# Employment and output effects of government spending: is government size important?

I. INTRODUCTION

In 1981 Barro proposed that temporary changes in government spending affect output more than permanent changes of the same size. He argued, this is because temporary changes have no wealth effects, and so have a larger impact on aggregate demand.(1) However, more recent research has shown that in a life-cycle model with an endogenous labor-leisure choice Barro's argument is reversed: permanent changes in government spending have a greater impact on employment and output than transitory changes of the same size, precisely because permanent changes are associated with wealth effects that influence the optimal supply of labor.(2)

The first objective in this paper is to examine possible differences in the effects of permanent and transitory changes in government spending. Government spending is decomposed for a number of countries into permanent and transitory components and their relative impacts on employment and output are examined.

My second objective is to investigate the relationship between public sector spending and economic growth and the relationship of economic growth to government size. Despite the importance of the subject, this is an area where no consensus exists as yet. For example, Landau |1983~ concluded that "growth of government hurts growth," whereas Kormendi and Meguire |1985~ found "no evidence that growth in the ratio of government consumption to output adversely affects economic growth," and Ram |1986~ reported that "government size has a positive effect on economic performance and growth."(3) Recently the above issue has become even more important because of the role played in some of the "new" growth theory models by externalities associated with the public sector. Barro |1990~ provides an example of such a model where government sector productivity and the size of government are important determinants of a country's growth rate. In the present paper, I isolate the "productivity effect" of government spending and then test its relationship to government size.

The empirical results suggest that permanent changes in government consumption generally have greater effects on output and employment than do transitory changes of the same size. This result, robust across all the different specifications examined, implies the existence of a negative wealth effect associated with permanent increases in government spending. It also reduces the potency of government spending as a stabilization tool. In addition, the empirical findings also support a negative relationship between the output effects of government spending and government size, even though the statistical significance of this relationship is sensitive to the choice of econometric methodology. Using Barro's |1990~ theoretical conclusions, the estimated equations imply that the optimal government size for the representative country is approximately 20 percent of GDP.

The rest of the paper is organized as follows. Section II presents the theoretical framework, and section III describes the econometric methodology. Section IV reports the empirical results. Finally, section V outlines the paper's main theoretical and policy implications.

II. THEORETICAL FRAMEWORK

Following Barro |1981~ and Aschauer |1989~, assume that government services are productive and let the aggregate production function take the Cobb-Douglas form:

(1) |y.sub.t~ = |a.sub.n~|n.sub.t~ + |a.sub.k~|k.sub.t~ + |a.sub.g~|g.sub.t~ + |u.sub.t~

with |u.sub.t~ = |Phi~ + |u.sub.t-1~ + |v.sub.t~, and where y is output, n is labor, k is the capital stock, g is government consumption (all variables in logarithms), and |Phi~ is the rate of technological progress. Government consumption enters as an input because it includes spending on the legal system, regulation, fire and police protection, correction, and national defense. To the extent that it allows for a more efficient allocation of property rights, g should have a positive marginal product. We want to estimate (1) for as many countries as possible, but because capital stock data exist only for a very small number of countries, k will be eliminated from the production function by using broadly accepted restrictions from economic theory. Under the assumption that the steady state capital-output ratio is constant, equation (1) becomes:(4)

(2) |Delta~|y.sub.t~ = (|Psi~ + |a.sub.n~|Delta~|n.sub.t~ + |a.sub.g~|Delta~|g.sub.t~ + |v.sub.t~)/(1-|a.sub.k~)

where |Delta~ is the difference operator. This equation will be estimated as

(3) |Delta~|y.sub.t~ = b + |b.sub.n~|Delta~|n.sub.t~ + |b.sub.g~|Delta~|g.sub.t~ + ||Epsilon~.sub.t~,

and under the additional assumption of constant returns to scale in capital and labor (|a.sub.n~ + |a.sub.k~ = 1), as(5)

(3|prime~) |Delta~|y.sub.t~ - |Delta~|n.sub.t~ = b + |b.sub.g~|g.sub.t~ + ||Epsilon~.sub.t~.

All estimated equations will be specified in first differences (growth rates) in order to avoid problems with nonstationarity and the spurious regression problem first emphasized by Granger and Newbold |1974~.

The production function implies that government spending affects output in two ways: first directly as an input (the "productivity" effect captured by |a.sub.g~ and |b.sub.g~), but also indirectly through its impact on employment, n. Equation (3), a structural equation, estimates the input effect. We can also estimate the overall effect by obtaining the reduced forms for y and n as functions of g.

Equation (2) and the assumption of profit maximization imply

(4) |Delta~|w.sub.t~ = ||Phi~ - (1-|a.sub.k~-|a.sub.n~)|Delta~|n.sub.t~ + |a.sub.g~|Delta~|g.sub.t~ + |v.sub.t~~/(1-|a.sub.k~)

where w is the logarithm of the real wage. Equation (4) is simply the demand for labor written in terms of the wage. Next, write the labor supply as

(5) |Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the permanent component of g. It is easy to show that in a life-cycle setting, labor supply is only affected by changes in government spending that are (or are perceived to be) permanent because only permanent changes in g have wealth effects.(6) If permanent increases in g are associated with a wealth loss, then |Zeta~ is positive.

At the labor market equilibrium, (4) and (5) can be combined to give the reduced form for employment and output. These will be estimated as

(6) |Mathematical Expression Omitted~

and

(7) |Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the transitory component of g. The appendix derives both equations and also shows that if there is a wealth loss (|Zeta~ |is greater than~ 0), permanent changes in g will affect employment and output more than transitory changes (|c.sub.1~|is greater than~|c.sub.2~ and |b.sub.1~|is greater than~|b.sub.2~), and thus the Barro effect is reversed. The reasons for this can be seen in Figures 1 and 2 that graphically examine the effects of transitory and permanent increases in government spending, respectively. In both figures, an increase in g shifts the demand for labor to the right because of the increase in labor productivity. The response of labor demand does not depend on whether the change is permanent or transitory. This is not true for labor supply, however. In Figure 1, where the increase in g is assumed to be temporary, there is no wealth effect and thus labor supply is unaffected. In Figure 2, where the increase in g is permanent and produces a wealth loss, labor supply shifts to the right. It follows that employment, and therefore output, will respond more to permanent than to transitory changes in government spending.(7)

III. EMPIRICAL METHODOLOGY

Our main sample consists of thirty-seven countries (Group I) for which data on output, the price level, and government consumption were available for at least thirty years. The data are from the I.M.F. International Financial Statistics. Employment data for sufficiently long time periods (at least twenty years) were available for only eighteen of the thirty-seven countries (Group II). The I.L.O. Yearbook of Labor Statistics was the source for employment data. Table I presents all the Group I countries along with a measure of government size and its variance for each.

Equation (3) will be estimated by two-stage least squares (2SLS) because of the possible endogeneity of n. Then, the relationship of the estimated |b.sub.g~'s to government size will be examined. Equations (6) and (7), the reduced forms for employment and output, and equation (3|prime~) can be consistently estimated by ordinary least squares (OLS).

The final issue that must be dealt with is the decomposition of government consumption into permanent and transitory components. Because every decomposition must be statistical, there are infinitely many ways to carry it out.(8) The traditional method of linear detrending is clearly not well-suited for our purposes because it generates a deterministic permanent component. In addition, since the influential work of Nelson and Plosser |1982~, it has been recognized that stochastic trends are useful in the modelling of economic time series. Guided by this, our first choice is the decomposition method first proposed by Beveridge and Nelson |1981~, and later modified by Cuddington and Winters |1987~, Miller |1988~, and Newbold |1990~. Watson |1986~ has showed that the Beveridge-Nelson method belongs to a class of optimal decompositions.

An alternative decomposition method, that will be employed in the next section, was the one proposed by Hodrick and Prescott |1980~ and used in the study of business cycles.(9) Kydland and Prescott |1990~ discuss this method, its selection criteria, and its uses. An additional attractive feature of the Hodrick-Prescott filter is that it does not so much rely on the "permanent vs transitory" nature of the series, as on the "persistent vs cyclical" distinction. Put differently, this approach, unlike the Beveridge-Nelson method, does not depend on the existence of a unit root in the series and, as a result, does not produce a trend component that is by construction a random walk.(10) This property is desirable in our present application, because it may not be permanence per se, but rather the persistence of government consumption that in fact affects output and labor supply decisions.

The next section presents the empirical results. All equations were estimated using both the Beveridge-Nelson (BN) and the Hodrick-Prescott (HP) decompositions; the results obtained are in most cases qualitatively similar. Because the Hodrick-Prescott technique has the advantage discussed in the previous paragraph, and in order to preserve space, I report only the Hodrick-Prescott results. In the single case where the two methods provide different results, both are presented. All results are available on request.

TABULAR DATA OMITTED

IV. EMPIRICAL RESULTS

Table II reports the estimated reduced-form equations for employment and output in terms of permanent and transitory government consumption. All variables are in logarithms. The output equations are generally estimated more successfully than the employment equations, which is not surprising given that most of the latter have fewer degrees of freedom. The coefficients usually have the expected sign (positive) and are often statistically significant. Note that all coefficients with the wrong sign are statistically insignificant. Our main interest at this point is to examine whether the effects of permanent and transitory government consumption are the same. In terms of equations (7) and (6), the null hypotheses are |b.sub.1~ = |b.sub.2~ and |c.sub.1~ = |c.sub.2~. These are tested by the F-statistic, whose significance level is reported in Table II. Recall that if a permanent increase in g leads to a wealth loss, the theory predicts that the permanent coefficients must be generally larger. In Table II, this is clearly verified for output: for seventeen countries, |c.sub.1~ = |c.sub.2~ can be rejected at the 10 percent level in favor of |c.sub.1~ |is greater than~ |c.sub.2~. In addition, there is no country where the transitory effect has a statistically significantly larger impact, and therefore no country where the Barro hypothesis is verified. Thus it seems wealth effects do exist, and permanent (or persistent) changes in government consumption tend to raise output more than do transitory changes of the same size.

The evidence is less conclusive on employment, however. There are only two countries (Australia and Japan) for which we can reject |b.sub.1~ = |b.sub.2~ in favor of |b.sub.1~ |is greater than~ |b.sub.2~ at the 10 percent level, and there are two other countries (Italy and Sweden) for which the inequality seems to hold in the opposite direction. It seems plausible, however, that this is a result of the limited number of observations we have for employment for most countries. As shown below, the ambiguity disappears when the data are pooled and the greater number of degrees of freedom allows for better identification of the parameters.

Next, we estimate equations (3) and (3|prime~) for the eighteen Group II countries. The method of estimation for equation (3) is two-stage least squares in order to account for the possible endogeneity of employment. The instruments include changes in the logarithm of contemporaneous and lagged population, |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, |Delta~|n.sub.t-1~, |Delta~|g.sub.t~, and |Delta~|g.sub.t-1~. Equation (3|prime~) can be consistently estimated with OLS. The results are reported in Table III. With the exception of the U.K., the estimates for |b.sub.g~ are positive and often significant. Once more, the small number of degrees of freedom may be preventing a tighter estimation of the |b.sub.g~ coefficient.

We also want to investigate the relationship between the output elasticity of government consumption and government size, s. The index s is defined as "average" government size, and it is calculated over the relevant time periods (dictated by employment data availability) as

|s.sub.i~ = |summation over t~|g.sub.it~ / |summation over t~|y.sub.it~

Figures 3 and 4 plot the estimates of |b.sub.g~ against s for the Hodrick-Prescott and Beveridge-Nelson decompositions, respectively.(11) The Pearson correlation coefficients between the two variables are (significance levels in parentheses):

Both methods of decomposition imply a negative relationship between the output elasticity of government consumption and government size, even though only for the Beveridge-Nelson estimates does this relationship appear to be statistically significant. There is evidence, therefore, that government consumption is productive, but its output elasticity falls with increases in government size.(12)

Evidence from Panel Data

A significant drawback of some of the regressions reported above is that their degrees of freedom were limited. An obvious way to remedy this within the present framework is to pool the data sets and conduct time series-cross section regression.(13) Table IV contains the panel regression results.

Panels A and B of Table IV estimate the reduced forms for output and employment, respectively. All coefficients have the right sign and are statistically significant, except for the coefficients of |Mathematical Expression Omitted~ on employment, which are negative but statistically not different from zero. The F-statistics test the hypothesis that the coefficients of permanent and transitory changes in g are equal. As expected given our previous results, this hypothesis can be rejected for output in favor of the alternative that permanent effects are greater. Interestingly, the null hypothesis is also rejected for employment, again in favor of the alternative that permanent changes in g have a greater impact. In addition, these findings are robust to different models for the error term.

We next examine the importance of government size. Equation (3) can be written as

(8) |Delta~|y.sub.it~ = |b.sub.0~ + |b.sub.n~|Delta~|n.sub.it~ + |b.sub.g~|Delta~|g.sub.it~ + ||Epsilon~.sub.it~,

and |b.sub.g~ can be allowed to vary across countries as a function of government size: |b.sub.gi~ = |Gamma~ + |Delta~|s.sub.i~. A negative |Delta~ would imply a negative relationship between the output elasticity of g and government size. Substituting the expression for |b.sub.g~ into (8), the parameter |Delta~ can be estimated from

(9) |Delta~|y.sub.it~ = |b.sub.0~ + |b.sub.n~|Delta~|n.sub.it~ + |Gamma~|Delta~|g.sub.it~ + |Delta~|s.sub.i~|Delta~|g.sub.it~ + ||Epsilon~.sub.it~.

Panels C and D of Table IV estimate several versions of equations (8) and (9). Instrumental variables estimation with the usual list of instruments is applied to the equations that include employment as an explanatory variable. The first interesting result is that the coefficients of |Delta~g are positive and statistically significant in all specifications. This means that government consumption has a positive marginal product. Equally interesting is that |Delta~ is estimated to be negative by the interaction terms of Panel D, even though it is statistically significant only in the last specification. Note also, that the last specification is in terms of labor productivity growth and estimated with OLS. This means that the decomposition method is irrelevant in this case (since |Mathematical Expression Omitted~ is only used as an instrument in some structural equations). TABULAR DATA OMITTED TABULAR DATA OMITTED TABULAR DATA OMITTED These results seem to suggest again, that while government consumption is productive, its output elasticity depends negatively on government size.

V. IMPLICATIONS AND CONCLUSIONS

The first finding of this paper is that permanent changes in government spending (like building more schools or enacting a new bureaucracy) have a greater impact on employment and output than transitory changes (like a commitment to send a man to Mars or a brief military engagement). From Table IV, a 1 percent permanent increase in g raises output in the representative country by 0.5 percent, whereas a 1 percent transitory increase raises output by only 0.1 percent. Theoretically, this means that, unlike transitory increases, permanent increases in g produce wealth losses that raise the optimal labor supply.

This finding also has implications for stabilization policy. Since permanent changes in government consumption have no cyclical effects, it is only transitory changes in g that can be used for stabilization purposes. Therefore, the paper's conclusion that the multipliers of transitory changes in g are generally small reduces the attractiveness of government consumption as a policy variable. In addition, the general imprecision with which the responses of output to transitory changes are estimated introduces significant uncertainty to the potency of g as a policy variable and thus reduces its countercyclical value (see Brainard |1967~).

The second finding of the paper is that government consumption is generally productive and thus belongs in the production function. Using the estimate of 0.281 for |b.sub.g~ from the second panel structural equation, and a value of 0.30 for the capital share (Maddison |1990~, Table 7), the output elasticity of government consumption for the representative country in our sample is |a.sub.g~ = |b.sub.g~(1-|a.sub.k~) = 0.20. This means that a 1 percent increase in g, holding all other inputs constant, produces an average 0.2 percent increase in output.

A third finding is that the output elasticity itself varies inversely with government size. To illustrate, consider three countries with different government sizes for the period examined. At the low end of the spectrum, Japan has a government size of less than 10 percent of GDP. At the other extreme, the government size in Sweden is 23 percent. For Germany, in the middle, it is 18 percent. Maintaining the 0.30 value for the capital share, and using |b.sub.g~ estimates from Table III, the output elasticities of government consumption in these countries can be calculated as follows: .32 in Japan, .22 in Germany (both statistically significant), and .11 in Sweden (not statistically significant). These numbers imply that a 1 percent increase in g, holding all other inputs constant, will raise output by 0.32 percent in Japan, 0.22 percent in Germany, and 0.11 percent in Sweden. In addition, these estimates have implications about the optimal government size in each of these countries. Using as a rule of thumb Barro's |1990~ conclusion that the optimal government size, s, is equal to the output elasticity of g, |a.sub.g~, one can conclude that the public sector is too small in Japan, too large in Sweden, and has about the right size in Germany. We can also go one step further. Writing |b.sub.g~ = |Gamma~ + |Beta~s, Barro's rule of s = |a.sub.g~ becomes s = (|Gamma~ + |Delta~s)(1 - |a.sub.k~) which gives

(10) s = (1-|a.sub.k~)|gamma~ / |1-|Delta~(1-|a.sub.k~)~

as the optimal government size for the representative country. Using our estimates for |Gamma~ and |Delta~ from the last two panel regressions, we obtain s = 0.19 (interestingly, both regressions imply almost the same s despite different estimates for |Delta~ and |Gamma~). Note also that this is almost identical to the 0.20 estimate for |a.sub.g~, as estimated independently above.

Another way to illustrate this result is to write the marginal product of government consumption as MPG = |a.sub.g~/s = (1-|a.sub.k~)(|Delta~+|Gamma~/s). Using |Delta~ = -.876 and |Gamma~ = .443 from Table IV, Figure 5 plots the marginal product of g against government size for |a.sub.k~ = 0.3 and |a.sub.k~ = 0.4. Barro's rule requires MPG = 1, and this is satisfied around s = 0.20 for both plausible values for capital's share. We conclude that the optimal government size for the representative country in our sample is approximately 20 percent.

APPENDIX

From (1) the logarithm of the marginal product of capital is

(11) log(|MPK.sub.t~) = log(|a.sub.k~) + |y.sub.t~ - |k.sub.t~.

Using (11) we can write (1) as

(12) |y.sub.t~ = ||a.sub.n~|n.sub.t~ + |a.sub.k~log(|a.sub.k~/|MPK.sub.t~) + |a.sub.g~|g.sub.t~ + |u.sub.t~~/(1-|a.sub.k~).

We assume that at the steady state the marginal product of capital is constant. From (11) this is equivalent to assuming that the steady state capital-output ratio is constant. Equation (12) can be then written as (2). We can also allow for deviations from the steady state by specifying |(1-|a.sub.k~).sup.-1~/||u.sub.t~ + |a.sub.k~log(|a.sub.k~/|MPK.sub.t~)~ as a random walk, in which case we again obtain (2) and (3). The parameter |a.sub.g~ is not identified in (2), but |b.sub.g~ is monotonically increasing in |a.sub.g~. Estimation of |b.sub.g~ provides a measure of the input effect of government spending on output. If the share of capital, |a.sub.k~, is not dramatically different across countries (or more precisely, if differences in |a.sub.k~ across countries do not depend on government size, which is a weaker requirement), then any relationship between |b.sub.g~ and government size would indicate the same relationship between |a.sub.g~ and government size.

Profit maximization requires that the log of the real wage be

(13) |w.sub.t~ = log(|a.sub.n~) + |y.sub.t~ - |n.sub.t~,

which using (12) gives (4). Combining (4) and (5) gives

(14) |Mathematical Expression Omitted~

where

|Psi~ = ||1+|Xi~(1-|a.sub.n~-|a.sub.k~)|(1-|a.sub.k~).sup.-1~~.sup.-1~ |is greater than~ 0.

This is estimated as (6). Note that if |Zeta~ |is greater than~ 0, (14) implies that |c.sub.1~ |is greater than~ |c.sub.2~. Finally, (6) can be used to eliminate n from (3). The reduced form for output is thus obtained:

(15) |Mathematical Expression Omitted~

which is estimated as (7). Again, if |Zeta~ |is greater than~ 0, (15) implies |b.sub.1~ |is greater than~ |b.sub.2~.

1. Ahmed |1986~ also developed and estimated a similar model for the United Kingdom.

2. See Christiano and Eichenbaum |1988~, Aiyagari, Christiano and Eichenbaum |1990~, and Karras |1990~.

3. Reversing the causal relationship, Conte and Darrat |1988~ found a feedback from real economic growth to the public sector size for approximately ten OECD countries (half of their sample).

4. Most of the following equations are derived in the appendix.

5. Romer |1990~ gives a convincing argument why production functions should exhibit constant returns to scale in rival factors of production.

6. It is plausible that in reality it may be persistence, rather than permanence, of changes in government spending that matters for labor supply decisions. In that case, the more persistent the government spending process, the larger labor effects will be. This issue is addressed by our choice of the Hodrick-Prescott filter as one of the decomposition methods in the empirical section.

7. It also follows from this analysis that the wage will respond more to transitory than to permanent changes in government spending. The only reason why this is not tested by estimating a reduced form for the wage is that wage data could not be obtained for a large number of countries. Karras |1990~ finds that U.S. data are consistent with the wage hypothesis.

8. See Christiano and Eichenbaum |1989~ for this and related issues. Durlauf and Phillips |1988~ present a different and less agnostic point of view.

9. The HP approach defines the trend component, |g.sup.P~, as the one that minimizes

|Mathematical Expression Omitted~

For the purposes of this study, a number of different values for |Lambda~ were tried, including |Lambda~ = 50, 100, 150, 200, and 500. They all gave very similar results. Results reported are for |Lambda~ = 100, the value suggested by Kydland and Prescott |1989~.

10. It may be worth noting here, however, that the Beveridge-Nelson identification assumption is not at all implausible, since stationarity tests (not reported here but available on request) on the levels and first differences of government consumption show convincingly that unit roots on the levels do exist.

11. For an example of a similar methodology see Kormendi and Meguire |1984~.

12. Note that this negative relationship is not simply one between government size, s, and the marginal product of government spending. From (1) it follows that the marginal product of government spending is |a.sub.g~(Y/G) = |a.sub.g~|s.sup.-1~, and it is therefore by construction negatively related to s. What the negative correlations of Figures 1 and 2 (and also evidence from the next section) suggest is that the parameter |a.sub.g~ itself varies inversely with government size, which is a more interesting and novel finding.

13. The estimated equations become |z.sub.it~ = f(|v.sub.it~) + |e.sub.it~, where z is the dependent variable, v is a vector of explanatory variables, and e the error term. This can be estimated with OLS if e is uncorrelated both across countries and time. It is likely, however, that e will have fixed effects (|e.sub.it~ = |u.sub.i~ + ||Epsilon~.sub.it~ or |e.sub.it~ = |u.sub.i~ + |h.sub.t~ + ||Epsilon~.sub.it~) in which case GLS is required. If, in addition, some of the right-hand-side variables are endogenous, then instrumental variables must be used. In addition, ||Epsilon~.sub.it~ was allowed to be autoregressive, ||Epsilon~.sub.it~ = |Rho~||Epsilon~.sub.it-1~, or ||Epsilon~.sub.it~ = ||Rho~.sub.i~||Epsilon~.sub.it-1~, but the estimated |Rho~'s were statistically insignificant.

REFERENCES

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Aiyagari, S. Rao, Lawrence J. Christiano and Martin Eichenbaum. "The Output, Employment, and Interest Rate Effects of Government Consumption." Discussion Paper 25, Federal Reserve Bank of Minneapolis, 1990.

Aschauer, David Alan. "Is Public Expenditure Productive?" Journal of Monetary Economics, March 1989, 177-200.

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-----. "Government Spending in a Simple Model of Endogenous Growth." Journal of Political Economy 98(5), 1990, S103-S125.

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Christiano, Lawrence J., and Martin Eichenbaum. "Is Theory Really Ahead of Measurement? Current Real Business Cycle Theories and Aggregate Labor Market Fluctuations." Working Paper No. 412, Federal Reserve Bank of Minneapolis, 1988.

-----. "Unit Roots in Real GNP: Do We Know, and Do We Care?" NBER Working Paper No. 3130, 1989.

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In 1981 Barro proposed that temporary changes in government spending affect output more than permanent changes of the same size. He argued, this is because temporary changes have no wealth effects, and so have a larger impact on aggregate demand.(1) However, more recent research has shown that in a life-cycle model with an endogenous labor-leisure choice Barro's argument is reversed: permanent changes in government spending have a greater impact on employment and output than transitory changes of the same size, precisely because permanent changes are associated with wealth effects that influence the optimal supply of labor.(2)

The first objective in this paper is to examine possible differences in the effects of permanent and transitory changes in government spending. Government spending is decomposed for a number of countries into permanent and transitory components and their relative impacts on employment and output are examined.

My second objective is to investigate the relationship between public sector spending and economic growth and the relationship of economic growth to government size. Despite the importance of the subject, this is an area where no consensus exists as yet. For example, Landau |1983~ concluded that "growth of government hurts growth," whereas Kormendi and Meguire |1985~ found "no evidence that growth in the ratio of government consumption to output adversely affects economic growth," and Ram |1986~ reported that "government size has a positive effect on economic performance and growth."(3) Recently the above issue has become even more important because of the role played in some of the "new" growth theory models by externalities associated with the public sector. Barro |1990~ provides an example of such a model where government sector productivity and the size of government are important determinants of a country's growth rate. In the present paper, I isolate the "productivity effect" of government spending and then test its relationship to government size.

The empirical results suggest that permanent changes in government consumption generally have greater effects on output and employment than do transitory changes of the same size. This result, robust across all the different specifications examined, implies the existence of a negative wealth effect associated with permanent increases in government spending. It also reduces the potency of government spending as a stabilization tool. In addition, the empirical findings also support a negative relationship between the output effects of government spending and government size, even though the statistical significance of this relationship is sensitive to the choice of econometric methodology. Using Barro's |1990~ theoretical conclusions, the estimated equations imply that the optimal government size for the representative country is approximately 20 percent of GDP.

The rest of the paper is organized as follows. Section II presents the theoretical framework, and section III describes the econometric methodology. Section IV reports the empirical results. Finally, section V outlines the paper's main theoretical and policy implications.

II. THEORETICAL FRAMEWORK

Following Barro |1981~ and Aschauer |1989~, assume that government services are productive and let the aggregate production function take the Cobb-Douglas form:

(1) |y.sub.t~ = |a.sub.n~|n.sub.t~ + |a.sub.k~|k.sub.t~ + |a.sub.g~|g.sub.t~ + |u.sub.t~

with |u.sub.t~ = |Phi~ + |u.sub.t-1~ + |v.sub.t~, and where y is output, n is labor, k is the capital stock, g is government consumption (all variables in logarithms), and |Phi~ is the rate of technological progress. Government consumption enters as an input because it includes spending on the legal system, regulation, fire and police protection, correction, and national defense. To the extent that it allows for a more efficient allocation of property rights, g should have a positive marginal product. We want to estimate (1) for as many countries as possible, but because capital stock data exist only for a very small number of countries, k will be eliminated from the production function by using broadly accepted restrictions from economic theory. Under the assumption that the steady state capital-output ratio is constant, equation (1) becomes:(4)

(2) |Delta~|y.sub.t~ = (|Psi~ + |a.sub.n~|Delta~|n.sub.t~ + |a.sub.g~|Delta~|g.sub.t~ + |v.sub.t~)/(1-|a.sub.k~)

where |Delta~ is the difference operator. This equation will be estimated as

(3) |Delta~|y.sub.t~ = b + |b.sub.n~|Delta~|n.sub.t~ + |b.sub.g~|Delta~|g.sub.t~ + ||Epsilon~.sub.t~,

and under the additional assumption of constant returns to scale in capital and labor (|a.sub.n~ + |a.sub.k~ = 1), as(5)

(3|prime~) |Delta~|y.sub.t~ - |Delta~|n.sub.t~ = b + |b.sub.g~|g.sub.t~ + ||Epsilon~.sub.t~.

All estimated equations will be specified in first differences (growth rates) in order to avoid problems with nonstationarity and the spurious regression problem first emphasized by Granger and Newbold |1974~.

The production function implies that government spending affects output in two ways: first directly as an input (the "productivity" effect captured by |a.sub.g~ and |b.sub.g~), but also indirectly through its impact on employment, n. Equation (3), a structural equation, estimates the input effect. We can also estimate the overall effect by obtaining the reduced forms for y and n as functions of g.

Equation (2) and the assumption of profit maximization imply

(4) |Delta~|w.sub.t~ = ||Phi~ - (1-|a.sub.k~-|a.sub.n~)|Delta~|n.sub.t~ + |a.sub.g~|Delta~|g.sub.t~ + |v.sub.t~~/(1-|a.sub.k~)

where w is the logarithm of the real wage. Equation (4) is simply the demand for labor written in terms of the wage. Next, write the labor supply as

(5) |Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the permanent component of g. It is easy to show that in a life-cycle setting, labor supply is only affected by changes in government spending that are (or are perceived to be) permanent because only permanent changes in g have wealth effects.(6) If permanent increases in g are associated with a wealth loss, then |Zeta~ is positive.

At the labor market equilibrium, (4) and (5) can be combined to give the reduced form for employment and output. These will be estimated as

(6) |Mathematical Expression Omitted~

and

(7) |Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the transitory component of g. The appendix derives both equations and also shows that if there is a wealth loss (|Zeta~ |is greater than~ 0), permanent changes in g will affect employment and output more than transitory changes (|c.sub.1~|is greater than~|c.sub.2~ and |b.sub.1~|is greater than~|b.sub.2~), and thus the Barro effect is reversed. The reasons for this can be seen in Figures 1 and 2 that graphically examine the effects of transitory and permanent increases in government spending, respectively. In both figures, an increase in g shifts the demand for labor to the right because of the increase in labor productivity. The response of labor demand does not depend on whether the change is permanent or transitory. This is not true for labor supply, however. In Figure 1, where the increase in g is assumed to be temporary, there is no wealth effect and thus labor supply is unaffected. In Figure 2, where the increase in g is permanent and produces a wealth loss, labor supply shifts to the right. It follows that employment, and therefore output, will respond more to permanent than to transitory changes in government spending.(7)

III. EMPIRICAL METHODOLOGY

Our main sample consists of thirty-seven countries (Group I) for which data on output, the price level, and government consumption were available for at least thirty years. The data are from the I.M.F. International Financial Statistics. Employment data for sufficiently long time periods (at least twenty years) were available for only eighteen of the thirty-seven countries (Group II). The I.L.O. Yearbook of Labor Statistics was the source for employment data. Table I presents all the Group I countries along with a measure of government size and its variance for each.

Equation (3) will be estimated by two-stage least squares (2SLS) because of the possible endogeneity of n. Then, the relationship of the estimated |b.sub.g~'s to government size will be examined. Equations (6) and (7), the reduced forms for employment and output, and equation (3|prime~) can be consistently estimated by ordinary least squares (OLS).

The final issue that must be dealt with is the decomposition of government consumption into permanent and transitory components. Because every decomposition must be statistical, there are infinitely many ways to carry it out.(8) The traditional method of linear detrending is clearly not well-suited for our purposes because it generates a deterministic permanent component. In addition, since the influential work of Nelson and Plosser |1982~, it has been recognized that stochastic trends are useful in the modelling of economic time series. Guided by this, our first choice is the decomposition method first proposed by Beveridge and Nelson |1981~, and later modified by Cuddington and Winters |1987~, Miller |1988~, and Newbold |1990~. Watson |1986~ has showed that the Beveridge-Nelson method belongs to a class of optimal decompositions.

An alternative decomposition method, that will be employed in the next section, was the one proposed by Hodrick and Prescott |1980~ and used in the study of business cycles.(9) Kydland and Prescott |1990~ discuss this method, its selection criteria, and its uses. An additional attractive feature of the Hodrick-Prescott filter is that it does not so much rely on the "permanent vs transitory" nature of the series, as on the "persistent vs cyclical" distinction. Put differently, this approach, unlike the Beveridge-Nelson method, does not depend on the existence of a unit root in the series and, as a result, does not produce a trend component that is by construction a random walk.(10) This property is desirable in our present application, because it may not be permanence per se, but rather the persistence of government consumption that in fact affects output and labor supply decisions.

The next section presents the empirical results. All equations were estimated using both the Beveridge-Nelson (BN) and the Hodrick-Prescott (HP) decompositions; the results obtained are in most cases qualitatively similar. Because the Hodrick-Prescott technique has the advantage discussed in the previous paragraph, and in order to preserve space, I report only the Hodrick-Prescott results. In the single case where the two methods provide different results, both are presented. All results are available on request.

TABULAR DATA OMITTED

IV. EMPIRICAL RESULTS

Table II reports the estimated reduced-form equations for employment and output in terms of permanent and transitory government consumption. All variables are in logarithms. The output equations are generally estimated more successfully than the employment equations, which is not surprising given that most of the latter have fewer degrees of freedom. The coefficients usually have the expected sign (positive) and are often statistically significant. Note that all coefficients with the wrong sign are statistically insignificant. Our main interest at this point is to examine whether the effects of permanent and transitory government consumption are the same. In terms of equations (7) and (6), the null hypotheses are |b.sub.1~ = |b.sub.2~ and |c.sub.1~ = |c.sub.2~. These are tested by the F-statistic, whose significance level is reported in Table II. Recall that if a permanent increase in g leads to a wealth loss, the theory predicts that the permanent coefficients must be generally larger. In Table II, this is clearly verified for output: for seventeen countries, |c.sub.1~ = |c.sub.2~ can be rejected at the 10 percent level in favor of |c.sub.1~ |is greater than~ |c.sub.2~. In addition, there is no country where the transitory effect has a statistically significantly larger impact, and therefore no country where the Barro hypothesis is verified. Thus it seems wealth effects do exist, and permanent (or persistent) changes in government consumption tend to raise output more than do transitory changes of the same size.

The evidence is less conclusive on employment, however. There are only two countries (Australia and Japan) for which we can reject |b.sub.1~ = |b.sub.2~ in favor of |b.sub.1~ |is greater than~ |b.sub.2~ at the 10 percent level, and there are two other countries (Italy and Sweden) for which the inequality seems to hold in the opposite direction. It seems plausible, however, that this is a result of the limited number of observations we have for employment for most countries. As shown below, the ambiguity disappears when the data are pooled and the greater number of degrees of freedom allows for better identification of the parameters.

Next, we estimate equations (3) and (3|prime~) for the eighteen Group II countries. The method of estimation for equation (3) is two-stage least squares in order to account for the possible endogeneity of employment. The instruments include changes in the logarithm of contemporaneous and lagged population, |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, |Delta~|n.sub.t-1~, |Delta~|g.sub.t~, and |Delta~|g.sub.t-1~. Equation (3|prime~) can be consistently estimated with OLS. The results are reported in Table III. With the exception of the U.K., the estimates for |b.sub.g~ are positive and often significant. Once more, the small number of degrees of freedom may be preventing a tighter estimation of the |b.sub.g~ coefficient.

We also want to investigate the relationship between the output elasticity of government consumption and government size, s. The index s is defined as "average" government size, and it is calculated over the relevant time periods (dictated by employment data availability) as

|s.sub.i~ = |summation over t~|g.sub.it~ / |summation over t~|y.sub.it~

Figures 3 and 4 plot the estimates of |b.sub.g~ against s for the Hodrick-Prescott and Beveridge-Nelson decompositions, respectively.(11) The Pearson correlation coefficients between the two variables are (significance levels in parentheses):

|b.sub.g~ |b.sub.g~ Hodrick-Prescott Beveridge-Nelson s -.232 -.517 (.355) (.028)

Both methods of decomposition imply a negative relationship between the output elasticity of government consumption and government size, even though only for the Beveridge-Nelson estimates does this relationship appear to be statistically significant. There is evidence, therefore, that government consumption is productive, but its output elasticity falls with increases in government size.(12)

Evidence from Panel Data

A significant drawback of some of the regressions reported above is that their degrees of freedom were limited. An obvious way to remedy this within the present framework is to pool the data sets and conduct time series-cross section regression.(13) Table IV contains the panel regression results.

Panels A and B of Table IV estimate the reduced forms for output and employment, respectively. All coefficients have the right sign and are statistically significant, except for the coefficients of |Mathematical Expression Omitted~ on employment, which are negative but statistically not different from zero. The F-statistics test the hypothesis that the coefficients of permanent and transitory changes in g are equal. As expected given our previous results, this hypothesis can be rejected for output in favor of the alternative that permanent effects are greater. Interestingly, the null hypothesis is also rejected for employment, again in favor of the alternative that permanent changes in g have a greater impact. In addition, these findings are robust to different models for the error term.

We next examine the importance of government size. Equation (3) can be written as

(8) |Delta~|y.sub.it~ = |b.sub.0~ + |b.sub.n~|Delta~|n.sub.it~ + |b.sub.g~|Delta~|g.sub.it~ + ||Epsilon~.sub.it~,

and |b.sub.g~ can be allowed to vary across countries as a function of government size: |b.sub.gi~ = |Gamma~ + |Delta~|s.sub.i~. A negative |Delta~ would imply a negative relationship between the output elasticity of g and government size. Substituting the expression for |b.sub.g~ into (8), the parameter |Delta~ can be estimated from

(9) |Delta~|y.sub.it~ = |b.sub.0~ + |b.sub.n~|Delta~|n.sub.it~ + |Gamma~|Delta~|g.sub.it~ + |Delta~|s.sub.i~|Delta~|g.sub.it~ + ||Epsilon~.sub.it~.

Panels C and D of Table IV estimate several versions of equations (8) and (9). Instrumental variables estimation with the usual list of instruments is applied to the equations that include employment as an explanatory variable. The first interesting result is that the coefficients of |Delta~g are positive and statistically significant in all specifications. This means that government consumption has a positive marginal product. Equally interesting is that |Delta~ is estimated to be negative by the interaction terms of Panel D, even though it is statistically significant only in the last specification. Note also, that the last specification is in terms of labor productivity growth and estimated with OLS. This means that the decomposition method is irrelevant in this case (since |Mathematical Expression Omitted~ is only used as an instrument in some structural equations). TABULAR DATA OMITTED TABULAR DATA OMITTED TABULAR DATA OMITTED These results seem to suggest again, that while government consumption is productive, its output elasticity depends negatively on government size.

V. IMPLICATIONS AND CONCLUSIONS

The first finding of this paper is that permanent changes in government spending (like building more schools or enacting a new bureaucracy) have a greater impact on employment and output than transitory changes (like a commitment to send a man to Mars or a brief military engagement). From Table IV, a 1 percent permanent increase in g raises output in the representative country by 0.5 percent, whereas a 1 percent transitory increase raises output by only 0.1 percent. Theoretically, this means that, unlike transitory increases, permanent increases in g produce wealth losses that raise the optimal labor supply.

This finding also has implications for stabilization policy. Since permanent changes in government consumption have no cyclical effects, it is only transitory changes in g that can be used for stabilization purposes. Therefore, the paper's conclusion that the multipliers of transitory changes in g are generally small reduces the attractiveness of government consumption as a policy variable. In addition, the general imprecision with which the responses of output to transitory changes are estimated introduces significant uncertainty to the potency of g as a policy variable and thus reduces its countercyclical value (see Brainard |1967~).

The second finding of the paper is that government consumption is generally productive and thus belongs in the production function. Using the estimate of 0.281 for |b.sub.g~ from the second panel structural equation, and a value of 0.30 for the capital share (Maddison |1990~, Table 7), the output elasticity of government consumption for the representative country in our sample is |a.sub.g~ = |b.sub.g~(1-|a.sub.k~) = 0.20. This means that a 1 percent increase in g, holding all other inputs constant, produces an average 0.2 percent increase in output.

A third finding is that the output elasticity itself varies inversely with government size. To illustrate, consider three countries with different government sizes for the period examined. At the low end of the spectrum, Japan has a government size of less than 10 percent of GDP. At the other extreme, the government size in Sweden is 23 percent. For Germany, in the middle, it is 18 percent. Maintaining the 0.30 value for the capital share, and using |b.sub.g~ estimates from Table III, the output elasticities of government consumption in these countries can be calculated as follows: .32 in Japan, .22 in Germany (both statistically significant), and .11 in Sweden (not statistically significant). These numbers imply that a 1 percent increase in g, holding all other inputs constant, will raise output by 0.32 percent in Japan, 0.22 percent in Germany, and 0.11 percent in Sweden. In addition, these estimates have implications about the optimal government size in each of these countries. Using as a rule of thumb Barro's |1990~ conclusion that the optimal government size, s, is equal to the output elasticity of g, |a.sub.g~, one can conclude that the public sector is too small in Japan, too large in Sweden, and has about the right size in Germany. We can also go one step further. Writing |b.sub.g~ = |Gamma~ + |Beta~s, Barro's rule of s = |a.sub.g~ becomes s = (|Gamma~ + |Delta~s)(1 - |a.sub.k~) which gives

(10) s = (1-|a.sub.k~)|gamma~ / |1-|Delta~(1-|a.sub.k~)~

as the optimal government size for the representative country. Using our estimates for |Gamma~ and |Delta~ from the last two panel regressions, we obtain s = 0.19 (interestingly, both regressions imply almost the same s despite different estimates for |Delta~ and |Gamma~). Note also that this is almost identical to the 0.20 estimate for |a.sub.g~, as estimated independently above.

Another way to illustrate this result is to write the marginal product of government consumption as MPG = |a.sub.g~/s = (1-|a.sub.k~)(|Delta~+|Gamma~/s). Using |Delta~ = -.876 and |Gamma~ = .443 from Table IV, Figure 5 plots the marginal product of g against government size for |a.sub.k~ = 0.3 and |a.sub.k~ = 0.4. Barro's rule requires MPG = 1, and this is satisfied around s = 0.20 for both plausible values for capital's share. We conclude that the optimal government size for the representative country in our sample is approximately 20 percent.

APPENDIX

From (1) the logarithm of the marginal product of capital is

(11) log(|MPK.sub.t~) = log(|a.sub.k~) + |y.sub.t~ - |k.sub.t~.

Using (11) we can write (1) as

(12) |y.sub.t~ = ||a.sub.n~|n.sub.t~ + |a.sub.k~log(|a.sub.k~/|MPK.sub.t~) + |a.sub.g~|g.sub.t~ + |u.sub.t~~/(1-|a.sub.k~).

We assume that at the steady state the marginal product of capital is constant. From (11) this is equivalent to assuming that the steady state capital-output ratio is constant. Equation (12) can be then written as (2). We can also allow for deviations from the steady state by specifying |(1-|a.sub.k~).sup.-1~/||u.sub.t~ + |a.sub.k~log(|a.sub.k~/|MPK.sub.t~)~ as a random walk, in which case we again obtain (2) and (3). The parameter |a.sub.g~ is not identified in (2), but |b.sub.g~ is monotonically increasing in |a.sub.g~. Estimation of |b.sub.g~ provides a measure of the input effect of government spending on output. If the share of capital, |a.sub.k~, is not dramatically different across countries (or more precisely, if differences in |a.sub.k~ across countries do not depend on government size, which is a weaker requirement), then any relationship between |b.sub.g~ and government size would indicate the same relationship between |a.sub.g~ and government size.

Profit maximization requires that the log of the real wage be

(13) |w.sub.t~ = log(|a.sub.n~) + |y.sub.t~ - |n.sub.t~,

which using (12) gives (4). Combining (4) and (5) gives

(14) |Mathematical Expression Omitted~

where

|Psi~ = ||1+|Xi~(1-|a.sub.n~-|a.sub.k~)|(1-|a.sub.k~).sup.-1~~.sup.-1~ |is greater than~ 0.

This is estimated as (6). Note that if |Zeta~ |is greater than~ 0, (14) implies that |c.sub.1~ |is greater than~ |c.sub.2~. Finally, (6) can be used to eliminate n from (3). The reduced form for output is thus obtained:

(15) |Mathematical Expression Omitted~

which is estimated as (7). Again, if |Zeta~ |is greater than~ 0, (15) implies |b.sub.1~ |is greater than~ |b.sub.2~.

1. Ahmed |1986~ also developed and estimated a similar model for the United Kingdom.

2. See Christiano and Eichenbaum |1988~, Aiyagari, Christiano and Eichenbaum |1990~, and Karras |1990~.

3. Reversing the causal relationship, Conte and Darrat |1988~ found a feedback from real economic growth to the public sector size for approximately ten OECD countries (half of their sample).

4. Most of the following equations are derived in the appendix.

5. Romer |1990~ gives a convincing argument why production functions should exhibit constant returns to scale in rival factors of production.

6. It is plausible that in reality it may be persistence, rather than permanence, of changes in government spending that matters for labor supply decisions. In that case, the more persistent the government spending process, the larger labor effects will be. This issue is addressed by our choice of the Hodrick-Prescott filter as one of the decomposition methods in the empirical section.

7. It also follows from this analysis that the wage will respond more to transitory than to permanent changes in government spending. The only reason why this is not tested by estimating a reduced form for the wage is that wage data could not be obtained for a large number of countries. Karras |1990~ finds that U.S. data are consistent with the wage hypothesis.

8. See Christiano and Eichenbaum |1989~ for this and related issues. Durlauf and Phillips |1988~ present a different and less agnostic point of view.

9. The HP approach defines the trend component, |g.sup.P~, as the one that minimizes

|Mathematical Expression Omitted~

For the purposes of this study, a number of different values for |Lambda~ were tried, including |Lambda~ = 50, 100, 150, 200, and 500. They all gave very similar results. Results reported are for |Lambda~ = 100, the value suggested by Kydland and Prescott |1989~.

10. It may be worth noting here, however, that the Beveridge-Nelson identification assumption is not at all implausible, since stationarity tests (not reported here but available on request) on the levels and first differences of government consumption show convincingly that unit roots on the levels do exist.

11. For an example of a similar methodology see Kormendi and Meguire |1984~.

12. Note that this negative relationship is not simply one between government size, s, and the marginal product of government spending. From (1) it follows that the marginal product of government spending is |a.sub.g~(Y/G) = |a.sub.g~|s.sup.-1~, and it is therefore by construction negatively related to s. What the negative correlations of Figures 1 and 2 (and also evidence from the next section) suggest is that the parameter |a.sub.g~ itself varies inversely with government size, which is a more interesting and novel finding.

13. The estimated equations become |z.sub.it~ = f(|v.sub.it~) + |e.sub.it~, where z is the dependent variable, v is a vector of explanatory variables, and e the error term. This can be estimated with OLS if e is uncorrelated both across countries and time. It is likely, however, that e will have fixed effects (|e.sub.it~ = |u.sub.i~ + ||Epsilon~.sub.it~ or |e.sub.it~ = |u.sub.i~ + |h.sub.t~ + ||Epsilon~.sub.it~) in which case GLS is required. If, in addition, some of the right-hand-side variables are endogenous, then instrumental variables must be used. In addition, ||Epsilon~.sub.it~ was allowed to be autoregressive, ||Epsilon~.sub.it~ = |Rho~||Epsilon~.sub.it-1~, or ||Epsilon~.sub.it~ = ||Rho~.sub.i~||Epsilon~.sub.it-1~, but the estimated |Rho~'s were statistically insignificant.

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Author: | Karras, Georgios |
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Publication: | Economic Inquiry |

Date: | Jul 1, 1993 |

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