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