We know that the first-order conditions cannot in themselves distinguish between maximum values, minimum values, points of inflection and saddle points, because they hold at each of these. We now develop second-order conditions which tell us when we can be sure that a point satisfying the first-order conditions is certainly a true maximum or minimum. We begin with an intuitive discussion and at the end of this section give a somewhat more rigorous account in terms of the Taylor series expansion.

Suppose that (x* v*) is a stationary point of the function /. If we make small changes in the x-vector in any direction from this point, and the result is to reduce the value of/, then this point must yield a local maximum of the function. Similarly, if we move away from this point a small distance in any direction and this increases the value of the function, this point must yield a local minimum. Finally, if moving in some directions increases the value of the function, while moving in other directions reduces the function value, then the stationary point must be a saddle point.

Note the emphasis on "in any direction." If simply moving in the x,-direction (parallel to the x;-axis), for each x, reduces the value of the function, this need not imply we have a maximum, since these are only a small subset of all the possible directions in which we could go. Figure 12.5 illustrates for a function of two variables. The bulge in the function means that (x*, x£) gives a stationary value which is not a local maximum, since moving in the direction indicated—not parallel to either axis—increases Lhe value of the function.

The requirement to take account of all possible directions of change in the x-vector is what complicates the algebra of the second-order conditions just as it complicates the determination of concavity and convexity (see section 11.4). The second-order partial J), tells us about the curvature of the function in the Xj-direction. If the function value decreases with a move from (x\. ... in

v.-direction v.-direction

Figure 12.5 A movement away from Orf, ) not in die a,-or .^-direction increases the value of the function

Figure 12.5 A movement away from Orf, ) not in die a,-or .^-direction increases the value of the function the a,-direction, as is necessary if this point is to he a maximum, then we have fii(x*,... .a**) < 0. However, it is not sufficient to state this as a condition for all i, because it takes no account of movements away from the x-vector that are not in any .v,-direction. For this, we require the total differential.

Given some function with continuous partial derivatives, its total differential for arbitrary changes dx, gives the change in the function in an arbitrary direction. Suppose this function is the total differential dy. At the stationary point

(aj* a,*), this total differential dy = df{xf a') = 0. If. for any (smalli movement away from this point dy becomes negative, that means the function is decreasing and (a J,..., a*) yields a maximum. If. for any (small) movement away, dy becomes positive, this means the function value is increasing and (a'______r,'j yields a minimum. Thus sufficient conditions for a local maximum or minimum can be expressed in terms of what happens to dy as we move away from (a,' a,')

in any direction, namely in terms of the second-order differential d2y. We can put this more formally as follows:

We have a function y — /(*t,...,xn) or y - /'(x). with x = (,Vi A„t.

At a point x* consider the total differential where the differentials dxj are to be interpreted as small given numbers. This is dy = l\(x*)dx\ + fi[x')dx2 + - • • + fj\')dx.

■n of course just a function of the vector x and as in chapter 11, we can lake the differential of this function to obtain the second-order differential

(Udy) s d2y = \d(j)(x-)dxO\ + \d{J2(x) dx2)\ + ■ • • + |rf(/B(**) dx„)\ = l/ii(x*)dxi dx\ + frjx')dx{ dx2 + •■■ + j]„(x')dx{ dx„\ + \f2i(x')dx2dx\ + f22(x')dx2dx2 + ■ •• + fin (x*) dx2 dxn j

+ 1 f„\(x*)dxndx, + f,a(x*)dx»dx2. + ••■ + f„„(x*)dxndx„\ i j

This expression is a quadratic form (see chapter 10). Now suppose that d2y < 0 for arbitrary vectors of values (dx,) (which may have to be very small). That means that in all directions the value of dy is decreasing around the point x*. But. ifrfy is /.ero at x4, as it must be if x* is an extreme value, then dy is negative in a neighborhood of x*. That means that the function is decreasing in the neighborhood of x*. and so if the first-order conditions are satisfied al x*. then this is a local maximum of the function. We refer to this as a local maximum because we only use information about the behavior of dy in a small neighborhood of x\ not over all possible vectors x. On the other hand, a global maximum would be the maximum of the function taken over all possible vectors x. We have

Theorem 12.3 It us sufficient for x" to yield a local maximum of the C2 function y = fix) that f,{x'\ = 0. jsl »

and the quadratic form d2y{x*) = (x')dxidxj < 0. i. j = I it i j

That is, d2\ is negative definite, or since d2y = dx' H dx, the Hessian matrix // is negative-definite.

Proof

By using the Taylor series expansion (definition I I.I in section I 1.6). we can expand fix) around the point x" to gel, for any x in the neighborhood of x* (i.e..

/(x) = f{K*) + dy(x')+d2y(l;) where £ lies between x* and x. Since, /,(x*) = 0 for all i = 1.2 n, we have

Moreover, since d2y(x) < 0, it must also be the case that for x sufficiently close to x", and hcnce S, close to x*, d2y(x) < 0 and d2y(tj) < 0. These results together give us

In other words, /(x) is less than /(x*) for any x near x* if /¡(x*) - 0 and i/2_v(x*) < 0. Thus x* yields a local maximum. ■

From this proof we see that if /(x) is a strictly concave function on x e R", then, if fi(x') = 0, /' = 1,2 «, it must be the case that x* is a unique global maximum. This follows clearly if d2f(x) < 0 for all x 6 R". a sufficient condition for strict concavity of /.

Theorem 12.4 |
Suppose that y = /(x) is a strictly concave function defined on xeR' |
\ If at |

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