Extreme Values of Multivariate Functions Professor Peter Cramton Economics 300Extreme 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 techniquesStationary points and tangent planes of bivariate functions 221 1 2 26 16 4g x x x x   221 2 1 2 1 24 2 16h x x x x x x    Slices of a bivariate function 221 1 2 26 16 4g x x x x   116 2 0gx  2216 8 0gx  Multivariate first-order condition • If is differentiable with respect to each of its arguments and reaches a maximum or a minimum at the stationary point, then each of the partial derivatives evaluated at that point equals zero, i.e. 12( , ,..., )nf x x x**1( ,..., )nxx**11**1( ,..., ) 0.........( ,..., ) 0nnnf x xf x xSecond-order condition in the bivariate case First total differential 121 1 2 1 2 1 2 21 1 2 2( , )( , ) ( , )i.e.y f x xdy f x x dx f x x dxdy f dx f dx12( , )f x xSecond-order condition in the bivariate case Second total differential 212121 1 2 2 1 1 2 212122211 1 12 1 2 22 2[ ] [ ][ ] [ ]( ) 2 ( ) ( )dy dyd y dx dxxxf dx f dx f dx f dxdx dxxxf dx f dx dx f dx         12( , )f x xExtreme 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 functionExtreme values and multivariate functions Sufficient Condition for a Local Minimum: Sufficient Condition for a Local Minimum: 221211 2211()0 if 0 and 0fd y f ff   2211 11 22 120 if 0 and d y f f f f  Extreme values and multivariate functions Sufficient Condition for a Local Maximum: Sufficient Condition for a Local Maximum: 221211 2211()0 if 0 and 0fd y f ff   2211 11 22 120 if 0 and d y f f f f  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 **12( , )xxSecond Order Conditions Local Minimum if ___________________ Local Maximum if  **11 1 22* * * * * *11 1 2 22 1 2 12 1 2( , ) 0( , ) ( , ) ( , )f x xandf x x f x x f x x **11 1 22* * * * * *11 1 2 22 1 2 12 1 2( , ) 0( , ) ( , ) ( , )f x xandf x x f x x f x xExercises • Choose (x1,x2) to minimize 221 2 1 2 1 2( , ) 4 2f x x x x x x   221 2 1 2 1 2*1 1 1*2 2 2( , ) 4 2:4 2 0 214 1 04f x x x x x xFOCf x xf x x            112211 12 22211 11 22 124241: We need to find , ,If 0 and . ( ) , then local minfxfxSOC f f ff f f f112211122211211 22 124241: 204Observe that 2 0 and . 2(4) 8 0 ( )1Hence, ( 2, ) is local minimum. 4fxfxSOCfffff f f   Exercise 2 • Find the local max and local min of 221 2 1 2 1 2( , ) 8 7 14f x x x x x x   221 2 1 2 1 2*1 1 1*2 2 2( , ) 8 7 14:8 2 0 414 14 0 1f x x x x x xFOCf x xf x x            112211122211211 22 128214 14: 2014Observe that 2 0 and . ( 2)( 14) 28 0 ( )Hence, (4,1) is local max. fxfxSOCfffff f f         Exercise 3 • Find the local max and local min of 221 2 1 2 1 2 1 2( , ) 2 4 16f x x x x x x x x     221 2 1 2 1 2 1 21 1 22 2 1( , ) 2 4 16:2 2 08 16 0f x x x x x x x xFOCf x xf x x            1 1 22 2 11 2 2 1*1 1 1*22 2 08 16 02 2 0 2 28(2 2 ) 16 0 02f x xf x xx x x xx x xx                  1 1 22 2 111122211211 22 12228 16: 218Observe that 2 0 and . (2)(8) 16 1 ( )Hence, (0,2) is local min. f x xf x xSOCfffff f f        Exercise 4 • Find the local max and local min of 221 2 1 2 1 2 1 211( , )82f x x x x x x x x     221 2 1 2 1 2 1 21 1 22 2 111( , )82:101104f x x x x x x x xFOCf x xf x x             1 1 22 2 11 2 2 11 1 1 1*1*21011041 0 11(1 ) 1 0 1 4 4 0410f x xf x xx x x xx x x xxx                       1 1 22 2 111122211211 22 121114: 1114Observe that 1 0 and 11. ( 1)( ) 1 ( )44Hence, no concl.f x xf x xSOCfffff f f             Exercise 6 • Find the local max and local min of 231 2 2 1 211( , )23f x x x x x   231 2 2 1 22*1 1 1*2 2 211( , )23:0 0 1 0 1f x x x x xFOCf x xf x x            2112211 112221: 201fxfxSOCfxff  11 1122211211 22 12At (0,1): 2001Observe that 0 and . (0)( 1) 0 0 ( )Hence, no conl.SOCfxffff f f      Exercise 7 • Find the local max and local min of 231 2 1 2 1 211( , )23f x x x x x x   231 2 1 2 1 22 * *1 1 1 1*2 2 …