Mathematical methods for economic theory

Martin J. Osborne
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4.3 Exercises on existence of an optimum

  1. For each of the following functions, determine (i) whether the extreme value theorem implies that the function has a maximum and a minimum and (ii) if the extreme value theorem does not apply, whether the function does in fact have a maximum and/or a minimum.
    1. x2 on the interval [−1,1]
    2. x2 on the interval (−1,1)
    3. |x| on the interval [−1,∞)
    4. f(x) defined by f(x) = 1 if x < 0 and f(x) = x if x ≥ 0, on the interval [−1,1].
    5. f(x) defined by f(x) = 1 if x < 0 and f(x) = x if x ≥ 0, on the interval (−∞,∞).
    6. f(x) defined by f(x) = x2 if x < 0 and f(x) = x if x ≥ 0 on the interval [−1,1].

    Solution

    1. EVT applies.
    2. EVT does not apply, because interval is not closed. Function has minimum of 0, but no maximum.
    3. EVT does not apply, because interval is not bounded. Function has minimum of 0, but no maximum.
    4. EVT does not apply, because function is not continuous. Function has minimum of 0 (attained at 0) and maximum of 1 (attained at all points in [−1,0) and at 1).
    5. EVT does not apply, because function is not continuous and interval is not bounded. Function has minimum of 0, but no maximum.
    6. EVT applies.
  2. Does the extreme value theorem imply that the problem
    maxxu(x) subject to p·x ≤ w and x ≥ 0,
    where x is an n-vector, u is a continuous function, and p > 0 (an n-vector) and w > 0 (a scalar) are parameters, has a solution? If not, specify a function u for which the problem has a solution, if there is such a function; and specify a function u for which the problem does not have a solution, if there is such a function. (The problem may be interpreted as the optimization problem of a consumer with a utility function u and income w, facing the price vector p.)

    Solution

    The objective function is continuous. The constraint set is the set of all nonnegative vectors on or below a hyperplane (the multidimensional analogue of a line in two dimensions and a plane in three dimensions). (Draw a diagram in the two dimensional case.) Thus it is compact. Hence the extreme value theorem implies that the problem has a solution.
  3. Does the extreme value theorem imply that the problem
    maxxpf(x) − w·x subject to x ≥ 0,
    where x is an n-vector, f is a continuous function, and p > 0 (a scalar) and w > 0 (an n-vector) are parameters, has a solution? If not, specify a function f and values of p and w for which the problem has a solution, if such exist; and specify a function f and values for p and w for which the problem does not have a solution, if such exist. (The problem may be interpreted as the optimization problem of a firm with production function f facing the input price vector w and the price of output p.)

    Solution

    The extreme value theorem does not imply that the problem has a solution, because the constraint set is not bounded.

    If p = 1, f(x) = x1/2, and w = 1, then the problem has a solution. (Draw a graph of the objective function.)

    If p = 1, f(x) = x2, and w = 1, then the problem does not have a solution. (Draw a graph of the objective function.)

  4. For each of the following problems, determine whether the extreme value theorem implies that a solution exists. (You need to give reasons for your answers! Note that for cases in which the extreme value theorem does not apply, you are not asked to determine whether there is in fact a solution.)
    1. maxx,yx2 + y2 subject to x2 + 2y2 ≤ 1.
    2. maxxf(x) subject to x ≥ 0, where f(x) = −x2 if x ≤ −1 and f(x) = −1 if x > −1.
    3. maxxf(x) subject to 0 ≤ x ≤ 1, where f(x) = x2 if 0 ≤ x ≤ 1/2 and f(x) = 1/2 if 1/2 < x ≤ 1.

    Solution

    1. The objective function is continuous, and the constraint set is closed and bounded, therefore the extreme value theorem implies the problem has a solution.
    2. The constraint set is not bounded, so the extreme value theorem does not apply.
    3. The objective function is not continuous, so the extreme value theorem does not apply.