Multifactor Productivity Estimates: A Sensitivity Analysis:

Multifactor Productivity Estimates: A Sensitivity Analysis John Baldwin, Wulong Gu, Tarek Harchaoui and Faouzi Tarkhani
Micro Economic Analysis Division
Statistics Canada October, 2005

Outline of Paper:

Outline of Paper

Productivity:

Productivity

What is MFP?:

What is MFP? Multifactor productivity is the difference between the increase in output and what would have been expected on the basis of existing technology.

How do we conceptualized it?:

How do we conceptualized it? Starting with the concept of a production function F that is a function of inputs Xi and a time variable t, we can be more specific. If Q=F(X1,X2,..Xn,t), then

How can we operationalize its measurement?:

How can we operationalize its measurement? The critical component is the marginal product in the previous equation. The task of the analyst is to find a way to measure the marginal product. In the simplest of worlds, it is done by assuming profit maximization that allows us to use the factor price as a proxy.

How do we measure MFP?:

How do we measure MFP? In a non-parametric world with relatively few variables—real output change, real labour and capital changes
Factor prices and factor shares
Where do the difficulties lie—particularly problematic is the measurement of changes in capital and its price

More complex extensions?:

More complex extensions? More recently, estimates of multifactor productivity have been moving to increase the number of factors considered—for example, to including different types of labour and different types of assets.
Where Wi is the share of factor i in the remuneration to all i factors (pi.Xi/Σ pi.Xi) and Si is the share of these factors in total output (Σ pi.Xi/PQ).

Measuring the capital component:

Measuring the capital component Measuring total capital
Measuring the price of capital

Measuring the price of capital services:

Measuring the price of capital services 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 (r) (either the opportunity cost of using capital or the financing costs), a term reflecting the depreciation of the asset (δ), and a term reflecting capital gains or losses from holding the asset (reflecting changes in the market price of an asset ,).
Jorgenson (1967) shows that the formula for the rental price of a unit of capital that costs q is

Variants on the theme:

Variants on the theme To take into account taxes, Christensen and Jorgenson (1969) developed the following user cost formula for the kth capital asset type in period t
The formula includes the effective rate of property taxes (nominal valued taxes assessed on the real stocks of land and structures), the corporate income tax rate, the present value of depreciation deductions for tax purposes on a dollar’s investment in capital type over the lifetime of the investment, the rate of the investment tax credit, and the expected capital gains.

What are the issues for the applied statistician:

What are the issues for the applied statistician The rate of return
Endogenous versus exogenous
Depreciation
Tax code or estimates
Capital gains
Existence of trading opportunities
Expected values
Over what time horizon
Tax Parameters
Level of Aggregation

Solving for the Endogenous Rate:

Solving for the Endogenous Rate

Choosing the Exogenous Rate:

Choosing the Exogenous Rate Choosing a rate—bewildering range of choice?
Cost of Capital Literature—combination of equity and debt returns?

Alternate Methods Adopted Here:

Alternate Methods Adopted Here Table 1.
Alternative Specifications of Capital Rental Price Formulas

Data:

Data Canadian Productivity Accounts
Integrated system of inputs and outputs
Capital stock built from perpetual inventory techniques
Rate of return derived endogenously from surplus
Estimated depreciation rates from two sources (used asset prices and length of life)
Asset prices from investment flows—28 asset classes
Long time series—back to 1961
Industry detail (for mfp about 100 industries)

Data:

Data For alternate exogenous method
Rate of return is weighted average of equity and debt
Same capital stock built from perpetual inventory techniques
Use same estimated depreciation rates from two sources (used asset prices and length of life)
Asset prices from investment flows—28 asset classes
To derive price of capital—take exogenous rate add in depreciation and subtract capital gains derived from used asset prices

Evaluation Procedure:

Evaluation Procedure Examine the relationship between the various components that are produced in each variant asking two questions
Does it accord with expectations?
Does it seem sensible in light of our understanding of the manner in which the economy functions?

The Components Examined are:

The Components Examined are Discount Rate=r
Cost of Capital=r+d+Δp
Factor Share=pX/PQ
Capital Composition=difference in growth of capital services using the price of capital across different assets and the growth of capital services assuming all assets yield the same return.
Multifactor Productivity Estimate

Nominal After Tax Rate of Return:

Nominal After Tax Rate of Return

Cost Share of Capital:

Cost Share of Capital

Price of Capital:

Price of Capital

Indices of Capital Quality:

Indices of Capital Quality

Indices of Capital Quality:

Indices of Capital Quality

Multifactor Productivity Estimates:

Multifactor Productivity Estimates

Multifactor Productivity Estimates:

Multifactor Productivity Estimates

A comparison of the results:

A comparison of the results Table 3.
A Comparison of Alternative Capital Rental Cost Formulas in the Business Sector, 1961-1981

The Effect of Ignoring Corporate Tax Provisions (61-81):

The Effect of Ignoring Corporate Tax Provisions (61-81)

The Effect of Ignoring Industry Detail:

The Effect of Ignoring Industry Detail

Conclusion:

Conclusion Choice of Options—endogenous versus exogenous, inclusion of capital gains as measured by asset price changes, instantaneous adjustment or smoothed series, tax rates, level of aggregation

Conclusion:

Conclusion Inclusion of capital gains (20-30%)
Decision on smoothing does not matter as much decision as to whether capital gains are included at all

Conclusion:

Conclusion Endogenous versus Exogenous
Exogenous method with capital gains gives most volatile user cost of capital
Exogenous method with capital gains gives most volatile cost share
Exogenous method sensitive to rate chosen
Exogenous method least likely to capture growth due to shifts across assets into those that are higher yielding
Exogenous rate allows scale estimates and monopoly profits to be considered—but so too does a parametric process

Conclusion:

Conclusion Endogenous versus Exogenous
In the end, it does not matter a great deal when the exogenous method recognizes that the cost of capital should contain an equity component.
The differences are about 20%--and statisticians should provide guidelines on this type of quality to users.
Finally, it should be noted that this difference may be ‘reasonable’ in quite a different sense. Since the two measures would be expected to be different—with the exogenous rate always yielding a result that is too high in a world where an asset has been omitted from the mfp calculation.

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