Multivariate Calculus

Professor Peter Cramton

Economics 300

Multivariate calculus

• Calculus with single variable (univariate)

• Calculus with many variables (multivariate)

1 2( , ,..., )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 once

Cobb-Douglas production function

• How does production change in L?

• Marginal product of labor (MPL)

• Partial derivatives use instead of d

1 12 220Q K L

1 12 210

QK L

L

Cobb-Douglas production function

• How does production change in K?

• Marginal product of capital (MPK)

• Partial derivatives use instead of d

1 12 220Q K L

1 12 210

QK L

K

Cobb-Douglas production function

• Note

• Produce more with more labor, holding capital fixed

• Produce more with more capital, holding labor fixed

1 12 220Q K L

1 12 210 0

QMPK K L

K

1 12 210 0

QMPL K L

L

Second-order partial derivatives

• Differentiate first-order partial derivatives

312 2

3 12 2

2

2

2

2

5

5

Q QK L

L L L

Q QK L

K K K

1 12 210

QK L

L

1 12 210

QK L

K

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

312 2

3 12 2

2

2

2

2

5 0

5 0

Q QK L

L L L

Q QK L

K 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

1 12 2

2

5 0Q Q

K LK L K L

1 12 2

2

5 0Q Q

K LL K L K

1 12 2

2 2

5 0Q Q

K LL K K L

1 12 210

QK L

L

1 12 210

QK L

K

1 12 220Q K L

Labor demand at different levels of capital

312 2

2

25 0

QK L

L

1 12 210 0

QK L

L

K = K0 K = K1

1 12 220Q K L

Contour plot • Isoquant is 2D projection in K-L space of Cobb-

Douglas production with output fixed at Q*

1 12 220Q 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)

1 12 2*Q AK L

2*

( )

Q

AK L

L

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 k0 0 0

1( , ,..., )ny x x0 0 0

1( , ,..., ) 0,y nF y x x

1( ,..., )ny f x x0 0

1( ,..., )nx x

1( , ,..., )nF y x x k0 0 0

1

0 0 0

1

0 0 0

10 0

1 0 0 0

1

( ,..., )

( , ,..., )

( , ,..., )( ,..., )

( , ,..., )

i

n

n

x n

i n

y n

y f x x

F y x x k

F y x xf x x

F 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

1 12 2*Q AK L

2*

( )

Q

AK L

L

2*

2 2

( )

Q

AdK L KL K

dL L L L

Implicit function theorem 0 0 0

10 0

1 0 0 0

1

( , ,..., )( ,..., )

( , ,..., )

ix n

i n

y n

F y x xf x x

F y x x

1 12 2Q AK L

1 12 210

QK L

L

1 12 210

QK L

K

1 12 2

1 12 2

10

10

Q

dK K L KLQdL LK LK

Implicit function theorem from differential

• For multivariate function

• Differential is

• Holding quantity fixed (along the isoquant)

• Thus

( , )Q F K L

Q QdQ dK dL

K L

0Q Q

dQ dK dLK L

Q

dK MPLLQdL MPK

K

Isoquants for Cobb-Douglas

1Q AK L

1 1

(1 ) 1

dK AK L K

dL 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

1 1QAK L

L

1 1 0 1 1( ) ( )

QA sK sL As K L

L

Isoquants for Cobb-Douglas

1Q AK L

1 1

(1 ) 1

dK AK L K

dL 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

Q QdQ dK dL

K L

Q QkQ K L

K L

1Q AK L

1QL AK L Q

L

(1 )

QK Q

K

In US, .67

Chain rule

1

1

1

1

( ,..., )

( ,..., )

...

n

i i m

n

i i n i

y f x x

x g t t

xy f x f

t x t x t

Chain rule exercise 2 23 2

8 18

4

What is dy/dz?

y x xw w

x z

w z

(6 2 )8 (2 2 )4

dy y x y w

dz x z w z

x w w x