1. Describing the production function: public capital goods

The model follows the suggestion of Barro and Sala-i-Martin (1992) by introducing the real-world feature of congestion in the consumption of publicly provided goods (both physical capital and transfer payments) by individual household-producers (firms), as here (as is the case for a substantial share of government productive expenditures) public goods are rivalrous but not excludable. In this model of congested public goods, every producer reaps the benefits from the provision of public capital. However, for a given stock of public capital (G(t)), the benefits to producers decline as they raise their individual k(t), and hence their output (and accordingly, aggregate K(t)). This is the nature of the congestion existing in the consumption of public capital. G(t) can represent, for example, the miles of highways provided, the number and size of airports, and law and order, and K(t) can represent highway traffic, air traffic, and the number of legal disputes.

Barro and Sala-i-Martin (1992) argue that national defense and domestic security services (police and prisons), which are often deemed to be prototypical nonrival and nonexcludable public goods, can also be considered to be subject to a form of congestion, using the argument set forward by Thompson (1974). The state of both national and local security depends on expenditures on defense and police services, G(t), in relation to the level of the perceived external threat, where that threat is a function of the “prize” potentially attainable by aggression, which is proportional to the private domestic capital stock, K(t). Hence, it can be argued that G(t)/K(t) represents the effective level of national and domestic security.

2. Describing the production function: transfers

In the traditional public finance literature, the role for government in affecting economic growth is limited. In those models where it is discussed, the government is usually required to decide the rate at which capital or output is taxed, and the distribution of the resulting revenue between productive expenditure on public investments and unproductive expenditure on government consumption or lump-sum transfer payments (see Alesina and Rodrick 1991, Lee 1992). A typical theoretical result is that for a given tax rate, economic growth is adversely affected by an increase in redistribution through public transfers. Running counter to the received theoretical wisdom are some cross-sectional empirical results which test the above finding and find the converse: that public transfers actually raise economic growth (Barro 1989, Sala-i-Martin 1992a).

The model of equation (3) includes productive public transfers as an input to private production functions. It is not argued here that transfers are a direct input to private production (as is public investment), but that they do raise the after-tax private return to capital by reducing the implicit “taxation” of output due to: (i) poor enforcement of private property rights, and (ii) workers possessing below average stocks of human capital (see below). Notwithstanding the possibility of overlaps, public transfer payments are argued to be of two broad types: intragenerational (such as from rich to poor) and intergenerational (such as from young to old), and both types raise economic growth by raising the marginal product of private capital.

Sala-i-Martin (1992a) provides a rationale for productive public transfers in arguing that intragenerational transfers assist in better enforcing private property rights, which reduces the extent of aggregate distortions in the economy, such as criminal activities. An increase in such transfers raises the probability of a household/producer receiving the marginal product of capital, and maintaining ownership of his capital stock. This in turn provides an incentive for households to accumulate capital and produce. Sala-i-Martin (1992a) also presents empirical evidence from cross-sectional (Summers and Heston 1991) data for a set of 75 countries, and finds that holding constant the overall size of government, public transfers are positively related to growth in per capita income.

Intergenerational transfers can also be regarded as positively affecting economic growth in that they induce relatively unproductive agents in the economy (typically the old) to leave the workforce, thus reducing the negative externalities produced by these relatively unproductive old on the productive young, raising the productivity of private capital, and so raising the rate of growth of output (see Sala-i-Martin 1992). The argument is that, given externalities in the average stock of human capital (such as those introduced by Lucas (1988)), and because skills deteriorate with age, these externalities produced by the old (with lower than average skills) have a detrimental effect on the productivity of the young (with higher than average skills). When the difference between the skill levels of the young and the old is large enough, it will be beneficial for the old to be induced to retire, as then aggregate output will be higher than if they remained in the workforce. Intergenerational transfers, in almost all countries paid on condition of retirement, are the means to achieve this end.

Both types of transfers can be thought of as a public input to private production which is subject to congestion, for much the same reasons as those outlined earlier for defense and domestic security. When an economic agent engages in more production, the protective role of a given level of transfer payments (improving the enforcement of private property rights and inducing workers with low human capital stocks to leave the workforce) is diminished by the associated increase in K(t).

1. Comparative statics of public finance variables and growth

Substituting (15) and (16) into (11) yields, after manipulation:

The left-hand side of (17) is the return to consumption, which is upward-sloping and linear in γ. The right-hand side of (17) can be rewritten as:

where ω-α/(1-α-β). Since A¹/(1-α-β) > 0, and ψ(.) > 0, then J(.) > 0. Accordingly, ∂J/∂γ = -ω(J(.)/γ) and ∂²J/∂γ² = ω(ω+1)(J(.)/γ²). We also know that (ω+1)=(1-β)/(1-α-β), and it is assumed that 0 < α,β < 1 and that α+β < 1.

While in this set up there is always a solution to $\overline{\gamma }$, the steady state growth rate, there are two cases to consider, depending on whether (1-α) ≷ β:

Case A is shown in Figure 1, and Case B in Figure 2. In both cases the ρ+σγ and J(.) lines always cross. An important implication of these Figures is that an increase in the marginal product of capital (a shift up in the J(.) locus) will increase (decrease) growth depending on whether (L-α) is greater (less) than β. Similarly, a fall in ρ or a fall in σ will increase (decrease) growth depending on whether (1-α) is greater (less) than β.

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Case A:[(1-α)>β]

Citation: IMF Working Papers 1994, 092; 10.5089/9781451951479.001.A001

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Case B: [(1-α)<β]

Citation: IMF Working Papers 1994, 092; 10.5089/9781451951479.001.A001

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Recognizing that r₁ = Ġ/Y ≡ IGOV, r₂ = T/Y ≡ SOCSEC and that (1-r₁-r₂) ≡ (1-CURREV), where IGOV is the ratio of public investment to gross domestic product (GDP), SOCSEC is the ratio of transfer payments to GDP, and CURREV is a measure of the ratio of current tax revenue to GDP (see Section IV and the Appendix for further details), then (17) can be rewritten as:

By the arguments given above, for (1-α)>β then: (i) $\partial \overline{\gamma }/\partial \left(1-\mathrm{CURREV}\right)>0,$, where $\overline{\gamma }$ is the steady-state growth rate implicit in (17’); (ii) $\partial \overline{\gamma }/\partial \left(\mathrm{IGOV}\right)>0,$ and (iii) $\partial \overline{\gamma }/\partial \left(\mathrm{SOCSEC}\right)>0,$.

For (1-α)>β, $\overline{\gamma }$ is also an increasing function of the willingness to forego consumption by saving (that is, a lower ρ (the more patient a society is) or higher σ^-1 (the more willing society is to substitute intertemporally) raises $\overline{\gamma }$). Here $\overline{\gamma }$ is also increasing in A, the productivity parameter of production technology. Linear panel regression equations, as an approximation to the nonlinear growth equation of (17’) will be run in Section V, using CURREV, IGOV and SOCSEC as explanatory variables, among others, and are expected to have the above signs.

2. Optimal growth and public finance variables

The optimal growth rate can also be calculated from (17). Using the implicit function theorem, where $\overline{\gamma }$ is the steady-state growth rate implicit in (17), gives:

where $\mathrm{f}\mathrm{\left(}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{r}}_{\mathrm{1}}\mathrm{-}{\mathrm{r}}_{\mathrm{2}}\mathrm{\right)}\mathrm{,}{\mathrm{r}}_{\mathrm{1}}\mathrm{,}{\mathrm{r}}_{\mathrm{2}}\mathrm{,}\tilde{\mathrm{\gamma }}\mathrm{\right)}\mathrm{=}\mathrm{0}$ is the implicit function, and Λ is one of (1-r₁-r₂),r₁,r₂. Examining first $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{r}}_{\mathrm{1}}\mathrm{-}{\mathrm{r}}_{\mathrm{2}}\mathrm{\right)}$ for given r₁ and r₂ gives:

and so the numerator of $\mathrm{\partial }\overline{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{r}}_{\mathrm{1}}\mathrm{-}{\mathrm{r}}_{\mathrm{2}}\mathrm{\right)}\mathrm{=}\mathrm{-}{\mathrm{f}}_{\mathrm{\left(}\mathrm{1}\mathrm{-}\mathrm{r}\mathrm{1}\mathrm{-}\mathrm{r}\mathrm{2}\mathrm{\right)}}\mathrm{/}{\mathrm{f}}_{\tilde{\mathrm{\gamma }}}$ is Positive, and given that from (21) ${\mathrm{f}}_{\tilde{\mathrm{\gamma }}}\mathrm{>}\mathrm{0}$, then unambiguously $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathrm{r}}_1\mathrm{-}{\mathrm{r}}_2\mathrm{\right)}>0$. Accordingly, as and r₁ and r₂ fall, that is, the taxes levied to fund the provision of public capital and transfers fall, then (1-r₁-r₂) rises and so does $\overline{\gamma }$.

Examining $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }{\gamma }_1$ for given r₂ and (1-r₁-r₂), gives:

and again from (21) we know ${\mathrm{f}}_{\tilde{\mathrm{\gamma }}}\mathrm{>}\mathrm{0}$, so $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }{\mathrm{r}}_{\mathrm{1}}\mathrm{=}\mathrm{-}{\mathrm{f}}_{\mathrm{r}\mathrm{1}}\mathrm{/}{\mathrm{f}}_{\tilde{\mathrm{\gamma }}}\mathrm{\gtrless }\mathrm{0}\mathrm{for}{\mathrm{r}}_{\mathrm{1}}\mathrm{\gtrless }\mathrm{\alpha }$. Examining $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }{\mathrm{r}}_{\mathrm{2}}$ for given r₁ and (1-r₁-r₂) yields:

and again from (21) we know that ${\mathrm{f}}_{\tilde{\mathrm{\gamma }}}\mathrm{>}\mathrm{0}$, so $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }{\mathrm{r}}_2\mathrm{=}\mathrm{-}{\mathrm{f}}_{\mathrm{r}2}\mathrm{/}{\mathrm{f}}_{\tilde{\mathrm{\gamma }}}\mathrm{\gtrless }\mathrm{0}\mathrm{for}{\mathrm{r}}_2\mathrm{\gtrless }\beta$ Accordingly, $\partial \tilde{\gamma }/\partial r_1=0$, given r₂ and (1-r₁-r₂), when $r_1=r_1^{*}=\alpha$ and $r_2=r_2^{*}=\beta$, as then the term in the square brackets in (22) equals zero. That is, when ${\mathrm{r}}_{\mathrm{1}}\mathrm{=}{\mathrm{r}}_{\mathrm{1}}^{\mathrm{*}}=\alpha$, then the growth rate is maximized, and the public capital sector is set to equal the share of output it would receive if it was a private input in competitive output and factor markets. Similar results are obtained from (23) for $\partial \tilde{\gamma }/\partial r_2=0$, given r₁ and (1-r₁-r₂). Further, when r₁ ≶ α for given r₂, or r₂ ≶ β for given r₁, then $\partial \tilde{\gamma }/\partial r_i\gtrless 0$, i=1,2.

From (22) also comes the implication that when r₁ (the initial, optimally-set tax to fund the public capital sector) is set such that r₁ = α, then if r₂ ≶ β, where β is the transfer sector of government’s optimal size, then $\mathrm{\partial }\tilde{\mathrm{\gamma }}\mathrm{/}\mathrm{\partial }{\mathrm{r}}_{\mathrm{1}}\mathrm{\gtrless }\mathrm{0}$. So, to move the rate of growth closer toward its maximum rate it will be necessary to set ${\hat{r}}_1\gtrless r_1$, where ${\hat{r}}_2$ is the revised tax on output used to fund the sector producing public capital goods. A related finding exists from (23) above: when r₂ (the initial, optimally-set tax to fund the transfer payments) is set such that r₂ = β, then if r₁ ≶ α, where a is the public-capital sector of government’s optimal size, then $\partial \tilde{\gamma }/\partial r_2\gtrless 0$. So, to move the rate of growth closer toward its maximum rate it will be necessary to set ${\hat{r}}_2\gtrless r_2$, where ${\hat{r}}_2$ is the revised tax on output used to fund the sector producing transfers.

The above results are similar to those obtained by Barro (1990): for Cobb-Douglas technology the size of G(t) and T(t) that maximizes the rate of growth is that which sets each equal to the respective shares of aggregate income they would attain if they were private inputs in competitive markets. As with all endogenous growth models, sustained growth is possible here as the marginal product of private capital does not approach zero in the limit (it approaches [(1-r₁-r₂)A(G(t)/K(t))^α(T(t)/K(t))^β]), so that the Inada conditions do not hold. Here, where the elasticity of substitution between K(t), G(t) and T(t) is unity in the aggregate production function, as in the case for Cobb-Douglas technology, then the relative size of government in each sector (r_i for i=1,2) that maximizes utility for the representative household also maximizes the rate of growth. ^11/

There are several reasons why we should expect that most governments are operating in the region where $\partial \tilde{\gamma }/\partial r_i>0$, (i=1,2). First, if in the real world (3) is not approximated by Cobb-Douglas technology, then as noted by Barro (1990), the utility-maximizing government will set a rate of growth which is less than the maximum. If that is the case, then $\partial \tilde{\gamma }/\partial r_i>0$, for i=1,2, and we should observe a positive effect on growth of public investment (Ġ/Y) and transfer payments (T/Y).

Second, if there are also some publicly provided goods which are not directly productive, then we should expect the size of government to be less than the growth-maximizing share of aggregate income for each sector of government, so then $\partial \tilde{\gamma }/\partial r_i>0$, i=1,2. It is clear that there are other activities of government beyond the taxing, transfers, and public investment activities modeled here. Such public activities as defense expenditure, education, government consumption spending, interest on national debt and international aid contributions will presumably also enter (1), along with per capita consumption. If that is the case, then the growth-maximizing share of G(t) and T(t) in aggregate income is smaller if the government is also using the income tax base to finance these other activities.

Of key importance is the model’s predictions for the effect on $\overline{\gamma }$ of changes in r₁ and r₂. It was shown above that $\overline{\gamma }$ is: (i) increasing in (1-r₁-r₂), which is the contribution of tax distortions; (ii) increasing in r₁ = Ġ/Y, the ratio of public investment to GDP; and (iii) increasing in r₂ = T/Y, the ratio of transfer payments to GDP. Accordingly, there is a nonmonotonic relationship between $\overline{\gamma }$ and taxes (r₁ and r₂), with $\overline{\gamma }$ increasing initially with r₁ = Ġ/Y and r₂ = T/Y, but later decreasing with increases in r₁ and r₂ beyond the point where ${\mathrm{r}}_{\mathrm{1}}\mathrm{=}{\mathrm{r}}_{\mathrm{1}}^{\mathrm{*}}=\alpha \mathrm{and}{\mathrm{r}}_{\mathrm{2}}\mathrm{=}{\mathrm{r}}_{\mathrm{2}}^{\mathrm{*}}=\beta$. This result emphasizes the trade-off between the growth-enhancing provision of transfers and public capital goods, and the growth-diminishing influence of the distortionary taxes which need to be raised to fund the provision of these same transfers and public capital goods. For small government (that is, low r₁ and low r₂) then the former two effects are likely to dominate; for large government (high r₁ and r₂) the latter, detrimental effect is likely to dominate. Of course when ${\mathrm{r}}_{\mathrm{1}}\mathrm{=}{\mathrm{r}}_{\mathrm{1}}^{\mathrm{*}}\mathrm{and}{\mathrm{r}}_{\mathrm{2}}\mathrm{=}{\mathrm{r}}_{\mathrm{2}}^{\mathrm{*}}$ and government is at its optimal size, (as given by the results from (22) and (23)), then the beneficial and the detrimental effects of the size of government exactly offset one another.

1. Model specification

The estimated equation for empirical testing is based on (17’):

where: the variables are as described in Sections III and IV, and in the Appendix; ε_it = α_i + v_it (i=1,…, N countries; t=1,…, T time); In is the natural logarithm; and β is a vector of coefficients associated with time-varying observable variables. ^14/ One key reason for combining time-series and cross - sectional data is to attempt to control for any unobservable country-specific effects which may be correlated with other explanatory variables (that is, the potential for E(ε_it|X_it) = E(α_i|X_it) ≠ 0). ^15/

There are three alternative econometric specifications of the model, each differing in their treatment of α_i. The “fixed effects” (FE) estimator treats α_i as a fixed (time invariant) but unknown constant differing across individual countries. In FE it is assumed that differences across countries can be captured in differences in the constant term, and so each α_i is an unknown parameter to be estimated. In effect, differences between countries are here viewed as parametric shifts of the regression function.

Alternatively, the “random effects” (RE) or variance components specification assumes that country-specific constant terms are randomly distributed across cross-sectional units (that is, countries). In RE, each ${\mathrm{\alpha }}_{\mathrm{i}}\mathrm{-}\mathrm{N}\mathrm{\left(}\mathrm{0}\mathrm{,}{\mathrm{\sigma }}_{\mathrm{\alpha }}^{\mathrm{2}}\mathrm{\right)}$, given that here α_i = α + η_i, where α is the group mean intercept and η_i are the errors associated with the ith country and which is constant over time, and are random (white noise) deviations (positive or negative) from this common mean. Then ε_it, the error term in (24), becomes ε_it = V_it + η_i. In the RE specification it is assumed that α_i is uncorrelated with v_it and the X_it. ^16/

The third approach, known as “between effects” (BE), undertakes the estimation without considering country-specific intercepts (a single, group mean intercept α is postulated). That is, it is assumed that the error term in (24) is ${\widetilde{\mathrm{ϵ}}}_{\mathrm{it}}\mathrm{=}{\mathrm{\eta }}_{\mathrm{i}}\mathrm{+}{\mathit{\nu }}_{\mathrm{i}}$, where v_i is the mean of the usual errors which vary over time and countries. BE is the OLS estimator based on the set of country (or group) means, which is an analogue to undertaking cross-sectional estimation, as BE simply asks whether poorer countries tend to grow faster than rich ones (that is, it looks at the growth experience between countries only). Hsiao (1986) demonstrates that RE is an optimal weighted average of both the FE (which uses variation within countries) and BE (which uses variation between countries) estimators.

In estimating these models a number of specification tests were carried out, to establish the efficacy of introducing unobserved, country-specific effects as opposed to OLS estimation. (Breusch and Pagan 1980), and to determine which of the FE, BE, or RE specifications is superior (Hausman 1978). On economic grounds neither OLS nor FE appear to be reasonable set ups, given that they both assume the same constant term for all years and for all countries (OLS), or different (yet fixed) constant terms for each country (FE). Neither assumption is particularly palatable in the context of an analysis of the causes of secular economic growth. ^17/

2. Estimation results

This section reports the results of estimating the TSCS regressions of (24), where the elements of the vector X_it are those discussed in Sections III and IV. The results from using ln(INIT) as the lone explanatory variable are reported first, then results for the full set of X_it variables, then results including EDUC to control for differences in countries’ steady states.

The results of all regressions are presented in Table 1, where in all cases the dependent variable is GRWKR, the rate of growth of GDP per worker. In column one a constant and ln(INIT) are the sole explanatory variables, where INIT is included to hold constant initial per-worker GDP. The estimated convergence coefficient is ${\hat{\mathrm{\beta }}}_{\mathrm{4}}\mathrm{=}\mathrm{-}\mathrm{0.0062}\mathrm{\left[}\mathrm{-}\mathrm{3.36}\mathrm{\right]}$, a result which confirms the findings of Dowrick and Nguyen (1989), Barro (1991), Mankiw, Romer, and Weil (1992), and Barro and Sala-i-Martin (1992a) of a statistically significant tendency toward convergence in OECD countries. It also has implications for the speed of convergence of OECD economies: the relatively low estimated convergence coefficient means that the OECD economies move halfway to steady state in about 113 years.

Table 1.

GDP Growth Regressions, 23 Developed Countries, 1971-88 ^1/

Dependent Variable: GRWKR Value of t-statistics in brackets

^1/All regressions use one-factor error structure for panel estimation, that is using group dummy variables. BE denotes between effects estimation; RE denotes random effects estimation; BEIV denotes between effects estimation and REIV denotes random effects estimation, both using the GIV technique. LOGL denotes the value of the log-likelihood function, where LOGL = - (N/2) [1+ln2π-ln((e‘e)/N)]. GRWKR is the rate of growth of real GDP per worker; IGOV is the mean ratio of public investment to GDP; INIT is the initial level of real GDP per worker; SOCSEC is the mean ratio of expenditure on social security and welfare to GDP; CURREV is the mean ratio of current revenue to GDP; EDUC is the mean ratio of secondary school enrollment to the population of children in that school age group. The sub-periods for analysis are: 1971-75, 1976-80, 1981-85, and 1986-88. For further details see the Appendix.

^2/The Breusch-Pagan (BP) test statistic had a value of 134.39 [1df, pval=1.54E-09], and the Hausman (H) test statistic comparing BE and RE a value of 8.12 [1df, pval=.004].

^3/The BP test statistic had a value of 134.43 [1df, pval=1.54E-09], and the H test statistic comparing BE and RE a value of 7.25 [1df, pval=.026].

^4/The BP test statistic had a value of 133.99 [1df, pval=1.54E-09], and the H test statistic comparing BEIV and REIV a value of 12.28 [4df, pval=.015].

^5/The H test for exogeneity of IGOV and SOCSEC had a value of 9.96 [2df, pval=0.041]; the independent variables used in the OLS reduced form estimation (to calculate predicted values for IGOV and SOCSEC) were R, CPI, AGE65, and YBAR.

^6/The BP test statistic had a value of 134.09 [1df, pval=1.54E-09], and the H test statistic comparing BEIV and REIV a value of 13.17 [5df, pval=.022]. The independent variables used in the OLS reduced form estimation (to calculate predicted values for IGOV and SOCSEC) were R, CPI, AGE65, and YBAR.

Table 1.

GDP Growth Regressions, 23 Developed Countries, 1971-88 ^1/

Dependent Variable: GRWKR Value of t-statistics in brackets

	BE 2/ (1)	BE 3/ (2)	BEIV 4/5/ (3)	BEIV 6/ (4)
Constant	0.0745	0.882	0.1297	0.1362
	[4.06]	[4.01]	[5.30]	[5.36]
ln(IGOV)			0.0084	0.0115
			[1.76]	[2.03]
ln(INIT)	-0.0062	-0.0081	-0.0093	-0.0097
	[-3.36]	[-3.24]	[-3.65]	[-3.77]
ln(SOCSEC)			0.0083	0.0083
			[5.78]	[5.87]
ln(CURREV)			-0.0190	-0.0209
			[-4.53]	[-4.63]
EDUC		0.0001		0.0001
		[1.13]		[0.80]
Sample size	92	92	92	92
LOGL	-602.54	-603.19	-137.54	-138.19

^1/All regressions use one-factor error structure for panel estimation, that is using group dummy variables. BE denotes between effects estimation; RE denotes random effects estimation; BEIV denotes between effects estimation and REIV denotes random effects estimation, both using the GIV technique. LOGL denotes the value of the log-likelihood function, where LOGL = - (N/2) [1+ln2π-ln((e‘e)/N)]. GRWKR is the rate of growth of real GDP per worker; IGOV is the mean ratio of public investment to GDP; INIT is the initial level of real GDP per worker; SOCSEC is the mean ratio of expenditure on social security and welfare to GDP; CURREV is the mean ratio of current revenue to GDP; EDUC is the mean ratio of secondary school enrollment to the population of children in that school age group. The sub-periods for analysis are: 1971-75, 1976-80, 1981-85, and 1986-88. For further details see the Appendix.

^2/The Breusch-Pagan (BP) test statistic had a value of 134.39 [1df, pval=1.54E-09], and the Hausman (H) test statistic comparing BE and RE a value of 8.12 [1df, pval=.004].

^3/The BP test statistic had a value of 134.43 [1df, pval=1.54E-09], and the H test statistic comparing BE and RE a value of 7.25 [1df, pval=.026].

^4/The BP test statistic had a value of 133.99 [1df, pval=1.54E-09], and the H test statistic comparing BEIV and REIV a value of 12.28 [4df, pval=.015].

^5/The H test for exogeneity of IGOV and SOCSEC had a value of 9.96 [2df, pval=0.041]; the independent variables used in the OLS reduced form estimation (to calculate predicted values for IGOV and SOCSEC) were R, CPI, AGE65, and YBAR.

^6/The BP test statistic had a value of 134.09 [1df, pval=1.54E-09], and the H test statistic comparing BEIV and REIV a value of 13.17 [5df, pval=.022]. The independent variables used in the OLS reduced form estimation (to calculate predicted values for IGOV and SOCSEC) were R, CPI, AGE65, and YBAR.

View Table

Table 1.

GDP Growth Regressions, 23 Developed Countries, 1971-88 ^1/

Dependent Variable: GRWKR Value of t-statistics in brackets

	BE 2/ (1)	BE 3/ (2)	BEIV 4/5/ (3)	BEIV 6/ (4)
Constant	0.0745	0.882	0.1297	0.1362
	[4.06]	[4.01]	[5.30]	[5.36]
ln(IGOV)			0.0084	0.0115
			[1.76]	[2.03]
ln(INIT)	-0.0062	-0.0081	-0.0093	-0.0097
	[-3.36]	[-3.24]	[-3.65]	[-3.77]
ln(SOCSEC)			0.0083	0.0083
			[5.78]	[5.87]
ln(CURREV)			-0.0190	-0.0209
			[-4.53]	[-4.63]
EDUC		0.0001		0.0001
		[1.13]		[0.80]
Sample size	92	92	92	92
LOGL	-602.54	-603.19	-137.54	-138.19

^1/All regressions use one-factor error structure for panel estimation, that is using group dummy variables. BE denotes between effects estimation; RE denotes random effects estimation; BEIV denotes between effects estimation and REIV denotes random effects estimation, both using the GIV technique. LOGL denotes the value of the log-likelihood function, where LOGL = - (N/2) [1+ln2π-ln((e‘e)/N)]. GRWKR is the rate of growth of real GDP per worker; IGOV is the mean ratio of public investment to GDP; INIT is the initial level of real GDP per worker; SOCSEC is the mean ratio of expenditure on social security and welfare to GDP; CURREV is the mean ratio of current revenue to GDP; EDUC is the mean ratio of secondary school enrollment to the population of children in that school age group. The sub-periods for analysis are: 1971-75, 1976-80, 1981-85, and 1986-88. For further details see the Appendix.

^2/The Breusch-Pagan (BP) test statistic had a value of 134.39 [1df, pval=1.54E-09], and the Hausman (H) test statistic comparing BE and RE a value of 8.12 [1df, pval=.004].

^3/The BP test statistic had a value of 134.43 [1df, pval=1.54E-09], and the H test statistic comparing BE and RE a value of 7.25 [1df, pval=.026].

^4/The BP test statistic had a value of 133.99 [1df, pval=1.54E-09], and the H test statistic comparing BEIV and REIV a value of 12.28 [4df, pval=.015].

^5/The H test for exogeneity of IGOV and SOCSEC had a value of 9.96 [2df, pval=0.041]; the independent variables used in the OLS reduced form estimation (to calculate predicted values for IGOV and SOCSEC) were R, CPI, AGE65, and YBAR.

^6/The BP test statistic had a value of 134.09 [1df, pval=1.54E-09], and the H test statistic comparing BEIV and REIV a value of 13.17 [5df, pval=.022]. The independent variables used in the OLS reduced form estimation (to calculate predicted values for IGOV and SOCSEC) were R, CPI, AGE65, and YBAR.

The most likely reason absolute convergence is observed in column 1 of Table 1 is that the correlation between INIT and both the steady-state real GDP per worker and the steady-state growth rate is relatively weak. Differences in ln(INIT) among OECD countries (due to the residual effects of World War II, agricultural shocks, and oil shocks) approaching similar steady states would show up as unconditional or absolute convergence. That is, for the relatively homogeneous group of OECD countries with similar technologies and preferences, absolute convergence is very close to conditional convergence.

Column two follows Barro (1991) and Mankiw, Romer, and Weil (1992), and adds EDUC (a proxy for the stock of human capital) as an explanatory variable. Human capital has played an important part in the endogenous growth literature, either as a direct input into research (Romer 1990), or because of positive externalities flowing from the average stock of human capital per person (Lucas 1988, Becker, Murphy, and Tamura 1990). For these reasons it is expected that for given INIT, GRWKR will be positively related to EDUC. Further, once additional variables (such as EDUC) are included that attempt to hold constant the cross-country variations in steady-state real GDP per worker and the steady-state growth rate, then the partial relation between GRWKR and INIT should become more negative. The results of column 2 $\left({\hat{\mathrm{\beta }}}_{\mathrm{4}}\mathrm{=}\mathrm{-}\mathrm{0.0081}\mathrm{\left[}\mathrm{-}3.24\mathrm{\right]}\right)$ show that the addition of EDUC does indeed result in INIT being even more strongly negatively-related to subsequent GRWKR, and accordingly conditional convergence is observed in the data. The speed of convergence here is such that the OECD economies now move halfway to steady state in about 86 years.

The explanatory variables in (24) are assumed to be exogenous, and to cause the consequent changes in economic growth. However, there is a strong likelihood of endogeneity with regard to some (or all) of these variables, and in order to correct for this potential simultaneity bias, generalized instrumental variable (GIV) estimation will be introduced. ^18/ Hausman (1978) tests verified this suspicion of E(α_i|X_it) ≠ 0 by finding that both IGOV and SOCSEC were likely to be luxury goods (and hence endogenous), as given an initial income, a higher GRWKR which raises the mean level of income over the sub-sample will lead to higher sub-sample mean values for both IGOV and SOCSEC. ^19/ Accordingly, GIV estimation was carried out using fitted values from reduced form estimation of IGOV and SOCSEC as instruments for actual IGOV and SOCSEC in the structural equation for growth (24). ^20/

The results of GIV estimation are presented in columns 3 and 4 of Table 1, for the assumed one-factor error structure, excluding and including EDUC, respectively. The preferred BEIV specification in column 3 yielded parameter estimates which are all statistically significant (IGOV at the 10 percent level), with a coefficient on INIT of ${\hat{\mathrm{\beta }}}_{\mathrm{4}}\mathrm{=}\mathrm{-}\mathrm{0.0093}\mathrm{\left[}\mathrm{-}\mathrm{3.65}\mathrm{\right]}$. ^21/ As predicted from the theoretical relation of equation (17’), GRWKR is positively related to SOCSEC and IGOV respectively, holding constant INIT and all other variables. The estimated coefficients on SOCSEC and IGOV are both individually statistically different from zero, with the coefficient on CURREV also significantly different from zero, and as predicted by (17’), is negatively related to GRWKR. Given that the explanatory variables enter in ln form, their coefficients indicate the percentage increase in GRWKR from a 1 percent increase in the explanatory variable (for example, a 1 percent increase in IGOV will raise GRWKR by 0.84 of a percentage point).

Similarly, evaluating the coefficients of the BEIV specification of column 4 (which controls for inter-country variation in the stock of human capital by including EDUC) reveals that all of the coefficients have the expected signs, with all except EDUC being statistically significant. The coefficient of convergence in column 4 is significant and slightly larger at ${\hat{\mathrm{\beta }}}_{\mathrm{4}}\mathrm{=}\mathrm{-}\mathrm{0.0097}\mathrm{\left[}\mathrm{-}3.77\mathrm{\right]}$. Again, as predicted by (17’), there are the strongly significant and positive effects of SOCSEC and IGOV on GRWKR, and also the strong, growth-inhibiting effect of CURREV on GRWKR. The rejection of FEIV and REIV in favor of BEIV indicates that most of the variation in the growth experience is occurring between countries, rather than within particular countries over time, and is consistent with recent work by Easterly, Kremer, Pritchett and Summers (1993). Such a result is perhaps not so surprising when the data reveal the relatively wide disparity in GRWKR among the OECD countries.

3. Comparison with earlier results

The above specification is innovative in that it allows for the simultaneous analysis of the differential partial contributions to growth of public investment, public transfers, and the levying of distortionary taxes, and allows for a nonmonotonic relationship between government and growth. Earlier work often examined the influence of public spending variables on growth in isolation from the effects of public taxation, and vice versa.

The finding of a strongly positive partial effect of public investment on growth echoes recent TSCS work by Skinner (1987) and Knight, Loayza, and Villanueva (1993), and the time-series results of Aschauer (1989). However, it contradicts the cross-sectional findings of Diamond (1989), Ford and Poret (1991) and Barro (1991) of a weak relation between public investment and growth, especially if the ratio of private investment to GDP is held constant. ^22/ Barro (1991) argues that his findings indicate either that public investment is not an important determinant of growth, or that governments are optimizing and going to the point where the marginal effect of such investment on the growth rate is close to zero. When private investment (IPRIV) was included in the regressions here (results not reported), all the coefficients were significant and had the expected signs, including IGOV (which becomes negative and insignificant in Barro’s (1991) analysis). It is likely that the failure of the above three cross-sectional studies to account for the endogeneity of IGOV, and for the potential for cross-country heterogeneity, generated their different findings. Moreover, in contradiction of Barro’s (1990) conclusions, my results for IGOV indicate that the typical OECD government was operating where the marginal effect of public investment on growth was positive. This is inconsistent with the optimizing behavior of government examined in Section III.2, where IGOV was chosen so as to maximize growth. However, in that section a number of caveats were given as to why utility-maximizing governments may choose to operate where ∂γ/∂(IGOV)>0.

The equally strong positive partial effect of public transfers on growth has not been previously analyzed using a TSCS framework, and contradicts the findings from earlier work, which has traditionally modeled governments as levying distortionary taxes and using these resources for government consumption spending or unproductive transfer payments (see Alesina and Rodrick 1991). Barro (1989) found evidence in his cross-sectional study that transfer payments were a luxury good, which he argued accounted for the positive correlation of mean transfer payments as a share of GDP with the growth rate. While such endogeneity was also found here, once GIV estimation was carried out a strong, positive relation between transfers and growth was brought out in the data. As for IGOV, this result indicates that the typical OECD government was operating where the marginal effect of transfers on growth was positive, that is where ∂γ/∂(SOCSEC)>0.

The detrimental partial effect of distortionary taxes on growth (∂γ/∂(CURREV)<0) has been observed and discussed previously by Marsden (1983), Skinner (1987), Barro (1989), Koester and Kormendi (1989), Martin and Fardmanesh (1990), Rebelo (1991), Dowrick (1992), and Engen and Skinner (1992), among others. The strong negative relation between taxes and growth found here concurs with the theoretical specification of Section III.1, where non-lump-sum taxes were shown to be growth-inhibiting. However, if the revenue from such taxes is used for productive government spending then growth can be promoted, if governments are operating (as shown above) in the range where the partial effects of such productive spending on growth are positive.

Unlike Barro (1991) and Mankiw, Romer, and Weil (1992), it is not clear that once endogeneity is taken into account, and given INIT, that GRWKR is significantly positively-related to the mean amount of human capital. ^23/ This result could reflect the often mediocre quality and usefulness of the human capital measures available to researchers; in particular, enrollment data do not control for the quality of schooling (see Behrman and Rosenzweig 1992). It could also reflect the possibility that EDUC, as a measure of school enrollment rates, may be proxying for the mean flow of investment in human capital rather than its mean stock. ^24/ However, the results of columns 1 and 3 are independent of doubts over the efficacy of EDUC.