Multivariate Calculus Professor Peter Cramton Economics 300 Multivariate calculus Calculus with single variable univariate y f x Calculus with many variables multivariate y f x1 x2 xn Partial derivatives With single variable derivative is change in y in response to an infinitesimal change in x With many variables partial derivative is change in y in response to an infinitesimal change in a single variable xi hold all else fixed Total derivative is change in all variables at once Cobb Douglas production function 1 2 1 2 Q 20 K L How does production change in L 1 1 Q 10 K 2 L 2 L Marginal product of labor MPL Partial derivatives use instead of d Cobb Douglas production function 1 2 1 2 Q 20 K L How does production change in K Q 12 12 10 K L K Marginal product of capital MPK Partial derivatives use instead of d Cobb Douglas production function 1 2 Note 1 2 Q 20 K L 1 1 Q MPL 10 K 2 L 2 0 L Q 12 12 MPK 10 K L 0 K Produce more with more labor holding capital fixed Produce more with more capital holding labor fixed Second order partial derivatives Differentiate first order partial derivatives Q 12 12 10 K L K 1 1 Q 10 K 2 L 2 L 1 3 Q Q 2 2 5 K L 2 L L L 2 Q Q 32 12 5 K L 2 K K K 2 Second order partial derivatives Note 1 3 Q Q 2 2 5 K L 0 2 L L L 2 Q Q 32 12 5 K L 0 2 K K K 2 MPL declines with more labor holding capital fixed MPK declines with more capital holding labor fixed Diminishing marginal product of labor and capital Second order cross partial derivatives 1 1 1 1 1 1 Q Q 10 K 2 L 2 10 K 2 L2 Q 20 K 2 L2 L K What happens to MPL when capital increases Q Q 12 12 5K L 0 K L K L 2 What happens to MPK when labor increases Q Q 12 12 5K L 0 L K L K 2 By Young s Theorem cross partials are equal Q Q 12 12 5K L 0 L K K L 2 2 Labor demand at different levels of capital 1 1 Q 10 K 2 L 2 0 L 1 3 Q 2 2 5 K L 0 2 L 2 K K0 K K1 1 2 1 2 Q 20 K L Contour plot Isoquant is 2D projection in K L space of CobbDouglas production with output fixed at Q 1 2 1 2 Q 20 K L Implicit function theorem Each isoquant is implicit function 1 2 1 2 Q AK L Implicit since K and L vary as dependent variable Q is a fixed parameter Solving for K as function of L results in explicit function 2 Q A K L L Use implicit function theorem when can t solve K L Implicit function theorem For an implicit function F y x1 xn k 0 0 0 defined at point y x1 xn with continuous 0 0 0 partial derivatives Fy y x1 xn 0 there is a function y f x1 xn defined in 0 0 neighborhood of x1 xn corresponding to F y x1 xn k such that y 0 f x10 xn0 F y x x k 0 0 1 0 n fi x x 0 1 0 n Fxi y 0 x10 xn0 Fy y 0 x10 xn0 Verifying implicit function theorem Cobb Douglas production 1 2 2 Q A K L L Derivative of isoquant 2 Q A dK L KL K 2 2 dL L L L Explicit function for isoquant 1 2 Q AK L Slope of isoquant marginal rate of technical substitution Essential in determining optimal mix of production inputs Implicit function theorem 0 fi x x 0 1 1 2 1 2 Q AK L 0 n 0 1 0 1 0 n 0 n Fxi y x x 0 Fy y x x 1 1 1 1 Q Q 10 K 2 L 2 10 K 2 L2 L K Q 1 1 2 2 dK 10 K L K L 1 1 2 2 Q dL L 10 K L K Implicit function theorem from differential For multivariate function Differential is Q F K L Q Q dQ dK dL K L Holding quantity fixed along the isoquant Thus Q Q dQ dK dL 0 K L Q dK MPL L Q dL MPK K Isoquants for Cobb Douglas Q AK 1 L 1 1 dK AK L K dL 1 AK L 1 L K bL Homogeneous functions When all independent variables increase by factor s what happens to output When production function is homogeneous of degree one output also changes by factor s A function is homogeneous of degree k if s Q F sK sL k Homogeneous functions Is Cobb Douglas production function homogeneous 1 Q AK L 1 1 1 A sK sL As s K L sQ Yes homogeneous of degree 1 Production function has constant returns to scale Doubling inputs doubles output Homogeneous functions Is production function homogeneous Q AK L A sK sL As K L s Yes homogeneous of degree For 1 constant returns to scale Doubling inputs doubles output For 1 increasing returns to scale Doubling inputs more than doubles output For 1 decreasing returns to scale Doubling inputs less than doubles output Q Properties of homogeneous functions Partial derivatives of a homogeneous of degree k function are homogeneous of degree k 1 1 Q AK L Q 1 1 AK L L Q 1 1 0 1 1 A sK sL As K L L Cobb Douglas partial derivatives don t change as you scale up production Isoquants for Cobb Douglas Q AK 1 L 1 1 dK AK L K dL 1 AK L 1 L K bL Euler s theorem Any function Q F K L has the differential Q Q dQ dK dL K L Euler s Theorem For a function homogeneous of degree k Q Q kQ K L K L 1 Let Q GDP and Cobb Douglas Q AK L Marginal products are associated with factor prices Q 1 L AK L Q L In US 67 Q K 1 Q K Chain rule y f x1 xn xi gi t1 tm y f x1 f xn ti x1 ti xn ti Chain rule exercise y 3 x 2 xw w x 8 z 18 w 4z What is dy dz 2 2 dy y x y w dz x z w z 6 x 2 w 8 2 w 2 x 4