# 4 The production function approach

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The production function approach was the first widely employed method for examining the impact of public capital in private production. It extends back at least as far as Meade (1952) who develops production function specifications that include public capital. More recently, Arrow and Kurz (1970) and Grossman and Lucas (1974) argue that the provision of public capital should be included in private-sector production functions. Despite these arguments, the impact of public capital did not receive widespread attention until empirically analysed by Aschauer (1989).

Aschauer (1989) uses production function estimates to ignite a debate about the role of public capital in private production, and its role in the productivity slowdown in the United States during the 1970s. Additional estimates are provided for the United States by Holtz-Eakin (1988) and Munnell (1990a, 1990b) that support the notion that output is responsive to the level of public capital.

For Canada, Wylie (1996) adopts the approach taken by Aschauer (1989) to estimate the elasticity of public capital in Canada. Using a trans-log production function, and Canadian aggregate data from 1946 to 1991, he finds that government capital has a positive elasticity. He concludes by arguing that his results support the finding for the United States that public capital plays an important role in business sector output and productivity growth.

These studies, particularly Aschaeur's, have been criticized for failing to account for non- stationarity and omitted variable bias (Tatom 1991a, 1993), simultaneity bias (Berndt and Hanson 1992), and because of the magnitude of the coefficient estimates, which critics claim are improbably large (Aaron 1990). The present study addresses many of these issues, presenting estimates that explicitly account for time series features of the data.

**Production function specification**

Consistent with previous studies, this paper assumes that
although public capital investment is financed through taxation,
the marginal quantities consumed by firms are 'free of
charge.' Firms view public capital as an unpaid factor of
production when maximizing profit. Their output is assumed to be a
function of private capital (K), labour input (L), public capital
(G), and multifactor productivity *MFP*(*t*) :

This paper uses the Cobb-Douglas function to generate parameter
estimates of *MFP*(*t*)*F*( *K*, *L
G*, ) . The Cobb-Douglas function is employed because it is
commonly used in the literature, making estimates comparable with
previous studies. Additionally, the resulting elasticity estimates
should be similar to labour and capitals' share of income,
which provides a transparent, widely understood set of prior
expectations against which the impact of including public capital
can be examined.

Because government capital is an unpaid factor, it is unclear what the production function's returns to scale should be. Viable alternatives present in the literature include constant returns to scale across all inputs and constant returns to scale across private inputs (See Aschauer 1989, Holtz-Eakin 1994). The first supposes that the public capital is taken as an input in the same fashion as private capital and labour. Implicitly, this functional form assumes that public capital's share of private sector income is captured by private sector inputs. However, there is little guidance about how public capital's share of income is distributed between the private sector inputs, and various authors have approached the problem in different ways.

The constant returns to scale across private sector inputs assumption posits that because public capital can increase the concentration of economic agents, and because it acts as a network, including public capital can lead to increasing returns to scale across all inputs while firms face constant returns to scale in private inputs. Because firms take variables such as land cost and transportation time into account when making investment decisions, public capital has a significant impact on where firms locate. However, once that choice is made, the influence of public capital does not affect its input decisions. Firms choose the amount of labour and private capital in order to minimize costs and maximize investor returns. While public capital is used in the production process (through shipping, for example), its marginal quantities are employed free of change. As a result, at the margin, public capital does not affect investment decisions about the optimal amounts of capital and labour that should be employed. The firm, therefore, may face constant returns to scale across private inputs with public capital leading to overall increasing returns to scale.

Because there is no consensus about which returns to scale
assumption is most valid, both specifications were analysed for
this paper. The results reported below examine only the function
with constant returns to scale across private inputs because it
provides estimates that appear to fit the data the most
reasonably.^{2}

The Cobb-Douglas function with constant returns to scale across
private inputs imposes the constraint that
*β _{l}* +

*β*= 1 so that the production function can be written as:

_{k}where a lower case letter denotes the log-level,
*β _{k}*,

*β*, and

_{l}*β*are the elasticities private capital, labour input, and public capital respectively,

_{g}*MFP*(

*t*) is a term that captures changes in intangible, difficult to measure, inputs such as management structure or research and development,

*p*indexes provinces and

*t*indexes time. Finally,

*e*is an identically and independently established error term.

_{t}## 4.1 Econometric methodology

**Pre-testing**

The provincial panel estimates for the 1981-to-2005 period contain trends over time. When variables have trends, it is important to ascertain the nature of the trends in order to avoid spurious results. Spurious results can arise in two situations. The first situation occurs when the levels of variables that do not share a stochastic trend are used in a regression. When this occurs, the changing levels over time can make it appear that a strong statistical relationship exists when there is no actual relationship. The second situation occurs when a variable with a deterministic trend is used in a regression on a variable with a stochastic trend. This often occurs in level regressions that include time trends. As with the first situation, a strong statistical relationship can be found between the variables because the levels change over time, not because of an economic relationship. Only when all variables have a deterministic trend, or all variables follow a common stochastic trend (are co-integrated), will a regression using levels provide meaningful inference. As a result, it is important to investigate the time series nature of the data prior to estimation.

Over the period in question, business sector gross domestic product (GDP), hours worked, private capital and public capital series have a particular relationship: labour input adjusts quickly to short-term economic fluctuations; private capital adjusts moderately; and, public capital tends to mimic trend GDP growth. It is important to note, however, that this is a generalization and that the pattern is not fixed. There are differences across provinces that can be exploited to increase the accuracy of the parameter estimates.

An IPS (Im, Pesaran and Shin 1997) unit root test is applied to gauge how the data should be treated. The IPS test aggregates individual augmented Dickey-Fuller unit root tests into a panel unit root test statistic. The test attempts to increase the power of the individual unit root tests by exploiting the cross sectional observations present in the panel. The null hypothesis is that all series contain a unit root, while the alternative hypothesis is that at least one series does not contain a unit root.

**
Table 1
IPS (Im, Pesaran and Shin) unit root test provincial panel data,
1981 to 2005
**

The IPS test statistic rejects the null hypothesis that all log-level GDP series contain a unit root while it fails to reject the null hypothesis of a unit root in labour input, private capital and public capital (Table 1).

The IPS tests imply that a mixture of processes are present in the underlying series. To avoid generating spurious results the series are transformed by taking first differences to remove their trends over time. The resulting equation, which is estimated below, is:

**Estimation strategy**

The provincial panel data provides a rich dataset; however, it will be necessary to control for fixed effects and contemporaneous shocks. Two system estimators are used to control for these effects. In order to make the impact of using the more sophisticated estimators clear, their results are compared with simple ordinary least squares (OLS) estimates.

The first set of estimates based on Equation (2) come from applying OLS to pooled log- differenced data. At this stage, the function is estimated without the public capital variable to generate a base case. The base case is then compared with OLS estimates from the pooled data that include public capital, and with a specification where multifactor productivity (MFP) is constrained to zero. Through imposing the constraint that MFP is zero, it is possible to examine the degree to which public capital and MFP are capturing similar features of real GDP growth in the sample.

Using OLS on pooled data potentially misses important fixed effects across provinces and covariances between provincial economic shocks. Two systems estimators that account for potential variance-covariance problems and fixed effects are, therefore, calculated to provide more efficient, consistent estimates.

The first system estimator (FGLS_{1}) treats each
province as a separate equation. It allows for province specific
MFP and variances as well as a contemporaneous covariance between
provinces. The equations are stacked as:

where the subscript *h* indexes provinces. Provincial
dummy variables are included to account for province-specific MFP
growth. The elasticities of private capital, labour and public
capital are constrained to be equal across provinces.

The system is then estimated as a generalized least squares problem. The variance-covariance matrix for the system, which allows for heterogeneous errors across provinces and contemporaneous correlation between provinces, is written as:

where *σ _{i,j}* is the covariance between
provinces

*i*and

*j*if

*i*≠

*j*, or the variance of province

*i*if

*i*=

*j*.

The second system estimator (FGLS_{2}) modifies
FGLS_{1} to account for residual serial correlation. It
adds an additional level of complexity by expanding the
variance-covariance matrix as:

where Ω is a matrix containing a first-order serial correlation correction (See Greene 2000 ch. 15).

**Evaluation of estimates**

The base case equation using pooled log-difference data to estimate (2) provides an estimate for the elasticity of capital of 0.31, similar to capital's share of income (Column (1) of Table 2). The elasticity of labour is 0.69 and is similar to labour's share of income. The estimate of MFP growth is 1.04% per year and is of a reasonable magnitude for the 1981-to-2005 period. All estimates are statistically significant and the data provide estimates that fit well with prior expectations.

When public capital is included in the production function, the
elasticities of hours worked and private capital remain unchanged
at 0.31 and 0.69, respectively (Column (2) of Table 2). The
estimate of MFP growth increases slightly to 1.15% per year. The
estimate for the elasticity of public capital, however, is negative
and statistically insignificant. When MFP growth is constrained to
zero, there is little change in the elasticity of hours worked and
private capital. The estimate for the elasticity of public capital,
however, increases to 0.41 and becomes statistically significant
(Column (3) of Table 2).^{3} The impact of public capital then is
difficult to disentangle from overall productivity growth. While
estimates of overall productivity growth probably incorporate some
influence of public capital when that variable is not included, it
is unlikely that when public capital is included, MFP growth should
be zero—unless all of it comes from public capital.

**
Table 2
Panel production function estimates
**

When the relationship is estimated using FGLS _{1,} the
estimate for the elasticity of capital is 0.31 and statistically
significant. The estimate for MFP growth is 1.01% and not
statistically different from zero when public capital is included
(Column (4) of Table 2). Estimates of provincial MFP do not differ
statistically from Ontario, which is used as the reference level.
If MFP growth is constrained to zero, the estimate for the
elasticity of private capital increases to 0.37 and the estimate
for the elasticity of public capital increases to 0.31 (Column (5)
of Table 2). Both estimates are statistically significant at the 5%
level. Moving to FGLS _{2} provides similar results. Once
more, it is difficult to disentangle the effects of public capital
from the estimates of MFP.

Panel estimates of the elasticity of private capital are close to private capital's share of income, regardless of the inclusion of public capital and the exclusion of MFP growth. Overall they appear to offer a robust set of private input elasticity estimates. Using this specification, estimates for the elasticity of public capital are intertwined with MFP growth. This supports the finding in Harchaoui and Tarkhani (2003) that MFP growth and public capital growth are related. While Harchaoui and Tarkhani (2003) argue that approximately 12% of MFP growth is accounted for by public capital using cost data, the production function data employed here do not permit the disentanglement of these two factors.

The clearest way to summarize the results is through a series of figures that plot point estimates with their 95% confidence intervals. Because estimates for the elasticity of capital are invariant to the inclusion of government capital and the suppression of MFP growth they provide a stable, and robust, set of estimates against which the impact of public capital can be analysed (Figure 3). The private input elasticity estimates are little changed across specifications and estimation techniques with relatively tight confidence intervals. The point estimate for the elasticity of public capital changes noticeably, moving from near zero to the 0.3 to 0.4 range depending on whether or not MFP growth is included in the equation (Figure 4). The confidence interval for public capital shrinks when MFP is constrained to zero. The MFP point estimate is little changed when government capital is included; however, its confidence interval expands and the estimates become statistically insignificant (Figure 5).

**
Figure 3
Production function estimates of the elasticity of private
capital
**

**
Figure 4
Production function estimates of the elasticity of public
capital
**

**
Figure 5
Production function estimates of multifactor productivity (MFP)
growth
**

**What does the production function approach tell
us?**

Production function results show that MFP estimates and the elasticity of public capital are capturing similar features of real GDP. Consequently, it is likely that, where the impact of public capital is excluded from the production function, MFP estimates are capturing the influence of public capital. However, it is unclear what fraction of MFP growth can be attributed to public capital. This is not an econometric or theoretical problem. It is a data problem that reflects the fact that changes in the provision of public capital do not vary greatly over time. In order to gain greater precision on the elasticity of public capital, it will be necessary to find different data or an alternate method for calculating public capital services.

This feature of public capital is not widely recognized in the literature sparked by Aschauer (1989). When panel estimates from the literature are compared with the results from this paper, it appears that the same data problem may be present in other studies (Table 3). Including MFP and public capital in the same equation generates results that suggest they may be capturing the same feature of GDP. Notably, including a trend tends to reduce the impact of public capital.

**
Table 3
Comparison of panel data results
**

It, therefore, seems reasonable that if one were to constrain MFP growth to zero and re-estimate the relationship in the other studies that the elasticity of public capital would increase. Unfortunately, this does not help to accurately estimate the elasticity of public capital in Canada or the rate of return associated with it. An alternative method for capturing the impact of public capital is necessary.

^{
2
} The alternative set of estimates is available
upon request.

^{
3
} Note that t-statistics are not exact when
multifactor productivity is constrained to be zero. The residual
sums are not zero so the inference is only approximate.

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