Euler's Theorem
It was developed by Swiss Mathematician Leonhard Euler.
It is mathematical relationship that applies to any homogeneous function.
If a production function is homogeneous of degree one (constant return to scale)
and factors are paid equal to marginal products, total product is exhausted with no
surlus or deficit.
Formula
If f(x) is said to be homogeneous of degree 't',
If α =1(Constant Return ¿ Scale)
α >1¿
α <1¿
Let f(x) be production function with two factors i.e. capital and labour.
Then,
Production function of degree 't' = ω t ×Y =f ( ωK , ωL )
Where,
Y = Output, K = Capital , L= Labour
t= parameter of return to scale
if t = 1 ( constant return to scale)
t > 1 Increasing return to scale
t < 1 Decreasing return to scale
∂Y ∂Y
∴Y = L . +K .
∂L ∂K
Where,
∂Y
=Marginal Product of Labour
∂L
∂Y
∂K
= Marginal Product of Capital
M.P. of Labour ×Amount of Labour + M.P. of Capital ×Amount of capital = Total
Product of firm
Homogeneous function ???
u = f(x,y) is said to be homogeneous function of degree 'n' if it can be expressible
in the form of :
ω n × f (x , y)=f ( ωx , ωy )
Example
f(x,y) = x2+y2+2xy
or, f ( ωx , ωy )=( ωx )2 +¿
= ω 2 x 2+ ω2 y 2+ 2 xy . ω2
= ω 2 ( x 2 + y 2+ 2 xy )
= ω2. f ( x , y )
A function is said to be homogeneous of degree n, if multiplication of each of its
independent variables by a constant ωwill alter the value of the function by
proportion ω n, that is, if ω n × f (x , y)=f ( ωx , ωy ) .
In Production Function
Note 1 : Homogeneous function of first degree often referred to as lenearly
homogeneous function.
Note 2 : Linear homogeneity means that raising all inputs (independent variables)
j-fold will always raise the output (vlaue of function) exactly j-fold also.
Proof
Q = f (K, L)
Capital - Labour Ratio (k) = K/L
j = 1/ L
Q j = Q × 1/L = Q/ L
f (K,L) .j = f (jK, jL) = f (K/L , L/L ) = f (k , 1)
Function of Capital Labour Ratio = ∅ (k)
APPL = Q/L = ∅ ( k )
1
APPK = Q/K = Q/L ×L/K = ∅ ( k ) × k =∅(k) /k
Therefore,
Total Product (Q) = L . ∅ ( k)
∂k
=
∂ ( KL ) = 1 × 1= 1
∂K ∂K L L
∂k
=
∂ ( KL ) = −K
2
∂L ∂L L
Now,
∂ Q ∂ ( L . ∅ (k )) ∂(∅ ( k ) ) ∂ k
= =L. ×
∂K ∂K ∂k ∂K
1
¿ L . ∅' ( k ) × =∅' ( k )
L
∂Q ∂ ( L . ∅ ( k ) )
= =L. ∂ ¿ ¿
∂L ∂L
∂Q
=L . ∂ ¿ ¿
∂L
−K
¿ L . ∅' ( k ) × 2
+ ∅ ( k )=−k . ∅ ' ( k )+ ∅ ( k )=∅ ( k )−k . ∅' ( k )
L
Euler's Theorem
∂Q ∂Q
K. + L. =Q
∂L ∂L
Proof:
=K. ∅ ' ( k ) + L. ( ∅ ( k ) −k . ∅ ' ( k ) )
= K. ∅ ' ( k ) + L. ¿
= K. ∅ ' ( k ) + L. ¿
= K. ∅ ' ( k ) + L. ¿
= L . ∅ ( k ) =Q
Important Notes:
Economically, this property means that under condition of constant return to
scale, if each factor is paid amount of its marginal product, the total product ,
the total product will be exactly exhausted by the distributive shares for all
the input factors or equivalently, the pure economic profit will be zero.
This situation is descriptive of the long run equilibrium under pure
competition.
The zero economic profit in the long run equilibrium is brought about by the
forces of competition through the entry and exit of firms.
This is not mandatory to have a production function that ensures product
exahustion for any and all (K,L) pairs.
When imperfect competition exists in the factor markets, the remuneration to
the factors may not be equal to the marginal products and consequently
Euler's theorem becomes irrelevant to the distribution picture.
