the marginal product. Using the rule for differentiating a power function,

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1 3 Augu 07 Chaper 3 Derivaive ha economi ue 3 Rule for differeniaion The chain rule Economi ofen work wih funcion of variable ha are hemelve funcion of oher variable For example, conider a monopoly elling in a marke wih demand price funcion p( q) 50 q Then oal revenue i R( q) 50q q and marginal revenue i MR( q) R( q) 50 q The oupu of he firm depend on he ize of he work-force z Suppoe ha wih z uni, he oupu i q / z The oupu i ofen called he produc of he firm and o he derivaive i he marginal produc Uing he rule for differeniaing a power funcion, MP( z) ( z ) 6z If z 4 he marginal produc i MR( q) 50 (4) 6(4) 3 and oupu i (4) 4 4 Then marginal revenue i In word, oupu increae hree ime a fa a he inpu and revenue increae wice a fa a oupu Then revenue increae 3 6 ime a fa a he inpu ha Le h( z) R( q( z)) be he funcion mapping inpu ino revenue We have ju concluded h( z) R( q) q( z) Thi i called he Chain Rule Chain Rule: If h( x) g( f ( x)), where f( x ) and g( y ) are differeniable funcion, hen h( z) g( f ( x) f ( x)

2 3 Augu 07 Proof (informal ): A long a you find he verbal argumen convincing (o ha you will remember he Chain Rule) you need no dwell on he following more formal derivaion Conider he mapping y f ( x) and z g( y) For any x define y f ( x x) f ( x) and z g( y y) g( y) h( x x) h( x) g( f ( x x) g( f ( x)) g( y y) g( y) x x x g( y y) g( y) y x y g( y y) g( y) y y x g( y y) g( y) f ( x x) f ( x) ( )( ) y x In he limi he lef hand ide i he derivaive, h ( x) and he righ hand ide i he produc of he derivaive g ( y) and f ( x) QED Example: We defined he naural logarihm ln( x ) o be he aniderivaive of x Conider h( x) ln f ( x) We define g( y) ln y Appealing o he Chain Rule, f( x) h( x) g( y) f ( x) f ( x) g( y) f ( x) y f ( x ) Produc Rule: The derivaive of h( x) f ( x) g( x) i h( x) f ( x) g( x) f ( x) g( x) To make hi a formal proof we would need o work wih a formal definiion of a limi However he meaning i clear

3 3 Augu 07 Proof: h( x x) h( x) f ( x x) g( x x) f ( x) g( x) Therefore f ( x x) g( x x) f ( x) g( x x) f ( x) g( x x) f ( x) g( x) ( f ( x x) f ( x)) g( x x) f ( x)( g( x x) g( x)) h( x x) h( x) f ( x x) f ( x) g( x x) g( x) g( x x) f ( x) x x x Finally we ake he limi a x 0 h( x) f ( x) g( x) f ( x) g( x) QED Quoien Rule: The derivaive of f( x) f ( x) g( x) f ( x) g( x) hx ( ) i h( x) gx ( ) gx ( ) Thi can be proved by applying he Produc Rule o he funcion f( x ) and k( x) g( x) gx ( ) Invere funcion rule For monopoly producion deciion, i wa helpful o work no wih he mapping from price o quaniy (he demand funcion q( p ) ) bu wih he revere mapping from quaniy o price (he demand price funcion pq ( )) Thi funcion i known a he invere mapping and i omeime wrien a follow: p( q) q ( p) Suppoe, ha he demand curve i q( p) p p 3

4 3 Augu 07 Then he lope i q( p) 00 p The demand curve i depiced below wih price on he horizonal axi A p 0 he lope i -0 If he quaniy fall 0 ime faer han he price rie, hen he price fall a fracion /0 of he rae a which he quaniy rie Generalizing, if he rae a which he quaniy change wih price i q ( p) hen he rae a which price change wih quaniy i / q ( p) Invere funcion rule: If he mapping x g( y) i he invere of he mapping y f ( x), hen g( y) / f ( x) / f ( g( y)) Uing hee rule, i i fairly eay o calculae he derivaive of almo any funcion ha you are likely o ee in an economic cla Bu olving for an aniderivaive (or indefinie inegral ) i a much rickier propoiion Fir of all, many funcion do no have an aniderivaive Second, even if omeone were o ell you ha a funcion did have an aniderivaive, hi doe no help in figuring ou wha i i Suppoe however ha he funcion o be inegraed, hx ( ), i he produc of wo oher funcion f( x ) and gx ( ) Suppoe alo ha you know he ani-derivaive of f( x ) Then he following reul ha appeal o he Produc Rule can be helpful 4

5 3 Augu 07 Inegraion by par f ( x) g( x) f ( ) G( ) f ( ) G( ) f ( x) G( x), where G( x) g( x) Proof: Fir define h( x) f ( x) G( x) Appealing o he produc rule, h( x) f ( x) G( x) f ( x) g( x) Rearranging erm, f ( x) g( x) h( x) f ( x) G( x) Inegraing over he inerval [, ] f ( x) g( x) h( x) f ( x) G( x) h( ) h( ) f ( x) G( x) f ( ) G( ) f ( ) G( ) f ( x) G( x) QED Remark: We ue he following hor-hand h( x) h( ) h( ) x Example: 3 ( x) 0 Define f ( x) Inegraing by par, x and gx ( ) 3 ( x) Then Gx ( ) ( x) 5

6 3 Augu 07 x 3 ( x) ( x) ( x) x ( ) ( x) 0 ( ) x 0 ( ) 6