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Managerial Economics, 8e
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Chapter 18
Optimization Techniques
Chapter Review
This chapter presents basic calculus and optimization techniques. It is not meant to be a substitute for a calculus class, but presents the necessary tools that can be applied to managerial economics.
The marginal value of a dependent variable is the change associated with a unit change in the independent variable. For example, marginal profit is the change in total profit associated with a one-unit change in output (quantity) or marginal cost is the change in total cost associated with a one-unit change in output (quantity).
Maximization occurs when a marginal value shifts from positive to negative. The relation between average and marginal is interesting. If marginal exceeds the average, it must pull the average up; likewise, if marginal is less than the average, it must pull the average down. Thus, the marginal must equal the average when the average is at a maximum or at a minimum.
Taking the derivative of Y with respect to X, allows a determination of the value of X that either maximizes or minimizes Y. Maxima or Minima can only be studied where the slope of the curve with Y on the vertical and X on the horizontal axis, is zero. That is, the rate of change is zero, or the first derivative is zero. Whether a maximum or minimum is reached is seen by an examination of the second derivative, which is a rate of change in the first derivative. If the second derivative is negative, a maximum is found, if the second derivative is positive, a minimum is found.
As an illustration, maximization of a profit function results in the well-known equality that Marginal Revenue (MR) must equal Marginal Cost (MC) at equilibrium. The second order condition results in the marginal cost curve cutting the marginal revenue curve from below.
Since managers often face constraints, the method of Lagrangian multipliers is used. The Lagrangian function combines the function and the constraints, so that its maximization (minimization) also results in the original function being maximized/minimized, and the constraints being satisfied. Importantly, the value of the Lagrangian multiplier measures the change in the variable that is being maximized or minimized when the constraint is relaxed by one unit!