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ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
BEE1024 Mathematics for EconomistsMultivariate Functions
Juliette Stephenson and Amr (Miro) AlgarhiAuthor: Dieter Balkenborg
Department of Economics, University of Exeter
Week 1
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level CurvesIsoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Objectives for the week
Functions in two independent variables.
Level curves ! indi¤erence curves or isoquants
Partial di¤erentiation ! partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Objectives for the week
Functions in two independent variables.
Level curves ! indi¤erence curves or isoquants
Partial di¤erentiation ! partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Objectives for the week
Functions in two independent variables.
Level curves ! indi¤erence curves or isoquants
Partial di¤erentiation ! partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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
8
6
4
0
2
4
6
z
2
2
y
1 1 2 3x
Exercise: Evaluate z = f (2, 1), z = f (3, 0), z = f (4,�4),z = f (4, 4)
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Example: production function
Q = 6pKpL = K
16 L
12
capital K � 0, labour L � 0, output Q � 0
0 2 4 614 16 18 20
L5
K
0
2
4
6
z
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Example: pro�t function
Assume that the �rm 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 pro�t of this �rm:
Π (K , L) = TR � TC= PQ � rK � wL= PK
16 L
12 � rK � wL
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Example: pro�t function
Assume that the �rm 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 pro�t of this �rm:
Π (K , L) = TR � TC= PQ � rK � wL= PK
16 L
12 � rK � wL
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Example: pro�t function
Assume that the �rm 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 pro�t of this �rm:
Π (K , L) = TR � TC= PQ � rK � wL= PK
16 L
12 � rK � wL
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Example: pro�t function
Assume that the �rm 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 pro�t of this �rm:
Π (K , L) = TR � TC= PQ � rK � wL= PK
16 L
12 � rK � wL
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
P = 12, r = 1, w = 3:
Π (K , L) = PK16 L
12 � rK � wL
= 12K16 L
12 �K � 3L
20K
0 515 20
L
10
5
0
5
10
15
20
z
Pro�ts is maximized at K = L = 8.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
P = 12, r = 1, w = 3:
Π (K , L) = PK16 L
12 � rK � wL
= 12K16 L
12 �K � 3L
20K
0 515 20
L
10
5
0
5
10
15
20
z
Pro�ts is maximized at K = L = 8.Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
5
2
1
2
y1 1 2 3x
2
1
1
2
y1
1 x
compare: topographic map
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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 2 4 614 16 18 20
L5
K
0
2
4
6
05
20
L0
15
20
K
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Finally, the linear function
z = 3x + 4y
has the graph and the level curves:
0 2 3 4 5x 0
y05
101520253035
0 2 3 4 5x
2
3
4
5
y
The level curves of a linear function form a family of parallel lines:
c = 3x + 4y 4y = c � 3x y =c4� 34x
slope � 34 , variable interceptc4 .
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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 =pKL
for the quantity Q = 2.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
Example: The cubic polynomialExample: production functionExample: pro�t 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 di¤erentlevels.
0
5
10
15
20
25
2
4L
1 2 3 4 5K 0
1
2
3
4
5
2
4L
1 2 3 4 5K
Q = KL Q =pKL
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
Level curves ! indi¤erence curves or isoquants
Partial di¤erentiation ! partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
Level curves ! indi¤erence curves or isoquants
Partial di¤erentiation ! partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Objectives for the week
Functions in two independent variables.
Level curves ! indi¤erence curves or isoquants
Partial di¤erentiation ! partial analysis in economics
The lecture should enable you for instance to calculate themarginal product of labour.
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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) at x = x0 is calledthe partial derivative of f with respect to x and denotedby
∂z∂x jx=x0,y=y0
=dFdx jx=x0
=dFdx(x0)
Notation: �d�=�dee�, �δ�=�delta�, �∂�=�del�It su¢ ces to think of y and all expressions containing only yas exogenously �xed constants. We can then use the familiarrules for di¤erentiating functions in one variable in order toobtain ∂z
∂x .Other common notations for partial derivatives are ∂f
∂x ,∂f∂y or
fx , fy orf 0x , f0y .
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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) at x = x0 is calledthe partial derivative of f with respect to x and denotedby
∂z∂x jx=x0,y=y0
=dFdx jx=x0
=dFdx(x0)
Notation: �d�=�dee�, �δ�=�delta�, �∂�=�del�
It su¢ ces to think of y and all expressions containing only yas exogenously �xed constants. We can then use the familiarrules for di¤erentiating functions in one variable in order toobtain ∂z
∂x .Other common notations for partial derivatives are ∂f
∂x ,∂f∂y or
fx , fy orf 0x , f0y .
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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) at x = x0 is calledthe partial derivative of f with respect to x and denotedby
∂z∂x jx=x0,y=y0
=dFdx jx=x0
=dFdx(x0)
Notation: �d�=�dee�, �δ�=�delta�, �∂�=�del�It su¢ ces to think of y and all expressions containing only yas exogenously �xed constants. We can then use the familiarrules for di¤erentiating functions in one variable in order toobtain ∂z
∂x .
Other common notations for partial derivatives are ∂f∂x ,
∂f∂y or
fx , fy orf 0x , f0y .
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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) at x = x0 is calledthe partial derivative of f with respect to x and denotedby
∂z∂x jx=x0,y=y0
=dFdx jx=x0
=dFdx(x0)
Notation: �d�=�dee�, �δ�=�delta�, �∂�=�del�It su¢ ces to think of y and all expressions containing only yas exogenously �xed constants. We can then use the familiarrules for di¤erentiating functions in one variable in order toobtain ∂z
∂x .Other common notations for partial derivatives are ∂f
∂x ,∂f∂y or
fx , fy orf 0x , f0y .
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: The example continued
Example: Letz (x , y) = y3x2
Then∂z∂x= 2y3x
∂z∂y= 3y2x2
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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 = 1 we obtain z = x3 + x2 + 1 and hence
∂z∂x jy=1
= 3x2 + 2x
∂z∂x jx=1,y=1
= 5
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Partial derivatives: A second example
For �xed, but arbitrary, y we obtain
∂z∂x= 3x2 + 2xy2
as follows: We can di¤erentiate the sum x3 + x2y2 + y4 withrespect to x term-by-term. Di¤erentiating x3 yields 3x2,di¤erentiating x2y2 yields 2xy2 because we think now of y2 as a
constant andd(ax 2)dx = 2ax holds for any constant a. Finally, the
derivative of any constant term is zero, so the derivative of y4 withrespect to x is zero.Similarly considering x as �xed and y variable we obtain
∂z∂y= 2x2y + 4y3
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Outline
1 Objectives
2 Lecture A: Functions in two variablesExample: The cubic polynomialExample: production functionExample: pro�t function.Level Curves
Isoquants
3 Lecture B: Partial derivativesA Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
A Basic ExampleNotationA Second ExampleThe Marginal Products of Labour and Capital
The Marginal Products of Labour and Capital
Example: The partial derivatives ∂q∂K and
∂q∂L of 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
Q
1 2 3 4 5L
Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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.
0 2 4 614 16 18 20
L5
K
0
2
4
6
The partial derivatives ∂q∂K and
∂q∂L describe geometrically the slope
of the function in the K - and, respectively, the L- direction.Balkenborg Multivariate Functions
ObjectivesLecture A: Functions in two variables
Lecture B: Partial derivatives
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=12K
16 L�
12 =
12
6pK
2pL> 0,
∂2q∂L2
=∂
∂L
�∂q∂L
�= �1
4K
16 L�
32 = �1
4
6pK
2pL3< 0,
Exercise: Find the partial derivatives of
z =�x2 + 2x
� �y3 � y2
�+ 10x + 3y
Balkenborg Multivariate Functions