Government consumption and growth.

Evans, Paul

I. INTRODUCTION

Interest in formulating and testing growth models has surged in recent years. Following the seminal articles of Romer [1986] and Lucas [1988], the theoretical growth literature has primarily sought to endogenize the trend growth rate, which had been treated as an exogenous parameter in the earlier vintage of growth models formulated by Solow [1956] and Cass [1965] inter alia. Although the goal of endogenizing trend growth rates is surely desirable, it is still important to test whether trend growth rates are indeed better characterized as exogenous or endogenous.

Many papers have attempted to do so. Unfortunately, virtually all of them have used ordinary least squares to fit cross-sectional regressions relating the average growth rate of per capita output over some period to its initial value and other country characteristics.(1) My 1996a paper shows that this method produces seriously biased estimates and unreliable inferences.

This paper makes two contributions. First, using a simple stochastic growth model that nests both exogenous and endogenous growth, it shows that the growth rate of output per capita should be mean stationary if growth is exogenous, and should be difference stationary if growth is endogenous and any difference-stationary variable affects investment. Specifically, if the share of output devoted to government consumption is difference stationary, the growth rate should be mean stationary or difference stationary depending on whether growth is exogenous or endogenous. Furthermore, the growth rate should be negatively related to the share of output devoted to exhaustive government consumption if, and only if, growth is endogenous.

The second contribution is to test whether growth is exogenous or endogenous. Data for 92 countries over the period 1960 to 1989 strongly support the mean stationarity of the growth rate of per capita output and are consistent with the difference stationarity of the share of output devoted to government consumption. Moreover, the growth rate is not significantly related to the government share. These two findings support the exogeneity of growth. Finally, the relationship is precisely estimated to be weak if not nonexistent. Hence, even if growth is endogenous, the degree of endogeneity is likely to be small.(2)

The rest of the paper is organized as follows. Section II formulates a simple stochastic growth model for demonstrating the basic theoretical results. Section III investigates whether growth rates are mean stationary and whether shares of output devoted to government consumption are difference stationary. Section IV examines whether and how growth rates and government shares are related to each other. Finally, section V summarizes the paper and offers a few conclusions.

II. THEORETICAL DISCUSSION

This section lays out a simple and tractable stochastic growth model that nests both exogenous and endogenous growth. The model is special in many ways because several restrictive assumptions are made for the sake of tractability. Nevertheless, results similar to the ones derived below should hold in a wide class of models.

Consider a closed economy inhabited by a government and a household that lives in periods t = 0, 1, 2, .... The sole activity of the government is exhaustively consuming G, which is financed by a proportional tax on gross output Y at the rate [Tau]. Its budget constraint is therefore

(1) [G.sub.t] = [[Tau].sub.t] [Y.sub.t].

I assume that government consumption is set so that

(2) [g.sub.t] = - ln[1 - ([G.sub.t] / [Y.sub.t])]

is a driftless difference-stationary process.

The household is endowed with one unit of labor, which it supplies completely inelastically each period; a positive initial capital stock; and a technology

(3) [Mathematical Expression Omitted]

0 [less than] [Alpha] [less than or equal to] 1, t = 0, 1, 2, ...,

which transforms the one unit of labor and the services of the [K.sub.t] units of capital on hand at the beginning of period t into [Y.sub.t] units of gross output during period t. In equation (3), [Alpha] is a parameter, a is a difference-stationary productivity shock, and b is a mean-stationary productivity shock. Because the capital variable K aggregates all reproducible factors including human capital and all technical knowledge available to the economy and not available elsewhere, its output elasticity a may conceivably be one; that is, K may be subject to constant, rather than decreasing, returns to scale.

The stochastic variables g, a, and b are generated by the process

(4) [Mathematical Expression Omitted]

where [Phi](L) is a 3 x 3 matrix of pth-order polynomials in the lag operator L for which the lead term is the 3 x 3 identity matrix and all of the roots of det[[Phi](L)] lie outside the unit circle, p is a nonnegative integer, [Gamma] and [Delta] are parameters that equal the unconditional means of [Delta]a and b and w is a serially uncorrelated error vector with a zero mean and a positive definite covariance matrix. Because g must lie on the interval, [0, [infinity]), the entries in the first row and column of the covariance matrix are assumed to shrink toward zero as g approaches zero. In other words, the first entry of w must be heteroskedastic in order for g to be difference stationary as is assumed. According to equations (3) and (4), the difference-stationary component to productivity is Harrod neutral and the stationary component is Hicks neutral. This assumption is necessary so that the economy can have a stochastic balanced growth path with growth in output whether [Alpha] [less than] 1 or [Alpha] = 1.

Given its endowments and knowledge of the process generating g, a, and b and of their current and past realizations, the household maximizes the objective function

(5) [E.sub.0] [summation of] [[Beta].sup.t] ln[C.sub.t] where t = 0 to infinity, 0 [less than] [Beta] [less than] 1,

subject to the sequence of budget constraints

(6) [C.sub.t] + [I.sub.t] = (1 - [[Tau].sub.t])[Y.sub.t], t = 0, 1, 2, ...,

the sequence of accumulation equations

(7) [K.sub.t+1] - [K.sub.t] = [I.sub.t] - [K.sub.t], t = 1, 2, 3, ...,

the sequence of technological constraints (3), and a sequence of solvency constraints. In equations (5) through (7), [C.sub.t] and [I.sub.t] are consumption and gross investment during period t and [Beta] is a parameter. In order to make the model tractable, I have made three noteworthy simplifying assumptions: the production function is Cobb-Douglas; momentary utility is logarithmic; and capital depreciates completely in a single period - see equation (7).

The first-order conditions for the household's problem are

(8) [Mathematical Expression Omitted]

and

(9) [C.sub.t] + [K.sub.t+1] = (1 - [[Tau].sub.t])[Y.sub.t].

In turn, equations (8) and (9) imply that(3)

(10) [C.sub.t] = (1 - [Alpha][Beta])(1 - [[Tau].sub.t])[Y.sub.t].

and

(11) [K.sub.t+1] = [Alpha][Beta](1 - [[Tau].sub.t])[Y.sub.t].

Finally, equations (10) and (11) imply that the household is always solvent since the transversality condition

(12) [Mathematical Expression Omitted]

is always satisfied.

Equilibrium is characterized by the stochastic process (4) for g, a, and b; a positive value for [K.sub.0]; and functions that map [K.sub.t], [g.sub.t], [g.sub.t-1], ..., [g.sub.t-p], [a.sub.t], [a.sub.t-1], ..., [a.sub.t-p], and [b.sub.t], [b.sub.t-1], ..., [b.sub.t-p+1] onto [Y.sub.t], [C.sub.t] and [K.sub.t+1] and that satisfy equations (3), (10), and (11). Note that equations (1), (7) and (9) imply the equilibrium condition

(13) [C.sub.t] + [I.sub.t] + [G.sub.t] = [Y.sub.t];

that is, the household and government always spend the gross output of the closed economy.

Updating equation (3) one period, taking logarithms of both members of the resulting equation, and substituting from equations (11), (1), and (2) yields

[Mathematical Expression Omitted]

or

(14) [Mathematical Expression Omitted]

where [y.sub.t] [equivalent to] ln[Y.sub.t].(4)

I consider the two cases of output elasticity [Alpha] = 1 and [Alpha] [less than] 1 separately. In the former case, equation (14) reduces to

(15) [Delta][y.sub.t+1] = ln [Beta] - [g.sub.t] + [b.sub.t+1].

Equations (15) and (4) have five important implications. First, the economy would have an average growth rate of [Delta] + In [Beta], were government consumption always zero. Second, the growth rate is higher, the more productive is the economy and the more patient is the household. Third, purely transitory shocks to productivity induce permanent changes in the level of output. Fourth, increasing g induces a lower growth rate. Fifth, if one abstracts from fluctuations in b, the growth rate wanders around the interval (-[infinity], [Delta] + In [[Beta]] a s g wanders around the interval [0, [infinity]). For this reason, the growth rate is difference stationary.

Letting [Alpha] [less than] 1, differencing equation (14), and rearranging the resulting equation yields

(16) [Mathematical Expression Omitted].

Equations (16) and (4) have five important implications, which differ markedly from those of the previous paragraph. First, the trend growth rate is [Gamma] since the terms in the right-hand side of equation (16) have zero means. Second, the trend growth rate depends on neither the level of productivity nor the subjective discount rate. Third, purely transitory shocks to productivity induce only transitory changes in the level of output. Fourth, increasing g induces a lower output level but does not affect the average growth rate. Fifth, the growth rate is mean stationary.

These results should hold in a much wider class of models than those considered here. The reason is plain to see. Without diminishing returns, no force leads households to stop accumulating (decumulating) reproducible factors following shocks to government consumption or to restore the initial level of reproducible factors following transitory shocks to productivity. By contrast, with diminishing returns, the return to reproducible factors falls (rises), inducing consumption to rise (fall) and reproducible factors to decumulate (accumulate). Given that this result should hold quite generally, one should be able to test whether growth is exogenous simply by testing whether growth rates are mean stationary and whether government consumption affects the growth rate permanently. Sections III and IV formulate and implement such tests.

The result that the growth rate should be difference stationary if growth is endogenous should generalize to cases in which other variables affecting investment are difference stationary. Specifically, equation (8) suggests that permanent shifts in [Tau] permanently affect the growth rate of per capita consumption and hence the growth rate of per capita output. Whether the shift occurs because government consumption changes or because transfer payments, government regulation, or the security of property rights changes should be immaterial.

III. TESTING ORDERS OF INTEGRATION

According to section II, [Delta][Y.sub.nt], the growth rate of per capita output in some country n, should be difference stationary if growth in that country is endogenous and if any variable affecting investment is difference stationary. In particular, if the growth is endogenous and [g.sub.nt], the share of output devoted to government consumption for country n, is difference stationary, [Delta][y.sub.nt] should also be difference stationary. By contrast, [Delta][y.sub.nt] should be mean stationary if growth is exogenous.

Because the theory of the previous section suggests that g and [Delta]y are heteroskedastic, I employ the method of Phillips and Perron [1988] in order to test whether they are mean stationary. Figure 1 plots the test statistics obtained for 92 countries over the period 1960-1989.(5) The data come from the Penn World Tables of Summers and Heston [1991] as updated in 1993. I selected a window length of six years. The results for moderately shorter and longer window lengths are qualitatively similar. According to the figure, the null hypothesis that g is difference stationary can be rejected for only three of the 92 countries at the .05 significance level. Because three is less than .05 x 92, it is sensible to posit that g is difference stationary for all 92 countries. By contrast, the null hypothesis that [Delta]y is difference stationary can be rejected for all but seven of the 92 countries. Because Phillips-Perron tests on 30 annual observations have little power to reject the null of difference stationarity even against alternatives far from the null, this finding constitutes strong evidence that [Delta]y is mean stationary for all 92 countries.

IV. A BIVARIATE TEST OF EXOGENOUS VS. ENDOGENOUS GROWTH

The theory of section II suggests that

(17) [Delta][y.sub.nt] = [[Mu].sub.n] + [[Theta].sub.n][g.sub.n,t-1] + [e.sub.nt],

n = 1, 2, ..., N, t = 1, 2, ..., T.

where the [Mu]s and [Theta]s are parameters, the es are error terms, N is the number of countries under consideration, and T is the sample size for each country.(6) The parameter [[Theta].sub.n] is zero or negative depending on whether growth is exogenous or endogenous in country n. Under the null hypothesis that ([[Theta].sub.n] = 0), the error term [e.sub.nt] has a zero mean and finite variance. Under the alternative hypothesis that ([[Theta].sub.n] [less than] 0), it also has a zero mean and finite variance so long as [g.sub.nt] is the only difference-stationary variable that affects investment in country n. If instead other such variables exist and growth is endogenous, then [e.sub.nt] must be a driftless difference stationary process. Variables that affect investment and that might conceivably be difference stationary include the ratio of real transfer payments to output, the degree of market power in product and factor markets, the extent to which the government regulates business, and the security of property rights.

Suppose that each e has the representation

(18) [Mathematical Expression Omitted],

where [Lambda] is a time-specific fixed effect, the [Psi]s are parameters, [Epsilon] is an error term that is uncorrelated across countries,(7) and m is a finite nonnegative integer large enough that [Epsilon] is essentially uncorrelated with all leads and lags of [Delta]g. Substituting equation (18) into equation (17) yields

(19) [Mathematical Expression Omitted].

Under the null hypothesis

[Mathematical Expression Omitted],

equation (19) can be rewritten in the form

(20) [Mathematical Expression Omitted]

n = 1, 2, ..., N, t = 1,2, ... T,

with [Theta] = 0 and [u.sub.nt] = [[Epsilon].sub.nt]

Furthermore, the error term u has a zero mean and a finite variance. The appendix shows that applying ordinary least squares to equation (20) yields an estimator [[micro]meter] which converges in probability to zero as T approaches infinity. Moreover, using the method of Newey and West [1987] produces a t-ratio that converges in distribution to standard normal. By contrast, under the alternative hypothesis

[Mathematical Expression Omitted],

the t-ratio of [Mathematical Expression Omitted] diverges to -[infinity] as T approaches infinity.(8) As a result, the null hypothesis that all of the countries 1, 2, ..., N grow exogenously can be easily tested against the alternative hypothesis that at least one of them grows endogenously.(9)

Using ordinary least squares, I fitted equation (20) to the data described in the previous section for values of m ranging from zero to five years. I then used the method of Newey and West with a window length of ten years to calculate standard errors, t-ratios, and marginal significance levels. Table I reports the estimates, standard errors, t-ratios, and marginal significance levels obtained. All of the estimates are statistically insignificant at the .05 level on a one-tailed test. For this reason, the hypothesis that all of the countries grow exogenously cannot be rejected. Furthermore, the estimates are small in magnitude and precisely estimated. Therefore, even if some countries do grow endogenously, the degree of endogeneity is likely to be small.

V. SUMMARY AND CONCLUSIONS

Using a simple stochastic growth model that nests exogenous and endogenous growth, I show that the growth rate of output per capita should be mean stationary if growth is exogenous. If instead growth is endogenous and any variable that affects investment is difference stationary, the growth rate should be difference stationary. The paper presents strong evidence that the growth rate is in fact mean stationary. Evidence is also found that is consistent with the difference stationarity [TABULAR DATA FOR TABLE I OMITTED] of the share of output devoted to government consumption, a variable affecting investment. These two pieces of evidence support the hypothesis that growth is exogenous for the 92 countries considered here.

The model formulated in this paper also indicates that permanent increases in the government's share of output appreciably and permanently reduce the growth rate of output per capita if growth is endogenous but only lower its level if growth is exogenous. No evidence is found that the government's share permanently affects the growth rate. The hypothesis that growth is exogenous is therefore further supported. Moreover, even if growth is endogenous, my estimates indicate that the degree of endogeneity is likely to be small.

APPENDIX

Without essential loss of generality, I suppose that

(A.1) [Delta][y.sub.nt] = [Theta][g.sub.n,t-1] + [e.sub.nt]

and

(A.2) [Mathematical Expression Omitted],

where [g.sub.n0] = 0 and [[e.sub.nt], [v.sub.nt]] is an error vector that is independently and identically distributed across economies and over time with a zero mean vector, a covariance matrix [[Mathematical Expression Omitted], [Mathematical Expression Omitted]], and finite fourth moments. Under the null hypothesis [H.sub.0], [Theta] = 0. For this reason, the estimator [Mathematical Expression Omitted] obtained by applying ordinary least squares to equation (A.1) for a sample consisting of data for countries 1, 2, ..., N and periods 1, 2 ..., T takes the form

(A.3) [Mathematical Expression Omitted].

Suppose that T approaches infinity. One then has

(A.4) [Mathematical Expression Omitted].

Furthermore, Hamilton [1994, Chapter 19, Section 3] has shown that conditional on [G.sub.nT] = {[g.sub.n0], [g.sub.n1], ..., [g.sub.n,T-1]},

(A.5) [Mathematical Expression Omitted].

Hence, conditional on [G.sub.T] [equivalent to] {[G.sub.1T], [G.sub.2T], ..., [G.sub.NT]},

(A.6) [Mathematical Expression Omitted].

[Mathematical Expression Omitted].

Combining equations (A.4) and (A.6) then produces

(A.7) [Mathematical Expression Omitted]

with

(A.8) [Mathematical Expression Omitted].

Equation (A.7) implies that for any natural number N, [Mathematical Expression Omitted] converges in probability to zero as T approaches infinity. Without loss of generality, I set the window length for the Newey-West standard error of [Mathematical Expression Omitted] to zero. It is then given by

(A.9) [Mathematical Expression Omitted],

where [Mathematical Expression Omitted]. Because [Mathematical Expression Omitted] converges in probability to 0, [e.sub.nt] converges in probability to [e.sub.nt]. Consequently, conditional on [G.sub.T],

(A.10) [Mathematical Expression Omitted]

as T approaches infinity. The standard normal distribution, however, does not depend on [G.sub.T]. For this

reason, [Mathematical Expression Omitted] must converge unconditionally in distribution to standard normal as T approaches infinity for any given natural number N.

Under [H.sub.1], one must allow for the possibility that the [Theta]s are not identical. In that case, equation (A.1) should be replaced by

(A.11) [Delta][y.sub.nt] = [[Theta].sub.n][g.sub.n,t-1] + [e.sub.nt],

and equation (A.3) takes the form

(A.12) [Mathematical Expression Omitted]

with

(A.13) [Mathematical Expression Omitted].

Under [H.sub.1], the quantity [N.sup.-1] [summation of] [M.sub.n][[Theta].sub.n] where n = 1 to N converges in probability to a negative quantity, which I denote [Theta], since the Ms are positive and at least some of the [Theta]s are negative. As a result, [Mathematical Expression Omitted] behaves like [N.sup.1/2] T [Theta] asymptotically; that is, it diverges to - [infinity]. Hence, [Mathematical Expression Omitted] diverges under [H.sub.1]. This result holds whether e is mean stationary or difference stationary. Finally, note that the convergence under [H.sub.0] and the divergence under [H.sub.1] is faster, the larger N is.

1. A far from exhaustive list is Kormendi and Meguire [1985]; Baumol [1986]; De Long [1988]; Barro [1991]; De Long and Summers [1991]; Barro and Sala-i-Martin [1992]; Mankiw, Romer, and Weil [1992]; King and Levine [1993]; De Gregorio [1993]; Easterly [1993]; Easterly, Kremer, Pritchett, and Summers [1993]; Alesina and Rodrik [1994]; and Persson and Tabellini [1994].

2. After writing this paper, I learned of papers by Jones [1995] and Kocherlakota and Yi [1995] that estimate models similar to mine. Because these papers consider much smaller samples of countries than I consider here and do not pool their data, their tests are likely to be much less powerful than mine. Furthermore, they appear not to account properly in their estimation for the possibility of a common trend in the data. My 1996b paper is also related to this paper.

3. Equations (8) and (9) become identities if equations (10) and (11) are substituted into them.

4. The model can be easily extended to permit government consumption to generate productive services as in Barro [1990]. If equation (3) is replaced by

[Mathematical Expression Omitted],

equation (1) implies that

[Mathematical Expression Omitted]

with [Alpha] [equivalent to] [[Alpha].sub.1] / (1 - [[Alpha].sub.2]) and [Zeta] [equivalent to] [[Alpha].sub.2] / (1 - [[Alpha].sub.2]). The parameter

[Alpha] is less than or equal to one depending on whether [[Alpha].sub.1] + [[Alpha].sub.2] is less than or equal to one. Manipulations similar to those in the text then lead to

[Mathematical Expression Omitted].

Empirical analysis of this equation using the procedure of section IV yields small and statistically insignificant estimates of [Xi]. Therefore, the data analyzed here are consistent with the assumption of the text that government consumption is not productive.

5. I included all countries in the 98-country sample of Mankiw, Romer, and Weil for which the requisite data are available over the entire period 1960-1989 plus Taiwan. These countries are Algeria, Angola, Argentina, Australia, Austria, Bangladesh, Belgium, Benin, Bolivia, Botswana, Brazil, Burkina Faso, Burundi, Cameroon, Canada, Central African Republic, Chad, Chile, Colombia, Congo, Costa Rica, Denmark, Dominican Republic, Ecuador, Egypt, El Salvador, Finland, France, Germany, Ghana, Greece, Guatemala, Haiti, Honduras, Hong Kong, India, Indonesia, Ireland, Israel, Italy, Ivory Coast, Jamaica, Japan, Jordan, Kenya, Korea, Madagascar, Malawi, Mali, Malaysia, Mauritania, Mauritius, Mexico, Morocco, Mozambique, Myanmar, Netherlands, New Zealand, Niger, Nigeria, Norway, Pakistan, Panama, Papua New Guinea, Paraguay, Peru, Philippines, Portugal, Rwanda, Senegal, Singapore, Somalia, South Africa, Spain, Sri Lanka, Sweden, Switzerland, Syria, Taiwan, Thailand, Togo, Trinidad and Tobago, Tunisia, Turkey, Uganda, United Kingdom, United States, Venezuela, Zaire, Zambia, and Zimbabwe.

6. Let [[Beta].sub.n], [[Gamma].sub.n], [a.sub.nt], and [b.sub.nt] be the analogues of [Beta], [Gamma], [a.sub.t], and [b.sub.t] for country n. If growth is endogenous, equation (15) implies that [[Mu].sub.n] [equivalent to] ln [[Beta].sub.n], [[Theta].sub.n] [equivalent to] -1, and [e.sub.nt] [equivalent to] [b.sub.nt]. If growth is exogenous, equation (16) implies that [[Mu].sub.n] = [[Gamma].sub.n], [[Theta].sub.n] = 0, and

[Mathematical Expression Omitted]

7. Ideally, the error term [Epsilon] should be allowed to be correlated across countries. In practice, doing so would create insurmountable problems. The inclusion of the time-specific fixed effect [Lambda] in equation (18) may be sufficient, however, to account for most cross-country correlation. Moreover, the estimates obtained below are consistent even if [Epsilon] is correlated across countries and the standard errors and t-ratios obtained may not be seriously biased.

8. Divergence occurs whether [Epsilon] is mean stationary or difference stationary. This result is fortunate since difference-stationary variables other than the share of output devoted to government consumption are likely to affect investment.

9. Note that the formulation (20) permits complete cross-country heterogeneity in the [Mu]s and [Psi]s under [H.sub.0] and in the [Mu]s, [Phi]s, and [Theta]s under [H.sub.1].

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