BEEM103OptimizationTechniquesforEconomistspeople.exeter.ac.uk/dgbalken/BEEM10309/week 02...
Transcript of BEEM103OptimizationTechniquesforEconomistspeople.exeter.ac.uk/dgbalken/BEEM10309/week 02...
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
BEEM103 Optimization Techniques for EconomistsMultivariate Functions
Dieter Balkenborg
Department of Economics, University of Exeter
Week 2
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level CurvesIsoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization4 Unconstrained Optimization
The first order conditionsExample 1Example 2: Maximizing profits
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example 3: Price Discrimination
5 Second order conditionsBalkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Functions in two variables
A functionz = f (x , y)
or simplyz (x , y)
in two independent variables with one dependent variable assignsto each pair (x , y) of (decimal) numbers from a certain domain Din the two-dimensional plane a number z = f (x , y).x and y are hereby the independent variablesz is the dependent variable.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example: The cubic polynomial
The graph of f is the surface in 3-dimensional space consisting ofall points (x , y , f (x , y)) with (x , y) in D.
z = f (x , y) = x3 − 3x2 − y2
1
-5
0
y
1
x
00-1
2
z
5
2 3-1
-2
Exercise: Evaluate z = f (2, 1), z = f (3, 0), z = f (4,−4),z = f (4, 4)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example: production function
Q =6√K√L = K
16 L
12
capital K ≥ 0, labour L ≥ 0, output Q ≥ 0
010
20
y
200
1015
x
50
2
4z
6
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example: profit function
Assume that the firm is a price taker in the product market and inboth factor markets.
P is the price of output
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example: profit function
Assume that the firm is a price taker in the product market and inboth factor markets.
P is the price of output
r the interest rate (= the price of capital)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example: profit function
Assume that the firm is a price taker in the product market and inboth factor markets.
P is the price of output
r the interest rate (= the price of capital)
w the wage rate (= the price of labour)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Example: profit function
Assume that the firm is a price taker in the product market and inboth factor markets.
P is the price of output
r the interest rate (= the price of capital)
w the wage rate (= the price of labour)
total profit of this firm:
Π (K , L) = TR − TC= PQ − rK − wL= PK
16 L
12 − rK − wL
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
P = 12, r = 1, w = 3:
Π (K , L) = PK16 L
12 − rK − wL
= 12K16 L
12 −K − 3L
20x
10
z
0
10
-10
0
20
y
2010
0
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
P = 12, r = 1, w = 3:
Π (K , L) = PK16 L
12 − rK − wL
= 12K16 L
12 −K − 3L
20x
10
z
0
10
-10
0
20
y
2010
0
Profits is maximized at K = L = 8.Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Level Curves
The level curve of the function z = f (x , y) for the level c isthe solution set to the equation
f (x , y) = c
where c is a given constant.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Level Curves
The level curve of the function z = f (x , y) for the level c isthe solution set to the equation
f (x , y) = c
where c is a given constant.
Geometrically, a level curve is obtained by intersecting thegraph of f with a horizontal plane z = c and then projectinginto the (x , y)-plane. This is illustrated on the next page forthe cubic polynomial discussed above:
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
0
1
-50
2
01
x
2 3
5
-1z
y-1
-2
-11 2 3
0
2
1
0
xz-1
y
-2
compare: topographic map
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
In the case of a production function the level curves are calledisoquants. An isoquant shows for a given output levelcapital-labour combinations which yield the same output.
0
x
2020
y100
0z
6
010
20
20
10
x
y
z0
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Finally, the linear function
z = 3x + 4y
has the graph and the level curves:
420
x
0
4
y
20
20z10
30
z
0
2
4
y
x
0 2 4
The level curves of a linear function form a family of parallel lines:
c = 3x + 4y 4y = c − 3x y =c
4− 34x
slope − 34 , variable intercept c4 .Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Exercise: Describe the isoquant of the production function
Q = KL
for the quantity Q = 4.Exercise: Describe the isoquant of the production function
Q =√KL
for the quantity Q = 2.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example: The cubic polynomialExample: production functionExample: profit function.Level Curves
Remark: The exercises illustrate the following general principle: Ifh (z) is an increasing (or decreasing) function in one variable, thenthe composite function h (f (x , y)) has the same level curves asthe given function f (x , y) . However, they correspond to differentlevels.
0
04
2
4
yx
20
20
z 10
0
04
2
4
yx
20
z 2
4
Q = KL Q =√KL
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
Level curves ←→ indifference curves or isoquants
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
Level curves ←→ indifference curves or isoquants
Partial differentiation ←→ partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial Derivatives: A Basic Example
Exercise: What is the derivative of
z (x) = a3x2
with respect to x when a is a given constant?
Exercise: What is the derivative of
z (y) = y3b2
with respect to y when b is a given constant?
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe 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).
The derivative of this function F (x) is called the partialderivative of f with respect to x and denoted by
∂z
∂x |y=y0=dF
dx
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe 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).
The derivative of this function F (x) is called the partialderivative of f with respect to x and denoted by
∂z
∂x |y=y0=dF
dx
Notation: “d”=“dee”, “δ”=“delta”, “∂”=“del”
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe 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).
The derivative of this function F (x) is called the partialderivative of f with respect to x and denoted by
∂z
∂x |y=y0=dF
dx
Notation: “d”=“dee”, “δ”=“delta”, “∂”=“del”
It suffices to think of y and all expressions containing only yas exogenously fixed constants. We can then use the familiarrules for differentiating functions in one variable in order toobtain ∂z
∂x.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: Notation continued
Other common notations for partial derivatives are ∂f∂x, ∂f
∂yor
fx , fy orf′x , f
′y .
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: Notation continued
Other common notations for partial derivatives are ∂f∂x, ∂f
∂yor
fx , fy orf′x , f
′y .
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
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: Notation continued
Other common notations for partial derivatives are ∂f∂x, ∂f
∂yor
fx , fy orf′x , f
′y .
Consider again function z = f (x , y). Fix y = y0, vary only x :z = F (x) = f (x , y0). Then evaluate at x = x0
The derivative of this function F (x) at x = x0 is calledthe partial derivative of f with respect to x and denoted
by∂z
∂x |x=x0,y=y0=dF
dx |x=x0=dF
dx(x0)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: The example continued
Example: Letz (x , y) = y3x2
Then
∂z
∂x= y32x = 2y3x
∂z
∂y= 3y2x2
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: A second example
Example: Letz = x3 + x2y2 + y4.
Setting e.g. y = y0 = 1 we obtain z = x3 + x2 + 1 and hence
∂z
∂x |y=1= 3x2 + 2x
Evaluating at x = x0 = 1 we then obtain:
∂z
∂x |x=1,y=1= 5
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: A second example
For fixed, but arbitrary, y we obtain
∂z
∂x= 3x2 + 2xy2
as follows: We can differentiate the sum x3+x2y2+y4 withrespect to x term-by-term. Differentiating x3 yields 3x2,differentiating x2y2 yields 2xy2 because we think now of y2 as a
constant andd(ax 2)dx
= 2ax holds for any constant a. Finally, thederivative of any constant term is zero, so the derivative of y4 withrespect to x is zero.Similarly considering x as fixed and y variable we obtain
∂z
∂y= 2x2y + 4y3
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
The Marginal Products of Labour and Capital
Example: The partial derivatives ∂q∂Kand ∂q
∂Lof a production
function q = f (K , L) are called the marginal product of capitaland (respectively) labour. They describe approximately by howmuch output increases if the input of capital (respectively labour)is increased by a small unit.Fix K = 64, then q = K
16 L
12 = 2L
12 which has the graph
0 1 2 3 4 50
1
2
3
4
x
Q
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
The Marginal Products
This graph is obtained from the graph of the function in twovariables by intersecting the latter with a vertical plane parallel toL-q-axes.
1020
z0246
y
20
x
15100 50
The partial derivatives ∂q∂Kand ∂q
∂Ldescribe geometrically the slope
of the function in the K - and, respectively, the L- direction.Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Diminishing productivity of labour:
The more labour is used, the less is the increase in output whenone more unit of labour is employed. Algebraically:
∂q
∂L=1
2K
16 L−
12 =
1
2
6√K
2√L> 0,
∂2q
∂L2=
∂
∂L
(∂q
∂L
)= −1
4K
16 L−
32 = −1
4
6√K
2√L3< 0,
Exercise: Find the partial derivatives of
z =(x2 + 2x
) (y3 − y2
)+ 10x + 3y
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
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
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Objectives
Subject: Optimization of multivariate functions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Objectives
Subject: Optimization of multivariate functions
Two basic types of problems:
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Objectives
Subject: Optimization of multivariate functions
Two basic types of problems:
1 unconstrained (e.g. profit maximization)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Objectives
Subject: Optimization of multivariate functions
Two basic types of problems:
1 unconstrained (e.g. profit maximization)2 constrained
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Unconstrained Optimization
-4-2
-4
-50
-40
-2
y x
-30
z
-20
-10
000
242
4
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Unconstrained Optimization
Objective: Find (absolute) maximum of function
z = f (x , y)
i.e., find a pair (x∗, y ∗) such that
f (x∗, y ∗) ≥ f (x , y)
holds for all pairs (x , y).Hereby both pairs of numbers (x∗, y ∗) and (x , y) must be in the domain ofthe function.
For an absolute minimum require f (x∗, y ∗) ≤ f (x , y).f (x , y) is called the “objective function”.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Example
The function
z = f (x , y) = − (x − 2)2 − (y − 3)2
has a maximum at (x∗, y ∗) = (2, 3) .
4
y
2
4
-15
-10z-5
0
x
200
0
42
yz0
2
x4
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
First order conditions
The following must hold: freeze the variable y at the optimal valuey ∗, vary only x then the function in one variable
F (x) = f (x , y ∗)
must have maximum at x∗: dFdx (x
∗) = 0. Thus we obtain the firstorder conditions
∂z
∂x |x=x ∗,y=y ∗= 0
∂z
∂y |x=x ∗,y=y ∗= 0
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Example 1
z = f (x , y) = − (x − 2)2 − (y − 3)2∂z
∂x= −2 (x − 2)× (+1) = 0 ∂z
∂y= −2 (y − 3)× (+1) = 0
x∗ = 2 y ∗ = 3
The maximum (at least the only critical or stationary point) is at
(x∗, y ∗) = (2, 3)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Maximizing profits
Production function Q (K , L).r interest ratew wage rateP price of outputprofit
Π (K , L) = PQ (K , L)− rK − wL.FOC for profit maximum:
∂Π
∂K= P
∂Q
∂K− r = 0 (1)
∂Π
∂L= P
∂Q
∂L− w = 0 (2)
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Intuition: Suppose P ∂Q∂K− r > 0. By using one unit of capital
more the firm could produce ∂Q∂Kunits of output more. The
revenue would increase by P ∂Q∂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. HenceP ∂Q
∂K− r = 0 must hold in optimum.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Rewrite FOC as
∂Q
∂K=
r
P(3)
∂Q
∂L=
w
P(4)
Division yields:
MRS = − dLdK
=∂Q
∂K
/∂Q
∂L=r
P
/ wP=r
w(5)
The marginal rate of substitution must equal the ratio of the inputprices!
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Why? If the firm uses one unit of capital less and ∂Q∂K
/∂Q∂Lunits of
labour more, output remains (approximately) the same. The firm
would save r on capital and spend ∂Q∂K
/∂Q∂L× w on more labour
while still making the same revenue. Profit would increase unless
r ≤ ∂Q∂K
/∂Q∂L× w . As symmetric argument interchanging the role
of capital and labour shows that r ≥ ∂Q∂K
/∂Q∂L× w must hold if
the firm optimizes profits.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Example
Q (K , L) = K16 L
12
∂Q
∂K=1
6K
16−1L
12
∂Q
∂L=1
2K
16 L
12−1
∂Q
∂K=1
6K−
56 L
12
∂Q
∂L=1
2K
16 L−
12
FOC:
1
6K−
56 L
12 =
r
P
1
2K
16 L−
12 =
w
P∂Q
∂K
/∂Q
∂L=
1
3K−
56− 1
6 L12−(− 1
2 ) =r
w
∂Q
∂K
/∂Q
∂L=
1
3
L
K=r
w
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Example
P = 12, r = 1 and w = 3; FOC:
1
6K−
56 L
12 =
1
12
1
2K
16 L−
12 =
3
12
2K−56 L
12 = 1 (*) 2K
16 L−
12 = 1
1
3
L
K=
1
3=⇒ K = L
Substituting L = K into (*) we get
2K−56K
12 = 2K−
26 = 2K−
13 = 1 =⇒ 2 = K
13 =⇒ K ∗ = L∗ = 23 = 8
Q∗ = K16 L
12 = 8
16+
12 = 8
46 = 8
23 = 22 = 4
Π∗ = 12Q∗ − 1K ∗ − 3L∗ = 48− 8− 24 = 16
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
20
15
x
105
0
10
z0
-10
20
10
2015
y5
0
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Outline
1 Functions in two variablesExample: The cubic polynomialExample: production functionExample: profit function.Level Curves
Isoquants
2 Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
3 Optimization
4 Unconstrained OptimizationThe first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
5 Second order conditions
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Example 3: Price Discrimination
A monopolist with total cost function TC (Q) = Q2 sells hisproduct in two different countries. When he sells QA units of thegood in country A he will obtain the price
PA = 22− 3QA
for each unit. When he sells QB units of the good in country B heobtains the price
PB = 34− 4QB .How much should the monopolist sell in the two countries in orderto maximize profits?
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Solution
Total revenue in country A:
TRA = PAQA = (22− 3QA)QA
Total revenue in country B:
TRB = PBQB = (34− 4QB )QB
Total production costs are:
TC = (QA +QB )2
Profit:
Π (QA,QB ) = (22− 3QA)QA + (34− 4QB )QB − (QA +QB )2
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The first order conditionsExample 1Example 2: Maximizing profitsExample 3: Price Discrimination
Profit:
Π (QA,QB ) = (22− 3QA)QA + (34− 4QB )QB − (QA +QB )2
FOC:
∂Π
∂QA= −3QA + (22− 3QA)− 2 (QA +QB ) = 22− 8QA − 2QB = 0
∂Π
∂QB= −4QB + (34− 4QB )− 2 (QA +QB ) = 34− 2QA − 10QB= 0
or
8QA + 2QB = 22 (6)
2QA + 10QB = 34.
linear simultaneous systemBalkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Second order conditions: A peak
critical point of function z = f (x , y) = −x2 − y2: (0, 0)
-4
-30
-20
0
z
-10
x y
0 2 4-2 -4
4
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Second order conditions: A trough
critical point of function z = f (x , y) = +x2 + y2: (0, 0)
-4
-4
-2
x
-2
y
0
02
z
30
20
10
402
4
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Second order conditions: A saddle point
critical point of function z = f (x , y) = +x2 − y2: (0, 0)
100
-10-5
50
10
z5
0
y
0
-50
0 -5 -10
105
x
-100
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Second order conditions: A monkey saddle
critical point of function z = f (x , y) = yx2 − y3: (0, 0)
This and more complex possibilities will be ignored.Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
The Hessian matrix
[∂2z∂x 2
∂2z∂y ∂x
∂2z∂x∂y
∂2z∂y 2
]
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Theorem
Suppose that the function z = f (x , y) has a critical point at(x∗, y ∗). If the determinant of the Hessian detH =∣∣∣∣∣
∂2z∂x 2
∂2z∂y ∂x
∂2z∂x∂y
∂2z∂y 2
∣∣∣∣∣=
∂2z
∂x2∂2z
∂y2− ∂2z
∂x∂y
∂2z
∂y∂x=
∂2z
∂x2∂2z
∂y2−(
∂2z
∂x∂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 ora trough. In this case the signs of ∂2z
∂x 2 |(x ∗,y ∗) and∂2z∂y 2 |(x ∗,y ∗) are the
same. If both signs are positive, then (x∗, y ∗) is a trough. If bothsigns are negative, then (x∗, y ∗) is a peak.
Nothing can be said if the determinant is zero.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Theorem
Suppose that the function z = f (x , y) has a critical point at(x∗, y ∗). If the determinant of the Hessian detH =∣∣∣∣∣
∂2z∂x 2
∂2z∂y ∂x
∂2z∂x∂y
∂2z∂y 2
∣∣∣∣∣=
∂2z
∂x2∂2z
∂y2− ∂2z
∂x∂y
∂2z
∂y∂x=
∂2z
∂x2∂2z
∂y2−(
∂2z
∂x∂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 ora trough. In this case the signs of ∂2z
∂x 2 |(x ∗,y ∗) and∂2z∂y 2 |(x ∗,y ∗) are the
same. If both signs are positive, then (x∗, y ∗) is a trough. If bothsigns are negative, then (x∗, y ∗) is a peak.
Nothing can be said if the determinant is zero.Notice that detH > 0 implies sign ∂2z
∂x 2= sign ∂2z
∂y 2
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
detH
< 0: saddle point
> 0:
{∂2z∂x 2> 0: trough
∂2z∂x 2< 0: peak
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example
z = x2 − y2. The partial derivatives are ∂z∂x= 2x and ∂z
∂y= −2y .
Clearly, (0, 0) is the only critical point. The determinant of theHessian is
detH =
∣∣∣∣∣
∂2z∂x 2
∂2z∂y ∂x
∂2z∂x∂y
∂2z∂y 2
∣∣∣∣∣=
∣∣∣∣2 00 −2
∣∣∣∣ = (2) (−2)− 0× 0 = −4 < 0
Hence the function has a saddle point at (0, 0).
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example
z = −x2 − y2. The partial derivatives are ∂z∂x= −2x and
∂z∂y= −2y . Clearly, (0, 0) is the only critical point. The
determinant of the Hessian is
detH =
∣∣∣∣∣
∂2z∂x 2
∂2z∂y ∂x
∂2z∂x∂y
∂2z∂y 2
∣∣∣∣∣=
∣∣∣∣−2 00 −2
∣∣∣∣ = 4 > 0
and ∂2z∂x 2< 0. Hence the function has a peak at (0, 0).
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Example
z = −x2 + 52xy − y2
∂z
∂x= −2x + 5
2y
∂z
∂y=
5
2x − 2y
(0, 0) critical point.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
Determinant of the Hessian is
detH =
∣∣∣∣∣
∂2z∂x 2
∂2z∂y ∂x
∂2z∂x∂y
∂2z∂y 2
∣∣∣∣∣=
∣∣∣∣−2 5
252 −2
∣∣∣∣ = (−2) (−2)−(5
2
)2
= −94< 0
Hence (0, 0) saddle point although both ∂2z∂x 2
and ∂2z∂y 2
negative.
Balkenborg Multivariate Functions
Functions in two variablesPartial derivatives
OptimizationUnconstrained Optimization
Second order conditions
-2
z -50
-100
x y
0-4-2
-40224
4 0
z
4
2
y
x-2
0
-4
-224 0 -4
Balkenborg Multivariate Functions