University of Nottingham

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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
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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
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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)
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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
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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
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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)
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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
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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
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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
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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
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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
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Multivariate Calculus
Example f(x1,x2) 4x12x2 - 2x22 - x1 - 1
D x1 ?0,1,x2 ?0,1
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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
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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
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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
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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
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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
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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
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Multivariate Calculus
Example
max f(x,y) x2 - y
s.t. x2 y2 ? 1
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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
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Multivariate Calculus
Example
max f(x,y) x2 - y
s.t. x2 y2 1 x y ? 0
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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)
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Multivariate Calculus
Comparative statics
f(x a) 0
F1(x1 , x2 a) 0 F2(x1 , x2 a) 0
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Multivariate Calculus
The envelope theorem
max f(x1 , x2 a) 0 s.t. h (x1 , x2 a) 0
Lagrangean
FOC
Derivative dL / da
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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
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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