Mathematical Economics: Lecture 9 Yu Ren WISE, Xiamen University October 17, 2011
Outline 1 Chapter 14: Calculus of Several Variables
New Section Chapter 14: Calculus of Several Variables
Partial Derivatives Definition: f : R n R 1 f x i (x 0 1,, x 0 n ) = lim h 0 f (x 0 1 x 0 i + h,, x 0 n ) f (x 0 1, x 0 i, x 0 n ) h
Partial Derivatives: Example Example 14.1 function f (x) = 3x 2 y 2 + 4xy 3 + 7y x (3x 2 y 2 ) = 2x 3y 2 = 6xy 2, x (4xy 3 ) = 4y 3, (7y) = 0. x partial derivative of x: x (3x 2 y 2 + 4xy 3 + 7y) = 6xy 2 + 4y 3.
Partial Derivatives: Example y (3x 2 y 2 ) = 3x 2 2y = 6x 2 y, y (4xy 3 ) = 3y 2 4x = 12xy 2, partial derivative of y: (7y) = 0. y y (3x 2 y 2 + 4xy 3 + 7y) = 6x 2 y + 12xy 2.
Economics interpretation Q = F (K, L) Marginal Product of capital (MPK ):( F/ K ) Marginal Product of labor (MPL ):( F / L)
Economics interpretation D = D(P 1, P 2, I) demand function for good 1 own Price elasticity of demand: P 1 D(P 1,P 2,I) D cross price elasticity of demand: P 2 D income elasticity of demand: I D D I P 1 D(P 1,P 2,I) P 2
Total Derivatives F (x + x, y ) F (x, y ) F x (x, y ) x F (x, y + y) F(x, y ) F y (x, y ) y F (x + x, y + y) F(x, y ) F x (x, y ) x + F y (x, y ) y
Total Derivatives F (x + x, y ) F (x, y ) F x (x, y ) x F (x, y + y) F(x, y ) F y (x, y ) y F (x + x, y + y) F(x, y ) F x (x, y ) x + F y (x, y ) y
Total Derivatives F (x + x, y ) F (x, y ) F x (x, y ) x F (x, y + y) F(x, y ) F y (x, y ) y F (x + x, y + y) F(x, y ) F x (x, y ) x + F y (x, y ) y
Total Derivatives df = F x (x, y )dx + F y (x, y )dy Example h = x 3 ln y dh = 3x 2 ln ydx + x 3 1 y dy
Total Derivatives df = F x (x, y )dx + F y (x, y )dy Example h = x 3 ln y dh = 3x 2 ln ydx + x 3 1 y dy
Jacobian df = F x 1 (x )dx 1 + + F x n (x )dx n Jacobian ( derivative of F at x) : F DF x = x 1 (x ),, F x n (x ) df = DF x dx, where dx = (dx 1, dx 2,, dx n )
Jacobian df = F x 1 (x )dx 1 + + F x n (x )dx n Jacobian ( derivative of F at x) : F DF x = x 1 (x ),, F x n (x ) df = DF x dx, where dx = (dx 1, dx 2,, dx n )
Jacobian df = F x 1 (x )dx 1 + + F x n (x )dx n Jacobian ( derivative of F at x) : F DF x = x 1 (x ),, F x n (x ) df = DF x dx, where dx = (dx 1, dx 2,, dx n )
Chain Rule A curve in R n : x(t) = (x 1 (t),, x n (t)). x i (t) are called coordinate function Example 14.5 The line segment connecting (0, 0) and (1, 1) is a curve.one possible parameterization is x(t) = t, y(t)=t, 0 t 1, Another parameterization is x(t) = t 2, y(t) = t 2, 0 t 1,
Chain Rule A curve in R n : x(t) = (x 1 (t),, x n (t)). x i (t) are called coordinate function Example 14.5 The line segment connecting (0, 0) and (1, 1) is a curve.one possible parameterization is x(t) = t, y(t)=t, 0 t 1, Another parameterization is x(t) = t 2, y(t) = t 2, 0 t 1,
Chain Rule Velocity vector (tangent vector): x (t) = (x 1 (t),, x n(t)) A curve x(t) = (x 1 (t),, x n (t)) is regular if and only if x i (t) is continuous and (x 1 (t),, x n(t)) (0,, 0) for all t. Continuously differentiable : f : R n R 1, ( f / x i ) exists and is continuous for all x i.
Chain Rule Velocity vector (tangent vector): x (t) = (x 1 (t),, x n(t)) A curve x(t) = (x 1 (t),, x n (t)) is regular if and only if x i (t) is continuous and (x 1 (t),, x n(t)) (0,, 0) for all t. Continuously differentiable : f : R n R 1, ( f / x i ) exists and is continuous for all x i.
Chain Rule Velocity vector (tangent vector): x (t) = (x 1 (t),, x n(t)) A curve x(t) = (x 1 (t),, x n (t)) is regular if and only if x i (t) is continuous and (x 1 (t),, x n(t)) (0,, 0) for all t. Continuously differentiable : f : R n R 1, ( f / x i ) exists and is continuous for all x i.
Chain Rule If X(t) = (x 1 (t),, x n (t)) is a C 1 curve on an interval about t 0 and f is a C 1 function on a ball about X(t 0 ), then g(t) f (x 1 (t),, x n (t)) is a C 1 function at t 0 and dg dt (t 0) = f (X(t 0 ))x x 1(t 0 )+ + f (X(t 0 ))x 1 x n(t 0 ) n
Chain Rule:Example 14.7 Example 14.7 f (x, y) = x 2 + y 2 let x(t)=t, y(t) = t g(t) = f (x(t), y(t)) g(t) = 2t 2 g (t) = 4t. g (1) = 4
Chain Rule: Example 14.7 the same with f x = 2x, f y = 2y, g (1) = f f (1, 1) 1 + (1, 1) 1 x y = 2 1 + 2 1 = 4.
Directional Derivative and Gradients X = X + tv g(t) = F(X +tv ) = F (x 1 + tv 1,, x n + tv n ) g (0) = F x 1 (X )v 1 + + F ( F x 1 (X ),, F x n (X ) ) x n (X )v n v 1. v n = DF x V the derivative of F at x in the direction of V.
Directional Derivative and Gradients Gradient: F x = F x 1 (X ). F x n (X )
Theorem Theorem 14.2 F(x) points at x into the direction in which F increases most rapidly.
Theorem Example 14.11 Consider again the production function: Q = F (K, L) = 4K 3/4 L 1/4. Current input bundles is (10000, 625). If we want to know in what proportions we should add K and L to increase production most rapidly, we compute the gradient vector ( ) 1.5 F(10, 000, 625) = 8
Explicit Function F = (f 1,, f m ) : R n R m f 1 (x + x) f 1 (x ) f 1 x 1 (x ) x 1 + + f 1 x n (x ) x n f 2 (x + x) f 2 (x ) f 2 x 1 (x ) x 1 + + f 2 x n (x ) x n f m (x + x) f m (x ) f m x 1 (x ) x 1 + + f m x n (x ) x n
Explicit Function F (X + X) F(X ) f 1 x 1 (X f ) 1 x n (X )... f m x 1 (X f ) m x n (X ) x 1. x n
Jacobian Derivative Jacobian Derivative of F at X f 1 x 1 (X f ) 1 x n (X ) DF x =... f m x 1 (X ) f m x n (X ) Example 14.13
Theorems Theorem 14.3 F : R n R m and a : R 1 R n, g(t) = F(a(t)), then g i (t) = Df i(a(t))a (t) and g (t) = DF (a(t))a (t)
Theorems Theorem 14.4 F : R n R m and A : R s R n, H = F A, then DH(s ) = DF(x )DA(s )
Theorems Theorem 14.5 2 f x i x j (x) = 2 f x j x i (x)
Higher Order derivatives & Hessian C 1 : continuously differentiable, f is continuous C 2 : twice continuously differentiable, f is continuous f : R n R 1 Hessian or Hessian Matrix: D 2 F x = 2 f (x 1 ) 2 (X ) 2 f x n x 1 (X )... 2 f x 1 x n (X ) 2 f (x n (X ) ) 2
Example 14.15 Example 14.15 Q = 4K 3/4 L 1/4. Q K = 3K 1/4 L 1/4, Q L = K 3/4 L 3/4
Example 14.15 then 2 Q L K = Q L ( Q K ) = L (3K 1/4 L 1/4 ) = 3 4 K 1/4 L 3/4 2 Q K L = Q K ( Q L ) = K (K 3/4 L 3/4 ) = 3 4 K 1/4 L 3/4 2 Q L = Q 2 L ( Q L ) = L (K 3/4 L 3/4 ) = 3 4 K 3/4 L 7/4 2 Q K = Q 2 K ( Q K ) = K (3K 1/4 L 1/4 ) = 3 4 K 5/4 L 1/4