1. An Evaluation of Alternative Methods of Estimating Capital Services
Joint work with John Baldwin and Yan
John Baldwin
Wulong Gu
Micro Economic Analysis Division
Statistics Canada May 14, 2008

2. Concept of capital services
A measure of capital input which would be consistent with production theory is the flow of services that capital goods provide over a time period.
The price of capital goods vs. the unit price of capital services (or user cost of capital):
While the price of the capital good is available, the price of the services that the capital good yields is not usually observed and needs to be inferred.
Statisticians need to estimate the price of capital services.
The price of capital goods differ from the unit price of capital services in a period that capital goods provide when capital is used over several periods.

3. User cost of capital services
Jorgenson and Griliches (1967) have developed a method for measuring the price and volume of capital services that is based on the economic theory of production.
The concept of capital services and capital rental prices is now well established with further developments due to Jorgenson (1995), Hulten (1990), Diewert (2004), and Schreyer, Diewert and Harrison (2005).
The price of capital goods differ from the unit price of capital services in a period that capital goods provide when capital is used over several periods.

4. User cost 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.
Jorgenson and Griliches (1967) show that the formula for the rental price of a unit of capital is
That price has three components:
a term reflecting the opportunity cost of capital (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, ).

5. User cost of capital services
The user cost of capital formula can be modified to take into account taxes (Christensen and Jorgenson, 1969; Schreyer, Diewert and Harrison, 2005).
The user costs by Christensen and Jorgenson are formulated in terms of nominal rates of return and nominal rates of asset price inflation.
Schreyer, Diewert and Harrison (2005) suggest that it is more practical to work with real rates as real rates are much more stable than nominal rates, particularly in high inflation periods.
The real rate of return is calculated as the nominal rate of return deflated by an overall price index (such as the gross domestic product [GDP] deflator or the consumer price index [CPI]), and
The real capital gains of the kth asset is difference between nominal capital gains and the overall inflation rate,
In this paper, we will use the user cost specification (14) that is based on real rates. To obtain a measure of the overall inflation rate , we construct a 5-year centered moving average of the rate of change of the consumer price index.

6. The issues in estimating the user cost of capital
The choice of the rate of return.
The choice of capital gains.
The treatment of corporate tax provisions in the user cost estimation.
While the method for measuring the price and volume of capital services is now well established with further developments due to Jorgenson (1995), Hulten (1990), Diewert (2004), and Schreyer, Diewert and Harrison (2005).
There are number of empirical issues facing statististicans (1) the choice of expected rate of return, (2) the choice of expected capital gains, and (3) the treatment of corporate tax provisions in the user cost estimation.

7. Endogenous vs. exogenous rate of returns
Endogenous (or ex post) rate of returns:
The rates of returns are calculated endogenously from data on capital income and capital stock from the system of national accounts.
Exogenous (ex ante) rates of returns:
The rates of returns are chosen exogenously from observed rates in financial markets.
The exogenous rate of returns in this paper is weighted average of debt costs and rate of returns on equity.
These nominal rates are then deflated by the consumer price index.
The resulting series of real exogenous rates are averaged over the period 1961 to 2001 to yield a constant rate of return of 5.1%. For the user cost specification (14) based on the exogenous rate of return, we will set the real rate of return to a constant, 5.1%.

8. Endogenous vs. exogenous rate of returns
The endogenous rate of returns approach was chosen at Statistics Canada to estimate the user cost of capital services:
It is provides a fully integrated set of accounts.
The assumptions that are required to make use of capital income in the national accounts in estimating capital services are fully compatible with the assumptions that underlie the growth accounting framework for estimating MFP.
It asks how sensitive the results are to the use of exogenous versus endogenous rates of return, to alternate ways of including capital gains, and to whether corrections are made for tax rates.
The paper also examines the effect of the various user cost formulae on the measured multifactor productivity growth.

9. Capital gains
There is little disagreement that, on theoretical grounds, capital gains should be included in the rental price of capital.
Question is how capital gains should be included.
Capital gains (estimated as change in asset prices) are volatile.
If there is no inexpensive way to realize capital gains, changes in asset prices derived from price indices are not a very useful way to measure the capital gains component of the rental price of capital.
The volatility in capital gains is not likely to be matched in the short run by changes in the marginal product of capital because of long gestation periods for capital projects .
If there is no inexpensive way to realize capital gains (high level of transaction costs that must be incurred in selling investment goods. ), changes in asset prices derived from price indices are not a very useful way to measure the capital gains component of the rental price of capital .

10. Capital gains
We will examine three alternative treatment of capital gains:
Instantaneous asset-specific real capital gains.
Smoothed asset-specific real capital gains
No asset-specific real capital gains: all assets have the same nominal capital gains (that are set equal to smoothed CPI).
The volatility in capital gains is not likely to be matched in the short run by changes in the marginal product of capital because of long gestation periods for capital projects .
If there is no inexpensive way to realize capital gains (high level of transaction costs that must be incurred in selling investment goods. ), changes in asset prices derived from price indices are not a very useful way to measure the capital gains component of the rental price of capital .

11. Tax parameters
The rental cost of capital formula requires us to estimate a number of tax parameters.
This formula is costly to implement and therefore sometimes ignored.
We will examine the effect of excluding tax parameters on estimated capital services and MFP growth.
The rental cost of capital formula requires us to estimate a number of tax parameters. This formula is costly to implement and therefore sometimes ignored.
Whether the omission is important will depend upon whether the tax code treats different assets and industries differently, whether the difference between the after-tax and the pre-tax rates differs across industries.

12. The alternative specifications of user cost formulae
Real capital gains
Yes, not smoothed
Yes, smoothed No
User cost with tax parameters
Endogenous rate of return M1 M2 M3
Exogenous rate of return M4 M5 M6
User cost without tax parameters M7 M8 M9
M10
M11
M12
We have a total of 12 alternative specifications of the user cost of capital estimation.
M1-M3 are endogenous rate of return specification with different treatment of capital gains.
M4-M6 are exogenous rate of returns specification with different treatment of capital gains.
m7_-12 differ from m1-m6 in that they exclude tax paramters.

13. The effects of alternative specifications
We will compare the effects of alternative user cost specifications on:
The average rates of return,
The cost share accruing to capital,
The increase in the price of capital services,
The increase in capital composition, and
The growth in multifactor productivity (MFP).
In order to assess the effect of the six alternate scenarios, we compare the average rates of return that are produced by each, the share accruing to capital, the increase in the price of capital services, the increase in capital composition (the difference between the growth of capital services and the growth in capital stock), and finally the growth in multifactor productivity (MFP).

14. Data for the paper The Canadian KLEMS database.
The data on investment and capital stock for 28 reproducible capital stock plus land and inventories at the industry level.
The rental cost of capital formula requires us to estimate a number of tax parameters. This formula is costly to implement and therefore sometimes ignored.
Whether the omission is important will depend upon whether the tax code treats different assets and industries differently, whether the difference between the after-tax and the pre-tax rates differs across industries.

15. The effects on rate of return
The endogenous rate of return is slightly higher than the exogenous rate of return over the entire period.
This difference could arise either because our exogenous rates are calculated using too few asset types, or because they contain a residual that is really due to scale economies.
The slightly higher endogenous rate in the late 1990s may reflect the importance of intangible capital and knowledge capital that are not included here.
The endogenous rate of return with instantanous capital gains are most volatile. This reflect the fact that instantaneous capital gains are volatile.

16. The cost share of capital
The cost share of capita is used to estimate to contribution of capital input to output growth.
The cost share of capital from endegonous rate is slightly higher than the cost share of capital from the exogenous rate, expecially before 1980.
The cost share of capital is more stable for the endogenous rate specifications than for the exogenous rate specifications. In particular, the cost share of capital for the endogenous rate specification with annual asset price change being used to represent capital gains (M1) is more stable than the cost share of capital for the exogenous rate specification. This occurs despite the fact that the former specification has the most volatile rate of return and the most volatile capital gains. The volatility of the rate of return and capital gain series for M1 is offsetting, which yield a more stable cost-share series.
The stability of the cost-share series arising from the endogenous alternative is an advantage of this technique if a proxy is desired to track the time path of the marginal product of capital that is regarded as not being highly volatile

17. The user cost of capital
The endogenous rate of return is expected to yield larger fluctuations in the user cost of capital than exogenous rate of return specifications.
This slides show the opposite. The user cost of capital using the endogenous rate of specification are more table that the one from exogenous rate of return.
The user cost of capital is most volatile for the exogenous rate specification that accounts for annual capital gains.
It is only when real capital gains are excluded that the exogenous rate method yields a stable series of the cost of capital.
The lower volatility of the endogenous rate favours its use by those who believe that the marginal product of capital changes only slowly over time.

18. Annual growth in capital quality and MFP, 1981-2001
1.1
0.9
Annual MFP growth (%)
0.3
0.4
For the period , the endogenous rate and exogenous rate methods yield similar estimates of capital quality growth and MFP growth.
For the period before 1981, the endogenous rate yields lower capital quality growth and higher MFP growth.
The growth in capital composition is lower for the case when capital gains, derived from asset price changes, are not included in the estimation of user cost. This result is due to (1) the long-run historical shift toward equipment (with relatively high depreciation and high user cost) and away from structures (with relatively low depreciation and low user cost), which increases the capital composition effect; and (2) the long-run tendencies for the price of structures (with low depreciation) to increase faster than the price of equipment (with high depreciation), causing the capital gains that are subtracted in the user cost of structures formulae to be larger than those subtracted in the user cost of equipment estimates. This increases the difference in the user cost of structures and equipment, and thus leads to an increase in the growth in capital composition.
The growth in MFP is higher for the case when capital gains, derived from asset price changes, are not included in the estimation of user cost.

19. The bias of excluding tax parameters in user cost
The bias of ignoring taxes in the annual MFP and capital composition estimates is small for both endogenous rate and exogenous rate methods.
The bias of ignoring taxes is much larger for the exogenous rate method than for the endogenous rate method.
This suggests that the endogenous method has an advantage over the exogenous method.
In order to assess the effect of the six alternate scenarios, we compare the average rates of return that are produced by each, the share accruing to capital, the increase in the price of capital services, the increase in capital composition (the difference between the growth of capital services and the growth in capital stock), and finally the growth in multifactor productivity (MFP).

20. The aggregation of capital service over industries
Top-down vs bottom-up approach.
The bottom-up approach are used by Statistics Canada and the U.S. BLS to estimate aggregate capital services.
Alternatively, the top-down approach can be used to estimate aggregate capital services.
Empirical evidence appears to favour the top-down approach:
Large difference in the rate of return. The rate of return is positively related to capital stock growth across industries.
The aggregate capital services at Statistics Canada and the BLS are calculated by aggregating capital service across industries (Jorgenson, Gollop and Fraumeni, 1987). This approach takes into account the difference in the rate of return across industries and avoids the assumption of perfect mobility of capital inputs across industries.
It has been argued that we should use the industry-specific return to calculate the user cost of capital and that aggregate capital services should then be calculated by aggregating capital service across industries (Jorgenson, Gollop and Fraumeni, 1987). This approach takes into account the difference in the rate of return across industries and avoids the assumption of perfect mobility of capital inputs across industries.
This suggests that the difference in the rate of return across industries is real and capital tends to move towards those industries that earn relatively high rates of return.

21. The difference in capital growth and MFP growth using the two approaches
Bottom-up
Top-down
Period 1981 to 2001
Capital services
3.55
3.09
Multifactor productivity growth
0.12
0.31
Reallocation of capital
0.46 …
The bottom-up approach yields higher capital growth and lower MFP growth than top-down approach.
The difference in capital service growth between the two approachs is called the effect of reallocation of capital.
The reallocation of capital services across industries accounts for 30% of capital composition growth and 15 percent of capital service growth over the 1981 to 2001 period.

22. Conclusions Comparison of exogenous rate and endogenous rates.
The endogenous rate has a number of advantages: more stable cost share of capital, more stable user cost of capital.
The effects of two approaches on MFP growth and capital service growth is small.
The rental cost of capital formula requires us to estimate a number of tax parameters. This formula is costly to implement and therefore sometimes ignored.
Whether the omission is important will depend upon whether the tax code treats different assets and industries differently, whether the difference between the after-tax and the pre-tax rates differs across industries.

23. Conclusions Comparison of alternative treatment of capital gains.
Moving from instantanous capital gains to moving average of capital has little effect on capital service growth and MFP growth.
What matters is whether we include asset-specific real capital gains. Capital service growth is higher and MFP growth is lower if we do not include asset-specific capital gains.
The rental cost of capital formula requires us to estimate a number of tax parameters. This formula is costly to implement and therefore sometimes ignored.
Whether the omission is important will depend upon whether the tax code treats different assets and industries differently, whether the difference between the after-tax and the pre-tax rates differs across industries.

24. Conclusions
The effect of excluding tax parameters on capital service growth and MFP growth.
The bias in MFP growth and capital service growth when we exclude tax parameters is small.
This suggests that it is probably a good idea to exclude tax parameters in gthe user cost calculation as the EU KLEMS did.
The rental cost of capital formula requires us to estimate a number of tax parameters. This formula is costly to implement and therefore sometimes ignored.
Whether the omission is important will depend upon whether the tax code treats different assets and industries differently, whether the difference between the after-tax and the pre-tax rates differs across industries.

25. Conclusions The difference between bottom-up and top-down approach.
Empirical evidence appears to favour the top-down approach.
The reallocation of capital accounts for a significant share of capital service growth.
The bottom-up approach yields higher capital growth and lower MFP growth than top-down approach.
The rental cost of capital formula requires us to estimate a number of tax parameters. This formula is costly to implement and therefore sometimes ignored.
Whether the omission is important will depend upon whether the tax code treats different assets and industries differently, whether the difference between the after-tax and the pre-tax rates differs across industries.