Question

1a. Define and discuss homogenous functions. 1b. Homogenous functions have interesting mathematical properties. Name and discuss two.

Answer 1

1a. If ƒ : V → W is a function between two vector spaces over a field F, and n is an integer, then ƒ is said to be homogeneous of degree n if

f(αv) = αⁿf(v)

M(x, y) = 3x 2 + xy is a homogeneous function when we add the powers of x and y in both the term it is the same (i.e. x² is x to power 2 and xy = x¹y¹ giving total power of 1 + 1 = 2). The degree of this homogeneous function is 2.

1b.Examples

(i) The average physical product of capital(K) and labour(L) can be expressed as a function of capital labour ratio.

Q = f(L,K)...(1)

Let us multuply equation 1 with a factor(p) = 1/L

f(L/L, K/L) = Q/L

⇒ f [(1,K/L)] = Q/L

⇒ g (K/L) = AP_L (Q/L = Average product of labor).....(2)

Similarly, we would have

Q/K = h(L/K)

⇒ AP_K = h (L/K) .......(3)

Eqns. (2) and (3) give us that if L and K are increased by the firm in the same proportion, keeping the K/L ratio constant, then there would be absolutely no change in AP_L and AP_K, i.e., the AP_L and AP_K functions are homogeneous of degree zero in L and K.

(ii) Euler's theorem

If u = f(x,y) then we can say that,

In a function of two independent variable x and y if we multiply x with the partial differentiation of that function with respect to x and add it to the multiplication of y with the partial diffrentiation of that function with respect to y it will be equal to dgree times( or, order) the function itself.

x *​ du/dx + y* du/dy = n* u [ n is the order of the function, u is the function itself)

I think this will be the answer. Thank you.