Measuring Capital and Capital Services: An Overview

1. NBS-OECD Workshop on National Accounts6-10 November 2006 Measuring Capital and Capital Services: An Overview Paul Schreyer OECD

2. Purpose of capital measurement • 2 main purposes: • Capital as a storage of wealth • Capital as a source of productive services • The two purposes correspond more or less to • Capital in the balance sheets and in the income side of the national accounts, e.g., value of fixed assets, depreciation, net and gross income… • Relevant questions: what is the value of wealth in the economy? Is current income sustainable?

3. Purpose of capital measurement • Capital in the production side of the national accounts • Relevant questions: capital and multi-factor productivity, growth accounting, competitiveness • In the SNA93, no explicit recognition of capital in the production account • In revision to SNA 93, explicit recognition of capital income and capital services in the production account

4. Stocks and flows – an integrated approach • Key objective of new capital manual: present integrated and consistent approach towards capital measurement to link relevant flows and relevant stocks: • Investment • Depreciation • Capital services • Net stock • Gross stock • Productive stock

5. System of capital in the SNA93 Age-price function Net value added Net stock CFC Gross stock Investment Retirement function

6. System of capital in the new SNA Age-price function Net value added Net stock Return on capital CFC User costs Gross stock Investment Retirement function Productive stock Age-efficiency function Capital services

7. Stocks and flows – an overview

8. Asset market equilibrium condition • Central economic relationship that links income and production perspective Walras (1874); Boehm-Bawerk (1888) • Stock value of an asset = discounted stream of future rental payments that the asset is expected to yield • P0t: price of new asset purchased at the beginning of period t • fnt: nominal rental payable at beginning of period t • (1+rt): nominal discount factor • P0t=f0t+1/(1+rt) + f1t+2/(1+rt)2 + f2t+3/(1+rt)3+…

9. Asset market equilibrium condition (2) In a functioning market, purchase price of an asset equals the discounted stream of expected rentals Purchasers will buy asset of flow of rental implies at least a rate of return that is as large as rt rt can also be considered the opportunity cost of investing in the asset = the return that the market would pay for investment of similar risk Central equation for integrated system of stocks and flows of capital

10. Rentals and asset prices – numerical example (1) • Assume: asset with • Service life of 8 years • Discount rate 5 % • Rental for a new asset is 10$ • For simplicity, no general inflation • Price of new asset and price of rentals are expected to rise by 2% per year: fnt+1=fnt*1.02 • Productive services of the asset decline by a constant amount over its service life:  linear age-efficiency pattern

11. Rentals and asset prices – numerical example (2) 10*0.88*1.02 =8.93 10.0/1.05=9.52

12. Rentals and asset prices – numerical example (3) 7.80/1.052 =7.08

13. Rentals and asset prices – numerical example (4) This example also shows a very important link between age-efficiency profile and the age-price profile For a given rate of interest, a given rate of price change of new assets, there will be exactly one sequence of asset prices for each age-efficiency profile Consider the following price history of an asset

14. Rentals and asset prices – numerical example (5), Price history of asset • Diagonal price movement: total change in value of asset between two years e.g. 40.12-32.12 = 8 • Vertical movement: price change of new asset (2%) = 40.92-40.12=0.80 • Horizontal movement: price difference due to age = 40.92-32.12=8.80

15. Rentals and asset prices – numerical example (6), Age-price profile 32.12/40.92=0.785

16. Rentals and asset prices – numerical example (7) Note: Age-price profile does not depend on time here because age-efficiency profile is not time-dependent and because the expected rates of asset price change and discount factors are given for any point in time However, as historical time moves on, it may well be that discount factors or price expectations change, in which case the age-price profiles of all assets would be affected.

17. Rentals and asset prices – numerical example (8), linear age-efficiency profile

18. Rentals and asset prices – numerical example (9), constant age-efficiency profile

19. Rentals and asset prices – numerical example (10), geometric age-efficiency profile

20. Rentals and asset prices – numerical example (11), hyperbolic age-efficiency profile

21. Retirement and survival functions (1) • Assets in a cohort are unlikely to retire all at the same moment • Typically, there is a retirement distribution around an average retirement age • To construct the age-efficiency function of a cohort, the age-efficiency function for a single asset has to be combined with a retirement distribution • This is shown in the following slides

22. Retirement and survival functions (2) • Probability density function of retirement: shows the (marginal) probability for an asset to retire at age T • For simplicity, log-normal distribution

23. Retirement and survival functions (3) • Call the probability that an asset retires at age T, FT • In our example with a linear age-efficiency function gs, we had gs=1-s/T. • For a single asset, the service life was assumed T=8 • With a retirement distribution, for each age s, there is a possibility that the asset retires at age s, or at age s+1 etc. • Calculate an average, with probability weighting where Tm is the maximum service life: • hs= ∑T=sTm[1-s/T]*FT • This creates a new age-efficiency profile for the cohort that reflects both efficiency loss and retirement {hs}

24. Retirement and survival functions (4), age-efficiency profiles • hs is non-linear even with linear gs!

25. Retirement and survival functions (5)A special case: one-hoss shay • Suppose a class of assets follows a one-hoss shay pattern of efficiency loss: efficiency is constant until the asset retires • Then, the combined age-efficiency/retirement pattern {hsG} becomes hsG= ∑T=sTmFT, i.e., only a retirement pattern. • More precisely, hsG is the cumulative probability density function that varies from h0G=1 for a new asset to hTmG=0 • It is now possible to construct the gross capital stock on the basis of the perpetual inventory method: • Gross capital stock = sum of past investments of a class of assets, with cohorts weighted by the retirement pattern {hsG}

26. Retirement and survival functions (6)A special case: one-hoss shay

27. Retirement and survival functions (7)Gross capital stock • Gross capital stock = sum of past investments of a class of assets, with cohorts weighted by the retirement pattern {hsG} • Gross capital stock = stock of assets surviving from past investments that ignores deterioration of assets and considers past investment ‘as new’ - only retirement is taken into account • Although the gross capital stock is often calculated in practice, it serves mainly as an intermediate step towards measuring depreciation and net stocks rather than as an analytical measure in itself. • Note: net stocks and depreciation can but do not have to be calculated via the gross stock. In fact, the usefulness of the gross stock is relatively limited. • Some countries (eg United States) do not publish gross stocks any more – they restrict themselves to net stocks and productive stocks (see later)

28. Retirement and survival functions (8)Gross capital stock • Note three types of valuation of stocks: • Historical prices = valuation in terms of prices of the year of acquisition • Constant prices = valuation in terms of a base year • Current prices = special case of constant prices = valuation in terms of the current (typically latest) year

29. Net or wealth stocks (1) • Net capital stock or wealth stock = market value of assets • Net capital stock = stock of assets surviving from past investments that has been corrected for retirement and for loss in value due to ageing • Net capital stock offers a wealth perspective. It is the capital stock that shows up in the balance sheets of the national accounts. • Calculation of net stocks: 2 possibilities: • Directly, as sum of past investments, weighted by age-price profile • Derived from gross stock and depreciation

30. Net or wealth stocks (2) • Starting point: age-price profile, derived from combined age-efficiency & retirement profile

31. Net or wealth stocks (3)

32. Depreciation (Consumption of fixed capital) (1) • Depreciation is the loss in value of an asset or a group of assets as they age • A flow concept • Economic meaning: deduction from gross income to account for the loss in capital value owing to the use of capital goods in production • SNA definition: « the decline, during the course of the accounting period, in the current value of the stock of fixed assets owned and used by a producer as a result of physical deterioration, normal obsolescence or normal accidental damage. » • Excluded: value losses due to acts of war or as a consequence of exceptional events such as major natural disasters

33. Depreciation (Consumption of fixed capital) (2) • Note: • Depreciation must be measured with reference to a given set of prices, ie the average prices of the period • “Used by producer” includes assets that are kept idle for whatever reasons • “Normal obsolescence” is included in depreciation but not “abnormal obsolescence”. Example: scrapping of energy-intensive machines following an oil-price shock

34. Computing depreciation (1) • Two avenues: directly and indirectly via net stock • Direct computation: main tool: age-price profile and investment series • Rate of depreciation of an s-year old asset = price difference between an s-year old asset and an s+1 year old asset divided by price of an s-year old asset: {P0t, P1t, P2t,…} is the age-price profile, so dst can be derived directly

35. Computing depreciation (2) • When applied to past investment, depreciation rates apply in a cumulative way: • Depreciation for a new capital good: d0It • Depreciation for a one-year old capital good: d1(1-d0)It-1 • Depreciation for a one-year old capital good: d2(1-d1)(1-d0)It-2 • Etc. • Total depreciation = d0It + d1(1-d0)It-1 + d2(1-d1)(1-d0)It-2 + … • Note: investment {It, It-1,…} is expressed in constant prices of a particular base year, therefore total depreciation is also in prices of this base year.

36. Computing depreciation (3) (46.01-37.56)/46.01=0.184

37. Computing depreciation (4)

38. Computing depreciation (5) Net capital stock for period 17 at prices of period 16

39. Computing depreciation (6) Indirect way of computing depreciation via net stocks and the following identity: Note: specific lags in this identity (depreciation year 16) disappear when everything is formulated in mid-year valuations

40. Computing depreciation (7) • Note: depreciation is in prices of period 16 • To obtain depreciation at current prices, apply investment goods deflator between years 16 and 17 • This is also an easy solution to split depreciation into price and volume components – the deflator for depreciation always equals the asset price deflator

41. Computing depreciation (8) • Empirical basis for depreciation rates • Derived from age-efficiency functions. This raises the issue of how age-efficiency parameters are constructed. Needed: average service lives and retirement distributions  surveys or assumptions • Measured directly. • Service lives and retirement distributions from surveys, combined with some assumptions about the functional form of age-price functions • Most frequent assumption: linear profile (constant amount of depreciation) • Ps/P0=1-s/T

42. Computing depreciation (9) • Geometric profile (constant rate of depreciation): • Ps/P0 = (1-)s where  = DBR/T • DBR: declining balance parameter, often set to equal 2 (“double declining balance”) • For example, BEA uses DBR=2.2 for computers, DBR=1.65 for machinery and equipment, DBR=0.9 for structures • T: expected mean service life • Econometric estimates based on used asset prices (Hulten Wykoff 1981)

43. Productive stocks and capital services (1) Net stock and depreciation have to do with the value side of capital and income Productive stocks and capital services have to do with the quantity side of capital and its role in production, i.e. as capital input The flow of capital services is normally assumed to be a constant proportion of the productive capital stock The productive stock is the stock of cumulative investment of a particular type, corrected for retirement and efficiency losses, as captured by the age-efficiency and retirement function

44. Productive stocks and capital services (3) Productive stock of asset type i at the beginning of period t+1, and in constant prices of period t: {hs} s=0,1,2,… is the combined age-efficiency/retirement profile By way of the numerical example used earlier, the productive stock is computed as follows:

45. Productive stocks and capital services (4)

46. Productive stocks and capital services (5)Example from Australia

47. User costs and its elements (1)Price of capital services • Productive stock and capital services are the quantity/volume of capital input to production • What is the price of capital services? • Price of capital services = user costs or rental price • Basic idea: how much would an owner of a capital good charge if he rented out the capital good for one period under competitive conditions? • The rental price/user cost should cover: • A ‘normal’ net return to the capital owner to account for opportunity costs • Depreciation • Expected revaluation

48. User costs and its elements (10) A reasonable approximation is: u0t = [r*t + d0] P0t. where the user cost is the sum of a real rate of return and depreciation This user cost formula, due essentially to Walras says that the user cost of capital is equal to the anticipated real interest rate plus the anticipated depreciation rate times the beginning of the period stock price of the asset.

49. User costs and its elements (11) • There are two broad options (Diewert 1980 and Harper, Berndt and Wood 1989): • Use of an endogenous (internal) rate of return (estimated capital services exactly corresponds to gross operating surplus and the capital element of gross mixed income) • Use of an exogenous (external) rate of return (estimated capital services is unlikely to be exactly equal to gross operating surplus and the capital element of gross mixed income)

50. Scope of assets and capital services • Note: new asset classification proposed in revised SNA • All fixed assets are within the scope of capital services • Some special cases: • Research and development – not presently recognised as fixed assets, but SNA revision will bring them into asset scope  statistical issues of how to measure R&D stocks, how to deflate them, how to depreciate them • Some assets are non-produced but sources of capital services, in particular land (see below for more extensive discussion) • 3. Some assets are produced but not fixed  inventories should they be part of the scope of capital services? • 4; Government assets