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School of Economics

University of Nottingham

Pre-sessional Mathematics Masters (MSc)

Dr Maria Montero Dr Alex Possajennikov

Topic 3 Multivariate Calculus

Multivariate Calculus

Functions of n variables

y f(x1, ,xn)

F Rn ? R

Example f(x,y) x2 - y2

In economics

Utility function u(x,y)

Production function y(K,L) K?L1-?

Limits

limx ? a f(x) b for any ? gt 0, there exists ? gt

0 such that f(x) - b lt ? for any x such that

x - a lt ?

x (x1, ,xn), a (a1, ,an)

Continuity

f(x) is continuous at x0 if limx ? x0 f(x) f(x0)

f(x) is continuous on D if it is continuous at

every point in D

Multivariate Calculus

Differentiation

xi0 ?xi xj0

Consider function f(x1, ,xn ). Fix x0 (x10,

,xn0).

The partial derivative of f with respect to xi

?f/?xi gives the slope of the tangent line to the

graph of the function in the hyperplane parallel

to xi axis and z

In economics

Marginal product of labour ?Y/?L

Marginal utility ?u/?xi

Partial derivatives are functions

If all partial derivatives of f(x) exist and

continuous, then f(x) is continuously

differentiable (C1)

Multivariate Calculus

Rules of differentiation

The same as for ordinary differentiation

The chain rule

f(x1, ,xn ) and xi(t1, ,tm )

Total derivative (gradient)

Approximation

f(x0 ?x) ? f(x0) Df(x0)?x

Total differential

Multivariate Calculus

Second order derivatives

?f/?xi(x1, ,xn ) is a function of n variables

Hessian

cross-partials

Example f(x) 4x12x2 - 2x22 - x1 - 1

Multivariate Calculus

Concavity and convexity

Convex set

D is a convex set if for any x,y ? D, (1-?)x?y ?

D, for any ? ? 0,1

Convex set

Non-convex set

f(x) is convex on convex D if f((1 - ?)x1 ?x2)

? (1 - ?)f(x1) ?f(x2)

f(x) is concave on convex D if f((1 - ?)x1 ?x2)

? (1 - ?)f(x1) ?f(x2)

Multivariate Calculus

Theorem f ? C2. f is concave on convex D when

D2f is negative semidefinite on D f is convex on

convex D when D2f is positive semidefinite on D

Example

f(x,y) x1/3y1/3 on D x gt 0, y gt 0

Multivariate Calculus

Implicit function

y f(x1,,xn) explicit function

F(x1,,xn,y) 0 implicit function

It may be difficult to solve F(x1,,xn,y) 0 to

get y f(x1,,xn)

Example y3 - xy ln x 0

Suppose we are interested in ?y/?xi

Multivariate Calculus

n 2

F(x,y) 0

Slopes of the level curves of the graph

Example utility function u(x,y) x2/3y1/3

Slope of indifference curve at x 1, y 1

Multivariate Calculus

Optimisation

x0 (x10, , xn0) is the global maximum of f(x)

if f(x0) ? f(x) for all x in D

x0 (x10, , xn0) is the local maximum of f(x)

if f(x0) ? f(x) for all x in small neighbourhood

of x0

Theorem f ? C1. If x0 is an interior local

maximum or minimum, then ?f/?xi(x0) 0 for all i.

Theorem f ? C2. ?f/?xi(x0) 0 for all i. If

D2f(x0) is negative definite, then x0 is a local

maximum. If D2f(x0) is positive definite, then x0

is a local minimum.

If D2f(x0) is indefinite, then x0 is neither

local maximum nor local minimum

Multivariate Calculus

Theorem f ? C2. ?f/?xi(x0) 0 for all i. If f is

concave (D2f is negative semidefinite) on D, then

x0 is a global maximum. If f is convex (D2f is

positive semidefinite) on D, then x0 is a global

minimum.

General approach to maximisation of C1 function

without constraints

1. Find all critical points of f(x) by solving

?f/?xi(x) 0 for all i.

2. Find values of f(x) at these points

3. Find values of f(x) at the boundary of D

4. Choose the highest value from 2 and 3

There may be no maximum

Multivariate Calculus

Example f(x1,x2) 4x12x2 - 2x22 - x1 - 1

D x1 ?0,1,x2 ?0,1

Multivariate Calculus

Constrained optimisation

max f(x1,,xn)

s.t. gi(x1,,xn) ? 0 i1,,m hj(x1,,xn)

0 j1,,k

Examples

utility maximisation subject to budget constraint

max u(x1,x2) s.t. p1x1 p2x2 ? I

profit maximisation subject to production

possibilities

welfare maximisation subject to individuals

reaction

Multivariate Calculus

Substitution method

max f(x1,x2)

s.t. h(x1,x2) 0

Solve h(x1,x2) 0 for x2 H(x1)

Substitute x2 H(x1) to get the problem max

f(x1,H(x1))

When there are inequality constraints g(x1,x2) ?

0, sometimes argument can be made that at maximum

g(x1,x2) 0

max f(x,y) x1/2y1/2 s.t. g(x,y) x y - 4 ? 0

Example

f is increasing in x and y

At maximum, g(x,y) 0

g is increasing in x and y

Multivariate Calculus

Lagrangean method

max f(x1,,xn)

s.t. gi(x1,,xn) ? 0 i1,,m hj(x1,,xn)

0 j1,,k

Lagrangean

Maximise the Lagrangean

?i, ?j are Lagrangean multipliers

Idea represent common tangent hyperplane of the

set defined by the constraints and the level

curves of function f

At maximum there is no way to move along

constraints without decreasing the value of f

Interpretation of the multipliers shadow price

of the constraint

Multivariate Calculus

Equality constraints

max f(x1,,xn)

s.t. hj(x1,,xn) 0 j1,,k

Lagrangean

Assume k ? n

Rows of Dh as vectors ?hi / ?x

Dh has rank m

Multivariate Calculus

Theorem f ,hj? C1. If x0 is a local maximum or

minimum and if rank Dh(x0) k then there exist

?j0 such that

Example

max f(x,y) x2 - y

s.t. x2 y2 1

Multivariate Calculus

Inequality constraints

max f(x1,,xn)

s.t. gi(x1,,xn) ? 0 i1,,m

Binding constraint gi(x0) 0

Non-binding constraint gi(x0) gt 0

Lagrangean

Theorem f ,hj? C1. x0 is a local maximum. g1(x0)

0,, gk(x0) 0 and gk1(x0) gt 0,, gm(x0) gt

0. rank Dg1,..,k(x0) k. Then there exist ?i0

such that

gi(x0) ? 0 for all i

?i0 gi(x0) 0 for all i

Complementary slackness condition

?i0 ? 0 for all i

At minimum ?i0 ? 0

Multivariate Calculus

Example

max f(x,y) x2 - y

s.t. x2 y2 ? 1

Multivariate Calculus

Mixed constraints

max f(x1,,xn)

s.t. gi(x1,,xn) ? 0 i1,,m hj(x1,,xn)

0 j1,,k

Lagrangean

Theorem f ,hj? C1. x0 is a local

maximum. Technical condition on rank of a

Jacobean. Then there exist ?i0,?j0 such that

hj(x0) 0 for all j

gi(x0) ? 0 for all i

?i0 gi(x0) 0 for all i

?i0 ? 0 for all i

Multivariate Calculus

Example

max f(x,y) x2 - y

s.t. x2 y2 1 x y ? 0

Multivariate Calculus

Sufficient conditions for local maximum to be

global maximum

L is concave

Can be written using concavity of f and h

General approach to maximisation of C1 function

with constraints

1. Form the Lagrangean

2. Find all critical points by solving the

appopriate system of equations and inequalities

3. Find values of f(x) at these points

4. Choose the highest value from Step 3

(boundary of D is taken care of by some of the

equations in the system)

Multivariate Calculus

Comparative statics

f(x a) 0

F1(x1 , x2 a) 0 F2(x1 , x2 a) 0

Multivariate Calculus

The envelope theorem

max f(x1 , x2 a) 0 s.t. h (x1 , x2 a) 0

Lagrangean

FOC

Derivative dL / da

Multivariate Calculus

The envelope theorem

max f(x1 , x2 a) 0 s.t. h (x1 , x2 a) 0

Let x10, x20 be a solution

FOC

Value function

Derivative dV / da

Derivative dh / da

Multivariate Calculus

The envelope theorem

max f(x1,,xn a)

s.t. hj(x1,,xna) 0 j1,,k

Suppose x0(a) is the solution

Technical conditions on smoothness of the

solution

V

V(a)

V(2)

Example

max u(x1,x2) s.t. p1x1 p2x2 I

V(1)

a,x