3.4 Exercises on quasiconcavity and quasiconvexity

  1. Draw the upper level sets of each of the following functions for the indicated values. In each case, say whether the set is consistent with the function's being quasiconcave. (Of course, the shape of one upper level set does not determine whether the function is quasiconcave, which is a property of all the upper level sets; the question is only whether the single upper level set your draw is consistent with quasiconcavity.)
    1.  f (x, y) = xy for the value 1.
    2.  f (x, y) = x2 + y2 for the value 1.
    3.  f (x, y) = −x2 − y2 for the value −1.

  2. For what values of the parameters a, b, c, and d is the function ax3 + bx2 + cx + d quasiconcave? (Use the characterization of quasiconcave functions of a single variable. There are several cases to work through.)

  3. Use the bordered Hessian condition to determine whether the function  f (x,y) = ye−x is quasiconcave for the region in which x ≥ 0 and y ≥ 0.

  4. Give an example to show that the sum of quasiconcave functions is not necessarily quasiconcave.

  5. The function  f  is concave and the function g is quasiconcave; neither is necessarily differentiable. Is the function h defined by h(x) =  f (x) + g(x) necessarily quasiconcave? (Either show it is, or show it isn't.)

  6. The functions  f  and g of a single variable are concave (but not necessarily differentiable). Is the function h defined by h(x) =  f (x)g(x) necessarily quasiconcave?

  7. Determine, if possible, which of the following properties each of the following functions satisfies: convexity, strict convexity, concavity, strict concavity, quasiconcavity, strict quasiconcavity.
    1.  f (x,y) = x2y2 for x ≥ 0 and y ≥ 0.
    2.  f (x,y) = x − ex − ex+y.

  8. The function  f  of two variables is defined by  f (x1, x2) = x1(x2)2. If possible, determine whether this function is quasiconcave for x1 > 0 and x2 > 0. If not possible, say why not.

  9. The three curves in the figure below are the sets of points for which the value of the function  f  (of two variables, x and y) is equal to 1, 2, and 4. Are these curves consistent or inconsistent with the function's being
    1. quasiconcave?

    2. concave?

  10. Show that a concave function is quasiconcave by using the fact that a function  f  is quasiconcave if and only if for all x ∈ S, all y ∈ S, and all λ ∈ [0,1] we have
    if  f (x) ≥  f (y) then  f ((1−λ)x + λy) ≥  f (y).

  11. Reminder: A function  f  is quasiconcave if and only if for every x and y and every λ with 0 ≤ λ ≤ 1, if  f (x) ≥  f (y) then  f ((1 − λ)x + λy) ≥  f (y).

    Suppose that the function U is quasiconcave and the function g is increasing. Show that the function  f  defined by  f (x) = g(U(x)) is quasiconcave.

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