public sector employment in this paper that excludes military personnel. The neoclassical growth model we develop will also not treat explicitly civilian unemployment. Since the latter is trendless over a period nearly half a century long, this omission is innocuous. Nevertheless, see Ho and Hoon (1997) for an attempt to link public sector employment to an endogenous natural rate of unemployment.

2. See Findlay and Wilson (1987) for a similar characterization of public sector demand for workers.

3. Note that this tax is non-distorting since labor supply is completely inelastic.

Appendix

In the simulation program to illustrate a temporary increase in productivity growth rate, a CES functional form is used:

f(k, [n.sub.p]) = [[[a.sub.1][k.sup.1-1/[Sigma]] + [a.sub.2][[n.sub.p].sup.1-1/[Sigma]]].sup.1/1-1/[Sigma]]

and the following parameters are used:

[Mu] = 0.014; [[Lambda].sub.0] = 0.015; [[Lambda].sub.1] = 0.02; [Rho] = 0.034; [Gamma] = 0.18; [Sigma] = 0.6; [a.sub.1] = 1/20; [a.sub.2] = 1.

Use (5) to express k = k([n.sub.p]), noting that [n.sub.g] = 1 - [n.sub.p]. Next, use (6) = 0 and (7) = 0 to eliminate c. Substituting k = k([n.sub.p]), we have an equation in [n.sub.p], which can be rearranged to define h([n.sub.p]) = 0. We then use Newton's Method to solve for the fixed point [Mathematical Expression Omitted]. We also compute [Mathematical Expression Omitted], [k.sup.*], and [c.sup.*].

Equations (6) and (7) describe the dynamics of [k.sub.t] and [c.sub.t]. The Jacobian can be calculated easily, and [n.sub.gt] is calculated from (5). We are now ready to simulate a temporary increase from [[Lambda].sub.0] to [[Lambda].sub.1].

We compute the dynamics in a backward manner. At date [t.sub.2] = 29.4089 (dates [t.sub.2] and [t.sub.1] are chosen such that the shock begins at to and k([t.sub.0]) = k([t.sub.2]), given a deviation, [Epsilon] = 0.00135, from the fixed point), perturb k and c by [Epsilon] in the direction given by the negative eigenvector from the Jacobian. Compute the corresponding [n.sub.g] from (5) numerically. Trace backward the time paths of [n.sub.g], k, and c under the laws of motion given 3.0 until [t.sub.1] = 9.4089. From [t.sub.1] to [t.sub.0], trace the time paths backward under the laws of motion given [[Lambda].sub.1]. Finally plot the time trends from [t.sub.0] to [t.sub.2].

The computer program is written in Matlab. Please contact the first-named author at kho@midway.uchicago.edu for the codes.

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