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Multivariate Calculus Professor Peter Cramton Economics 300Multivariate calculus • Calculus with single variable (univariate) • Calculus with many variables (multivariate) 12( , ,..., )ny f x x x()y f x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 onceCobb-Douglas production function • How does production change in L? • Marginal product of labor (MPL) • Partial derivatives use  instead of d 112220Q K L112210QKLLCobb-Douglas production function • How does production change in K? • Marginal product of capital (MPK) • Partial derivatives use  instead of d 112220Q K L112210QKLKCobb-Douglas production function • Note • Produce more with more labor, holding capital fixed • Produce more with more capital, holding labor fixed 112220Q K L112210 0QMPK K LK  112210 0QMPL K LL  Second-order partial derivatives • Differentiate first-order partial derivatives 31223122222255QQKLL L LQQKLK K K            112210QKLL112210QKLKSecond-order partial derivatives • Note • MPL declines with more labor, holding capital fixed • MPK declines with more capital, holding labor fixed • Diminishing marginal product of labor and capital 3122312222225050QQKLL L LQQKLK K K              Second-order cross partial derivatives • What happens to MPL when capital increases? • What happens to MPK when labor increases? • By Young’s Theorem cross-partials are equal 1122250QQKLK L K L       1122250QQKLL K L K       11222250QQKLL K K L     112210QKLL112210QKLK112220Q K LLabor demand at different levels of capital 31222250QKLL  112210 0QKLLK = K0 K = K1112220Q K LContour plot • Isoquant is 2D projection in K-L space of Cobb-Douglas production with output fixed at Q* 112220Q K LImplicit function theorem • Each isoquant is implicit function • 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 • Use implicit function theorem when can’t solve K(L) 1122*Q AK L2*()QAKLLImplicit function theorem For an implicit function defined at point with continuous partial derivatives there is a function defined in neighborhood of corresponding to such that 1( , ,..., )nF y x x k0 0 01( , ,..., )ny x x0 0 01( , ,..., ) 0,ynF y x x 1( ,..., )ny f x x001( ,..., )nxx1( , ,..., )nF y x x k0 0 010 0 010 0 010010 0 01( ,..., )( , ,..., )( , ,..., )( ,..., )( , ,..., )innxninyny f x xF y x x kF y x xf x xF y x xVerifying implicit function theorem • Cobb-Douglas production • Explicit function for isoquant • Derivative of isoquant • Slope of isoquant = marginal rate of technical substitution – Essential in determining optimal mix of production inputs 1122*Q AK L2*()QAKLL2*22()QAdK L KL KdL L L L     Implicit function theorem 0 0 010010 0 01( , ,..., )( ,..., )( , ,..., )ixninynF y x xf x xF y x x1122Q AK L112210QKLL112210QKLK112211221010QdK K L KLQdL LKLK     Implicit function theorem from differential • For multivariate function • Differential is • Holding quantity fixed (along the isoquant) • Thus ( , )Q F K LQQdQ dK dLKL0QQdQ dK dLKL  QdK MPLLQdL MPKK   Isoquants for Cobb-Douglas 1Q AK L11(1 ) 1dK AK L KdL AK L 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 ( , )ks Q F sK sLHomogeneous functions • Is Cobb-Douglas production function homogeneous? • Yes, homogeneous of degree 1 • Production function has constant returns to scale – Doubling inputs, doubles output 1Q AK L1 1 1( ) ( )A sK sL As s K L sQ       Homogeneous functions • Is production function homogeneous? • 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 ( ) ( )A sK sL As K L s Q       Q AK LProperties of homogeneous functions • Partial derivatives of a homogeneous of degree k function are homogeneous of degree k-1 • Cobb-Douglas partial derivatives don’t change as you scale up production 1Q AK L11QAK LL1 1 0 1 1( ) ( )QA sK sL As K LL      Isoquants for Cobb-Douglas 1Q AK L11(1 ) 1dK AK L KdL AK L L   K bLEuler’s theorem • Any function Q = F(K,L) has the differential • Euler’s Theorem: For a function homogeneous of degree k, • Let Q = GDP and Cobb-Douglas • Marginal products are associated with factor prices QQdQ dK dLKLQQkQ K LKL1Q AK L1QL AK L QL(1 )QKQKIn US,   .67Chain rule 1111( ,..., )( ,..., )...ni i mni i n iy f x xx g t txy f x ft x t x t         Chain rule exercise 22328 184What is dy/dz?y x xw wxzwz  (6 2 )8 (2 2 )4dy y x y wdz x z w zx w w


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UMD ECON 300 - Multivariate Calculus

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