Mathematical methods for economic theory

Martin J. Osborne
Previous section Previous page Contents Text Exercises Search Next page Next section

1.6 Exercises on calculus: many variables

  1. Determine whether each of the following sets is open, closed, both open and closed, or neither open nor closed.
    1. {(x, y): x2 + y2 < 1}
    2. {x: x is an integer}
    3. {(x, y): 0 < x < 1 and y = 0}.

    Solution

    1. Every point in this set is an interior point, so the set is open. The boundary of the set is {(x, y): x2 + y2 = 1}; no point in the boundary is thus a member of the set, so the set is not closed.
    2. No point in the set is an interior point, so the set is not open. Every point in the set is a boundary point, so the set is closed.
    3. Note that the set is a line segment in two-dimensional space. Thus no member of the set is an interior point, and hence the set is not open. (No disk centered at a point in the set is contained entirely within the set.) The set of boundary points of the set is {(x, y): 0 ≤ x ≤ 1 and y = 0}. The points (0, 0) and (1, 0) are not members of the set, so the set is not closed.

  2. For each of the following functions, find the partial derivatives f'1, f'2, and f"12.
    1. f(x1, x2) = 2x3
      1       + x1x2
    2. f(x1, x2) = (x1 + 2)/(x2 + 1)

    Solution

    1. f'1(x1, x2) = 6x2
      1       + x2; f'2(x1, x2) = x1; f"12(x1, x2) = 1.
    2. f'1(x1, x2) = 1/(x2 + 1); f'2(x1, x2) = −(x1+2)/(x2 + 1)2; f"12(x1, x2) = −1/(x2 + 1)2.

  3. For the production function f(K, L) = 9K1/3L2/3, find the marginal products of K and L (i.e. the partial derivatives of the function with respect to K and with respect to L).

    Solution

    fK(K, L) = 3K−2/3L2/3; fL(K, L) = 6K1/3L−1/3.