The Reconciliation of Industry Productivity Measures with

National Measures: An Exact Translog Approach

EMG Workshop 2006

December 13, 2006

Erwin Diewert and Denis Lawrence

Introduction• We want to generalize our earlier work on

national productivity measurement to show the industry sources of productivity gains

• Unfortunately we have not been able to put together a data set to illustrate our new methodology

• Hence we will illustrate the national methodology for Australia, drawing on our earlier work with the Productivity Commission and then show algebraically how the contribution analysis works

The Basic Framework

• Market sector GDP function: gt(P,x) max y {Py : (y,x) belongs to St}

• Value of outputs equals value of inputs in period t:

gt(Pt,xt) = Ptyt = Wtxt ; yt is output; xt is input;

• Real income generated by market sector in period t is

t Wtxt/PCt = wtxt = gt(pt, xt) = Ptyt/PC

t = ptyt

where PCt is consumption price

• This is the amount of consumption period t income can buy

and this will be our suggested economic welfare measure.

Identifying the Contributions

• The main determinants of growth in real income generated by the market sector of the economy are:

– Technical progress or improvements in Total Factor Productivity;

– Growth in domestic output prices or the prices of internationally traded goods and services relative to the price of consumption; and

– Growth in primary inputs.• We need a way of identifying the effect of each of these

factors in isolation, ie what would have happened to real income if only each of these changes had occurred separately and all else remained the same?

Productivity Growth

• Definition of a family of period t productivity growth factors:

(p,x,t) gt(p,x)/gt-1(p,x)

• Laspeyres type measure: Lt (pt-1,xt-1,t) gt(pt-1,xt-1)/gt-1(pt-1,xt-1)

• Paasche type measure: Pt (pt,xt,t) gt(pt,xt)/gt-1(pt,xt)

• Fisher type measure: t [Lt P

t]1/2

• But how can we empirically implement the above theoretical definitions? It can be done by assuming a translog technology.

Real Output Price Growth Factors• Definition of a family of period t real output price

growth factors:

(pt-1,pt,x,s) gs(pt,x)/gs(pt-1,x)

• Laspeyres type measure: Lt (pt-1,pt,xt-1,t-1)

gt-1(pt,xt-1)/gt-1(pt-1,xt-1).

• Paasche type measure: Pt (pt-1,pt,xt,t)

gt(pt,xt)/gt(pt-1,xt).

• Fisher type measure: t [Lt P

t]1/2

• Gives increase in real income due to changes in real output prices

Input Quantity Growth Factors

• Definition of a family of period t input quantity growth factors:

(xt-1,xt,p,s) gs(p,xt)/gs(p,xt-1)

• Laspeyres type measure: Lt (xt-1,xt,pt-1,t-1)

gt-1(pt-1,xt)/gt-1(pt-1,xt-1).

• Paasche type measure: Pt (xt-1,xt,pt,t)

gt(pt,xt)/gt(pt,xt-1).

• Fisher type measure: t [Lt P

t]1/2

• Gives the increase in real income due to input growth alone

Real Income Growth Decomposition

• The input growth and real output price contribution factors (to real income growth) can be broken down into separate effects that are defined in similar ways.

• With the assumption of a translog technology, we can get the following exact decomposition of real income growth into contribution factors:

t/t-1 t = t t t where t = wtxt/ wt-1xt-1 is observable and

ln t = ln PT(pt-1,pt,yt-1,yt) and ln t = ln QT(wt-1,wt,xt-1,xt);

where PT is the Törnqvist (real) output price index and QT is the Törnqvist input quantity index.

• We cumulate the now observable relationships

t/t-1 = t t t

into the “levels” relationships t/t-1 = Tt At Bt

Terms of Trade Contribution Factors• The terms of trade contribution factors are made up of

two separate effects (which we combine in the following figures):

• A real export price effect which adds to real income growth if the price of exports increases more rapidly than the price of consumption and

• A real import price effect which adds to real income growth if the price of imports falls compared to the price of consumption

• In the present setup, the entire value of investment is converted into consumption equivalents and added to actual consumption and the price of capital is the usual user cost of capital which includes a depreciation term.

• But this framework overstates real (sustainable) consumption by the amount of depreciation

The Real Net Income Approach

• In our Productivity Commission study, we take depreciation out of user cost and instead subtract it from gross investment.

• Now investment is converted to consumption equivalents only if it is positive after netting out depreciation; thus, we have moved from real GDP (GDP deflated by the consumption price index) to real NDP (NDP deflated by the consumption price index).

• The remaining user cost term is the reward for waiting or postponing consumption; thus, income is now labour income plus the net return to capital.

• In the net framework, the role of TFP growth is magnified and in the Australian data, the role of capital deepening is diminished as we shall see.

Diewert-Lawrence Database

• Initially developed for DCITA• Extended market sector coverage – covers 16 of the 17

sectors in the National Accounts instead of the ABS MFP’s 12 sectors

• Builds up an output measure from final consumption components rather than sectoral gross value added

• Outputs and inputs are measured in terms of producer prices rather than consumer prices

• Constructs consistent capital and inventory input series and measures inventory change in a consistent manner

• Runs from 1959-60 to 2003-04• This version includes a balancing real rate of return and

improved capital tax treatment

Price Indexes

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Government

Consumption

Investment

ExportsImports

Price Indexes (cont’d)

0

2

4

6

8

10

12

14

16

18

20

22

24

26

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Labour

Capital

GDP

Domestic Output

Individual Contributors to Real Income - GDP

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Productivity

Labour Input

Capital Input

Terms of Trade

Domestic Output Price

Cumulative Contributions to Real Income - GDP

0

1

2

3

4

5

6

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Dom. Output Price

Dom. Output Price + ToT

Dom. Output Price + ToT + Capital Input

Dom. Output Price + ToT + Capital Input + LabourInput

Dom. Output Price + ToT + Capital Input + LabourInput + Productivity = Real Income

Investment Price Indexes

0

1

2

3

4

5

6

7

8

9

10

11

12

13

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Gross Investment

Waiting Capital Services

Depreciation

Net Investment

Investment Quantities

0

5,000

10,000

15,000

20,000

25,000

30,000

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Gross Investment

Net Investment

Waiting Capital Services

Depreciation

$1960m

Individual Contributors to Real Income - NDP

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Productivity

Labour Input

Capital Input

Terms of Trade

Domestic Output Price

Cumulative Contributions to Real Income - NDP

0

1

2

3

4

5

6

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Dom. Output Price

Dom. Output Price + ToT

Dom. Output Price + ToT + Capital Input

Dom. Output Price + ToT + Capital Input + LabourInput

Dom. Output Price + ToT + Capital Input + LabourInput + Productivity = Real Income

Alternative TFP Indexes

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

ABS

D-L GDP

D-L NDP

Individual Contributors to Real Income - NDP

0.9

1.0

1.1

1.2

1.3

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Productivity

Labour Input

Terms of Trade

Capital Input

Domestic Output Price

Cumulative Contributions to Real Income - NDP

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Dom. Output Price

Dom. Output Price + ToT

Dom. Output Price + ToT + Capital Input

Dom. Output Price + ToT + Capital Input + LabourInputDom. Output Price + ToT + Capital Input + LabourInput + Productivity = Real Income

Empirical Conclusions

• For Australia, we find that changes in the terms of trade, while important over a few short periods (including recent years), are not a long run explanation for the improvement in Australian living standards over the period 1960–2004.

• When we move to a net domestic product framework from a gross domestic market sector framework, the role of capital deepening as an explanatory factor for improving living standards is reduced and the role of technical progress (or TFP growth) and labour growth is increased.

• Now we turn to the algebra for establishing an industry contributions framework to national productivity growth

Notation and the industry setup

• We assume that there are I industries in the market sector of the economy.

• As in section 2, we assume that there is a common list of M (net) outputs which each industry produces or uses as intermediate inputs. The net output vector for industry i in period t is yit [y1

it,...,yMit], which are sold at the positive

producer prices for industry i in period t, Pit [P1it,...,PM

it],

for i = 1,...,I. • There is also a common list of N primary inputs used by

each industry. In period t, we assume that industry i uses nonnegative quantities of N primary inputs, xit [x1

it,...,xNit]

which are purchased at the positive primary input prices Wit [W1

it,...,WNit] for i = 1,...,I.

Define the industry i period t net product function, git(Pit,xit), as follows:

(81) git(Pit,xit) max y {Pity : (y,xit) belongs to Sit} = Pityit

Constant returns to scale and industry adding up implies:

(82) Pityit = Witxit

Define the period t, industry i real input and output price vectors, wit and pit :

(83) wit Wit/PCt ; pit Pit/PC

t ;

As in section 2, we can define the real income generated by industry i in period t, it, as the nominal income generated by industry i in period t, Witxit, divided by the consumption price deflator for period t, PC

t.

Using (81)-(83), we have:

(84) it Witxit/PCt i = 1,...,I ; t = 0,1,2, ... .

= witxit

= Pityit/PCt

= pityit

= git(pit,xit)

Define it as the period t chain link rate of growth factor for the real income generated by industry i:

(85) it it/it1 ;

Now assume that the industry i, period t (deflated) GDP function, g it(p,x), has a translog functional form. Repeat the analysis at the national level except now apply it at the industry level. We can derive the following industry counterparts to the national equation (42):

(86) pityit/pit1yit1 = it/it1 = it = it it it

Define the industry i level of total factor productivity in period t relative to period 0 as Tit, the industry i level of real output prices in period t relative to period 0 as Ait and the industry i level of primary input in period t relative to period 0 as Bit. These industry levels can be defined in terms of the corresponding industry chain link factors, it,it and it as follows:

(90) Ti0 1 ; Tit Tit1 it ;

(91) Ai0 1 ; Ait Ait1it ;

(92) Bi0 1 ; Bit Bit1it .

Since equations (86) hold as exact identities under our present assumptions, the following cumulated counterparts to these equations will also hold as exact decompositions:

(93) pityit/pi0yi0 = it/i0 = Tit Ait Bit ;

The nominal value of market sector output in period t is the corresponding sum of industry nominal values, i=1

I Pityit, which can be converted into the period t real income generated by the market sector, t, by dividing this sum by the period t consumption price deflator, PC

t:

(94) t i=1I Pityit/PC

t = i=1I pityit = i=1

I it ; t = 0,1,...

where the last equality follows using (84). Define industry i’s share of market sector nominal (or real) net output in period 0 as

(95) si0 i0/0 ; i = 1,...,I.

Using the above definitions, we can decompose the growth in market sector real income, going from period 0 to t, as follows:

(96) t/0 = [i=1I it]/0 using (94)

= i=1I [it/i0][i0/0]

= i=1I si

0 [it/i0] using (95)

= i=1I si

0 Tit Ait Bit using (93).

(96) t/0 = i=1I si

0 Tit Ait Bit

There are four sets of factors at work:

The industrial structure of net product in the base period; i.e., the base period industry shares of market sector net output, si

0; The total factor productivity performance of industry i cumulated

from the base period to the current period; i.e., the industry productivity factors, Tit;

The growth in industry output prices (deflated by the price of the consumption aggregate) going from period 0 to t; i.e., the industry real output price factors, Ait and

The growth in primary inputs utilized by industry i going from period 0 to t; i.e., the industry primary input growth factors, Bit.

Note that if high productivity industries absorb more primary inputs over time relative to low productivity industries, this composition effect will make a positive contribution to overall real income growth. Note also that if an industry i experiences growth in its (net) output prices relative to the price of consumption, then the corresponding real output price factor Ait will be greater than one and this effect will contribute to overall real income growth. It is this type of factor that is missing in traditional Total Factor Productivity decompositions; i.e., the traditional analysis ignores favourable (or unfavourable) output price effects.