Question

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

Answer 1

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) \\eq (1-\lambda)f(x)g(x) + \lambda f(y)g(y)`

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