This is a PowerPoint presentation on fundamental math

fall ‘ 97 Principles of Microeconomics Slide 1

This is a PowerPoint presentation on fundamental mathtools that are useful in principles of economics.

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R. Larry Reynolds (Boise State University)


Math Review

· Mathematics is a very precise language that is useful to express the relationships between related variables

· Economics is the study of the relationships between resources and the alternative outputs

· Therefore, math is a useful tool to express economic relationships


Relationships

· A relationship between two or more variables can be expressed as an equation, table or graph· equations & graphs are “continuous”· tables contain “discrete”

information· tables are less complete than equations· it is more difficult to see patterns in tabular

data than it is with a graph -- economists prefer equations and graphs


Equations

· a relationship between two variables can be expressed as an equation

· the value of the “dependent variable” is determined by the equation and the value of the “independent variable.”

· the value of the independent variable is determined outside the equation, i.e. it is “exogenous”


Equations [cont . . .]

· An equation is a statement about a relationship between two or more variables

· Y = fi (X) says the value of Y is determined by the value of X ; Y is a “function of X.”

· Y is the dependent variable · X is the independent variable

· A linear relationship may be specified: Y = a

mX [the function will graph as a straight line]

· When X = 0, then Y is “a”· for every 1 unit change in X, Y changes by “

m”


Y = 6 - 2X

· The relationship between Y and X is determined; for each value of X there is one and only one value of Y [function]

· Substitute a value of X into the equation to determine the value of Y

· Values of X and Y may be positive or negative, for many uses in economics the values are positive [we use the NE quadrant]


Equations -- Graphs [Cartesian system]

The X axis[horizontal]

The Y axis [vertical]

The North East Quadrant(NE), where X > 0, Y > 0 {both X and Y are positive numbers}

X > 0

+1 +2 +3

Y>0

+1

+2

+3(X,Y) where X<0, Y>0

X<0

-3 -2 -1

(X,Y) where X<0 and Y<0

Y<0

-1

-2

-3

(X,Y) where X>0 and Y<0

(Left click mouse to add material)


When the values of the independent and dependent variablesare positive, we use the North East quadrant

1 2 3 4 5 6

1

2

3

5

6 (X, Y)

(3, 5)

Go to the right {+3} units andup {+5} units!

(1,6)

Right {+1} oneand up {+6} six

(5, 1)

Right 5 and up 1

(2.5, 3.2)

to the right 2.5 unitsand up 3.2 units



Given the relationship, Y = 6 - 2X,

1 2 3 4 5 6

1

2

3

4

5

6sets of (X, Y)

(0, 6)

when X = 0 then Y = 6[this is Y-intercept]

(1, 4)

when X = 1then Y = 4

(2, 2)

When X = 2, then Y = 2

(3, 0)

When X = 3, Y = 0,[this is X-intercept]

The relationship for all positivevalues of X and Y can be illustrated by the line AB

A

B

A line that slopes fromupper left to lower right represents an inverse or negative relationship, when thevalue of X increases, Y decreases!

X

Y(Left click mouse to add material)


1 2 3 4 5 6

1

2

3

4

5

6

Y

X

Given a relationship, Y = 6 - .5X

(0,6)

(1,5.5)

(2, 5)

(4,4)

(6,3)For every one unit increase inthe value of X, Y decreases byone half unit. The slope of thisfunction is -.5! The Y-intercept is 6.

What is the X-intercept?Y when X

0 12

6

512

, ,

.



1 2 3 4 5 6

1

2

3

4

5

6

Y

X

For a relationship, Y = 1 + 2X

When X=0, Y=1 (0,1)When X = 1, Y = 3

(1,3)When X = 2, Y = 5(2,5)

This function illustrates a positive relationshipbetween X and Y. For every one unit increasein X, Y increases by 2 !

run+1

rise+2

slope = +2

for a relationship Y = -1 + .5X

-1

This function shows that for a 1unit increase in X, Y increases one half unit

run+2

rise+1

slope = +12



Problem

· Graph the equation: Y = 9 - 3X· What is the Y intercept? The slope?· What is the X intercept? Is this a

positive (direct) relationship or negative (inverse)?

· Graph the equation Y = -5 + 2X· What is the Y intercept? The slope?· What is the X intercept? Is this a

positive (direct) relationship or negative (inverse)?


Equations in Economics

· The quantity [Q] of a good that a person will buy is determined partly by the price [P] of the good. [Note that there are other factors that determine Q.]

· Q is a function of P, given a Price the quantity of goods purchased is determined. Q = fp (P)

· A function is relationship between two sets in which there is one and only one element in the second set determined by each element in the first set.


Relationship [cont . . . ]

· Q = fp (P) {Q is a function of P}

· Example: Q = 220 - 5P· If P = 0, then Q = 220· If P = 1, then Q = 215· for each one unit increase in

the value of P, the value of Q decreases by 5


Q = 220 - 5P

· This is an inverse or negative relationship· as the value of P increases, the value of Q decreases

· the “Y intercept” is 220, this is the value of Q when; P = 0

· the “X intercept” is 44, this is the value of P when Q = 0

· This is a “linear function,” i.e. a straight line· The “slope” of the function is -5

· for every 1 unit change in P, Q changes by 5 in the opposite direction


The equation provides the information to construct a table.However, it is not possible to make a table to include everypossible value of P. The table contains “discrete” data and doesnot show all possible values!

Q = 220 -5PValue of P Value of Q

combination A 0 220

combination B 1 215

combination C 2 210

combination D 3 205

combination E 4 200

combination F 10 170

combination G 44 0


For the relationship, Q = 220 - 5P, the relationship can be graphed ...

$5

10

1520

2530

35

40

45

50

55PR

ICE

QUANTITY

44

40 80 120 160 200 240 280

When the price is $44, 0 unit will be bought;at a price of $0, 220 units will be bought.

Demand

Notice that we have drawn the graph “backwards,” P{independent}variable is placed on the Y-axis. This is done because we eventuallywant to put supply on the samegraph and one or the other must be

reversed! Sorry!

70

At P=$30,Q = 70

170

At a price of $10, thethe quantity is



Slopes and Shifts

· Economists are interested in how one variable {the independent} “causes” changes in another variable {the dependent}

· this is measured by the slope of the function

· Economists are also interested in changes in the relationship between the variables· this is measured by “shifts” of the function


Slope of a function or “line”

· The slope measures the change in the dependent variable that will be “caused” by a change in the independent variable

· When, Y = a m X; m is the slope m

Y

X

rise

run


1 2 3 4 5 6

1

2

3

4

56

Y

X

Y = 6 -.5X

as the value of X increases from 2 to 4,

X = 2 the value of Ydecreases from5 to 4

Y= -1

X is the run {+2},

Y is the rise [or change in Y caused by X]{in this case, -1}

slope is riserun

so, slope is -1/2 or -.5

Slope of a Line


Shifts of function

· When the relationship between two variables changes, the function or line “shifts”

· This shift is caused by a change in some variable not included in the equation

· [the equation is a polynomial]

· A shift of the function will change the intercepts [and in some cases the slope]


1 2 3 4 5 6

1

2

3

4

56

Y

X

Given the function Y = 6 - .5X,

A decrease in the

function would be Y’ = 4 - .5X

shiftsleft

Shiftsright

an increase in thefunction would represent an increase in the intercept [from 6to a larger number]

the function shifts and itsslope also changes

Just the slopechanges {in this case, an increase in the absolutevalue of .5 to -1.8}Y” = 6 - 1.8X [x intercept = 3.3]



Shifts in functions

· In Principles of Economics most functions are graphed in 2-dimensions, this means we have 2 variables. [The dependent and independent]

· Most dependent variables are determined by several or many variables, this requires polynomials to express the relationships

· a change in one of these variables which is not shown on a 2-D graph causes the function to “shift”


Slope and Production

· The output of a good is determined by the amounts of inputs and technology used in production

· example of a case where land is fixed and fertilizer is added to the production of tomatoes.

· with no fertilizer some tomatoes, too much fertilizer and it destroys tomatoes


FERTILIZER

ton

s o

f to

mato

es

1

2

3

4

5

6

7

8

9

10

11

12

With no fertilizer we get 3 tons of tomatoes

With 1 unit of Fertilizer [F], we get6 tons

The increase in tomatoes [T] “caused” by Fis +3, this is the slope

With 2 units of F, the output of Tincreases to 8

With the 3rd unit of F,T increases to 9

The maximum output of T possible with all inputsand existing technology is 10 units with 6 units of F

use of more F causes the tomatoes to “burn” and output declines

TPf

1 2 3 4 5 6 7 8 9



Slope and Marginal Product

· Since the output of tomatoes [T] is a function of Fertilizer [F] , the other inputs and technology we are able to graph the total product of Fertilizer [TPf]

· From the TPf, we can calculate the marginal product of fertilizer [MPf]

· MPf is the TPf “caused” by theF


1

2

3

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5

6

7

8

9

10

11

12

TPf

Fertilizer [F]Tomatoes [T]0 3

1 6

TPf = +3, F = +1; +3/+1 = 3 [slope = +3]

run=1

rise = +3

rise/run =+3+3

3

MPf [slope]

3 {technically,this is between 0 and the first unit of F}

2 8

TPf = +2, F = +1; +2/+1 = 2

Given: T = f (F, . . . ), MPf = [TPf/F]

3 9

TPf = +1, F = +1; +1/+1 = 1

16 10

TPf = +1, F = +3; +1/+3 .33 [this is an approximation because F>1]

.33

8 9

TPf = -1, F = +2; -1/+2 = -.5

-.5 [a negative slope!]

1 2 3 4 5 6 7 8 9

2


Given a functional relationship such as: Q = 220 - 5P, we can express the equation for P as a function of QThink of an equation as a “balance scale,” what you do to one sideof the equation you must do to the other in order to maintain balance

Q = 220 - 5Psubtract 220 from both sides -220 -220

-220 + Q = -5P divide every term in both sidesby -5 -5 -5 -5

44 - 15

Q = 1P

or, P = 44 - .2 Q

The equation P = 44 - .2Q is the same as Q = 220 - .5P



How do economists estimate relationships?

· Humans behavioral relationships are:· modeled on the basis of theories· models are verified through empirical

observations and statistical methods

· The relationships are estimates that represent populations {or distributions} not specific individuals or elements


An Example

· Hypothesis: the amount of good X [Q] that Susan purchases is determined by the price of the good [Px], Susans’s income [Y], prices of other related goods [Pr] and Susan’s preferences.

· Q = fi (Px,Y, Pr, preferences, . . .)· [. . . indicates there are other variables

that are not included in the equation]


Model of Relationship

· Q = fi (Px,Y, Pr, preferences, . . .) acts a a model to represent the relationships of each independent variable to Q [dependent variable]

· For simplicity, the relationship is described as “linear.” If the relationship were believed not to be linear, with a bit more effort we might construct a “nonlinear model.”


Empirical verification

· To test the model, we would like to observe Susan’s buying pattern.

· If Px,Y, Pr and preferences were all changing at the same time, we would use a multivariate analysis called “multiple regression.” For simplicity we have been lucky enough to find a period where only Px has changed. Y, Pr and preferences have remained unchanged over the period in which we observe Susan’s purchases

fall ‘ 97 Principles of Microeconomics Slide 33Quantity per week

Pri

ce o

f g

ood

X

2

4

6

8

10

12

14

16

18

Susan’s purchases each week

week price of goodduring the week [P]

quantity Susanpurchased [Q]

1 $10 20 units X

2 $15 10 units X

3 $11 15 units X

4 $ 7 22 units X

5 $6 22 units X

Data from these observations can be plotted on the graph

During a 5 week period,Susan was observedmaking the followingpurchases

Clearly there is a pattern, however it is not a perfect relationship.Through statistical inference we can estimate some general characteristics about the relationship

2 4 6 8 10 12 14 16 18 20 22


2 4 6 8 10 12 14 16 18 20 22 24 26Quantity per week

Pri

ce o

f g

ood

X

2

4

6

8

10

12

14

16

18

Given the observed data about Susan’s purchases:

( Q= 10, P= $15)

(15, 11)(20,10)

(22,7)

(22,6)

We can estimate a line that minimizes the square of the difference that each point [that represents two variables]lies off the estimated line.

No single point may liethe line, but the line is anestimate of the relationship

P = 23 - .75Q is our estimateof the relationship between theprice and the quantity that Susanpurchases each week, ceteris paribus orall other things equal

P = 23 - .75Q may be writtenQ = 30.667- 1.333P


2 4 6 8 10 12 14 16 18 20 22 24 26Quantity per week

Pri

ce o

f g

ood

X

2

4

6

8

10

12

14

16

18

Given the observed data about Susan’s purchases:and our estimated function: P = 23 - .75Q or Q = 30.67 - 1.33P,

we would predict that at a price of $10 Susan would purchase about 17.37 units, [Q = 30.67 - 1 .33 P,

P = 10 so Q =17.37]

Q = 17.37

P = 10

We observed that Susan bought 20 units when the price was $10 so estimate is off by a small amount [-2.63 units]

At a price of $6 our equation predictsthat 22.67 units will be purchased

P = 6

Q = 22.67

Since we observed that she purchased 22, we are off

by .67 units

our estimates are not perfect, but theygive an approximation of the relationship


Statistical Estimates

· The estimates are not “perfect” but they provide reasonable estimates

· There are many statistical tools that measure the confidence that we have in out predictions· these include such things as correlation,

coefficient of determination, standard errors, t-scores and F-ratios


Slope & Calculus

· In economics we are interested in how a change in one variable changes another· How a change in price changes sales. How a change

in an input changes output. How a change in output changes cost. etc.

· The rate of change is measured by the slope of the functional relationship· by subtraction the slope was

calculated as rise over run where rise = Y = Y1 - Y2 and run = X = X1 - X2,

slope YX


DerivativeThere are still more slides on this topic

· When we have a nonlinear function, a simple derivative can be used to calculate the slope of the tangent to the function at any value of the independent variable

· The notation for a derivative is written:dY

dXis the change in Y caused bya change in X" "


Summary

· a derivative is the slope of a tangent at a point on a function

· is the rate of change, it measures the change in Y caused by a change in X as the change in X approaches 0

· in economics jargon, [the slope or rate of change] is the “marginal”

dYdX

dYdX

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This is a PowerPoint presentation on fundamental math