Question

Problem 5. Suppose a consumer has a continuous and quasiconcave utility function
u(x1, x2). Then show that set of solutions to the expenditure minimization problem is
a convex set in R2

Answer 1

Solution:

A consumer has utility function u(x1, x2) =,

Therefore u(x1,x2)= min{v1(x1, x2), v2(x1, x2)}

Where, v1 and v2 are both quasi-concave functions

A function f(x1, x2) is quasi-concave if and only if for all (x1, x2) in its domain, the upper-contour set      f +(x1, x2) = {(x 0 1 , x2)|f(x 0 1 , x0 2 ) ≥ f(x1, x2)} is a convex set. Since v1 and v2 are quasi-concave, it must be that the upper contour sets v + i (x1, x2) = {(x 0 1, x0 2) |vi(x 0 1 , x0 2 ) ≥ vi(x1, x2)} are convex. From the definition of the function u, we see that for every (x1, x2) in the domain, the upper contour set of u will be the set u +(x1, x2) = v + 1 (x1, x2)∩V + 2 (x1, x2). Since the intersection of convex sets is convex, it follows that u is a quasi-concave function.

Now, showing expenditure minimization problem is a convex set in R2