Capital stocks, capital services, and depreciation:

an integrated framework

Nicholas OultonLondon School of Economics

Sylaja SrinivasanBank of England

Paper to be presented to the 1st EUKLEMS Consortium Meeting, London, 26-27 October, 2004

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This paper

• Presents an integrated framework– Approach to measuring capital and depreciation is

consistent– Common dataset

• Illustrates empirical differences in growth rates, levels of aggregate capital and depreciation when assumptions are changed on– Asset composition – Asset level depreciation rates– Asset prices– Method of aggregation

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Roadmap

• Theory

capital services versus capital stocks

rental prices versus asset prices

• Obsolescence and ICT: does the basic theory still apply?

• Some evidence for the UK

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• In a production function, the measure of capital needed is one that captures the flow of capital services (Jorgenson and Griliches(1967))

• This measure is called a Volume Index of Capital Services (OECD (2001)) or VICS

),( LKfy

Wealth or VICS?

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Aggregate Capital: Wealth

•Stock measure

•Growth of aggregate capital is weighted average of growth rates of individual assets where weights are shares of each asset in the value of wealth

•Since value of each asset is its price times quantity, we refer to these weights as “asset price” weights

•Asset price is the price you pay for the asset

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Aggregate Capital: VICS

•Flow measure : measures the flow of capital services derived from all capital assets

•Growth of aggregate capital is weighted average of growth rates of individual assets where weights are shares of each asset in the value of profits

•Since value of each asset in value of profits is its rental price times quantity, we refer to these weights as “rental price” weights

•Rental price is the (notional) price you pay to hire the asset

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Rental price/user cost of capital

Definition:

What a firm would have to pay to rent an asset for 1 period

Why does it matter? If firms optimise, User cost = MR times Marginal Physical Product

of the asset

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,0 1,0 1,0 ,1

,0

1,0

BASIC EQUATION

Relationship between user cost/rental price and asset price:

( )

: user cost/rental price of new (age zero) asset at end of t

: asset price o

K A A At t t t t

Kt

At

p r p p p

p

p

f new (age zero) asset at end of period t-1

: rate of return in period ttr

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,0

,0 1,0 ,0 ,1 ,0 1,0

,0 ,1

,0 1,0

Adding and subtracting :

( ) ( )

where ( ) is depreciation (pure effect of ageing) and

( ) is revaluation (pure effect of time)

At

K A A A A At t t t t t t

A At t

A At t

p

p r p p p p p

p p

p p

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,0 ,1 ,0

,0 1,0 ,0 ,0 1,0

Define the rate of depreciation, ( ) / :

( )

Now drop age subscripts, add subscript i for asset type, and assume

(1) is constant over time for

A A At t t t

K A A A At t t t t t t

t

p p p

p r p p p p

, 1 , 1

a given asset

(2) The rate of return is the same for all assets (cost minimisation)

( )K A A A Ait t i t i it it i tp r p p p p

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0 01,0 ,

We can also show from the basic equation:

The asset price equals the present value of the

future stream of rental prices:

(1 )

assuming the asset is valueless after peri

znA K

zt t z z tp p r

od n

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Accounting relationship

1Aggregate profit

m Kit iti

p K

13

, 1

, 1

Capital stocks and capital services

(1) (1 )

: rate of decay (geometric)

(2) , 0

(3) Set 1 [Berndt and Fuss, 1986]

it it i i t

i

it it i t it

it

A I d A

d

K h A h

h

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Relationship between decay and depreciation

If decay is geometric, then the depreciation rate equals the decay rate:

The converse is also true

i id

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1 , 1 , , 11

1

1 2 , 1 , 2

Growth of VICS:

1ln[ / ] [ ]ln[ / ]

2

where , 1,..., [Jorgenson and Griliches, 1967]

Growth of wealth stock:

1ln[ / ] [ ]

2

m

t t it i t i t i ti

Kit it

it m Kit iti

t t i t i t

K K w w K K

p Kw i m

p K

A A v v

, 1 , 21

, 1 , 1, 1

, 1 , 11

ln[ / ]

where , 1,...,

m

i t i ti

Ai t i t

i t m Ai t i ti

A A

p Av i m

p A

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Empirical requirements

• Asset prices

• Investment (real)

• Aggregate profit (needed to derive the rate of return)

• Tax adjustment factors

• Depreciation rates

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Obsolescence: does it make a difference?

The answer is no.

1. Scrapping is a form of decay (and its prospect causes depreciation). If an asset is scrapped because of obsolescence, then it can’t deliver any services

2. It doesn’t matter why an asset decays, just that it does decay

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Panel approach to estimating depreciation

where• p(i,s,t) is the transactions price (not quality

adjusted) of the ith computer that is s years old in year t

• z(i) is some characteristic (say speed) which affects the perceived quality of computers

• YD: year dummy

0 1 2 3log ( , , ) ( ) ( , ) ( , ),

1, 2

p i s t z i age i t YD i t

t

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Panel approach to estimating depreciation

Coefficient on age (beta2):

measures depreciation

Coefficient on year dummy (beta3):

measures quality-adjusted price change

0 1 2 3log ( , , ) ( ) ( , ) ( , ),

1, 2

p i s t z i age i t YD i t

t

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UK national accountsAssets

Traditional

Buildings and other structures

Other machinery and equipment (“plant”)

Transport equipment (“vehicles”)

Intangibles

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Assets

Traditional ICT related

Buildings and other structures Computers

Other machinery and equipment (“plant”)

= rest + computers + part of software

Software

Transport equipment (“vehicles”)

Intangibles

= rest + part of software

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Aggregate Capital Growth: Does separating out ICT matter?

Basic ----- Separate out computers and software -----

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Growth rates, per cent per quarter

1979 Q1-2002 Q2

Variant Wealth VICS

BEA 0.76 0.81

ICT1 0.67 0.78

ICT3 0.72 1.04

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Guide: Basic Longer asset lives Separate out computers and software (US prices) US private non-residential

Ratio of Depreciation to GDP (nominal)

6

7

8

9

10

11

12

13

14

1979 1982 1985 1988 1991 1994 1997 2000

BEA ONS1 ICT3 US pvt nonresidential

(% p.a.)

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Conclusions• Separating out ICT matters to growth rate of capital; effect is larger for VICS measure

• Asset level depreciation rate assumptions matter for the level of wealth

• Aggregate depreciation rate rises a bit in the 1990s when ICT is treated separately (though not as much as in U.S.) • However, depreciation to GDP ratio is almost flat except in the last few quarters (even when ICT is treated separately)

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THE END