the adding up problem product exhaustion theorem yohannes mengesha

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The Adding up problem: “Product Exhaustion Theorem” By Yohannes Mengesha W/Michael PhD Fellow,

Department of Agri-Economics

Haramaya University

Content

1. The adding up problem 2. Product Exhaustion and Euler’s Theorem

2.1. The Euler construction is pure mathematics 2.2. Postulations 2.3. Explanation

3. Criticisms of the exhausion theory 4. Clark’s Product Exhaustion Theorem

4.1. Postulations 4.2. Explanation

5. Reference

1. The Adding up Problem

The product exhaustion theorem states that since factors of production are rewarded

equal to their marginal product, they will exhaust the total product. The way this

proposition is solved has been called the adding up problem. Wick steed in The

Coordination of the Laws of Distribution demonstrated with the help of Euler’s

Theorem, that payment in accordance with marginal productivity to each factor

exactly exhausts the total product.

The adding up problem states that in a competitive factor market when every factor

employed in the production process is paid equal to the value of its marginal product,

then pay in the production process is paid a price equal to the value of its marginal

product, then payments to the factors exhaust the total value of the product. It can be

represented numerically as under:

Q = (MPL) L + (MPc) C

Where, Q is total output, MP is marginal product, L is labour and K is capital. To find

out the value of output, multiply through P (Price). Thus

The Adding up problem: “Product Exhaustion Theorem” 2013

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PxQ=(MPLXP)L+(MPcxP)C

(MPL X P) = VMPL and (MPc x P) = VMPc

PQ = VMPL x VMPc

Where, VMPL is the value of marginal product of labour and VMPc is the value of

marginal product of capital.

2. Product Exhaustion and Euler’s Theorem

Central to the marginalist revolution of the 1870s was the recognition that values are

set by the simultaneous working of forces on two sides of potential exchanges. The

one-way, supply-side causation of classical economics was rejected. First applied to

product market (to resolve the diamond-water paradox), the extension of the

explanatory model to input or factor markets now seems an inevitable next step.

Resource inputs are valued both because they involve opportunity costs and

generate potential final product value. No prospective supplier of an input unit will

accept less than the unit’s opportunity cost, and no demander will pay more than the

anticipated increment to value promised from use of the unit.

Closure seemed to have been accomplished; the explanatory model seemed

complete. But a dangling question disturbed the early neoclassical converts. How can

we know that the product value paid out to input owners, on the basis of marginal

contributions, exhausts the total value placed by users on final output? The adding-

up problem commanded much attention until a relatively straightforward resolution

was attained by the understanding of stylized interaction processes made possible

through application of Euler’s theorem.

2.1. The Euler construction is pure mathematics

The theorem states that when a function exhibits certain properties (i.e., homogeneity

of degree one) then certain consequences follow. Specifically, the theorem states

that when a function, y = F(K, L), that relates a dependent variable y to one or more

independent variables, K and L, is homogeneous of degree one, the sum of the


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separate partial derivatives multiplied by the corresponding independent variables is

equal to the total value of the function or the dependent variable: y = FKK + FLL

where FK is the partial derivative of F with respect to K, etc.

In its distribution theory application, Euler’s theorem states that payments to all

inputs, in accordance with separate marginal value products, exhausts the total

product value when the production function that relates inputs to output exhibits the

properties indicated.

Economists are familiar with these properties under the postulate of constant returns.

If equiproportional changes in all inputs generate equiproportional change in output,

the required conditions are satisfied. Marginal productivity payment, in value units,

exhausts total value of product.

2.2. Postulations

1. It assumes a linear standardised production of first degree which implies

invariable returns to scale.

2. It assumes that the factors are complementary, i.e. if a variable factor

increases; it increases the marginal productivity of the fixed factor.

3. It assumes that factors of production are perfectly divisible.

4. The relative shares of the factors are invariable and independent of the level of

the product.

5. There is a stationary, reckless economy where there are no profits.

6. There is perfect competition.

7. It is applicable only in the long run.

2.3. Explanation


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Based on these postulations of Euler,

Wicksteed proved his theorem that

when each factor was paid according to

its marginal product, the total product

would be exactly exhausted. This is

based on the postulation of a linear

standardised function. Few economists

criticised his work and pointed out that

the production function does not yield a

horizontal long run average cost curve

LRAC but a U Shaped LRAC curve.

The U shaped LRAC curve first shows

decreasing returns to scale, then

constant and in the end increasing

returns to scale.

The solution of the product exhaustion theorem is based on a profitless long run,

perfectly competitive equilibrium position of an industry which operates at the

minimum point, E of its LRAC curve as represented in the Diagram (1).

At this point the firm is in full

equilibrium, the marginal

revenue productivity MRP

of the factors being equal to

the combined marginal cost

of the factors MFC. This is

represented in the Diagram

2

Where, MRP = MFC at

point A. It is at point A that

the total product OQ is

exactly distributed to OM


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factors and nothing is left

over.

The product exhaustion problem is solved with a linear standardised production

function:

P = δ P C + δ P L

δC δL

Nevertheless there are diminishing returns to scale, less than the total product will be

paid to the factors:

P > δ P C + δ P L

δC δL

In such a condition, there will be super normal profits in the industry. They will attract

new firms into the industry. Consequently output will increase, price will fall and

profits will be eradicated in the long run. In this way, the distributive shares of the

factors as determined by their marginal productivities will absolutely exhaust the total

product.

3. Criticisms of the Exhausion Theory

Neoclassical theory assumes that the total product Q is exactly exhausted

when the factors of production have received their marginal products; this is

written symbolically as Q = (∂Q/∂L) · L + (∂Q/∂K) · K.

This relationship is only true if the production function satisfies the condition

that when L and K are multiplied by a given constant then Q will increase

correspondingly. In economics this is known as constant returns to scale. If an

increase in the scale of production were to increase overall productivity, there would

be too little product to remunerate all factors according to their marginal

productivities; likewise, under diminishing returns to scale, the product would be more

than enough to remunerate all factors according to their marginal productivities.

Research has indicated that for countries as a whole the assumption of constant

returns to scale is not unrealistic. For particular industries, however, it does not hold;


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in some cases increasing returns can be expected, and in others decreasing returns.

This situation means that the neoclassical theory furnishes at best only a rough

explanation of reality.

One difficulty in assessing the realism of the neoclassical theory lies in the definition

and measurement of labour, capital, and land, more specifically in the problem of

assessing differences in quality. In macroeconomic reasoning one usually deals with

the labour force as a whole, irrespective of the skills of the workers, and to do so

leaves enormous statistical discrepancies.

The ideal solution is to take every kind and quality of labour as a separate productive

factor, and likewise with capital. When the historical development of production is

analyzed it must be concluded that by far the greater part of the growth in

output is attributable not to the growth of labour and capital as such but to

improvements in their quality. The stock of capital goods is now often seen as

consisting, like wine, of vintages, each with its own productivity. The fact that a good

deal of production growth stems from improvements in the quality of the productive

inputs leads to considerable flexibility in the distribution of the national income. It also

helps to explain the existence of profits.

In support of this issue, _____________,2013 stipulated that, constant returns

to scale are, in reality, incompatible with competitive equilibrium. For if long cost

curve of the firm is horizontal and coincides with the price line the size of the firm is

indeterminate, if it below the price line the firm will become a monopoly concern and

if it is above the price line, the firm will cease to exist.

The entire study is based on the postulation that factors are fully divisible. Since the

entrepreneur cannot be varied, we have not taken him as a separate factor. In fact,

entrepreneurship disappears in the stationary economy. When there is full equilibrium

at the minimum point of the LRAC curve, there is no uncertainty and profits disappear

altogether.

So hypothesis of an entrepreneur-less economy is justified for the solution of the

adding up problem. But once uncertainty appears, the entrepreneur becomes a

residual claimant and the exhaustion of the production problem disappears.

http://www.britannica.com/EBchecked/topic/178400/economic-growth


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Under imperfect or

monopolistic competition,

the total product adds up to

more than the share paid to

each factor, that is P is

greater than C and L. taking

an imperfect labour market,

the average and marginal

wage curve (AW and MW)

incline upward and the

average and marginal

revenue product curves

(ARP and MRP) are inverted

U shaped as represented in

the diagram 3.

Equilibrium is established at point E where the MRP curve cuts the MW curve from

above. The firm employs OQ units of labour by paying QA wage which is less than

the marginal productivity when there is imperfect competition. This argument applies

not only to labour but all shares even under constant returns to scale in the industry.

4. Clark’s Product Exhaustion Theorem

4.1. Postulations

1. There is free competition in both the product markets and factor markets.

2. Prices and Wages are not manipulated either by government action or

collusive agreements.

3. The quantity of each factor is given.

4. There are no changes in the tastes and of consumers or techniques of

production. It means that the same goods are produced in the same quantities

and by the same methods.

5. The quantity of capital equipment is fixed. But the form of capital equipment

can be changed to co-operate with the quantity of labour available. It means


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that in the long run plants can be adapted and replaced in keeping with the

availability of labour.

6. Workers are interchangeable and of equal efficiency. It means that there is a

single wage rate for all occupations for the reason that there is perfect labour

mobility.

4.2. Explanation

Based on these hypotheses Clark

summarised the working of such an

economy in figures as represented

below. There are two factors of

production: labour, land and capital.

Units of labour are shown on the

horizontal axis and the MP of labour

on the vertical axis in diagram 4 of

the representation. The MPL curve is

the marginal physical product of

labour which falls steadily as more

workers are employed with fixed

capital.

OL represents the number of workers available for employment. If all are employed

the MP of the last worker is LA who paid LA wage. Since all workers receive the

same wage rate, the total wage bill is the number of workers OL multiplied by the

wage rate LA (=OB), i.e. the area OLAB.

The triangular area ABM goes to the owners of capital as the residual interest. Thus

the total product is the area OMAL which has been distributed as PLAB as wages to

workers and ABM is interest to owners of capital i.e. between labour and capital, the

two factors in the economy.


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If labour is a fixed factor

and capital is a variable

factor in the economy, it

can again be

represented that the

total product is fully

exhausted. Now units

of capital are taken on

the horizontal axis and

the MP of capital on the

vertical axis and MPc

as the marginal

physical product curve

of capital in diagram 5.

Given the same quantity of labour, OK of capital is employed with KC as its marginal

product so that the rate of interest is OD = KC per unit of capital. This gives the area

OKCD as the total interest income on capital. The workers receive wages equal to

the area CDM.

Thus the total product of the economy is the area OKCD and workers as wages

equal to the area CDM so that OMCK = OKCD + CDM. It must be noted that for the

product exhaustion theorem, to be valid in Clark’s view, area OLAB in diagram 4

must equal DCDM in diagram 5 and area OKCD of diagram 5 must equal DABM in

diagram 4 of the representation.

the adding up problem product exhaustion theorem yohannes mengesha

Economy & Finance