# 5 Cost function approach

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The cost function approach uses input prices as explanatory variables. These are more likely to be exogenous than the input variables used in a production function. Thus, the cost function is viewed by many economists as a better way to estimate the impact of public capital. In the cost function approach, public capital is assumed to be an unpaid factor of production that affects the level of the variable cost curve. Typically a flexible functional form such as a trans-log or generalized Leonteif specification is employed. Estimates using cost functions continue to generate a positive impact from public capital (see, for example, Morrison and Schwartz 1996, Nadiri and Mamuneas 1994, Conrad and Seitz 1994, Lynde and Richmond 1992, Shah 1992, and Berndt and Hanson 1992). The magnitude of the returns are smaller than those derived from the primal approach leading authors to conclude that the cost function estimates are more realistic.

For Canada, Harchaoui (1997), Harchaoui and Tarkhani (2003) and Brox and Fader (2005) provide estimates of the cost savings associated with public capital using industry data sets. Their models assume that the real level of public capital enters the cost function as an unpaid factor of production. Firms are represented as minimizing costs over private capital and labour, which constitute the variable cost function for the firm, but take public capital as given in the total cost function. Changes in public capital are assumed to change the height of the variable cost curve. Their approach also includes a demand function.

Using this approach, Harchaoui (1997) finds that the impact of
public capital is significant, accounting for about 12% of overall
business sector productivity growth. Harchaoui and Tarkhani (2003)
re-examine the relationship using an expanded data set, reporting
that on average a one-dollar increase in public capital reduces
private production costs by 17 cents. Brox and Fader (2005) perform
a similar exercise, arguing that the elasticity of public capital
with respect to private costs is -0.48.^{4}

We extend these analyses with an alternate approach to modelling the impact of public capital in the cost function.

**Cost function specification**

The relationship between costs and public capital that follows is based on the relationship outlined in Fernald (1999). Suppose a representative firm faces a standard cost minimization problem:

where *w* is a vector of input prices, *z* is a
vector of input quantities, *f*(*z*) is the
transformation function describing the level of output that is
produced with *z* inputs and *q* is the minimum
desired level of production The vectors *w* and *z*
contain information on capital and labour. Under standard first
order conditions, a cost function and conditional factor demand
functions can be described.

Public capital is introduced by assuming that firms use public capital in the production process. It is, therefore, a factor of production in the transformation function:

Where MFP, *K* and *L* are standard inputs and
*T*(*V,R*) is a combinatorial function that generates
transportation services from public capital (*R*) and
vehicles (*V*). Note that the majority of public capital
consists of roads (Baldwin and Dixon 2008). Unlike functional forms
where the stock of public capital is assumed to enter directly,
this paper follows Fernald (1999) and assumes that
*T*(*V,R*) combines roads and vehicles, which allows
the impact of public capital to be approximated as proportional to
the input share of transportation services.

As a result, it is possible to write down equilibrium conditions and a cost function where per unit cost is a function of capital, labour, multifactor productivity (MFP) and the reflection of public capital in transport cost shares. In this setting, private inputs exhaust the economic surplus so that the sum of private inputs is one. Public capital is an unpaid factor that is assumed to affect the height of the total cost curve.

By assuming that public capital usage is proportional to the transportation cost share, it is possible to generate from Equation (5) a measure of the elasticity of public capital that varies over time and across industries. Moreover, the measure exhibits enough variability that it will capture a distinct feature of cost changes due to public capital that differs from MFP.

After taking logarithms, the following Cobb-Douglas cost equation is employed for estimation:

where *c* is total cost, *y* is output, *p*
is the price of capital ( *k* ), labour ( *l* ) or
public capital use (*Tc*), *i* indexes industries and
*t* indexes time. Homogeneity of degree 1 in prices and
constant returns to scale across private inputs are imposed on the
function.

## 5.1 Econometric methodology

The Capital, Labour, Energy, Materials and Services (KLEMS) database contains volatile, disaggregated data that pose a number of challenges for researchers wishing to use it (see Macdonald 2007). Because the data are noisy and contain unusual, or aberrant, observations, less common estimators can be useful for generating parameter estimates.

The most common estimator employed by economists is ordinary least squares (OLS), which is sensitive to aberrant observations. Aberrant observations can influence OLS slope and intercept estimates because they are formed from sample averages, variances and covariances. These measures are inflated by aberrant observations and can lead to poor inference in their presence.

In the presence of aberrant observations, estimation techniques that minimize the impact of unusual observations can be useful for describing statistical relationships. In this paper, Rousseeuw and Yohai's (1984) S-estimator is employed to juxtapose OLS estimates, and to provide insight into how public capital affects private costs. The S-estimator searches through sub-samples of the data to find the one that produces estimates where the dispersion of the residuals is the smallest (for more information, see Rousseeuw and Yohai 1984, and, Chen 2002).

By minimizing the estimate of the residual variance across sub-samples, the algorithm chooses parameter estimates that represent a majority of the observations. Aberrant observations, which increase the residual variance, are not employed and can be identified from diagnostic measures for further evaluation. Moreover, once identified, the aberrant observations can be re-weighted to remove their influence and OLS can be applied to the remaining sample.

**Pre-testing**

In the presence of level shifts and trend changes, unit root testing is challenging. These types of observations can make a stationary series appear to follow a unit root process and lead to poor inference (Madalla and Kim 2003). The KLEMS dataset is affected by both of these types of events (Macdonald 2007). As a result, testing the unit root hypothesis is problematic and caution needs to be exercised.

Panel unit root tests are available that combine the cross sectional information from the panel in an attempt to increase the power of the tests. These tests should lead to better inference about the presence of a unit root. However, in the presence of aberrant observations and trend changes, it is not clear that this is accomplished.

**
Table 4
Hadri LM panel unit root tests
**

Inspection of the price series suggests that Hadri's LM test may provide sufficient inference. However, the dataset does contain aberrant observations that can affect results. Hadri's LM test is a panel unit root test that follows the same vein as the univariate KPSS test (Kwaitkowski, Phillips, Schmidt and Shin 1992). The null hypothesis assumes the series follow a linear trend over time.

The results imply that all series follow a unit root process (Table 3). However, the underlying data are subject to unusual observations that can affect results. The data are, therefore, subjected to a second unit root examination.

**
Figure 6
ADF unit root test statistics for Ln(C/Y)
**

**
Figure 7
ADF unit root test statistics for Ln(P _{L}
)
**

For each of the industries, the augmented Dickey-Fuller (ADF) regression with one lag is run using OLS and the S-estimator. Histograms of the resulting test statistics are presented in Figures 6 through 9. Since the S-estimator is insensitive to aberrant observations, if the aberrant observations are affecting the OLS test statistics, the S-estimator test statistics should give a different outcome.

**
Figure 8
ADF unit root test statistics for Ln(P _{K}
)
**

**
Figure 9
ADF unit root test statistics for public capital cost
proxy
**

The distribution of responses implies that, for the majority of series, the unit root hypothesis is not rejected for all variables. The result is consistent across estimation strategies; however, in all price series, the OLS distribution is to the left of the S-estimator test distribution, implying that aberrant observations are affecting the OLS test statistics. In this case, the effect is not strong enough to change the test result.

The individual ADF unit root tests and Hadri's LM test imply the series follow unit root processes. The results appear robust to estimation strategy and test type.

**Estimation strategy**

The unit root tests imply that a unit root process is present in the data. However, it is unclear whether the series are cointegrated or should be first differenced. The presence of aberrant observations in the dataset makes residual distributions non-normal and many cointegration tests unreliable. Rather than attempt to generate panel cointegration tests in this setting, which is less than ideal for them, two approaches are employed and the results compared.

The first approach log-differences Equation (6). It assumes that the contemporaneous log- differences are sufficient for capturing the relationship between the explanatory and response variables. Because the cost share of capital and labour are fairly stable over time, the log- differenced cost function is not estimated as a system. Rather, OLS and S-estimator estimates are formed from the single log-difference equation.

The second approach assumes the data are co-integrated and estimates the level equation including a trend for MFP but asks whether problems appear in the estimates. The data are estimated first as a seemingly unrelated regression (SUR) system using OLS. A second set of estimates is formed using reweighed least squares (RLS). The RLS estimator gives aberrant observations a weight of zero. They are identified by applying Rousseeuw and van Driesen's (1999) minimum covariance determinant algorithm and the S-estimator to the input variables and Equation (6), respectively.

For both approaches, the units in the panel are treated as distinct entities. They are not pooled, nor are the error processes constrained to have the same variance, which is a commonly applied assumption for panel data models. The units employed here are drawn from a range of industries that have considerable differences between them. Variation in capital intensity makes it difficult to argue that the elasticity of capital, or labour, is similar across industries. Moreover, the small size of some industries makes it difficult to argue that the stochastic processes affecting industries have similar magnitudes.

Therefore, Equation (6) is estimated separately for each industry. This is akin to allowing for unit-specific elasticities of capital and labour, multifactor productivity (MFP), cost savings from public capital and error processes. Once the estimates are generated, their average and individual values are examined.

**Parameter estimate evaluation**

The average log-difference estimates of MFP accord well with
expectations (Tables 5 and 6). The largest MFP estimates are found
in primary, construction and manufacturing industries, while
services experience the smallest gains.^{5} For all industries,
the S-estimator and OLS estimates of MFP are similar, and have an
average of -1.63% and -1.43%, respectively.^{6} For both
estimators, manufacturing sector MFP growth is larger than MFP
growth in the total economy.

The elasticity of labour estimate is near one for many industries, and notably different from labour's share of income on average. The result does not conform with prior expectations about the elasticity of labour in a level cost function, nor non-parametric estimates of labour's share of income in the aggregate economy. The discrepancy arises from the nature of year-to-year labour and capital cost changes.

The cost of capital is a forward looking variable that amortizes capital expenditures over a number of periods. As a result, it changes less rapidly than wage rates. The log-difference equation estimates reflect the difference by indicating that year-to-year output price changes reflect changes in wage rates to a greater extent than their share of income would suggest. This result tends to occur in cost functions when log-differenced prices are used. In a number of industries, the S-estimate is different from the OLS estimate suggesting that aberrant observations are present.

Public capital generates estimates of the elasticity of cost savings across industries that accord well with prior expectations. Those industries that are most reliant on moving goods experience the largest reductions in costs, due to changing public capital provision over time. The service industries experience small, or zero, reductions. For the majority of industries, the S-estimator and OLS provide similar results. On average, OLS suggests that the elasticity of unit prices with respect to public capital is around -0.15 for the total economy, and around -0.22 for the manufacturing sector.

The averages assume that all industries receive the same benefit from public capital, which is likely to be incorrect, given different industry propensities to transport goods. When nominal gross domestic product (GDP) weights are used instead, the calculated cost savings falls by around three to four percentage points for all industries, and by around four to six percentage points in the manufacturing sector. Nevertheless, the results continue to show that public capital has an important impact on private sector costs.

The log-level equation provides slightly lower results for MFP growth compared with the log- difference equation. The OLS and RLS estimates suggest that, on average, MFP growth is -0.92 or -0.81, respectively. In each case, manufacturing is the source of most MFP growth, with a large MFP estimate for computer and peripheral equipment. Service industries experience minimal MFP growth.

**
Figure 10
Log-difference equation MFP (multifactor
productivity)
**

**
Figure 11
Log-difference equation elasticity of labour
**

**
Figure 12
Log-difference equation marginal cost savings of public
capital
**

The estimates for the elasticity of capital and labour from the level equation closely accord with their respective income shares on average. OLS yields estimates of 0.37 and 0.63, respectively, while the S-estimates imply that the elasticity of labour is 0.56 and the elasticity of capital is 0.44.

**
Figure 13
Log-level equation MFP (multifactor productivity)
growth
**

**
Figure 14
Log-level equation elasticity of labour
**

To this point the results do not differ greatly between the OLS and alternative method that takes into account aberrant observations—the RLS or S-estimator estimates. However, the estimate of the elasticity of the public capital is sensitive to the estimation method. OLS provides estimates that are similar on average and across industries to the first-difference specification, while the

RLS-estimates in the log level approach suggest that the provision of public capital does not lower per unit costs over time.

**
Figure 15
Log-level equation elasticity of capital
**

**
Figure 16
Log-level equation marginal cost saving of public
capital
**

**
Table 5
Cost function parameter estimate averages (business
sector)
**

**
Table 6
Cost function parameter estimate averages (manufacturing
only)
**

When no corrections are made for aberrant observations, both methods (log-difference and log-level) produce comparable estimates for the elasticity of cost savings from public capital, whether or not a simple or a weighted average is calculated.

The results are similar to the results found in the literature for Canada, in that they provide a negative elasticity of public capital on the cost of the business sector—though they are slightly higher than produced previously. In particular, Harchaoui and Tarkhani (2003) find an elasticity of cost savings of 0.06 for the business sector using a trans-log cost function. Lastly, with the exception of the RLS estimates of the log-level SUR (seemingly unrelated regression) system, the marginal cost savings are robust to aberrant observations.

**
Figure 17
Aberrant observations by year
**

Although the RLS estimates for the log level formulation imply there is no impact of public capital when aberrant observations are removed from the sample, this does not mean that an economic relationship is not present. Rather it implies that if the log-level equation is believed, then the marginal cost savings from public capital are generated at particular points in time (Figure 17). For the log-difference and log-level equations, aberrant observations are clustered around the first oil shock and the two recessions in the early 1980s. The log-level equation also has a rising number of aberrant observations toward the end of the period.

The timing of aberrant observations implies that industries that are most able to make use of infrastructure inputs during periods of economic hardship are generating the cost savings seen in the level equation. In other words, during periods of stress, industries that are able to make better use of public capital generate greater cost savings. The explanation may be reasonable during the early part of the period; however, the increase in the number of aberrant observations after 1995 in the log-level equation is difficult to explain economically. Unlike the first oil shock and around the recessions of the early 1980s, there is not an accompanying increase in aberrant observations from the log-difference equation, nor a macroeconomic shock. The discrepancy casts doubt on the veracity of the results for the level equation. In particular, it is possible that the increasing numbers of aberrant observations at the end of the period are due to a misspecification of the trend. Because of this, additional evidence is provided in the next section that the log-level equations suffer other problems.

**Robustness of the estimates**

Incorrectly specifying the trend in a time series regression can lead to spurious results. In particular, if a series has a stochastic trend, which is probable for the price series, and a linear tend is fitted, a relationship between the two variables will only exist because they both increase over time. This is the problem that arises when the SUR and RLS estimates are examined.

The hallmark of a spurious regression is an R-squared statistic that is larger than the Durbin- Watson statistic. This relationship signals a spurious relationship because it can be shown that, in a spurious regression, the R-squared statistic converges to a random variable while the Durbin- Watson statistic converges to zero (for a relatively non-technical discussion, see Granger 2001). The Durbin-Watson and R-squared statistics from the log-difference and log-level models are plotted in Figures 18 and 19, respectively. The level equations have high R-squared values and low Durbin-Watson statistics, making them appear spurious, and supporting Tatom's (1991a, 1993) argument that level regressions are not providing adequate inference.

The log-difference equations have Durbin-Watson statistics that are near two on average, suggesting that, while serial correlation may be an issue for particular industries, it is not a concern overall. The log-difference equations exhibit lower R-squared statistics than the level equations and, for a number of industries, the regressions explain minimal amounts of the variation in the dependent variables.

**
Figure 18
Durbin-Watson D-statistic histograms
**

**
Figure 19
R-squared statistic histograms
**

The data suggest that the log-difference cost function estimates are more appropriate than the level equations. After accounting for the stochastic trend, the marginal cost saving estimates appear robust to trend specification errors, have a magnitude consistent with previous estimates for Canada and correspond to a statistically valid estimate from the production function. Moreover, for the chosen functional form, the simple and nominal GDP-weighted averages for the marginal cost saving estimates fall within the confidence intervals from all production function estimates.

**What does the cost function tell us?**

The cost function estimates suggest that, over time, additional public capital reduces private costs. The impact differs importantly across industries, where goods-producing industries tend to experience cost reductions while service industries experience minimal effects.

It is important to note that the way public capital is measured here influences the results. Only industries that pay for transportation services, which tend to be goods-producing industries, appear to be making use of public infrastructure. Service industries also make use of public infrastructure, however, they benefit from the density of urban environments and agglomeration effects, which are not captured in transportation costs. Service industries benefit indirectly, rather than directly.

Nevertheless, the cost-function estimates generate results similar to those previously obtained for Canada. The average cost savings from public capital for the business sector reported here are around 0.11 when nominal GDP weights are used and the preferable log difference approach is used. The elasticity would be -.08 if the log level equation were used and no corrections for aberrant observations made. The latter is similar to the 0.06 cost savings for the business sector from public capital reported in Harchaoui and Tarkhani (2003) where they use industry data sets, log level equations and estimate the impact of public capital directly, using a partial rather than a total cost function, as is done here.

^{
4
} Harchaoui (1997) and Harchaoui and Tarkhani
(2003) use internally consistent databases developed at Statistics
Canada. Brox and Fader (2005) use their own economic time series
for analysis.

^{
5
} The weighted average of multifactor productivity
(MFP) growth across industries is lower than the unweighted average
regardless of the estimation technique employed.

Log-level | Log-difference | |||
---|---|---|---|---|

OLS | RLS | OLS | RLS | |

Nominal GDP, weighted average of MFP growth | -0.27 | -0.42 | -1.05 | -1.29 |

Note: GDP stands for gross domestic product; OLS stands for ordinary least squares; and RLS stands for reweighed least squares.

^{
6
} On the cost side multifactor productivity
captures the extent to which increases in input prices are not
passed on to unit costs.

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