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```									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 {Py : (y,x) belongs to St}

• Value of outputs equals value of inputs in period t:
gt(Pt,xt) = Ptyt = Wtxt ; yt is output; xt is input;
• Real income generated by market sector in period t is
t  Wtxt/PCt = wtxt = gt(pt, xt) = Ptyt/PCt = ptyt
where PCt is consumption price
• This is the amount of consumption period t income
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 Pt]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 Pt]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 Pt]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 = wtxt/ wt-1xt-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
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
• 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
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
Price Indexes

1982
1984
1986
1988
1990
1992
Imports

1994
1996
1998
Exports

2000
Investment
Government

Consumption

2002
2004
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
Price Indexes (cont’d)

1986
1988
1990
1992
GDP

1994
Output

1996
Domestic

1998
2000
Capital
Labour

2002
2004
Individual Contributors to
Real Income - GDP
2.0
Productivity
1.8
Labour
Input
1.6

1.4                                                                                                                                   Capital
Input

1.2
Terms of
1.0

Domestic Output
0.8                                                                                                                               Price

0.6
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Cumulative Contributions to
Real Income - GDP
6
Dom. Output Price

5          Dom. Output Price + ToT

Dom. Output Price + ToT + Capital Input

4
Dom. Output Price + ToT + Capital Input + Labour
Input
Dom. Output Price + ToT + Capital Input + Labour
3          Input + Productivity = Real Income

2

1

0
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Investment Price Indexes

13
12
11                                                                                                               Waiting Capital Services
10
9
8                                                                                Gross Investment
7                                                                     Depreciation
6
5                                                                                                                                  Net Investment

4
3
2
1
0
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Investment Quantities

\$1960m
30,000

Gross Investment
25,000

20,000

Net
15,000                                                                                                                                    Investment
Waiting Capital Services

10,000

5,000
Depreciation

0
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Individual Contributors to
Real Income - NDP
2.2

Productivity
2.0

1.8

Labour
1.6
Input

1.4
Capital
Input
1.2

1.0

Domestic Output Price
0.8
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Cumulative Contributions to
Real Income - NDP
6
Dom. Output Price

5          Dom. Output Price + ToT

Dom. Output Price + ToT + Capital Input
4
Dom. Output Price + ToT + Capital Input + Labour
Input
Dom. Output Price + ToT + Capital Input + Labour
3          Input + Productivity = Real Income

2

1

0
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
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
Alternative TFP Indexes

1989

1991

1993

1995

1997

1999

2001
ABS
D-L NDP

D-L GDP

2003
Individual Contributors to
Real Income - NDP
1.3

Productivity
1.2

Labour Input
1.1

Capital Input
1.0
Domestic Output Price

0.9
1995   1996   1997   1998   1999   2000   2001      2002      2003        2004
Cumulative Contributions to
Real Income - NDP
1.6
Dom. Output Price

1.5      Dom. Output Price + ToT

Dom. Output Price + ToT + Capital Input
1.4
Dom. Output Price + ToT + Capital Input + Labour
Input
Dom. Output Price + ToT + Capital Input + Labour
1.3
Input + Productivity = Real Income

1.2

1.1

1.0

0.9
1995      1996      1997       1998      1999      2000   2001   2002   2003   2004
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  [y1it,...,yMit], which are sold at the positive
producer prices for industry i in period t, Pit  [P1it,...,PMit],
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  [x1it,...,xNit]
which are purchased at the positive primary input prices
Wit  [W1it,...,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 {Pity : (y,xit) belongs to Sit} = Pityit

Constant returns to scale and industry adding up implies:

(82) Pityit = Witxit

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

(83) wit  Wit/PCt ; pit  Pit/PCt ;
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, Witxit, divided by the consumption price deflator for period t, PCt.
Using (81)-(83), we have:
(84) it  Witxit/PCt                            i = 1,...,I ; t = 0,1,2, ... .
= witxit
= Pityit/PCt
= pityit
= 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/it1 ;

Now assume that the industry i, period t (deflated) GDP function, git(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) pityit/pit1yit1 = it/it1 = 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  Tit1 it ;
(91) Ai0  1 ; Ait  Ait1it ;
(92) Bi0  1 ; Bit  Bit1it .

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) pityit/pi0yi0 = it/i0 = Tit Ait Bit ;
The nominal value of market sector output in period t is the corresponding
sum of industry nominal values, i=1I Pityit, 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, PCt:

(94) t  i=1I Pityit/PCt = i=1I pityit = i=1I 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 si0 [it/i0]                    using (95)
= i=1I si0 Tit Ait Bit                 using (93).
(96) t/0 = i=1I si0 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, si0;
 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.

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