Question

Consider the function 0.20lnx+0.40lny , is it homothetic? Explain

Answer 1

Q. Consider the function f(x, y) = 0.20 ln x + 0.40 ln y, is it homothetic? EXPLAIN.

Ans: In mathematics, a homothetic function is a monotonic transformation of a function which is homogeneous of degree n.

If a function g(x, y) is defined on |R. And for ƛ > 0 , if g(ƛ x, ƛ y) = ƛⁿ g(x, y), then the function g(x, y) is called homogeneous function of degree n.

Now, the given function is-

f(x, y) = 0.20 ln x + 0.40 ln y .........................................................(1)

or, f(x, y) = ln x^0.20 + ln y^0.40

or, f(x, y) = ln (x^0.20 y^0.40)

Now, h(x, y) = e^{f(x, y)} = e^{ln (x0.20 y0.40)} = x^0.20 y^0.40

Here, f(x, y) is a monotonic transformation of h(x, y).

Monotonic transformation is a way of transforming a set of numbers into another set that preserves the order of the original set, it is a function mapping real numbers into real numbers, which satisfies the property, that if x>y, then f(x)>f(y), simply it is a strictly increasing function.

Now, h(ƛ x, ƛ y) = (ƛ x)^0.20 (ƛ y)^0.40 = ƛ^{(0.20 + 0.40)} x^0.20 y^0.40 = ƛ^0.60 h(x, y)[ here n = 0.60]

So, the function h(x, y) is homogeneous function of degree 0.60

And, f(x, y) is a monotonic transformation of h(x, y).

So, the function f(x, y) = 0.20 ln x + 0.40 ln y is it homothetic.