    # Eulers Product Exhaustion Theorem Meaning and Solution

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

P x Q = (MPL X P) L + (MPc x P) 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.

Euler’s Product Exhaustion Theorem

Postulations

1. It assumes a linear standardised production of first degree which implies invariable returns to scale.
1. It assumes that the factors are complementary, i.e. if a variable factor increases; it increases the marginal productivity of the fixed factor.
1. It assumes that factors of production are perfectly divisible.
1. The relative shares of the factors are invariable and independent of the level of the product.
1. There is a stationary, reckless economy where there are no profits.
1. There is perfect competition.
1. It is applicable only in the long run.

Explanation

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 a 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 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.

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