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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

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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.

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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.