Concavity and quasiconcavity

a. Let () = for ++. Notice that x here is a single variable, ++, not a vector. Using calculus, determine for which values of the parameter the function is a concave function over the convex set ++.

b. Let and be concave functions over a convex set . Let , + 0, 0. Define the function + over to be ( + )() = () + (). Prove that the function + is a concave function over .

General case: If 1 , 2 , ... , are concave functions over the convex set and = (1, 2, ... , ) + then 1 1 + 2 2 + is a concave function over the set . Don't need to prove. Here 2 , 3 , just means a second function, third function, etc, not the square of a function, cube, etc. Similar results apply to convex functions.

c. Concave functions in and in , < .

i. Let and () = 1 2 . Show is a concave function over the convex set .

ii. Now let us embed into 2 . Let = (1, 2 ) 2 and (1, 2 ) = 1 1 2 . (f does not depend on 2). Show is a concave function over the convex set 2 . In general, for > , if (1, ... , ) is concave in (1, ... , ) then (1, ... , , ... , ) = (1, ... , ) is concave in (1, ... , ).

d. Let be a concave function over a convex set . Prove that is also a quasiconcave function over .

e. Let be a quasiconcave function over a convex set . Let be a strictly increasing function. Prove that a monotonic transformation of a quasiconcave function over is a quasiconcave function over ie prove that = is a quasiconcave function over . Hint: , , ({, }) = min{(), ()}.

f. Let ( 1, ... , ) ++ and = (1, ... , ) ++ . Let > 0, > 0, 0. (10) Define the generalised CES function over the convex set ++ . ++ , () = (11 + 22 + )

Using the results from this question, show that this generalised CES function is a quasiconcave function over ++ when (0, 1].