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)
fall ‘ 97 Principles of Microeconomics Slide 2
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
fall ‘ 97 Principles of Microeconomics Slide 3
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
fall ‘ 97 Principles of Microeconomics Slide 4
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”
fall ‘ 97 Principles of Microeconomics Slide 5
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”
fall ‘ 97 Principles of Microeconomics Slide 6
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]
fall ‘ 97 Principles of Microeconomics Slide 7
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
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fall ‘ 97 Principles of Microeconomics Slide 8
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
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fall ‘ 97 Principles of Microeconomics Slide 9
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)
fall ‘ 97 Principles of Microeconomics Slide 10
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
, ,
.
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fall ‘ 97 Principles of Microeconomics Slide 11
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
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fall ‘ 97 Principles of Microeconomics Slide 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)?
fall ‘ 97 Principles of Microeconomics Slide 13
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.
fall ‘ 97 Principles of Microeconomics Slide 14
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
fall ‘ 97 Principles of Microeconomics Slide 15
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
fall ‘ 97 Principles of Microeconomics Slide 16
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
fall ‘ 97 Principles of Microeconomics Slide 17
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
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fall ‘ 97 Principles of Microeconomics Slide 18
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
fall ‘ 97 Principles of Microeconomics Slide 19
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
fall ‘ 97 Principles of Microeconomics Slide 20
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
fall ‘ 97 Principles of Microeconomics Slide 21
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]
fall ‘ 97 Principles of Microeconomics Slide 22
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]
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fall ‘ 97 Principles of Microeconomics Slide 23
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”
fall ‘ 97 Principles of Microeconomics Slide 24
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
fall ‘ 97 Principles of Microeconomics Slide 25
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
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fall ‘ 97 Principles of Microeconomics Slide 26
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
fall ‘ 97 Principles of Microeconomics Slide 27
1
2
3
4
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
fall ‘ 97 Principles of Microeconomics Slide 28
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
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fall ‘ 97 Principles of Microeconomics Slide 29
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
fall ‘ 97 Principles of Microeconomics Slide 30
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]
fall ‘ 97 Principles of Microeconomics Slide 31
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.”
fall ‘ 97 Principles of Microeconomics Slide 32
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
fall ‘ 97 Principles of Microeconomics Slide 34
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
fall ‘ 97 Principles of Microeconomics Slide 35
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
fall ‘ 97 Principles of Microeconomics Slide 36
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
fall ‘ 97 Principles of Microeconomics Slide 37
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
fall ‘ 97 Principles of Microeconomics Slide 38
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" "
fall ‘ 97 Principles of Microeconomics Slide 39
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