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 CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 Secondorder partial derivatives
• Differentiate firstorder 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 Secondorder 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? Secondorder 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 crosspartials 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 KL 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
• CobbDouglas 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 CobbDouglas
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 CobbDouglas 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 k1 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 • CobbDouglas partial derivatives don’t change as you scale up production Isoquants for CobbDouglas
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 CobbDouglas 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 ...
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 Spring '08
 Hayes
 Derivative, MPl, wL, wK wL

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