Integrated industry-level and aggregate TFP-measures: Different approaches Pirkko Aulin-Ahmavaara,...

Integrated industry-level and aggregate TFP-measures: Different approaches

Pirkko Aulin-Ahmavaara, Perttu Pakarinen, Sami Toivola

Statistics Finland October , 2004

Economy as a unit of production

Economy is here treated as a unit of production that consists of sub-units called industries.

The output of an industry

• Gross output at basic prices?

• Gross value added at basic prices?

• “Sectoral” output at basic prices?

Economy as a unit of production

Tthe output of an economy?

• Gross value added at basic prices?

• Deliveries to final demand (of domestic products) at basic prices?

• Gross domestic product at market prices?

• Deliveries to final demand at market prices?

• Net value added at basic prices?

• Etc.

Two different approaches in this paper

• Deliveries to final demand (of domestic products at basic prices) approach

• Value added (at basic prices) approach

Deriving TFP measures from accounting equations

vpzq '' q is the price vector of outputs

z is the vector of output quantities

p is the price vector of inputs v is the vector of input quantities

ii

ij

ij

jjj

iii

pdqd

vdzdtd

loglog

logloglog

Rate of TFP -growth:

yd log is the logarithmic time derivative of y

(1)

(2)

(Jorgenson and Griliches, 1967)

Price concepts

is the basic price received by the producer. Basic price of a product is assumed to be identical in all its uses

jq

jji qp

is the purchaser’s price paid by the user minus any trade and transport margins, which are treated as separate products. Thus

Taxes minus subsidies on products paid by the user

jip

Industry-level

llj

Ll

kkj

Kk

i

Mij

Mi

i

Mij

Mij

i

Mij

Mi

iiji

iijij

iijijj

LpKp

MqMpMq

MqMpMqQq

)(

)(

)]loglog

log)(log

log)(log

log[)(log 1

lijlj

Lljkj

kkj

Kkj

Mij

i

Mij

Mi

Mij

Mij

i

Mij

Mi

iji

ijiijiji

iji

jjjjjj

LdLpKdKp

MdMqpMdMq

MdMqpMdMq

QdQqQqtd

(3)

(4)

Deliveries to final demand approach

.

)()(

ll

Ll

kk

Kk

i

Mi

Mi

Mi

i

Mi

Mi

iiii

j ijij

jjj

jjj

LpKp

MqpMqMqp

MqQqYq

.]loglog

log)(log

log)(log[)(log 1

lll

Llk

kk

Kk

Mi

i

Mi

Mi

Mii

i

Mi

Mi

ii

iiij

jjjj

jj

LdLpKdKp

MdMqpMdMq

MdMqpYdYqYqTd

Economy level, economy as a single unit of production

(7)

(8)

Aggregation from industry level

)]loglog(

)loglog(

)log)(log)((

)log)(log)((

log[)(log 1

lll

Ll

j lijlj

Llj

kk

kKk

jkj

kkj

Kkj

Mi

i

Mi

Mi

Mi

j

Mij

i

Mij

Mi

Mij

ii

iiij

iji

ijiij

jjjjj

jj

LdLpLdLp

KdKpKdKp

MdMqpMdMqp

MdMqpMdMqp

tdQqYqTd

(10)

ZZj pp if for all values of j then

ZdZpdZp

dZpZdZp

ZZ

jj

Zjj

j

Zj

log

log

and the reallocation term disappears

(11)


Value added approach

Industry level

llj

Llj

kkj

Kkj

i

Mij

Mijij

iijjjjj

LpKp

MpMpQqVv

]loglog

log[)(log 1

lijlj

Lljkj

kkj

Kkj

jjjjjvj

LdLpKdKp

VdVvVvtd

Value added

Rate of TFP growth

(12)

(15)

)]loglog

log[()(log 1

ll

lLlk

kk

Kk

jjjjj

jj

v

LdLpKdKp

VdVvVvTd

Economy-level

.

ll

Ll

kk

Kk

j i

Mij

Mij

j iijij

jjj

jjj

LpKp

MpMpQqVvValue added

Rate of TFP growth

(13)

(16)


)]loglog(

)loglog(

log[)(log 1

ll

lLl

jkj

kkj

Kkj

kk

kKk

jkj

kkj

Kkj

j

vjjjj

jj

v

LdLpKdKp

KdKpKdKp

tdVvVvTd


i

Mi

Mi

Mi

i

Mi

Mi

iiii

jjj

jjj

MqpMqMqp

VvYq

)()(

Obs.

(14)

(20)


Relationship between the measures based on the two approaches

Industry-level

Economy-level

jjjjjvj tdQqVvtd log)()(log 1

)]log)(log)((

)log)(log)((

log)[()(log 1

j

Mij

i

Mij

Mi

Mij

Mi

i

Mi

Mi

Mi

jij

iijiiji

iiii

vj

jjj

jj

MdMqpMdMqp

MdMqpMdMqp

TdVvYqTd

(21)

(19)

Value added approach 2, economy level value added

Price and quantity of economy-level value added

Accounting identity

j

jjj

j VvVvvV

l

lLl

kk

Kk LpKpvV

)loglog

log()(log 1

ll

lLlk

kk

Kk

W

LdLpKdKp

VvVdvVTd

Rate of TFP-growth

(23)

(24)

(25)

)]loglog(

)loglog(

)loglog(

log[)(log 1

j lljlj

Lljl

ll

Ll

jkj

kkj

Kkjk

kk

Kk

jjjj

jjjj

W

LdLpLdLp

KdKpKdKp

VdVvVvVd

tdQqvVTd


(26)

Value added approach 2, economy level value added

Application based on the Laspeyres indices

It was possible to replicate the system based on the theoretical Divisia indices.

Application based on the Törnqvist indices

The value added approach:

It was possible to have similar formulas for both industry-level and economy level TFP growth as those in the theoretical system.

Also the aggregation rule has the same terms as those in the theoretical case.


))log((log(

))log((log(

)log(log

1

1

jjj

k kjkj

jj

jj

jj

jj

jj

j jj

k kk

jjj

l ljlj

j jj

jj

jjj

jj

j jj

l ll

vj

jjj

jj

j

v

KQq

Kp

Qq

Vv

Vv

VvK

Vq

Kp

LQq

Lp

Qq

Vv

Vv

VvL

Vv

Lp

tVv

VvT


But the reallocation terms do not disappear if all the industries pay identical prices for their capital and labour inputs.

They only disappear if rates of growth of labour and capital inputs are identical in all the industries.


Let the price p of an input x to be equal to one in both of the years 0 and 1. Furthermore

))/()/)((2/1(1100

jjjjj xxxxw

For the reallocation term to disappear we should have

)ln(lnlnln 0101jjjjj xxwxx



The formulas for TFP-growth included the same terms as those in the theoretical system and in the system based on Laspeyres indices.

But additional terms representing the reallocation of industries’ output between intermediate and final uses and the reallocation of intermediate inputs at basic prices between industries were needed in the aggregation equation.

Calculations using data for Finland

Choice 1. SUTs or SIOTs?

We were using SIOTs to keep the formulas somewhat more simple.


Choice 2. Is the level of aggregation used in the deflation the same as the one used in the calculation?

Since product baskets can be different in different uses deflation at lower level of aggregation means relaxing the assumption of identical basic prices in different uses.

This again means that in the deliveries to final demand approach terms representing the price differences are not any more only about taxes and subsidies or that additional terms representing the differences in basic prices are needed both in economy level TFP equation and in the aggregation equation.


Choice 3. Are the deflators based on basic prices or basic prices plus net taxes on products?

If basic prices then we can assume that the growth rates of the volumes of taxes and subsidies are equal to those of output in respective uses.

If basic prices plus net taxes on products then we assume that the share of net taxes on a product is the same in all of its uses.

But this does not hold in a VAT-system. And also not necessarily for other product taxes since the uses consist of different product baskets.

Results 2001/2000Table 1. Economy level TFP-growth and contributions to the economy level growth of output

Deliveries to final demand approach Value added approachLaspeyres

(29)*Törnqvist 1

(39)*Törnqvist 2

(44)*Laspeyres

(36)*Törnqvist(50*)

Growth of output 0,954 1,036 0,975 1,609 1,528Net taxes on domestic products inintermediate uses**

0,081 0,060 0,060

Imported intermediate inputs -0,322 -0,338 -0,338Net taxes on imported products inintermediate uses

0,008 0,004 0,004

Labour 0,262 0,266 0,266 0,343 0,345Capital 0,721 0,702 0,702 0,942 0,909Economy level TFP growth 0,204 0,342 0,281 0,325 0,274

Results 2001/2000Table 2. Contributions to the economy level TFP growth

Deliveries to final demand approach Value added approachLaspeyres

(32)*Törnqvist 1

(42)*Törnqvist 2

(45)*Laspeyres

(37)*Törnqvist

(51)*Industry level TFP growth -0,270 -0,196 -0,196 -0,353 -0,263Reallocation of- Industry output between final andintermediate uses

0,061

- Intermediate uses of domesticproducts

0,116 0,116

- Net taxes on domestic products inintermediate uses**

-0,037 -0,031 -0,031

- Intermediate uses of importedproducts

-0,009 -0,009

- Net taxes on imported productsintermediate uses

-0,007 -0,012 -0,012

- Labour 0,224 0,215 0,215 0,292 0,280- Capital 0,295 0,198 0,198 0,386 0,259Economy level rate of TFP growth 0,204 0,342 0,281 0,325 0,275

Results 2000/1999

Contribution to the economy level growth of outputDeliveries to final demand approach Value added approachLaspeyres Törnqvist 1 Törnqvist 2 Laspeyres

Growth of output 8,456 8,175 8,298 5,336Net taxes on domestic products in intermediate uses -0,171 -0,221 -0,221Imported intermediate inputs 4,215 4,030 4,030Net taxes on imported products in intermadiate uses 0,090 0,100 0,100Labour 0,672 0,648 0,648 0,843Capital 0,474 0,471 0,471 0,595Economy level TFP growth 3,176 3,149 3,272 3,897

3,176 3,149 3,269 3,897

Results 2000/1999Contributions to the economy level TFP growth

Deliveries to final demand approach Value added approachLaspeyres Törnqvist 1 Törnqvist 2 Laspeyres

Industry level TFP growth 2,273 2,860 2,860 2,854Reallocation of Industry output between final and intermediate uses -0,123Intermediate uses of domestic products -0,930 -0,930Net taxes on domestic products in intermediate uses 0,013 0,001 0,001Intermediate uses of imported products 0,320 0,320Net taxes on imported products intermediate uses 0,059 0,101 0,101Labour 0,298 0,276 0,276 0,374Capital 0,534 0,645 0,645 0,670Economy level rate of TFP growth 3,176 3,149 3,272 3,897

Integrated industry-level and aggregate TFP-measures: Different approaches Pirkko Aulin-Ahmavaara,...

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