The Canadian Productivity Review

Volume 2007, Number 009

Multifactor Productivity in Canada: An Evaluation of Alternative Methods of Estimating Capital Services

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Multifactor Productivity in Canada: An Evaluation of Alternative Methods of Estimating Capital Services

John R. Baldwin and Wulong Gu

Executive summary

Multifactor productivity growth measures have been developed as summary statistics to measure improvements in the efficiency of the production process — that come from technological progress and organizational change. They do so by comparing actual growth rates in output with the increase in output that would have been expected from an increase in inputs using pre-existing or current production techniques.

At any point in time, existing techniques allow additional factor inputs (labour, capital) that are applied to the production process to produce additional output. Additional factors that are added to the production process multiplied by the existing marginal product of those factors provides an estimate of the expected amount of output in a given period that should have resulted from the use of these factor inputs. If actual output exceeds this, productivity is said to have increased.

Multifactor productivity is measured as the difference between growth in output and the increase in output that would have been expected  because of the use of additional units of inputs . This is calculated as the difference in the rate of change of output minus a weighted average of the change in inputs (labour, capital), where the weighted average is calculated to give a proxy for the change that should have occurred if inputs had been as productive as they were in the past.

The weights that are generally used to aggregate changes in a type of factor are the relative shares of each type of factor in the total compensation received by that factor. The unit price of each type of factor is needed to estimate these shares. In the case of prices for labour, the task is relatively straightforward. Transactions are observed continuously in labour markets that can be used for this purpose. In the case of capital, the unit price of capital needs to be estimated. While the price of the capital good is available, the price of the services that the capital good should command, when it is used over a period that is shorter than its length of life, is not usually observed and needs to be inferred.

Economic theory is used to suggest a formula that can be used to infer the price of capital. The user cost of capital can be thought of as the price that a well functioning market would produce for an asset that is being rented by an owner to a user of that asset. That price would comprise a term reflecting the opportunity cost of capital, a term reflecting the depreciation of the asset, and a term reflecting capital gains or losses from holding the asset. This formulation requires data on the rate of return, depreciation, capital gains from holding assets, tax rates on capital, and the price of the asset.

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In this paper, we discuss the methodology that has been employed to estimate rental prices of different assets and the problems involved in their implementation. We then present a set of alternate estimates of multifactor productivity growth using different methods to generate the required data on the rate of return, the depreciation rate, capital gains and tax rates.

Several issues that are raised in the literature are discussed and addressed in turn in this paper. The first is whether rates of return should be calculated exogenously or endogenously. The second is the source of depreciation rates. The third is whether and how capital gains should be included in the user cost of capital formula. The fourth is how tax rates are included in the formula.

The differences in the estimates of multifactor productivity that have been provided in this paper outline the extent to which alternate assumptions matter. The alternate methods considered here yield a range of estimates. Inclusion of capital gains has a significant impact not only on the estimated growth in capital services but also on its volatility. Our method of including capital gains in most of the variants investigated in this paper is the conventional one — that involves calculating the rate of changes in the prices of investment goods over time.

There is considerable volatility in the capital gains term when this is done. Since this term implicitly captures the expectation of capital gains and the volatility in this term is large, it is difficult to choose a method of averaging that properly incorporates expectations.

Since it is essentially arbitrary whether we include or exclude capital gains, we can place a bound on our estimates of the growth in multifactor productivity by comparing the difference between estimates with and without this term. For the period 1981 to 2001, these differences are 0.1 percentage point or about 23% to 30% of the multifactor productivity estimate. For the period from 1961 to 1981, the ranges are smaller and are around 4% to 10% of the multifactor productivity estimate.

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We also find that ignoring corporate taxes leads to an upward bias in the growth of capital services for the endogenous rate method for both the 1961 to 1981 and 1981 to 2001 periods.

The other major choice that has to be made is between endogenous as opposed to exogenous rates. The Canadian data do not show large differences in the actual rate of returns — where the endogenous rate is calculated from National Accounts industry data and the exogenous rate of return is calculated from a weighted average of the equity and the bond markets. It is worth pointing out that the differences between the two multifactor productivity estimates using each are not large. The endogenous multifactor productivity rate is about 12% to 14% below the exogenous multifactor productivity rate — within the margin of error that one would expect from misstating the exogenous rate of return by 25%.

The quality of any statistical program depends upon the extent to which the summary statistics are fit for use. Meeting this objective requires that agencies give users guidance on appropriate use. Generally, only point estimates of multifactor productivity are provided by statistical programs. The effect of using alternate assumptions for the production of summary statistics is rarely investigated. This paper addresses this gap for multifactor productivity estimates.

Doing so is important when users employ multifactor productivity measures in cross-country comparisons. Users need to be aware that the multifactor productivity estimates are point estimates with a confidence interval around them. That confidence interval cannot easily be calculated using classical statistical techniques. But we can give users an idea of how our estimates would change if we employed alternate assumptions.

We have shown here that the point estimates that we produce could very well vary by 20%, either because of the choice of the endogenous over the exogenous technique or because of the way in which capital gains are measured. When users employ multifactor productivity estimates for cross-country studies, these types of errors should be kept in mind.

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