Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
1
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Optimization
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
BEEM103 Optimization Techniques for Economists
Multivariate Functions
Isoquants
Dieter Balkenborg
Department of Economics, University of Exeter
Week 2
3
4
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example 3: Price Discrimination
Multivariate Functions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Functions in two variables
A function
z = f (x, y )
or simply
z (x, y )
in two independent variables with one dependent variable assigns
to each pair (x, y ) of (decimal) numbers from a certain domain D
in the two-dimensional plane a number z = f (x, y ).
x and y are hereby the independent variables
z is the dependent variable.
5
Second order conditions
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Outline
1
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
The graph of f is the surface in 3-dimensional space consisting of
all points (x, y , f (x, y )) with (x, y ) in D.
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
z = f (x, y ) = x 3 − 3x 2 − y 2
Isoquants
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
5
2 0
1
00
-1
y -5 -2
z
-1
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Outline
1
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
x
Exercise: Evaluate z = f (2, 1), z = f (3, 0), z = f (4, −4),
z = f (4, 4)
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Q=
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
√
√
1
1
6
K L = K 6 L2
capital K ≥ 0, labour L ≥ 0, output Q ≥ 0
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
5
3
2
1
Example: production function
Isoquants
2
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
z
6
4
2
0
20
10
y
0 0
5
10
15
x
Second order conditions
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
20
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Outline
1
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example: profit function
Assume that the firm is a price taker in the product market and in
both factor markets.
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
P is the price of output
Isoquants
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example: profit function
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example: profit function
Assume that the firm is a price taker in the product market and in
both factor markets.
Assume that the firm is a price taker in the product market and in
both factor markets.
P is the price of output
P is the price of output
r the interest rate (= the price of capital)
r the interest rate (= the price of capital)
w the wage rate (= the price of labour)
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example: profit function
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
P = 12, r = 1, w = 3:
1
Assume that the firm is a price taker in the product market and in
both factor markets.
1
20
r the interest rate (= the price of capital)
10
w the wage rate (= the price of labour)
z
total profit of this firm:
0
-10
0
Π (K , L) = TR − TC
= PQ − rK − wL
1
1
= PK 6 L 2 − rK − wL
10
x
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
20
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
0
20
10
y
Multivariate Functions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Outline
P = 12, r = 1, w = 3:
1
1
Π (K , L) = PK 6 L 2 − rK − wL
1
1
1
= 12K 6 L 2 − K − 3L
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Isoquants
20
10
z
1
= 12K 6 L 2 − K − 3L
P is the price of output
Balkenborg
1
Π (K , L) = PK 6 L 2 − rK − wL
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
0
-10
0
10
x
20
y
Profits is maximized at K = L = 8.
Balkenborg
20
10
0
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Level Curves
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Level Curves
The level curve of the function z = f (x, y ) for the level c is
the solution set to the equation
The level curve of the function z = f (x, y ) for the level c is
the solution set to the equation
f (x, y ) = c
f (x, y ) = c
where c is a given constant.
where c is a given constant.
Geometrically, a level curve is obtained by intersecting the
graph of f with a horizontal plane z = c and then projecting
into the (x, y )-plane. This is illustrated on the next page for
the cubic polynomial discussed above:
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Multivariate Functions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Isoquants
In the case of a production function the level curves are called
isoquants. An isoquant shows for a given output level
capital-labour combinations which yield the same output.
2
z
y
25
10
0
-1-5 0
-1
1
1
3
2
y
x
-2
0
-1 0
z-1
-2
3
2
6
x
z
20
0
20
10
0 0
compare: topographic map
Balkenborg
1
y
Multivariate Functions
10
20
x
y
z
Balkenborg
0
Multivariate Functions
0
10
20
x
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Finally, the linear function
z = 3x + 4y
has the graph and the level curves:
Exercise: Describe the isoquant of the production function
Q = KL
4
z
30
20
10
0
y
0
2
4
4
2
0
x
for the quantity Q = 4.
Exercise: Describe the isoquant of the production function
√
Q = KL
2
0
0
y
2
z4
x
for the quantity Q = 2.
The level curves of a linear function form a family of parallel lines:
c
3
c = 3x + 4y
4y = c − 3x
y= − x
4 4
c
3
slope − 4 , variable intercept 4 .
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Remark: The exercises illustrate the following general principle: If
h (z ) is an increasing (or decreasing) function in one variable, then
the composite function h (f (x, y )) has the same level curves as
the given function f (x, y ) . However, they correspond to different
levels.
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
z
y
20
410
20
0 0
4
z4
2
4
y
x
Q = KL
Q=
Balkenborg
√
2
20
0 0
2
4
The lecture should enable you for instance to calculate the
marginal product of labour.
x
KL
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Objectives for the week
Objectives for the week
Functions in two independent variables.
Level curves ←→ indifference curves or isoquants
Level curves ←→ indifference curves or isoquants
The lecture should enable you for instance to calculate the
marginal product of labour.
The lecture should enable you for instance to calculate the
marginal product of labour.
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Outline
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Exercise: What is the derivative of
z (x ) = a3 x 2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Partial differentiation ←→ partial analysis in economics
Partial Derivatives: A Basic Example
Isoquants
2
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Functions in two independent variables.
Balkenborg
1
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
with respect to x when a is a given constant?
Exercise: What is the derivative of
z (y ) = y 3 b 2
with respect to y when b is a given constant?
Second order conditions
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Outline
1
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: Notation
Consider function z = f (x, y ). Fix y = y0 , vary only x:
z = F (x ) = f (x, y0 ).
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
The derivative of this function F (x ) is called the partial
derivative of f with respect to x and denoted by
Isoquants
2
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
∂z
dF
=
∂x |y =y0
dx
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: Notation
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: Notation
Consider function z = f (x, y ). Fix y = y0 , vary only x:
z = F (x ) = f (x, y0 ).
Consider function z = f (x, y ). Fix y = y0 , vary only x:
z = F (x ) = f (x, y0 ).
The derivative of this function F (x ) is called the partial
derivative of f with respect to x and denoted by
The derivative of this function F (x ) is called the partial
derivative of f with respect to x and denoted by
∂z
dF
=
∂x |y =y0
dx
∂z
dF
=
∂x |y =y0
dx
Notation: “d”=“dee”, “δ”=“delta”, “∂”=“del”
Notation: “d”=“dee”, “δ”=“delta”, “∂”=“del”
It suffices to think of y and all expressions containing only y
as exogenously fixed constants. We can then use the familiar
rules for differentiating functions in one variable in order to
∂z
obtain ∂x
.
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: Notation continued
Other common notations for partial derivatives are
fx , fy orfx′ , fy′ .
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: Notation continued
∂f ∂f
∂x , ∂y
or
Other common notations for partial derivatives are
fx , fy orfx′ , fy′ .
∂f ∂f
∂x , ∂y
or
Consider again function z = f (x, y ). Fix y = y0 , vary only x:
z = F (x ) = f (x, y0 ). Then evaluate at x = x0
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: Notation continued
Other common notations for partial derivatives are
fx , fy orfx′ , fy′ .
∂f ∂f
∂x , ∂y
or
The derivative of this function F (x ) at x = x0 is called
the partial derivative of f with respect to x and denoted
by
∂z
dF
dF
=
=
( x0 )
∂x |x =x0 ,y =y0
dx |x =x0
dx
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: The example continued
Consider again function z = f (x, y ). Fix y = y0 , vary only x:
z = F (x ) = f (x, y0 ). Then evaluate at x = x0
Balkenborg
Multivariate Functions
Example: Let
z (x, y ) = y 3 x 2
Then
∂z
∂x
∂z
∂y
= y 3 2x = 2y 3 x
= 3y 2 x 2
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Outline
1
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: A second example
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Example: Let
z = x 3 + x 2y 2 + y 4.
Isoquants
2
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Setting e.g. y = y0 = 1 we obtain z = x 3 + x 2 + 1 and hence
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
∂z
= 3x 2 + 2x
∂x |y =1
Evaluating at x = x0 = 1 we then obtain:
∂z
=5
∂x |x =1,y =1
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Partial derivatives: A second example
For fixed, but arbitrary, y we obtain
∂z
= 2x 2 y + 4y 3
∂y
Balkenborg
Multivariate Functions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Outline
1
∂z
= 3x 2 + 2xy 2
∂x
as follows: We can differentiate the sum x 3 +x 2 y 2 +y 4 with
respect to x term-by-term. Differentiating x 3 yields 3x 2 ,
differentiating x 2 y 2 yields 2xy 2 because we think now of y 2 as a
d (ax 2 )
constant and dx = 2ax holds for any constant a. Finally, the
derivative of any constant term is zero, so the derivative of y 4 with
respect to x is zero.
Similarly considering x as fixed and y variable we obtain
Multivariate Functions
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Isoquants
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
The Marginal Products of Labour and Capital
∂q
and ∂q
Example: The partial derivatives ∂K
∂L of a production
function q = f (K , L) are called the marginal product of capital
and (respectively) labour. They describe approximately by how
much output increases if the input of capital (respectively labour)
is increased by a small unit.
1
1
1
Fix K = 64, then q = K 6 L 2 = 2L 2 which has the graph
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
The Marginal Products
This graph is obtained from the graph of the function in two
variables by intersecting the latter with a vertical plane parallel to
L-q-axes.
z
Q4
6
4
2
0
20
3
y
10
0 0
5
10
15
20
x
2
1
0
0
1
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
2
3
4
5
x
Multivariate Functions
∂q
The partial derivatives ∂K
and ∂q
∂L describe geometrically the slope
of the function in the K - and, respectively, the L- direction.
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
Diminishing productivity of labour:
The more labour is used, the less is the increase in output when
one more unit of labour is employed. Algebraically:
√
1 1 −1
1 6K
∂q
> 0,
= K 6L 2 = √
∂L
2
2 2L
√
∂2 q
1 1 −3
∂ ∂q
1 6K
6
2
=− K L =− √
< 0,
=
∂L2
∂L ∂L
4
4 2 L3
Multivariate Functions
Optimization
“Since the fabric of the universe is most perfect, and
is the work of a most perfect creator, nothing whatsoever
takes place in the universe in which some form of
maximum or minimum does not appear.”
Leonhard Euler, 1744
Exercise: Find the partial derivatives of
z = x 2 + 2x y 3 − y 2 + 10x + 3y
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Objectives
Objectives
Subject: Optimization of multivariate functions
Subject: Optimization of multivariate functions
Two basic types of problems:
Balkenborg
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Objectives
Objectives
Subject: Optimization of multivariate functions
Subject: Optimization of multivariate functions
Two basic types of problems:
Two basic types of problems:
1
unconstrained (e.g. profit maximization)
1
2
Balkenborg
Multivariate Functions
unconstrained (e.g. profit maximization)
constrained
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
The first
Example
Example
Example
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Unconstrained Optimization
-4
-2
2
4
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Unconstrained Optimization
-2
2
0
0 0
y -10
-4
Objective: Find (absolute) maximum of function
4
x
z = f (x, y )
i.e., find a pair (x ∗ , y ∗ ) such that
-20
z
f (x ∗ , y ∗ ) ≥ f (x, y )
-30
holds for all pairs (x, y ).
-40
Hereby both pairs of numbers
(x ∗ , y ∗ ) and (x, y ) must be in the domain of
the function.
-50
For an absolute minimum require
f (x ∗ , y ∗ ) ≤ f (x, y ).
f (x, y ) is called the “objective function”.
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
The first
Example
Example
Example
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Example
1
z = f (x, y ) = − (x − 2)2 − (y − 3)2
order conditions
1
2: Maximizing profits
3: Price Discrimination
2
4
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
4
y
z
0 0
2
x
Balkenborg
Multivariate Functions
2
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Isoquants
has a maximum at (x ∗ , y ∗ ) = (2, 3) .
z
The first
Example
Example
Example
Outline
The function
0
-50 0 2
-10
-15 x
Multivariate Functions
4
y
4
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
The first
Example
Example
Example
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
First order conditions
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Outline
1
The following must hold: freeze the variable y at the optimal value
y ∗ , vary only x then the function in one variable
∗
F (x ) = f (x, y )
must have maximum at x ∗ :
order conditions
dF
dx
∂z
=0
∂x |x =x ∗ ,y =y ∗
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Isoquants
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
(x ∗ ) = 0. Thus we obtain the first
∂z
=0
∂y |x =x ∗ ,y =y ∗
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Multivariate Functions
The first
Example
Example
Example
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Example 1
Multivariate Functions
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Outline
1
z = f (x, y ) = − (x − 2)2 − (y − 3)2
∂z
∂z
= −2 (x − 2) × (+1) = 0
= −2 (y − 3) × (+1) = 0
∂x
∂y
y∗ = 3
x∗ = 2
The maximum (at least the only critical or stationary point) is at
(x ∗ , y ∗ ) = (2, 3)
Balkenborg
Multivariate Functions
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Isoquants
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
The first
Example
Example
Example
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Maximizing profits
Production function Q (K , L).
r interest rate
w wage rate
P price of output
profit
Π (K , L) = PQ (K , L) − rK − wL.
Intuition: Suppose P ∂Q
∂K − r > 0. By using one unit of capital
more the firm could produce ∂Q
∂K units of output more. The
∂Q
revenue would increase by P ∂K , the cost by r and so profit would
increase. Thus we cannot have a profit optimum. If P ∂Q
∂K − r < 0
it would symmetrically pay to reduce capital input. Hence
P ∂Q
∂K − r = 0 must hold in optimum.
FOC for profit maximum:
∂Π
∂K
∂Π
∂L
∂Q
−r = 0
∂K
∂Q
= P
−w = 0
∂L
= P
Balkenborg
(1)
(2)
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
The first
Example
Example
Example
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Multivariate Functions
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Rewrite FOC as
∂Q
∂K
∂Q
∂L
=
=
r
P
w
P
(3)
(4)
Division yields:
dL
∂Q
MRS = −
=
dK
∂K
∂Q
r.w
r
=
=
∂L
P P
w
(5)
.
∂Q
Why? If the firm uses one unit of capital less and ∂Q
∂K
∂L units of
labour more, output remains (approximately)
the same. The firm
.
∂Q
∂Q
would save r on capital and spend ∂K ∂L × w on more labour
while still
. making the same revenue. Profit would increase unless
∂Q
r ≤ ∂K ∂Q
∂L × w . As symmetric argument interchanging the role
.
∂Q
of capital and labour shows that r ≥ ∂Q
∂K
∂L × w must hold if
the firm optimizes profits.
The marginal rate of substitution must equal the ratio of the input
prices!
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
The first
Example
Example
Example
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Example
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Example
1
P = 12, r = 1 and w = 3; FOC:
1
Q (K , L) = K 6 L 2
1
1 1
1 1 1
∂Q
∂Q
= K 6 −1 L 2
= K 6 L 2 −1
∂K
6
∂L
2
∂Q
∂Q
1 −5 1
1 1 1
= K 6 L2
= K 6 L− 2
∂K
6
∂L
2
1 −5 1
K 6 L2
6
5
1
2K − 6 L 2
1L
3K
FOC:
1 −5 1
K 6 L2
6 ∂Q
∂Q
∂K
∂L
∂Q
∂Q
∂K
∂L
=
=
=
1 1 −1
r
w
K 6L 2 =
P
2
P
r
1 − 5 − 1 12 −(− 12 )
=
K 6 6L
3
w
1L
r
=
3K
w
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Substituting L = K into (*) we get
5
1
2K − 6 K 2
Q∗
Π∗
2
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
1
1
= 2K − 6 = 2K − 3 = 1 =⇒ 2 = K 3 =⇒ K ∗ = L∗ = 23 = 8
1
1
1
4
2
1
= K 6 L 2 = 8 6 + 2 = 8 6 = 8 3 = 22 = 4
= 12Q ∗ − 1K ∗ − 3L∗ = 48 − 8 − 24 = 16
Multivariate Functions
The first
Example
Example
Example
1 1 −1
1
3
K 6L 2 =
12
2
12
1
− 12
6
= 1 (*)
2K L = 1
1
=
=⇒ K = L
3
=
Multivariate Functions
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Outline
1
20
Functions in two variables
Example: The cubic polynomial
Example: production function
Example: profit function.
Level Curves
Isoquants
10
z
2
Partial derivatives
A Basic Example
Notation
A Second Example
The Marginal Products of Labour and Capital
3
Optimization
4
Unconstrained Optimization
The first order conditions
Example 1
Example 2: Maximizing profits
Example 3: Price Discrimination
5
Second order conditions
0
-10
0
5
10
15
x
20
0
Balkenborg
5
15
10
y
Multivariate Functions
20
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
The first
Example
Example
Example
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Example 3: Price Discrimination
The first
Example
Example
Example
order conditions
1
2: Maximizing profits
3: Price Discrimination
Solution
Total revenue in country A:
Q2
A monopolist with total cost function TC (Q ) =
sells his
product in two different countries. When he sells QA units of the
good in country A he will obtain the price
TRA = PA QA = (22 − 3QA ) QA
Total revenue in country B:
PA = 22 − 3QA
TRB = PB QB = (34 − 4QB ) QB
for each unit. When he sells QB units of the good in country B he
obtains the price
PB = 34 − 4QB .
Total production costs are:
How much should the monopolist sell in the two countries in order
to maximize profits?
Profit:
TC = (QA + QB )2
Π (QA , QB ) = (22 − 3QA ) QA + (34 − 4QB ) QB − (QA + QB )2
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
The first
Example
Example
Example
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
order conditions
1
2: Maximizing profits
3: Price Discrimination
Second order conditions: A peak
Profit:
Π (QA , QB ) = (22 − 3QA ) QA + (34 − 4QB ) QB − (QA + QB )2
critical point of function z = f (x, y ) = −x 2 − y 2 : (0, 0)
FOC:
∂Π
∂QA
∂Π
∂QB
= −3QA + (22 − 3QA ) − 2 (QA + QB ) = 22 − 8QA − 2QB = 0
0
-4
-2
x
= −4QB + (34 − 4QB ) − 2 (QA + QB ) = 34 − 2QA − 10QB
= 0
-4
0
y
2
4
-10
z
-20
or
8QA + 2QB
2QA + 10QB
= 22
= 34.
(6)
-30
linear simultaneous system
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
4
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Second order conditions: A trough
Second order conditions: A saddle point
critical point of function z = f (x, y ) = +x 2 + y 2 : (0, 0)
critical point of function z = f (x, y ) = +x 2 − y 2 : (0, 0)
100
30
50
20
-10
z
10
10
-4
-2
y
-4
-2 0
00
2
x
Balkenborg
z-5
5
2
0
00
-50
-5
5
x
-10
10
-100
4
y4
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Second order conditions: A monkey saddle
The Hessian matrix
critical point of function z = f (x, y ) = yx 2 − y 3 : (0, 0)
"
Balkenborg
Multivariate Functions
This and more complex possibilities will be ignored.
∂2 z
∂x 2
∂2 z
∂x ∂y
Balkenborg
∂2 z
∂y ∂x
∂2 z
∂y 2
#
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Theorem
Theorem
Suppose that the function z = f (x, y ) has a critical point at
(x ∗ , y ∗ ). If the determinant of the Hessian det H =
2
2 2
∂z
∂2 z ∂ z
∂2 z ∂2 z
∂2 z ∂2 z
∂2 z ∂2 z
∂x 2 ∂y ∂x ∂2 z
∂2 z = ∂x 2 ∂y 2 − ∂x ∂y ∂y ∂x = ∂x 2 ∂y 2 − ∂x ∂y
∂x ∂y ∂y
2
Suppose that the function z = f (x, y ) has a critical point at
(x ∗ , y ∗ ). If the determinant of the Hessian det H =
2
2 2
∂z
∂2 z ∂ z
∂2 z ∂2 z
∂2 z ∂2 z
∂2 z ∂2 z
∂x 2 ∂y ∂x ∂2 z
∂2 z = ∂x 2 ∂y 2 − ∂x ∂y ∂y ∂x = ∂x 2 ∂y 2 − ∂x ∂y
∂x ∂y ∂y
2
is negative at (x ∗ , y ∗ ) then (x ∗ , y ∗ ) is a saddle point.
If this determinant is positive at (x ∗ , y ∗ ) then (x ∗ , y ∗ ) is a peak or
∂2 z
∂2 z
are the
a trough. In this case the signs of ∂x
2 |(x ∗ ,y ∗ ) and ∂y 2
∗ ∗
|(x ,y )
same. If both signs are positive, then (x ∗ , y ∗ ) is a trough. If both
signs are negative, then (x ∗ , y ∗ ) is a peak.
Nothing can be said if the determinant is zero.
Balkenborg
is negative at (x ∗ , y ∗ ) then (x ∗ , y ∗ ) is a saddle point.
If this determinant is positive at (x ∗ , y ∗ ) then (x ∗ , y ∗ ) is a peak or
∂2 z
∂2 z
are the
a trough. In this case the signs of ∂x
2 |(x ∗ ,y ∗ ) and ∂y 2
∗ ∗
|(x ,y )
same. If both signs are positive, then (x ∗ , y ∗ ) is a trough. If both
signs are negative, then (x ∗ , y ∗ ) is a peak.
Nothing can be said if the determinant is zero.
∂2 z
∂2 z
Notice that det H > 0 implies sign ∂x
2 = sign ∂y 2
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example
det H


 < 0: (

 > 0:
saddle point
> 0: trough
< 0: peak
∂2 z
∂x 2
∂2 z
∂x 2
∂z
∂z
= 2x and ∂y
= −2y .
z = x 2 − y 2 . The partial derivatives are ∂x
Clearly, (0, 0) is the only critical point. The determinant of the
Hessian is
2
∂z
∂2 z ∂x 2 ∂y ∂x 2 0 det H = ∂2 z
= (2) (−2) − 0 × 0 = −4 < 0
=
2
∂x ∂y ∂ z2 0 −2 ∂y
Hence the function has a saddle point at (0, 0).
Balkenborg
Multivariate Functions
Balkenborg
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Example
Example
∂z
= −2x and
z = −x 2 − y 2 . The partial derivatives are ∂x
∂z
∂y = −2y . Clearly, (0, 0) is the only critical point. The
determinant of the Hessian is
2
∂z
∂2 z ∂x 2 ∂y ∂x −2 0 =4>0
det H = ∂2 z
=
2
∂x ∂y ∂ z2 0 −2 z
∂z
∂x
∂z
∂y
∂y
and
∂2 z
∂x 2
< 0. Hence the function has a peak at (0, 0).
Balkenborg
5
= −x 2 + xy − y 2
2
5
= −2x + y
2
5
=
x − 2y
2
(0, 0) critical point.
Multivariate Functions
Balkenborg
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Multivariate Functions
Functions in two variables
Partial derivatives
Optimization
Unconstrained Optimization
Second order conditions
Determinant of the Hessian is
2
∂z
∂2 z ∂x 2 ∂y ∂x −2 25
det H = ∂2 z
∂2 z = 5
−2
∂x ∂y ∂y
2
2
9
= − <0
4
2
= (−2) (−2) − 5
2
4
2
z
0
-4
0 0 -2
-2
-4
4 2
-50 x
y
-4
-2
4
2
-100
Hence (0, 0) saddle point although both
Balkenborg
∂2 z
∂x 2
and
Multivariate Functions
∂2 z
∂y 2
2
y z0
negative.
4
Balkenborg
Multivariate Functions
x
0
-2
-4