Measuring Productivity Change (Without Neoclassical Assumptions)

Bert M. BalkStatistics Netherlands

andRotterdam 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 spin-free 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 Unit Output

Input value: Output value:

Cost 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 K

Labour L Output Y

Energy E Unit (Goods &

Materials M Services)

Services S

Input: (Gross) Output:

Cost + Profit = Revenue

KL-VA model

Capital K Revenue

Unit minus

Labour L E,M,S cost

Input: Output:

Primary inputs cost + Profit = Value added

K-CF model

Value added

Capital K Unit minus

L cost

Input: Output:

Capital input cost + Profit = 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 Kt

ij + ∑i ∑j vtij It

ij ,

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 K Value added

cost) Unit - ex post

Labour L t-s depreciation

Input: Output:

Partial primary + Profit = Net value added

inputs cost (ex post)

KL-NNVA model

Capital

(waiting K Value added

cost) Unit - anticipated

Labour L t-s depreciation

Input: Output:

Partial primary + Profit* = Net value added

inputs cost (normal)

KL-NNVA model

Profit* ≡

Profit + Unanticipated asset revaluation

K-NCF model

Capital Net value added

(waiting K Unit (ex post)

cost) - L cost

Input: Output:

Waiting cost of + Profit = Net cash flow

(owned) capital (ex post)

K-NNCF model

Capital Net value added

(waiting K Unit (normal)

cost) - L cost

Input: Output:

Waiting cost of + Profit* = Net cash flow

(owned) capital (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 It

ij )

+ 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 It

ij )

+ 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

0

0,5

1

1,5

2

2,5

3

3,5

4

96/00 00/05 2004 2005 2006

KLEMS-Y

KL-VA

Explanation (1): From macro to microProfitability 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!