Economics 2301

Lecture 31

Univariate Optimization

Second-Order Conditions

The second-order condition provides a sufficient condition but, as we will see, not a necessary condition, for characterizing a stationary point as a local maximum or a local minimum.

Second-Order Conditions

Local Maximum: If the second derivative of the differentiable function y=f(x) is negative when evaluated at a stationary point x* (that is, f”(x*)<0), then that stationary point represents a local maximum.

Local Minimum: If the second derivative of the differentiable function y=f(x) is positive when evaluated at a stationary point x* (that is, f”(x*)>0), then that stationary point represents a local minimum.

Second-Order Conditions for Our Examples

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Graph of Example 1

y=-6X2+24X-30

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Graph of Example 2

Y=3X2-30X+4

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Second-Order Conditions for Example 3

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Graph of Example 3

y=0.2X3-5X2+15X-4

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Failure of Second-Order Condition

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Stationary Point of a strictly concave function

If the function f(x) is strictly concave on the interval (m,n) and has the stationary point x*, where m<x*<n, the x* is a local maximum in that interval. If a function is strictly concave everywhere, then it has, at most, one stationary point, and that stationary point is a global maximum.

Note that example 1 satisfies this condition.

Stationary Point of a Strictly Convex Function

If the function f(x) is strictly convex on the interval (m,n) and has the stationary point x*, where m<x*<n, the x* is a local minimum in that interval. If a function is strictly convex everywhere, then it has, at most, one stationary point, and that stationary point is a global minimum.

Note that example 2 satisfies this condition.

Inflection Point

The twice-differentiable function f(x) has an inflection point at x* if and only if the sign of the second derivative switches from negative in some interval (m,x*) to positive in some interval (x*,n), in which case the function switches from concave to convex at x*, or the sign of the second derivative switches from positive in some interval (m,x*) to negative in some interval (x*,n), in which case the function switches from convex to concave at x*. Note that, in either case, m<x<n.

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Derivatives of Our Cubic Function

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Graph of Example 3

y=0.2X3-5X2+15X-4

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0 5 10 15 20 25

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Concave

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Inflection point

Optimal Excise Tax

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