Methods of Economic Nalysis II

8/11/2019 Methods of Economic Nalysis II

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EC115 - Methods of Economic AnalysisLecture 2

Functions with more than one variable

Renshaw - Chapter 14 & Chapter 17

University of Essex - Department of Economics

Week 17

Domenico Tabasso (Universityof Essex - Department of Economics)

Lecture 2 Week 17 1 / 38
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Topics for this week

Definition of a function with more than one variable

Graphs of functions with two variables: level curves

Linear functionsLeontief functions

The Cobb-Douglas function

Homogeneous functions and and returns to scale


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Definition

A function of several variables is a relation between

some independent variables x1, x2, x3,...xn and somedependent variable zsuch that

z=f(x1, x2, x3,...xn)

specifies the value ofzgiven the values ofx1, x2, x3,...xn.

For example, we might encounter functions of thefollowing form

z = 1002x1+5x2+3x357x4

z = 3x21 9x2x3+10x34

z = e

2x1+3x2


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A few examples - 1

Domenico Tabasso (Universityof Essex - Department of Economics)Lecture 2 Week 17 4 / 38
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A few examples - 2

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A few examples - 3

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A few examples - 4



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Graph of a function

How can we graph these functions?

Only functions of at mostTWOindependent variablescan be graphed!

Consider any function

z=f(x,y)

This type of function can be graphed in a threedimensional space.

Hence we have to extend the techniques we learnedabout functions of the form z=f(x)and try to usethem for this new set of functions.

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Three-dimensional graph: z=f(x,y)

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Example:

Consider the following linear functionz=f(x,y) =3x+2y10

This is an example of aplaneor a flat surface.This is the extension of a straight line into threedimensions or R3.

Example of coordinates:

(x1,y1, z1), (1,1,-5);(x2,y2, z2), (2,-0.5,-5);(x3,y3, z3), (10,-12.5,-5).

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Here is the graph: z=3x+2y10

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Sections (slices) through the surface of a function

Three dimensional graphs sometimes are very hard tograph and could lead to confusion.

In Economics we normally analyze these functions by

looking at their sections.To obtain a particular section of the function we shouldask the following question:

What are the values of x,y such thatz=a, whereaisa constant?

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Why is this useful?

For instance, consider the previous equation:

z=3x+2y10

and assume z=a. Then we have:

a=3x+2y10but sinceais a constant we can rewrite our equation as:

y=

3

2x+

a+10

2

So what we have here is our old friend y=f(x), whichcan be graphed on a x,y plan!

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All we have to do now is to impose some values for aso

that we can see how the relation between x and y changeswhen achanges. So, for example ifa=0we get

y=3

2

x+5

while if we impose a=10what we get is:

y=3

2

x+10

and so on.

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z=3x+2y10 A two dimensional graph

14

16Y

12

a=20

10

6 a=10

4

0

2

a=0

2

0 1 2 3 4 5X

4



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What are the values ofx, zsuch that y=c,where c isa constant? These can be represented in the x, zplaneby

x=1

3

z+102c

3for example ify=0,

x=1

3

z+10

3


l
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Level curves - 1

Hence, by holding constant a variable (x, y orz) wehave reduced a three dimensional function into a twodimensional function (a function of one-variable).

Here these are represented by straight lines since theoriginal function was linearin a two dimensional space.

In general, we call level curvesany two dimensional

representations ofz=f(x,y)when holding a variableconstant.


L l 2
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Level curves - 2

Can we hold zconstant?

Sure! Thats exactly what we do for drawing theindifference curves of a utility function.

Imagine:

U=100xy

Indifference curves? Fix U=Uwhere U is a constant and

the solve for yy=

U

100x

As Uvaries we obtain a map of indifference curves.


L l 2 2
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Level curves: z=x2 +y2 holding z constant


L l 2 + 2
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Level curves: z=x2 +y2 holding z constant


E i A li ti


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Economic Applications:

Consumer Theory and the Utility Function

= u=U(x,y)(indifference curves)

Producer Theory and the Production Function= y=F(L, K) (isoquants)

The typical functions we encounter in economics are:

1 Cobb-Douglas functions

2 Linear Functions3 Quasi-linearFunctions

4 LeontiefFunctions

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Cobb-Douglas function:

u = Axy Utility functiony = ALK Production function

where A, and areusuallypositive constants.

u=100x1/2y1/2 y= z2

100x

u=100x7y3 y=

z100x7

1/3


Cobb Douglas functions level curves u 100x 1/2y 1/2
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Cobb-Douglas functions level curves - u=100x1/2y1/2

100

70

80

50

60

30

40

10

20

1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 1 5 16 1 7 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30

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Linear Function:

u = ax+by,y = aL+bK,

where aand bareusuallypositive constants.

Quasi-Linearfunction:

u = Ay+v(x),

y = AK+v(L),

where Ais usuallya positive constant and v()is afunction of one variable.


Quasi linear function an example: z = 100x + y 3
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Quasi-linear function, an example: z=100x+y

3.5

4

2.5

3

1

1.5

2

0

0.5

1 4 7 10 13 16 19 22 25 28

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Leontief functions examples


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Leontief functions, examples

In order to produce 1 mobile phone (P) we need 1 keyboard(K) and 1 software (S). The production function is:

P=min {K, S}

In order to produce 1 bike (B) we need 1 frame (F) and 2

tyres (T). So the production function is:

B=min

F, 12

T


Leontief functions level curves (isoquants)
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Leontief functions level curves(isoquants)

Y

1

1

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NB important


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NB important

inputs must increase in the same proportion

Otherwise we would be changing not only the scaleof theinputs but also the relative amounts of different inputsused.


Cobb-Douglas functions level curves
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Cobb Douglas functions level curves


Homogeneous Functions
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g u u

Functions whose return to scale rate does not varies are

called homogeneous. A functionf

(x,y

)is said to behomogeneous of degree k if

f(tx, ty) =tkf(x,y)

for all x,yand all t>0,where k is a scalar scalar.

In words:a function is homogeneous of degree k if we obtain

the same result whether we multiply each

independent variable by t or if the whole function is

multiplied by tk


Example 1
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p

z=f(x,y) =x2y+3xy2 +y3

What is its degree of homogeneity?

f(tx, ty) = (tx)2(ty) +3(tx)(ty)2 + (ty)3

= t2x2ty+3txt2y2 +t3y3

= t3

(x2y

+3xy2

+y3

)= t3f(x, y)


Example 2: Cobb-Douglas function


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p g

y=F(L, K) =AL

K

What is its degree of homogeneity?

F(tL, tK) = A(tL)(tK)

= t+(ALaK)

= t+F(L, K).

The degree of homogeneity is just = +


Returns to Scale of a Cobb-Douglas
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g

If+=1,the function is homogeneous of degree1=Constant Returns to Scale.

If

+ >

1,the function is homogeneous of a degree>1=Increasing Returns to Scale.

If+


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Partial Differentiation
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Consider a function z=f(x,y)

partial derivative ofz wrt x zx

partial derivative ofz wrt y z

y

We use the usual rule of derivation, because when we dothe partial derivative wrt xwe held the rest of the variablesfixed.

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Methods of Economic Nalysis II