Measuring Productivity Change
(Without Neoclassical Assumptions)
Bert M. Balk
Statistics Netherlands
and
Rotterdam School of Management
Erasmus University
Washington DC, 14 May 2008
Opening quote
“… productivity measurement is all about
comparing outputs with inputs, …”
Ross Gittins, “Productivity should be a spinfree zone”, The Sydney Morning Herald, June
23-24, 2007.
Usual neo-classical
assumptions
– Technology exhibits constant returns to scale
(CRS).
– Competitive input and output markets
(exogenously given prices).
– Production unit maximizes profit.
– Production unit has perfect foresight (ex post
prices used as if ex ante known).
Can we dispense with such assumptions?
Generic model (market)
Input
Input value:
Cost
Unit
Output
Output value:
Revenue
What is productivity (change)?
Usual financial performance measures are:
– profit = revenue minus cost (positive, negative, or zero);
– profitability = revenue divided by cost (greater than 1, less
than 1, or equal to 1).
Profit and profitability change can be decomposed in price and
quantity components.
Natural decomposition for profit is additive (indicators) and for
profitability multiplicative (indices).
The quantity component of profit (-ability) change, or real
profit (-ability) change, is called (total factor) productivity
change.
Formal definitions
TFP index: IPROD ≡ Q(output) / Q(input)
Good choice: Fisher
TFP indicator: DPROD ≡ Q(output) - Q(input)
Good choice: Bennet
Interpretation of difference-type measures requires that
all prices be deflated by some general inflation index (for
example: headline CPI).
This sounds simple, but …
… what precisely is to be considered as input
and output?
KLEMS-Y model
Capital
Labour
Energy
Materials
Services
K
L
E
M
S
Input:
Cost
Unit
+ Profit =
Output Y
(Goods &
Services)
(Gross) Output:
Revenue
KL-VA model
Capital
K
Unit
Labour
L
Input:
Primary inputs cost
+ Profit =
Revenue
minus
E,M,S cost
Output:
Value added
K-CF model
Capital
K
Input:
Capital input cost
Unit
+ Profit =
Value added
minus
L cost
Output:
Cash flow
TFP indices (1)
IPROD-Y ≡ Q(gross output) / Q(KLEMS)
IPROD-VA ≡ Q(value added) / Q(KL)
IPROD-CF ≡ Q(cash flow) / Q(K)
If Profit = 0 then
ln(IPROD-CF) ≥ ln(IPROD-VA) ≥
ln(IPROD-Y).
TFP indicators (1)
But for indicators we get
DPROD-CF = DPROD-VA = DPROD-Y .
Thus the three models are not really different!
Capital input cost
Total user cost of all asset types i and ages j is
C(capital)t = ∑i ∑j utij Ktij + ∑i ∑j vtij Itij ,
where K and I are quantities of asset-type by
age (available at the beginning of year t and
invested at midyear respectively).
Unit user cost (ex post) (1)
For asset of age j (at midyear):
utj = rtPt-j-0.5 + (Pt-j-0.5 – Et-Pt+j+0.5) +
(Et-Pt+j+0.5 - Pt+j+0.5)
(j=1,…,J)
where rt denotes a certain nominal interest rate, the P’s
are prices (valuations), and E is the expectation
operator.
Unit user cost (ex post) (2)
There are three components
–“waiting cost” (interest rate times asset price);
–anticipated time-series depreciation (effect of
time and ageing);
–unanticipated revaluation.
This leads to four additional input - output
models.
KL-NVA model
Capital
(waiting
cost)
Labour
K
Unit
L
Input:
Partial primary
inputs cost
+ Profit =
Value added
- ex post
t-s depreciation
Output:
Net value added
(ex post)
KL-NNVA model
Capital
(waiting
cost)
Labour
K
Unit
L
Value added
- anticipated
t-s depreciation
Input:
Partial primary
inputs cost
+ Profit* =
Output:
Net value added
(normal)
KL-NNVA model
Profit* ≡
Profit + Unanticipated asset revaluation
K-NCF model
Capital
(waiting K
cost)
Input:
Waiting cost of
(owned) capital
Unit
+ Profit =
Net value added
(ex post)
- L cost
Output:
Net cash flow
(ex post)
K-NNCF model
Capital
(waiting K
cost)
Input:
Waiting cost of
(owned) capital
Unit
+ Profit* =
Net value added
(normal)
- L cost
Output:
Net cash flow
(normal)
TFP indices (2)
IPROD-NVA ≡ Q(net value added) / Q(KwL)
IPROD-NCF ≡ Q(net cash flow) / Q(Kw)
If Profit = 0 then
ln(IPROD-NCF) ≥ ln(IPROD-NVA) ≥
ln(IPROD-VA)
ln(IPROD-NCF) ≥ ln(IPROD-CF) ≥
ln(IPROD-VA)
TFP indicators (2)
DPROD-NCF = DPROD-NVA = DPROD-CF =
DPROD-VA = DPROD-Y
but
DPROD-NNVA = DPROD-NNCF are different.
The rate of return (1)
Net cash flow (ex post) can be seen as the return to the
capital stock. The accounting identity is
rt (∑i ∑j Pt-i,j-0.5 Kt-i,j-0.5 + (1/2)∑i ∑j Ptij Itij )
+ Profit = Net cash flow .
Setting Profit = 0 and solving for rt delivers the so-called
“endogenous (or internal, or balancing) rate of return”.
Specific for unit.
Alternative: use some reasonable, exogenous rate.
The rate of return (2)
Alternatively, normal net cash flow can be seen as the
return on the capital stock. The accounting identity is
rt (∑i ∑j Pt-i,j-0.5 Kt-i,j-0.5 + (1/2)∑i ∑j Ptij Itij )
+ Profit* = Normal net cash flow .
Setting Profit* = 0 and solving for rt delivers the “normal
endogenous rate of return”. This rate absorbs also
unanticipated asset revaluations.
Alternative: use some reasonable, exogenous rate.
Implementation by Statistics
Netherlands
Details in Van den Bergen et al. (2007) and
Balk & Van den Bergen (2006).
TFP change (%): Commercial
sector
4
3,5
3
2,5
KLEMS-Y
KL-VA
2
1,5
1
0,5
0
96/00 00/05
2004 2005 2006
Explanation (1): From macro
to micro
Profitability change of an aggregate unit is the
outcome of:
– Lower level profitability change, which can
be decomposed in
– Productivity change
– Differential price change
– Expansion and contraction of units
– The entry of units
– The exit of units
Explanation (2): What is
productivity change?
Technological change (incl. management
technics).
Efficiency change (technical).
Scale effects.
Input- and/or output-mix change (for example
due to relaxation of capacity restrictions,
movement from / toward perfect competition).
And here come the neoclassical
assumptions …
When efficiency change, scale effects, and
input-/output-mix effects are assumed away,
and the unit is assumed to have perfect
foresight (so that ex post user cost = ex ante
user cost), then …
… productivity change = technological change.
But all these assumptions are highly unlikely!