Measuring Productivity in the Public
Sector: Some Conceptual Problems
W. Erwin Diewert
Department of Economics
University of British Columbia
Vancouver, Canada
ONS-UKCeMGA and NIESR International Conference on Public
Service Measurement held at Cardiff, November 11-13, 2009
1
Introduction
How can we measure the outputs and inputs of public sector
producers and their productivity?
• It is fairly straightforward to measure the price and
quantity of inputs used by public producers (but there
are some problems associated with the measurement of
capital services: the depreciation and cost of capital
problem).
• The present paper will focus on the problems associated
with the measurement of public sector outputs.
• Some attention will be paid to the problems associated
with measuring quality change in the public sector.
2
General Methods for Valuing Govt Outputs
Hierarchy for valuing government outputs:
• First best: valuation at market prices or purchaser’s
valuations;
• Second best: valuations at producer unit costs of
production;
• Third best: government establishment output growth is set
equal to real input growth and the aggregate
establishment output price growth is set equal to an index
of input price growth
• Cases 1-4 below are first best methods of valuation;
• Cases 5-8 are second best methods of valuation;
• Case 9 is the third best method of valuation
3
First Best Methods of Valuation
Case1: Price and quantity information on outputs is
available; no quality change
This is the most straightforward and simple case.
Some notation: the vector of inputs used by a government
establishment in period t is xt ≡ [x1t,...,xNt] and the
corresponding vector of market outputs produced during
period t is yt ≡ [y1t,...,yMt] for t = 0,1. We also assume that
we can observe the corresponding period t input and
output price vectors, wt ≡ [w1t,...,wNt] and pt ≡ [p1t,...,pMt]
for t = 0,1.
More notation: p⋅y ≡ ∑n=1N pnyn is the inner product of the
vectors p and y.
4
First Best Methods: Case 1 (cont)
Aggregate output and input growth can be measured by the
Fisher (1922) output index QF and the Fisher input index
QF* defined as follows:
(1) QF ≡ [(p0⋅y1/p0⋅y0)(p1⋅y1/p1⋅y0)]1/2 ;
(2) QF* ≡ [(w0⋅x1/w0⋅x0)(w1⋅x1/w1⋅x0)]1/2.
The corresponding measures of aggregate output and input
price growth are the Fisher output and input price indexes
defined as follows:
(3) PF ≡ [(p1⋅y0/p0⋅y0)(p1⋅y1/p0⋅y1)]1/2 ;
(4) PF* ≡ [(w1⋅x0/w0⋅x0)(w1⋅x1/w0⋅x1)]1/2.
Finally, the productivity growth of the government
establishment can be defined as the Fisher index of output
growth divided by the Fisher index of input growth,
QF/QF*.
5
First Best Methods: Case 2
Case 2: Quantity but not price information on
outputs is available; no quality change; but
comparable market sector prices are available
• It is assumed that there are private sector producers in the
same marketplace that are producing outputs that closely
resemble the outputs being produced by the government
establishment.
• In this case, we simply use these comparable prices, say pt*
for period t, in place of the missing government
establishment output prices, pt.
• Now apply the same methodology as was used in Case 1.
6
First Best Methods: Case 3
Case 3: Quantity but not price information on
outputs is available; there is quality change over
the two periods; somewhat comparable market
sector prices are available
•
•
•
This method requires the existence of market sector
producers who are supplying somewhat comparable
services;
The prices of these “comparable” services for govt
output n are say pnjt for periods t = 0,1 and market
sector suppliers j = 1,…,J.
We also assume that for each market sector provider j of
the “comparable” service n, we can observe a vector of
quality characteristics, znjt , for each time period t.
7
First Best Methods: Case 3 (cont)
• Assume that there is a hedonic function, hnt(z), that relates
the market prices pnjt to the amounts of the various
quality determining characteristics; i.e., we have the
following hedonic regression model:
(5) pnjt = hnt(znjt) + εnjt ;
j = 1,...,J
• Once the unknown parameters that determine the hedonic
function hnt(z) have been estimated, we can insert the
vector of public production unit characteristics znt* into
this function and obtain an imputed price pnt* for the nth
output of the government production unit;
(6) pnt* ≡ hnt(znt*).
• Now we can apply the same methodology as was used in
Cases 1 and 2.
8
First Best Methods: Case 3 (cont)
Problems with the hedonic regression methodology:
• How should the functional form for the hedonic function be
determined?
• Should the individual observations in the hedonic
regression (5) be weighted by the economic importance of
each observation j or not and if so, should value or quantity
weights be used?
• Should single period hedonic regressions of the type defined
by (5) be run or should the data for periods 0 and 1 be
combined into a single regression? (Do not combine!)
• Which characteristics should be included in the regression?
In my view, it is the characteristics that purchasers desire
that should be included (and not cost determining factors).
9
First Best Methods: Case 3 (cont)
More problems with the hedonic regression methodology:
• There may be very few or no market sector producers of
outputs that are comparable to the outputs produced by
the government establishment;
• Even if there is an adequate sample of market sector
producers of “similar” outputs, the quality characteristics
of the market sector producers may be so different from
the quality characteristics of the public producer that it
would be extremely hazardous to extrapolate from the
market sector part of the characteristics space into the
public sector part and thus the imputed prices that would
be obtained using equation (6) would be meaningless.
However, this is a viable method and it is our first method
that can deal with the quality change problem.
10
First Best Methods: Case 4
Case 4: Quantity but no price information on
outputs is available; there are no comparable
market sector output prices
The basic idea is to imbed the government producer in a
general equilibrium model. Then the vectors of
government nonmarket production will be explanatory
variables in the production functions for the market
sector of the economy as well as in the utility functions for
the households which are resident in the economy. Now it
is possible to adapt the methodology initially developed by
Allais (1977), Boiteux (1951), Debreu (1951) and Diewert
(1983b) and work out a measure of the value of the change
in government nonmarket production and to determine
approximate prices for these government outputs.
11
First Best Methods: Case 4 (cont)
• Diewert (1986) worked out this methodology in some detail.
• But it is totally impractical! It is just too difficult to engage
in such a massive econometric estimation exercise. (We
would need to estimate production functions or dual cost or
profit functions for all major sectors in the economy plus
estimate utility functions or dual expenditure functions for
all major household groups and this is just too big a job).
• The only advantage of this methodology is that it shows
that it is not impossible to measure the benefits of
nonmarket public production in principle!
• We now turn to second best methods for valuing
government nonmarket outputs; methods that are based on
a complete or partial knowledge of producer costs.
12
Second Best Methods: Case 5
Case 5: Cost functions Ct(y,w) are available for both
periods; no quality changes
• We assume that a cost function has been econometrically
estimated for each period, say Ct(y,w) for t = 0,1.
• Define a family of theoretical output quantity indexes,
α(y0,y1,w,t), as follows:
(7) α(y0,y1,w,t) ≡ Ct(y1,w)/Ct(y0,w). (Note only the y’s change)
• Special cases of (7):
(8) α0 ≡ C0(y1,w0)/C0(y0,w0) ;
(Laspeyres type index)
(9) α1 ≡ C1(y1,w1)/C1(y0,w1).
(Paasche type index)
(10) αF ≡ [α0α1]1/2.
(Fisher type index)
Given that we know the cost functions, all of the above can be
calculated.
13
Second Best Methods: Case 5 (cont)
• Define a family of input price indexes, β(w0,w1,y,t), as
follows:
(11) β(w0,w1,y,t) ≡ Ct(y,w1)/Ct(y,w0).
• Two special cases are:
(12) β0 ≡ C0(y0,w1)/C0(y0,w0); (Laspeyres type price index)
(13) β1 ≡ C1(y1,w1)/C1(y1,w0). (Paasche type input price index)
• Each of the above indexes is equally plausible; take the
geometric average of them:
(14) βF ≡ [β0β1]1/2
(Fisher type input price index).
• We need to define one more family of theoretical indexes
based on cost functions; namely productivity indexes.
14
Second Best Methods: Case 5 (cont)
• Define a family of reciprocal indexes of technical progress,
γ(y,w), as follows:
(15) γ(y,w) ≡ C1(y,w)/C0(y,w).
• This is an inverse measure of technical progress or
productivity growth because costs should go down over
time if outputs y and input prices w are held fixed. Thus
there is (regular) technical progress if γ(y,w) < 1.
• Two special cases of (15) are:
(18) γ(y0,w1) ≡ C1(y0,w1)/C0(y0,w1) ≡ γ0 ; (Odd looking!)
(19) γ(y1,w0) ≡ C1(y1,w0)/C0(y1,w0) ≡ γ1
• As usual, take the geometric average:
(20) γF ≡ [γ0 γ1]1/2.
15
Second Best Methods: Case 5 (cont)
• We have the following theoretical decomposition of the
ratio of period 1 total costs to period 0 total costs into
explanatory factors:
(21) C1(y1,w1)/C0(y0,w0) = w1⋅x1/w0⋅x0 = αF βF γF.
• The exact decomposition of (one plus) cost growth over
the two periods under consideration given by (21) is our
preferred decomposition of cost growth into explanatory
factors which can be implemented if the economic
statistician has estimates for the period 0 and 1 cost
functions at hand.
• Note that there is no quality change in this model.
• The explanatory factors are output growth, input price
growth and (reciprocal) technical progress (or reciprocal
productivity growth).
16
Second Best Methods: Case 6
Case 6: Output quantity data and input price and
quantity data available for both periods;
estimates of marginal costs or incremental costs
are available; no quality changes
• In this model, we do not assume a complete knowledge of
the two cost functions but we do assume that estimates of
marginal costs (or incremental accounting costs), pt ≡
∇yCt(yt,wt) for t = 0,1, are available for each period.
• Our basic strategy here is to obtain first order
approximations to the various unobserved costs that
appeared in the theoretical cost function based α’s, β’s
and γ’s defined in the previous section and then use these
approximations in order to obtain simple index numbers.
17
Second Best Methods: Case 6 (cont)
•
In this case, we obtain the following approximations to the Fisher cost
function based indexes defined in the previous model:
(22) αF ≅ [(p0⋅y1/p0⋅y0)(p1⋅y1/p1⋅y0)]1/2 ≡ QF :
(23) βF ≅ [(w1⋅x0/w0⋅x0)(w1⋅x1/w0⋅x1)]1/2 ≡ PF* ;
(24) γF ≅ [QF/QF*]−1.
Using the fact that PF*QF* = w1⋅x1/w0⋅x0, it can be shown that we have
the following exact decomposition of the cost ratio using the
approximations (22)-(24) in place of their theoretical counterparts:
(25) w1⋅x1/w0⋅x0 = QF PF*[QF/QF*]−1.
•
•
Thus in this case, normal index number theory can be used to provide
input, output and productivity indexes provided that we have estimates
of marginal (or incremental) costs for each period that we can use to
value the nonmarket outputs of the government establishment; i.e., we
can use the Case 1 methodology for this case.
18
Second Best Methods: Case 7
Case 7: Cost functions Ct(y,w,z) with quality
variables z are available for both periods
• This case is similar to Case 5 in that we assume that the
two establishment cost functions Ct have been
econometrically estimated but now, we also allow for
quality change in the outputs produced by the government
establishment over the two periods. The vector z is a vector
of quality characteristics which affects the cost of
producing the vector of outputs y.
• The analysis of this case is very similar to the analysis for
Case 5 above.
19
Second Best Methods: Case 7 (cont)
• The main difference is that we package together changes
in the quality characteristics, z0 and z1, along with changes
in the (unadjusted) output vectors, y0 and y1, into a
composite index of output growth, which of course,
includes changes in quality.
• Thus the quality changes are valued from a cost
perspective rather than from a demander or user
perspective.
• The details for Case 7 are explained in the Appendix to
the paper. (This case is very similar to Case 5).
• Usually we do not have econometrically estimated cost
functions at our disposal. Hence in the following Case 8,
we will adapt the first order approximation methodology
developed in Case 6 to the present situation where there is
quality change.
20
Second Best Methods: Case 8
Case 8: Output quantity data and input price and
quantity data available for both periods; estimates
of marginal costs are available; quality variables
are available along with estimates of marginal
costs of changes in the quality variables for each
period
• We assume that information on y0 and y1 (quality
unadjusted output quantities), x0 and x1 (input quantities),
w0 and w1 (input prices) is available along with information
on marginal costs for each period, p0 and p1.
• Information on the period 0 and 1 characteristics vectors z0
and z1, which describe the qualities of the outputs produced
in each period, are also assumed to be available.
21
Second Best Methods: Case 8 (cont)
•
We assume that estimates of the marginal or incremental costs of
changing the quality variables in each period are available. Thus we
assume that estimates of the cost based characteristics prices, ω0 and ω1,
are available where
(26) ω0 ≡ ∇zC0(y0,w0,z0) ; ω0 ≡ ∇zC1(y1,w1,z1).
• We can derive the following approximations to the theoretical cost
function based indexes defined in the Appendix by (A14), (A18) and
(A29) :
(27)αF ≅ [{p0⋅y1 + ω0⋅(z1−z0)}/p0⋅y0][p1⋅y1/{p1⋅y0 − ω1⋅(z1−z0)}]1/2 ≡ QAF :
(28) βF ≅ [(w1⋅x0/w0⋅x0)(w1⋅x1/w0⋅x1)]1/2 ≡ PF* ;
(29) γF ≅ [QAF/QF*]−1
where QAF is defined in (27) and is the quality adjusted Fisher ideal
index of outputs (using marginal costs as prices pt in this case), QF* is
the Fisher index of inputs and PF* is the Fisher input price index.
• QAF makes intuitive sense! Note that QAF equals QF if z1 = z0 .
22
Second Best Methods: Case 8 (cont)
• If there is no quality change in the outputs produced so
that z1 = z0, then all of the indexes in this case reduce to the
Case 6 indexes, which are quite conventional.
• If there is quality change, then the above analysis shows
that with a few adjustments, normal index number theory
can be used to provide input, output and productivity
indexes in a consistent theoretical framework.
• But we do require estimates of marginal (or incremental)
costs for each period (p0 and p1) and we require estimates
of the incremental changes in cost (ω0 and ω1) due to
incremental changes in the quality vector z for each period
and finally, we require an estimate of the change in the two
quality vectors, z1−z0.
• But keep in mind that this is a second best method!
(Because we are valuing from a producer cost perspective
instead of from a value to the purchaser perspective.)
23
Conclusion
• We suspect it will be difficult to implement any of the first
best methodologies for measuring the values of nonmarket
government outputs, particularly when there are quality
changes over time; i.e., user valuations in nonmarket
situations are inherently difficult to obtain.
• However, it will certainly be possible to implement the two
second best methodologies that do not rely on a complete
knowledge of the period cost functions (Cases 6 and 8). For
Case 6, we require only “reasonable” cost allocations for
the various government outputs and for Case 8, we require
in addition, information on how costs vary as quality
varies.
• Finally, in many cases, we will simply have to settle for the
Case 9 methodology; i.e., the existing SNA methodology.
24