Question

Show that the sum of two concave functions is concave. Is the product of two concave functions also concave?

Answer 1

Answer:

1.

Let f and g concave functions. Then for a in (0,1), $f( (1-\lambda)x + \lambda y ) \geq (1-\lambda)f(x) + \lambda f(y)$ and $g( (1-\lambda)x + \lambda y ) \geq (1-\lambda)g(x) + \lambda g(y)$

Now, $(f+g)( (1-\lambda)x + \lambda y ) = f( (1-\lambda)x + \lambda y ) + g( (1-\lambda)x + \lambda y )\geq (1-\lambda)f(x) + \lambda f(y) + (1-\lambda)g(x) + \lambda g(y) = (1-\lambda)(f(x)+g(x)) + \lambda (f(x)+g(x))= (1-\lambda)(f+g)(x) + \lambda(f+g)(x)$

This shows that the function f+g is concave.

2. Let f and g concave functions.

$(fg)( (1-a)x + \lambda y )= f( (1-\lambda)x + \lambda y )g( (1-\lambda)x + \lambda y )  \geq [(1-a)f(x) + \lambda f(y)] [(1-\lambda)g(x) + \lambda g(y)] = (1-\lambda)^2 f(x)g(x) + \lambda (1-\lambda) f(x)g(y)+ \lambda (1-\lambda) f(y)g(x)+ \lambda ^2 f(y)g(y) \neq (1-\lambda)f(x)g(x) + \lambda f(y)g(y)$

This shows that the product of two concave functions isn't concave.