Bahan Mata Kuliah Matematika Ekonomi Gratis Terbaru - Kosngosan Situs Anak Kost Mahasiswa Pelajar Course 3
EQUATION IN ECONOMICS
(Course 3)
OLEH
SYAIFUL HADI
DJAIMI BAKCE
JURUSAN AGRIBISNIS
FAKULTAS PERTANIAN
UNIVERSITAS RIAU
INTRODUCTION
An equation is a statement that two expressions
are equal to one another.
In economic modelling we express relationships
are equations and then use them to obtain
analytical result. Solving the equations gives us
values for which the equations are true.
We can express the condition for market
equilibrium as an equation in terms of price, P,
solving the equation for P tells us the price at
which the market is in equlibrium
REWRITING AND SOLVING
EQUATIONS
When rewriting equation:
1. Add to or subtract from both sides.
2. Multiply or divide through the whole or each side (but don’t
divide by 0).
3. Square or take the square root of each side.
4. Use as many stages as you wish.
5. Take care to get all the signs correct.
Example: Plot the equations y = -5 + 2x and y = 30 - 3x. At what
values of x and y do they cross ? Find also the algebraic solution by
setting the two expressions in x equal to one other.
We are asked to plot two linier functions, so plotting two points on
each then connecting them will suffice.
The table shows x value of 0 and 10 and the corresponding y values
for each line. These points are used to plot the lines shown in figure.
Notice that the line cross at x = 7, y = 9 (yang cara subtitusi sperti
dibalik)
x
y=-5+2x
y=30-3x
0
2
4
6
8
10
12
-5
30
-1
24
3
18
7
12
11
6
15
0
19
-6
For algebraic solution, the
two y value are equal so
equate the right hand sides of
the expressions and solve for
x:
-5 + 2x = 30 – 3x
We want term in ix on the
left-hand side but not on the
right, so add 3x to both sides
since -3x + 3x = 0. We then
have:
or
-5 + 2x + 3x = 30
-5 + 5x = 30
To remove the constant term
from the left side we now add
5 to each side, giving:
y=-5+2x
y=-30-3x
Lines intersect at (7, 9)
5x = 35
And so, dividing by 5, we have x
=7
We then find the value for y by
substi tuting x=7 in either of the
equations. Using y=30 – 3x gives:
y = 30 – 21 = 9
Which confirms the graphical
result. The solution is x = 7 and y
=9
SOLUTION IN TERMS OF OTHER
VARIABLES
Not all the equations you deal with have numerical solutions.
Sometimes when you solve and equation for x you obtain and
expression containing other variables.
Use same rules to transpose the equation.
Remember that in the solution x will not occur on the right-hand side
and will be on its own the left-hand side.
If you are given a relationship in the form y=f(x), rewriting the
equation in the form x=g(y) is called finding the inverse function.
To be able to find the inverse there must be just one x value
corresponding to each y value.
For non linier function there can be difficulties in finding an inverse,
but we may be able to do so for restricted set values.
The function y=x2 has two x values (one positive and one negative)
corresponding to every y value, but if we consider the restricted
function y=x2, x>0 this function has the inverse x=y.
For the linier functions often use in economic models inverse functions
can always be found. One reason for finding the inverse function can
always be found. One by y is conventionally plotted in economic on
the horizontal axis.
Demand and supply equations provide examples of this.
Solve for x in term of z
x = 60 + 0.8x + 7z
At first glance you seem to already have a solution for x, but
notice that x occurs also on the right-hand side of the
equation. We must collect terms in x on the left-hand, so we
subtract 0.8x from both sides and obtain:
x - 0.8x = 60 + 7z
since both left-hand side terms contain x we may write:
(1 – 0.8)x = 60 + 7z
which gives
0.2x = 60 + 7z
To get x with a coefficient of 1 we divide both sides by 0.2 =
1/5, which is the same thing as multiplying both sides by 5.
This gives:
x = 300 + 35z
Given y = x + 5, obtain an expression for x in term of y.
Begin by interchanging the side so that the sides with x is on
the left of the equation. We then have:
x + 5 = y
Next subtract 5 from both sides, giving:
x = y – 5
To find x we must square both sides. This means that the whole
of the right-hand side is multiplied by itself, so use brackets.
We obtain:
x = (y – 5)2
Squaring out the bracket we may also write:
x = y2 – 10y + 25
SUBSTITUTION
When two expression are equal to one another, either can
be substituted for the other.
The technique is used the effect of the imposition of a per
unit tax on a good and to solve simultaneous equations.
When substituting, always be sure to substitute the whole
of the new expression and combine it with the other term in
exactly the same way the expression it replaces was
combined with them.
For example, if y = x2 + 6 and x = 30 - , find an
expression for y in term of . Substituting 30 - for x we
obtain:
y = (30 - )2 + 6
which on multiplying out and collecting term becomes
y = 900 – 54 + 2
DEMAND AND SUPPLY
Demand and supply function in economics express
the quantity demanded or supplied as a function of
price, Q = f (P).
According to mathematical convention the dependent
variable (Q) should be plotted on the vertical axis.
Economic analysis, however, use the horizontal axis
as the Q and for consistency we follow that approach.
So that we can determine the points on graph in the
usual way, before plotting a demand or supply
function we first find its inverse function giving P as a
function of Q.
Find the inverse function for the demand equation Q = 80 – 2P
and sketch the demand curve.
Adding 2P to both sides of the demand equation we get
2P + Q = 80
Subtracting Q from both sides we obtain:
2P = 80 – Q
Dividing each side by 2 gives the inverse function
P = (80 – Q)/2 = 40 – (Q/2)
The demand function is linier, so it suffices to plot two point.
Selected values of Q are shown in the table together with
corresponding value for P.
Q
P = 40 - (Q/2)
0
40
20
30
40
20
60
10
80
0
MARKET EQULIBRIUM
Market equilibrium occurs when the quantity supplied
equals the quantity demanded of a good.
The supply and demand curves cross at the equilibrium
price and quantity.
If you plot both the demanded and supply curves you
can of approximate equilibrium values from the graph.
Another approach is to solve algebraically for the point
where the demand and supply equation are equal. This
gives exact value. Suppose we wish to find the
equilibrium price and quantity when demand is given by
Demand: Q = 96 – 4P
And the supply equation is
Supply : Q = 8P
For an algebraic solution we can use the equation in this form.
Since in equilibrium then quantity supplied equals the quantity
demanded, the right-hand side of the supply equation must
equal the right-hand side of the demand equation. This gives an
equation in P:
Supply Q = Demand Q (in equilibrium), so
8P = 96 – 4P
Adding 4P to both sides gives
12P = 96
Dividing by 12 we find P = 8 which is the equilibrium price. We
can then substitute this into either equation, say the equation.
This gives:
Q = 8 x 8 = 64
Which is the quantity supplied in equilibrium and therefore also
the quantity demanded. Market equilibrium occurs at Q = 64, P
= 8.
Jawaban QUIS I
1.
Sketch the total cost function: TC = 300 + 40Q – 10Q2 + Q3,
write expressions for AC, FC, VC and AVC !
AC = 300/Q + 40 – 10Q +
Q2
0Q
4
+
00 3
3
Q
=
TC Q2 +
10
FC = 300
VC = 40Q – 10Q2 + Q3
AVC = 40 – 10Q + Q2
Q
TC
0
300
1
331
2
348
3
4
357 364
5
375
6
396
7
433
8
492
9
579
10
700
11
861
12
1068
Jawaban QUIS I
2. If the firm in question 1 faces the demand curve P = 100 – 0.5Q
Find an expression for the firm’s profit function and sketch
the curve !
TR = P . Q = 100Q – 0.5Q2
Profit = TR – TC = 100Q – 0.5Q2 – (300 + 40Q – 10Q2 + Q3)
= -300 + 60Q + 9.5Q2 – Q3
400
300
200
Profit
100
=-300+60Q+9.5Q2Q3
0
0
1
2
3
4
5
6
7
-100
-200
-300
-400
Q
8
9
10
11
12
13
14
Jawaban QUIS I
3. A firm in perfect competition sells it output at a price Rp
12. Plot it total revenue function (TR) = 12Q !
Price constant, TR a line through the origin
-
1
12
2
24
3
36
4
48
5
60
6
72
7
84
8
96
9
108
10
120
11
132
12
144
13
156
14
168
180
160
140
2
=1
R
T
120
100
TR
Q
TR
Q
80
60
40
20
-
1
2
3
4
5
6
7
Q
8
9
10
11
12
13
14
QUIS II
Rewriting these equations expressing
P as a function of Q then plot them on
a graph
Supply : Q = 4P
Demand : Q = 280 – 10P
(Course 3)
OLEH
SYAIFUL HADI
DJAIMI BAKCE
JURUSAN AGRIBISNIS
FAKULTAS PERTANIAN
UNIVERSITAS RIAU
INTRODUCTION
An equation is a statement that two expressions
are equal to one another.
In economic modelling we express relationships
are equations and then use them to obtain
analytical result. Solving the equations gives us
values for which the equations are true.
We can express the condition for market
equilibrium as an equation in terms of price, P,
solving the equation for P tells us the price at
which the market is in equlibrium
REWRITING AND SOLVING
EQUATIONS
When rewriting equation:
1. Add to or subtract from both sides.
2. Multiply or divide through the whole or each side (but don’t
divide by 0).
3. Square or take the square root of each side.
4. Use as many stages as you wish.
5. Take care to get all the signs correct.
Example: Plot the equations y = -5 + 2x and y = 30 - 3x. At what
values of x and y do they cross ? Find also the algebraic solution by
setting the two expressions in x equal to one other.
We are asked to plot two linier functions, so plotting two points on
each then connecting them will suffice.
The table shows x value of 0 and 10 and the corresponding y values
for each line. These points are used to plot the lines shown in figure.
Notice that the line cross at x = 7, y = 9 (yang cara subtitusi sperti
dibalik)
x
y=-5+2x
y=30-3x
0
2
4
6
8
10
12
-5
30
-1
24
3
18
7
12
11
6
15
0
19
-6
For algebraic solution, the
two y value are equal so
equate the right hand sides of
the expressions and solve for
x:
-5 + 2x = 30 – 3x
We want term in ix on the
left-hand side but not on the
right, so add 3x to both sides
since -3x + 3x = 0. We then
have:
or
-5 + 2x + 3x = 30
-5 + 5x = 30
To remove the constant term
from the left side we now add
5 to each side, giving:
y=-5+2x
y=-30-3x
Lines intersect at (7, 9)
5x = 35
And so, dividing by 5, we have x
=7
We then find the value for y by
substi tuting x=7 in either of the
equations. Using y=30 – 3x gives:
y = 30 – 21 = 9
Which confirms the graphical
result. The solution is x = 7 and y
=9
SOLUTION IN TERMS OF OTHER
VARIABLES
Not all the equations you deal with have numerical solutions.
Sometimes when you solve and equation for x you obtain and
expression containing other variables.
Use same rules to transpose the equation.
Remember that in the solution x will not occur on the right-hand side
and will be on its own the left-hand side.
If you are given a relationship in the form y=f(x), rewriting the
equation in the form x=g(y) is called finding the inverse function.
To be able to find the inverse there must be just one x value
corresponding to each y value.
For non linier function there can be difficulties in finding an inverse,
but we may be able to do so for restricted set values.
The function y=x2 has two x values (one positive and one negative)
corresponding to every y value, but if we consider the restricted
function y=x2, x>0 this function has the inverse x=y.
For the linier functions often use in economic models inverse functions
can always be found. One reason for finding the inverse function can
always be found. One by y is conventionally plotted in economic on
the horizontal axis.
Demand and supply equations provide examples of this.
Solve for x in term of z
x = 60 + 0.8x + 7z
At first glance you seem to already have a solution for x, but
notice that x occurs also on the right-hand side of the
equation. We must collect terms in x on the left-hand, so we
subtract 0.8x from both sides and obtain:
x - 0.8x = 60 + 7z
since both left-hand side terms contain x we may write:
(1 – 0.8)x = 60 + 7z
which gives
0.2x = 60 + 7z
To get x with a coefficient of 1 we divide both sides by 0.2 =
1/5, which is the same thing as multiplying both sides by 5.
This gives:
x = 300 + 35z
Given y = x + 5, obtain an expression for x in term of y.
Begin by interchanging the side so that the sides with x is on
the left of the equation. We then have:
x + 5 = y
Next subtract 5 from both sides, giving:
x = y – 5
To find x we must square both sides. This means that the whole
of the right-hand side is multiplied by itself, so use brackets.
We obtain:
x = (y – 5)2
Squaring out the bracket we may also write:
x = y2 – 10y + 25
SUBSTITUTION
When two expression are equal to one another, either can
be substituted for the other.
The technique is used the effect of the imposition of a per
unit tax on a good and to solve simultaneous equations.
When substituting, always be sure to substitute the whole
of the new expression and combine it with the other term in
exactly the same way the expression it replaces was
combined with them.
For example, if y = x2 + 6 and x = 30 - , find an
expression for y in term of . Substituting 30 - for x we
obtain:
y = (30 - )2 + 6
which on multiplying out and collecting term becomes
y = 900 – 54 + 2
DEMAND AND SUPPLY
Demand and supply function in economics express
the quantity demanded or supplied as a function of
price, Q = f (P).
According to mathematical convention the dependent
variable (Q) should be plotted on the vertical axis.
Economic analysis, however, use the horizontal axis
as the Q and for consistency we follow that approach.
So that we can determine the points on graph in the
usual way, before plotting a demand or supply
function we first find its inverse function giving P as a
function of Q.
Find the inverse function for the demand equation Q = 80 – 2P
and sketch the demand curve.
Adding 2P to both sides of the demand equation we get
2P + Q = 80
Subtracting Q from both sides we obtain:
2P = 80 – Q
Dividing each side by 2 gives the inverse function
P = (80 – Q)/2 = 40 – (Q/2)
The demand function is linier, so it suffices to plot two point.
Selected values of Q are shown in the table together with
corresponding value for P.
Q
P = 40 - (Q/2)
0
40
20
30
40
20
60
10
80
0
MARKET EQULIBRIUM
Market equilibrium occurs when the quantity supplied
equals the quantity demanded of a good.
The supply and demand curves cross at the equilibrium
price and quantity.
If you plot both the demanded and supply curves you
can of approximate equilibrium values from the graph.
Another approach is to solve algebraically for the point
where the demand and supply equation are equal. This
gives exact value. Suppose we wish to find the
equilibrium price and quantity when demand is given by
Demand: Q = 96 – 4P
And the supply equation is
Supply : Q = 8P
For an algebraic solution we can use the equation in this form.
Since in equilibrium then quantity supplied equals the quantity
demanded, the right-hand side of the supply equation must
equal the right-hand side of the demand equation. This gives an
equation in P:
Supply Q = Demand Q (in equilibrium), so
8P = 96 – 4P
Adding 4P to both sides gives
12P = 96
Dividing by 12 we find P = 8 which is the equilibrium price. We
can then substitute this into either equation, say the equation.
This gives:
Q = 8 x 8 = 64
Which is the quantity supplied in equilibrium and therefore also
the quantity demanded. Market equilibrium occurs at Q = 64, P
= 8.
Jawaban QUIS I
1.
Sketch the total cost function: TC = 300 + 40Q – 10Q2 + Q3,
write expressions for AC, FC, VC and AVC !
AC = 300/Q + 40 – 10Q +
Q2
0Q
4
+
00 3
3
Q
=
TC Q2 +
10
FC = 300
VC = 40Q – 10Q2 + Q3
AVC = 40 – 10Q + Q2
Q
TC
0
300
1
331
2
348
3
4
357 364
5
375
6
396
7
433
8
492
9
579
10
700
11
861
12
1068
Jawaban QUIS I
2. If the firm in question 1 faces the demand curve P = 100 – 0.5Q
Find an expression for the firm’s profit function and sketch
the curve !
TR = P . Q = 100Q – 0.5Q2
Profit = TR – TC = 100Q – 0.5Q2 – (300 + 40Q – 10Q2 + Q3)
= -300 + 60Q + 9.5Q2 – Q3
400
300
200
Profit
100
=-300+60Q+9.5Q2Q3
0
0
1
2
3
4
5
6
7
-100
-200
-300
-400
Q
8
9
10
11
12
13
14
Jawaban QUIS I
3. A firm in perfect competition sells it output at a price Rp
12. Plot it total revenue function (TR) = 12Q !
Price constant, TR a line through the origin
-
1
12
2
24
3
36
4
48
5
60
6
72
7
84
8
96
9
108
10
120
11
132
12
144
13
156
14
168
180
160
140
2
=1
R
T
120
100
TR
Q
TR
Q
80
60
40
20
-
1
2
3
4
5
6
7
Q
8
9
10
11
12
13
14
QUIS II
Rewriting these equations expressing
P as a function of Q then plot them on
a graph
Supply : Q = 4P
Demand : Q = 280 – 10P