Setting the partial derivatives equal to 0 and solving gives — Show that variance minimized for fixed cost with function when NhSh/ Hint

(Show that the variance is minimized for a fixed cost with the cost function in (3.12) when nh ∝ NhSh/√ch, as in (3.13). Hint: Use Lagrange multipliers.

SOLUTION

We wish to minimize

subject to the constraint

Lagrange multipliers are often used for such problems. (See, for example, Thomas, G.B. and Finney, R. L. (1982). Calculus and Analytic Geometry, Fifth edition. Reading, MA: Addison-Wesley, p. 617.)

Define

Then

k = 1… H, and