Extreme Values of Multivariate Functions Professor Peter Cramton Economics 300 Extreme values of multivariate functions In economics many problems reflect a need to choose among multiple alternatives Consumers decide on consumption bundles Producers choose a set of inputs Policy makers may choose several instruments to motivate behavior We now generalize the univariate techniques Stationary points and tangent planes of bivariate functions g 6 x1 x12 16 x2 4 x22 h x12 4 x22 2 x1 16 x2 x1 x2 Slices of a bivariate function g 6 x1 x 16 x2 4 x 2 1 g1 6 2 x1 0 2 2 g 2 16 8 x2 0 Multivariate first order condition If f x1 x2 xn is differentiable with respect to each of its arguments and reaches a maximum or a minimum at the stationary point x1 xn then each of the partial derivatives evaluated at that point equals zero i e f1 x x 0 f n x1 xn 0 1 n Second order condition in the bivariate case f x1 x2 First total differential y f x1 x2 dy f1 x1 x2 dx1 f 2 x1 x2 dx2 i e dy f1dx1 f 2 dx2 Second order condition in the bivariate case f x1 x2 Second total differential dy dy d y dx1 dx2 x1 x2 2 f1dx1 f 2 dx2 f1dx1 f 2 dx2 dx1 dx2 x1 x2 f11 dx1 2 2 f12 dx1 dx2 f 22 dx2 2 Extreme values and multivariate functions Sufficient condition for a local maximum minimum If the second total derivative evaluated at a stationary point of a function f x1 x2 is negative positive for any dx1 and dx2 then that stationary point represents a local maximum minimum of the function Extreme values and multivariate functions Sufficient Condition for a Local Minimum 2 f12 d y 0 if f11 0 and f 22 0 f11 2 Sufficient Condition for a Local Minimum d y 0 if f11 0 and f11 f 22 f 2 2 12 Extreme values and multivariate functions Sufficient Condition for a Local Maximum 2 f12 d y 0 if f11 0 and f 22 0 f11 2 Sufficient Condition for a Local Maximum d y 0 if f11 0 and f11 f 22 f 2 2 12 Extreme values of multivariate functions bivariate case Choose x1 x2 to maximize or to minimize f x1 x2 First Order Conditions f1 x1 x2 0 and f2 x1 x2 0 stationary points 1 2 x x Second Order Conditions Local Minimum if f11 x1 x2 0 and f11 x x f 22 x x f12 x x 1 2 1 2 1 2 Local Maximum if f11 x1 x2 0 and f11 x x f 22 x x f12 x x 1 2 1 2 1 2 2 2 Exercises Choose x1 x2 to minimize f x1 x2 4 x1 2 x x x2 2 2 2 1 f x1 x2 4 x1 2 x22 x12 x2 FOC f1 4 2 x1 0 x1 2 1 f 2 4 x2 1 0 x 4 2 f1 4 2 x1 f 2 4 x2 1 SOC We need to find f11 f12 f 22 If f11 0 and f11 f 22 f12 then local min 2 f1 4 2 x1 f 2 4 x2 1 SOC f11 2 f12 0 f 22 4 Observe that f11 2 0 and f11 f 22 2 4 8 0 f12 2 1 Hence 2 is local minimum 4 Exercise 2 Find the local max and local min of f x1 x2 8x1 7 x x 14 x2 2 2 2 1 f x1 x2 8 x1 7 x x 14 x2 2 2 2 1 FOC f1 8 2 x1 0 x 4 1 f 2 14 x2 14 0 x 1 2 f1 8 2 x1 f 2 14 x2 14 SOC f11 2 f12 0 f 22 14 Observe that f11 2 0 and f11 f 22 2 14 28 0 f12 2 Hence 4 1 is local max Exercise 3 Find the local max and local min of f x1 x2 2 x1 4 x x 16 x2 x1 x2 2 2 2 1 f x1 x2 2 x1 4 x x 16 x2 x1 x2 2 2 FOC f1 2 2 x1 x2 0 f 2 8 x2 16 x1 0 2 1 f1 2 2 x1 x2 0 f 2 8 x2 16 x1 0 2 2 x1 x2 0 x2 2 2 x1 8 2 2 x1 16 x1 0 x 0 1 x 2 2 f1 2 2 x1 x2 f 2 8 x2 16 x1 SOC f11 2 f12 1 f 22 8 Observe that f11 2 0 and f11 f 22 2 8 16 1 f12 2 Hence 0 2 is local min Exercise 4 Find the local max and local min of 1 2 1 2 f x1 x2 x1 x2 x1 x2 x1 x2 8 2 1 2 1 2 f x1 x2 x1 x2 x1 x2 x1 x2 8 2 FOC f1 1 x1 x2 0 1 f 2 x2 1 x1 0 4 f1 1 x1 x2 0 1 f 2 x2 1 x1 0 4 1 x1 x2 0 x2 1 x1 1 1 x1 1 x1 0 1 x1 4 4 x1 0 4 x1 1 x2 0 f1 1 x1 x2 1 f 2 x2 1 x1 4 SOC f11 1 f12 1 1 f 22 4 Observe that f11 1 0 and 1 1 f11 f 22 1 1 f12 2 4 4 Hence no concl Exercise 6 Find the local max and local min of 1 2 1 3 f x1 x2 x2 x1 x2 2 3 1 2 1 3 f x1 x2 x2 x1 x2 2 3 FOC f1 x 0 x 0 2 1 1 f 2 x2 1 0 x 1 2 f1 x 2 1 f 2 x2 1 SOC f11 2 x1 f12 0 f 22 1 At 0 1 SOC f11 2 x1 0 f12 0 f 22 1 Observe that f11 0 and f11 f 22 0 1 0 0 f12 2 Hence no conl Exercise 7 Find the local max and local min of 1 2 1 3 f x1 x2 x1 x2 x1 x2 2 3 1 2 1 3 f x1 x2 x1 x2 x1 x2 2 3 FOC f1 1 x 0 x 1 or x 1 2 1 1 1 f 2 x2 1 0 x 1 2 Two stationary points 1 1 and 1 1 f1 1 x 2 1 f 2 x2 1 SOC f11 2 x1 f12 0 f 22 1 At 1 1 SOC f11 2 x1 2 f12 0 f 22 1 Observe that f11 2 0 and f11 f 22 2 1 2 0 f12 2 Hence 1 1 is local max At 1 1 SOC f11 2 x1 2 1 2 f12 0 f 22 1 Observe that f11 2 0 and f11 f 22 2 1 2 0 f12 2 Hence at 1 1 no concl