BEE1024 Mathematics for Economists - Exeterpeople.exeter.ac.uk/dgbalken/ME08/week1handout.pdf ·...

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



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




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.




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.





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)





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





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





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




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:





5

2

1

2

y1 1 2 3x

2

1

1

2

y1

1 x

compare: topographic map





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





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 .





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.





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




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.





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?





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 .





Partial derivatives: The example continued

Example: Letz (x , y) = y3x2

Then∂z∂x= 2y3x

∂z∂y= 3y2x2





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





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





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





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




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


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