Econ 300_Summer 2009_Slides 6 - Multivariate Calculus[1]

Econ 300_Summer 2009_Slides 6 - Multivariate Calculus[1] -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Multivariate calculus • Calculus with single variable (univariate) y y f ( x) • Calculus with many variables (multivariate) f ( x1 , x2 ,..., xn ) Partial derivatives • With single variable, derivative is change in y in response 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 Q 20 K L 1 2 1 2 • How does production change in L? 1 1 wQ 10 K 2 L 2 wL • Marginal product of labor (MPL) • Partial derivatives use w instead of d Cobb-Douglas production function Q 20 K L 1 2 1 2 • How does production change in K? wQ 1 1 10 K 2 L2 wK • Marginal product of capital (MPK) • Partial derivatives use w instead of d Cobb-Douglas production function Q • Note 20 K L 1 1 wQ MPL 10 K 2 L 2 ! 0 wL wQ 1 1 MPK 10 K 2 L2 ! 0 wK 1 2 1 2 • 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 1 1 wQ 10 K 2 L 2 wL wQ 1 1 10 K 2 L2 wK 5 K L 3 2 1 2 3 2 w § wQ · ¨ ¸ wL © wL ¹ w § wQ · ¨ ¸ wK © wK ¹ wQ 2 wL 2 wQ 2 wK 2 5 K L 1 2 Second-order partial derivatives • Note w § wQ · ¨ ¸ wL © wL ¹ w § wQ · ¨ ¸ wK © wK ¹ wQ 2 wL 2 5 K L  0 5 K L  0 3 2 1 2 1 2 3 2 wQ 2 wK 2 • MPL declines with more labor, holding capital fixed • MPK declines with more capital, holding labor fixed • Diminishing marginal product of labor and capital • What happens to MPL when capital increases? Second-order cross partial derivatives wQ 1 1 1 1 wQ 2 2 10 K 2 L2 10 K L wK wL • What happens to MPK when labor increases? w § wQ · ¨ ¸ wK © wL ¹ wQ wK wL 2 5K L ! 0 5K L ! 0 1 2 1 2 1 2 1 2 1 2 1 2 • By Young’s Theorem cross-partials are equal w § wQ · ¨ ¸ wL © wK ¹ wQ wLwK 2 wQ wLwK 2 wQ wK wL 2 5K L ! 0 Labor demand at different levels of capital 1 1 wQ 10 K 2 L 2 ! 0 wL wQ 2 wL 2 K = K0 5 K L  0 K = K1 1 2 3 2 Q 20 K L 1 2 1 2 Contour plot • Isoquant is 2D projection in K-L space of CobbDouglas production with output fixed at Q* Q 20 K L 1 2 1 2 Implicit function theorem • Each isoquant is implicit function Q * AK L 1 2 1 2 • 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 K ( L) • Use implicit function theorem when can’t solve K(L) §Q · ¨ A¸ ©¹ 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 ) z 0, f ( x1 ,..., xn ) defined in there there is a function y 0 0 neighborhood of ( x1 ,..., xn ) corresponding to F ( y, x1 ,..., xn ) k such that y0 0 0 f ( x10 ,..., xn ) 0 1 0 n F ( y , x ,..., x ) k fi ( x ,..., x )  0 1 0 n 0 Fxi ( y 0 , x10 ,..., xn ) 0 Fy ( y 0 , x10 ,..., xn ) Verifying implicit function theorem • Cobb-Douglas production • Explicit function for isoquant Q * AK L 1 2 1 2 K ( L) • Derivative of isoquant dK ( L) dL §Q · ¨ A¸ ©¹  2 L * 2 §Q · ¨ A¸ ©¹ L * 2 KL 2 L K  L • Slope of isoquant = marginal rate of technical substitution – Essential in determining optimal mix of production inputs Implicit function theorem f i ( x ,..., x )  0 1 0 n Fxi ( y , x ,..., x ) 0 0 Q AK L 1 2 1 2 Fy ( y , x ,..., x ) 1 1 wQ wQ 1 1 10 10 K 2 L 2 10 K 2 L2 wL wK  10 K L 1 2 1 2 0 1 0 1 0 n 0 n dK dL wQ  wL wQ wK 1 2 1 2 10 K L K  L Implicit function theorem from differential • For multivariate function Differential • Differential is Q F ( K , L) dQ wQ wQ dK  dL wK wL • Holding quantity fixed (along the isoquant) • Thus wQ wQ dQ dK  dL 0 wK wL wQ dK wL  MPL  wQ dL MPK wK Isoquants for Cobb-Douglas Q dK dL AK 1D D L D AK L  D D (1  D ) AK L K 1D D 1 DK  1D L 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 sQ k F ( s K , sL ) Homogeneous functions • Is Cobb-Douglas production function homogeneous? homogeneous? 1D D Q AK L 1D D 1D D 1D D 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 As AK L D E D D E A( sK ) ( sL) D E KL E s D E Q • Yes, homogeneous of degree D+E • For D+E = 1, constant returns to scale – Doubling inputs, doubles output • For D+E > 1, increasing returns to scale – Doubling inputs, more than doubles output • For D+E < 1, decreasing returns to scale – Doubling inputs, less than doubles output Properties of homogeneous functions • Partial derivatives of a homogeneous of degree k function function are homogeneous of degree k-1 Q AK L wQ 1D D 1 D AK L wL 1D D 1 0 1D D 1 D A( sK ) ( sL) D As K L 1D D wQ wL • Cobb-Douglas partial derivatives don’t change as you scale up production Isoquants for Cobb-Douglas Q dK dL AK 1D D L D AK L  D D (1  D ) AK L K 1D D 1 DK  1D L bL Euler’s theorem • Any function Q = F(K,L) has the differential dQ wQ wQ dK  dL wK wL • Euler’s Theorem: For a function homogeneous of degree degree k, AK L • • Marginal products are associated with factor prices wQ 1D D L D AK L wL In US, D | .67 wQ wQ kQ K L wK wL Let Q = GDP and Cobb-Douglas Q 1D D DQ wQ K wK (1  D )Q Chain rule y xi wy wti f ( x1 ,..., xn ) gi (t1 ,..., tm ) wf wx1 wf wxn  ...  wx1 wti wxn wti Chain rule, exercise 8.3.1 y 3 x  2 xw  w x 8 z  18 w 4z What is dy/dz? 2 2 dy dz wy wx wy ww  wx wz ww wz (6 x  2 w)8  (2 w  2 x)4 ...
View Full Document

This note was uploaded on 11/14/2010 for the course ECON Econ 326 taught by Professor Hayes during the Spring '08 term at University of Maryland Baltimore.

Ask a homework question - tutors are online