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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
  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 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
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
       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
                                                                                                                                       Trade
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

                                                                                                                                Terms of Trade
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

                                                   Terms of Trade

                                                         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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