technological progress, productivity and profit rate in macroeconomics

7/29/2019 Technological Progress, Productivity and Profit Rate in Macroeconomics.

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TECHNOLOGICAL PROGRESS, PRODUCTIVITY

AND PROFIT RATE IN MACROECONOMICS

1. PRODUCTIVITY AS INDEX OF TECHNOLOGICAL PROGRESS

The relation of net domestic product versus employment in an economy varying

with time can be approximated within a production period by a bilinear production

function Y Y N ruled by the economys total capital stock K and the level of

technology within the production period under consideration. Owing to changes in

technology, the bilinear production function Y Y N may differ from period to

period. The approximation is depicted in the following Fig. 1

where K stands for the capital stock of the economy, i.e. the total capital in-

vested in means of production,

Ystands for the income as the net domestic product (value added),

FEY stands for the income at full employment of the capital stock K ,

which is the maximum value of income, i.e.FE

Y Y ,

N stands for the number of labour (or employment) units (i.e. hours of

work of a lay man) entered as input in the net domestic product Y ,

ZIN stands for the number of labour units corresponding to zero income

0Y and the initiation of positive income 0Y ,

FEN stands for the number of labour units corresponding to full employ-

ment of the capital stock K .

According to Fig.1, no additional income (net domestic product) Y can be pro-duced if the labour units N exceed

FEN .

Fig. 1: Bilinear production function Y Y N for given capital K

Y Y N

N

Y

FEN

FEY

ZIN


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Technological progress represents but an increase of the efficiency of the capital

stock K , which can be measured as the ratio of incomeFE

Y to capital stock K ,

thereby representing a higher income-capital ratio /FE

Y K. This ratio therefore is

characteristic of the real potential of technological progress inherent in a given

capital stock K, and is herein called characteristic technological productivity,

/FE

Y KCharacteristic Technological Productivity . (1.1)

Further, if the level of income Y falls below the full-employment incomeFE

Y , then

the ratio /Y K represents an active technological productivity smaller than the

characteristic technological productivity /FE

Y K. For given capital stock K , the

ratio of the active technological productivity /Y K to the characteristic technologi-

cal productivity /FE

Y K equals the income activity ratio /FE

Y Y ,

//

/FE

FE

Y KY Y

Y K Income Activity Ratio , (1.2)

which implies

/ / /FE FEY K Y Y Y K Active Technological Productivity . (1.3)

In view of Fig. 1, the income activity ratio /FE

Y Y can be expressed in terms of la-

bour units as below

/ ZIFE

FE ZI

N NY Y

N N

Income Activity Ratio , (1.4)

and hence, the employment ratio /FE

N N can be expressed as follows

/ 1 / / /FE ZI FE FE ZI FEN N N N Y Y N N , (1.5)

which indicates that the employment ratio / FEN N is an increasing function of the

income activity ratio /FE

Y Y , with the increase rate of the former being smaller

than the increase rate of the latter forFE

N N .

For the capital stock K measured in constant prices of a given base year, techno-

logical progress reasonably leads to a higher capital-labour ratio /FE

K N , whose

product with technological productivity /FE

Y K defines the labour productivity

/FE FE

Y N at full employment,

/ / /FE FE FE FEY N Y K K N Full -Employment Labour Productivity . (1.6)


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Evidently, the labour productivity /FE FE

Y N refers to prices of the given base year.

On account of the fact that both of the technological productivity /FE

Y K and the

capital-labour ratio /FE

K N increase with technological progress, it follows that

the full-employment labour productivity /FE FE

Y N must increase with technologi-

cal progress, thereby being an index characteristic of the technological progress.

Further, if the level of employment N falls below the full employmentFE

N , then

the ratio /Y N represents a real labour productivity smaller than the full-employ-

ment one, due to the bilinear production function Y Y N used in Fig. 1.

By means of the income activity ratio /FE

Y Y and the employment ratio /FE

N N ,

the labour productivity /Y N becomes equal to

/

/ //

FE

FE FE

FE

Y YY N Y N

N N Labour Productivity , (1.7)

which also refers to prices of the given base year, and due to equation (1.5) can

be put in the formulation

1/ /

1 / / /FE FE

ZI FE ZI FE FE

Y N Y NN N N N Y Y

. (1.8)

Equation (1.8) implies that for given capital stock K and production function

Y Y N , the labour productivity /Y N increases with increasing income activity

ratio /FE

Y Y up to its maximum value /FE FE

Y N .

Further, equation (1.8) after rearranging terms in the denominator and replacing

the labour productivity /FE FE

Y N according to equation (1.6) becomes

1/ / /

1 / 1 / 1FE FE

ZI FE FE

Y N Y K K N

N N Y Y

, (1.9)

which discloses that the labour productivity /Y N , and hence, its growth, can be

attributed to only four sources:

1) The technological productivity /FE

Y K, which is exclusively dependent on the

technological level of the means of production used, and hence, on the techno-

logical progress. With all the others remaining constant, the larger the techno-

logical productivity /FE

Y K is, the larger the labour productivity /Y N becomes

2) The capital-labour ratio /FE

K N , which also is exclusively dependent on the

technological level of the means of production used, and hence, on the techno-

logical progress. With all the others remaining constant, the larger the capital-

labour ratio /FE

K N is, the larger the labour productivity /Y N becomes.

3) The income activity ratio /FE

Y Y of the economy. With all the others remaining


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constant, the larger the income activity ratio /FE

Y Y is, the larger the labour

productivity /Y N becomes.

4) The labour ratio /ZI FE

N N , which exclusively depends on the bilinear produc-

tion function Y Y N corresponding to the production period considered.

With all the others remaining constant, the larger the labour ratio /ZI FE

N N is,

the smaller the labour productivity /Y N becomes.

In line with the relations (1.7) andFE

Y Y , the labour productivity /Y N becomes

smaller than its full-employment value /FE FE

Y N by adding unproductive (unnec-

essary) labour unitsUNN to the productive (necessary) labour units FEN ,

/ /FE FE FE UN FE

Y N Y N N N N N for . (1.10)

On the other hand, equation (1.3) implies that the active technological productivity

/Y K remains equal to the characteristic technological productivity /FE

Y K, for

FE UN FEN N N N ,

/ /FE FE UN FE

Y K Y K N N N N for . (1.11)

2. INCOME DISTRIBUTION AND TECHNOLOGICAL PROGRESS

The income Y can be analysed into the profits K N w r of its capital input

and the real wages N w of its labour input as below

Y K N w r N w , (2.1)

where w stands for the real wage of the labour unit used for measuring the total

labour (or employment), and

r stands for the profit rate in aggregate terms (i.e. for all the economy).

The distribution of the income Y between capital and labour is described by thefollowing ratio e

1

/K N w r Y N w Y Ne

N w N w w

, (2.2)

which corresponds with the Marxist surplus-value rate or exploitation rate.

Combining equations (2.1) and (2.2) yields

1Y e N w , (2.3)

which for full employment can be given the form


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1FE FE EF EFY e N w , (2.3a)

and define

/ 1FE FE FE FEY N e w

Full -Employment Labour Productivity , (2.4)

whereFE

e andFEw stand for the surplus-value rate and the real wage, respec-

tively, at full employment.

Recalling that the full-employment labour productivity /FE FE

Y N increases with

technological progress, equation (2.4) implies that the magnitude 1 FE FEe w

increases with technological progress. This allows both the real wageFEw and

the income distribution ratio (i.e. the Marxist surplus-value rate or exploitation rate)

FEe to rise with technological progress, on the condition that each of them in-creases less than the full-employment labour productivity /

FE FEY N .

Now, equation (1.7) in view of equation (2.4) can be given the form

/

/ 1/

FE

FE FE

FE

Y YY N e w

N N Labour Productivity , (2.5)

which implies that for given capital stock K , income activity ratio /FE

Y Y , and

production function Y Y N , with the given income activity ratio / FEY Y and

production function Y Y N implying given employment ratio / FEN N , the la-

bour productivity /Y N increases with the magnitude 1 FE FEe w .

Equation (2.3a) in view of equation (1.1) can be written as

1/

FE

FE

FE FE

eY K

K N w

Technological Productivity , (2.6)

where ( )/FE FE

K N w is a percentage which corresponds with the Marxist organic

composition of capital at full employment.

On account of equation (2.6) it follows that for constant surplus-value rateFE

e , the

organic composition of capital ( )/FE FE

K N w decreases with increasing techno-

logical productivity /FE

Y K, that is, with technological progress, which contradicts

Marxs assertion about the opposite.

Within this frame and taking into account that equations (1.3) and (2.6) yield

1

/ /

FE

FEFE FE

eY K Y Y

K N w

Active Technological Productivity , (2.7)


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it is deduced that for constant surplus-value rateFE

e and given income activity ra-

tio /FE

Y Y , the active technological productivity /Y K increases with technologi-

cal productivity /FE

Y K, that is, with technological progress.

3. PROFIT RATE AND TECHNOLOGICAL PROGRESS

The profit rate at full employmentFE

r can by equation (2.1) be defined as

1

FE

FE

FE FE FE FE

FE

FEFE FE

FE

wY K

Y N w K Nr

wK N w

K N

Full - Employment Profit Rate , (3.1)

which, recalling that both ofFE

Y K and /FE

K N increase with technological pro-

gress, allows the full-employment profit rateFE

r to increase with technologicalprogress, on the sufficient condition that the ratio of the full-employment real wage

FEw to the full-employment capital-labour ratio /

FEK N , that is, the inverse of the

full-employment organic composition of capital ( )/FE FE

K N w , will not increase

far more than in proportion to the characteristic technological productivityFE

Y K.

For constantFE

Y K and /FE

K N , which represents technological stagnation, the

only way for an increase of the full-employment profit rateFE

r is a decrease of the

full-employment real wage FEw . By doing so, an economy of lower technologicallevel (lower

FEY K and /

FEK N ) tries to become more attractive to investments

and able to sell the productFE

Y at a pricelower than its real value without loss of

the initial profits, thereby becoming competitive in comparison with economies of

higher technological level. However, such a counter-labour policy meets the rea-

sonable reactions of the labour class, and may set in danger the cohesion of the

society of the production factors.

Equation (3.1), in view of equations (2.2) and (2.3), can be put in the equivalent

formulation

11 1

FE FE

FE

FE

FE FE FE

e er

K e

N w Y K

Full - Employment Profit Rate , (3.2)

which, recalling that the technological productivityFE

Y K increases with techno-

logical progress, implies that the full-employment profit rateFE

r increases with

technological progress under constant full-employment surplus-value rateFE

e .

This outcome actually overturns the Marxist law of the tendency of the profit rate

FEr to fall with technological progress under constant surplus-value rate

FEe .


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The profit rate r in the general case of partially (i.e. not fully) employed capital

stock can by means of equations (1.3), (2.1) and (2.2) be defined as

1 1

1 1 1/ /FE FE

e e er

K e e

N w Y K Y Y Y K

Profit Rate , (3.3)

which indicates that for given income activity ratio /FE

Y Y , the profit rate r in-

creases with the technological productivityFE

Y K , and hence, with technological

progress, under constant surplus-value rate e . This outcome also overturns the

Marxist law of the tendency of the profit rate r to fall with technological progress

under constant surplus-value rate e .

Only if the income activity ratio /FE

Y Y incidentally tends to fall with technological

progress so that the product ( / ) ( / )FE FE

Y Y Y K will decrease, the profit rate r

incidentally tends to fall with technological progress under constant surplus-value

rate e . In short, the Marxist law of the falling tendency of the profit rate r can

only comply with a falling income activity ratio /FE

Y Y .

technological progress, productivity and profit rate in macroeconomics

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