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EECM 3714

Lecture 2: Unit 2
Linear equations

Renshaw, Ch
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• When manipulating equations, any operation must be performed on both sides of the
equation
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If any operation performed on the whole of both sides of an equation, the equation remains a
true statement
...

Variables and parameters
An equation may contain one or more unknown values, called variables (x, y, z) and one or
more known values, called parameters (coefficients or constants, a, b, c)

GENERAL LINEAR EQUATION
• Equation is linear if none of the variables that appear in it is raised to any power
except power 1
...


• The solution to this equation is 𝑥 =

𝑐−𝑏
𝑎

• Check by replacing x with its solution:

( )

a c a−b + b = c  c − b + b = c

(an identity)

• Note that any equation becomes an identity when we replace the unknown with
its solution
...
In this function:
• 𝑦 is the dependent variable
• 𝑥 is the independent variable
• 𝑎 is the slope → Shows the change in 𝑦 for one-unit change in 𝑥,
• i
...
𝑎 =

Δ𝑦
𝛥𝑥

• 𝑏 is the y-intercept → Shows the value of 𝑦 if 𝑥 is 0
• 𝑥-intercept → The 𝑥-intercept shows the value of 𝑥 if 𝑦 is 0,
• i
...
𝑥 = − 𝑏Τ𝑎

INVERSE FUNCTIONS AND IMPLICIT VS EXPLICIT FUNCTIONS
𝑦−𝑏
𝑎

• Inverse function → if 𝑦 = 𝑎𝑥 + 𝑏, then 𝑥 =
is the inverse function
...

• Any pair of values of 𝑥 and 𝑦 that satisfies one question automatically satisfy the other
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It’s clear which variable is the dependent, and
which is the independent
• Implicit function → 𝑎𝑥 + 𝑏𝑦 − 𝑐 = 0
...


SIMULTANEOUS EQUATIONS: 2 BY 2
• The solution to systems of linear equations involves finding values for all of the
variables that satisfy all of the equations in the system simultaneously
...
5

• Plug this solution into the expression for 𝑦 → 𝑦 = 2(−1
...
5
• Now plug 𝑥 = −1
...
5) + 𝑦 = 9 → ∴ 𝑦 = 6

• Note that these answers are the same (as they should be!)

SIMULTANEOUS EQUATIONS: 3 BY 3
• We can again solve for these using either elimination or substitution
• There are just a few more steps to follow

• Consider the following system of equations:
• 𝑥 + 𝑦 − 𝑧 = −4

• 2𝑥 − 𝑦 − 𝑧 = 0
• 3𝑥 − 4𝑦 − 𝑧 = 0

SUBSTITUTION
• Rewrite the first equation, making 𝑧 the subject: 𝑥 + 𝑦 + 4 = 𝑧
...

• 3rd equation: 3𝑥 − 4𝑦 − 𝑥 + 𝑦 + 4 = 0 ⟹ 2𝑥 − 5𝑦 − 4 = 0
...
t
...
𝑥 → 𝑥 = 2𝑦 + 4

SUBSTITUTION
• Plug the expression 𝑥 = 2𝑦 + 4 into 2𝑥 − 5𝑦 − 4 = 0
• 2 2𝑦 + 4 − 5𝑦 − 4 = 0 ⟹ −𝑦 + 4 = 0; ∴ 𝑦 = 4
...


• Then plug 𝑥 = 12; 𝑦 = 4 into 𝑥 + 𝑦 + 4 = 𝑧
• 𝑧 = 12 + 4 + 4 = 20

ELIMINATION
• Again consider our system of equations:
o 𝑥 + 𝑦 − 𝑧 = −4
...
(2)
o 3𝑥 − 4𝑦 − 𝑧 = 0
...

• Multiply the first equation by 2 and multiply the second equation by 1
...
The result is:
• 3𝑥 + 3𝑦 − 3𝑧 = −12
...
(3a)
• Now subtract (3a) from (1a):
• 3𝑥 + 3𝑦 − 3𝑧 − 3𝑥 − 4𝑦 − 𝑧 = −12 − 0 ⟹ 7𝑦 − 2𝑧 = −12

ELIMINATION 3
• Now you have a 2 by 2 system:
o 3𝑦 − 𝑧 = −8
...
(5)
• Now get rid of 𝑦

• Multiply (4) by 7 and (5) by 3
...
(4a)

• 21𝑦 − 6𝑧 = −36
...


ECONOMIC APPLICATIONS
• Demand and supply analysis
• National Income

• IS-LM analysis

DEMAND AND SUPPLY ANALYSIS
• Demand and supply are the basic building blocks of economic analysis
...

• Supply is based on the theory of the behaviour of the firm
...
g
...
Thus, the market
demand curve is negatively sloped
...
g
...
Thus, the market supply
curve is positively sloped
...
13

• In a perfectly competitive market, demand and supply gives us the equilibrium price and
quantity via the equilibrium condition, 𝑞 𝑑 = 𝑞 𝑠
...

•
•

When the given price is below the equilibrium price, more goods will be demanded than what is
supplied, and we will have excess demand
...


INVERSE DEMAND AND SUPPLY FUNCTIONS
• In economics price is mostly of the vertical axis
...

• Inverse demand function: price is subject of equation: 𝑝 =
• Inverse supply function: price is subject of equation: 𝑝 =
• See Figure 3
...
17

𝑞𝑠
𝑐

𝑏
𝑎

1 𝑑
− 𝑞
𝑎

−

𝑑
𝑐

DEMAND AND SUPPLY ANALYSIS: EXAMPLES
1
...


Solve for the equilibrium prices, 𝑝𝑖 , as well as the equilibrium quantities, 𝑞𝑖 , (recall
that in equilibrium, 𝑞𝑖𝐷 = 𝑞𝑖𝑆 )
...
Consider the following system of equations, depicting the equilibrium conditions
for three interdependent commodities:
2𝑝1 = 77 − 4𝑝2 − 𝑝3 ; 3𝑝2 = 114 − 4𝑝1 − 7𝑝3 ; 3𝑝3 = 48 − 2𝑝1 − 𝑝2
...


EXAMPLE 1
• In equilibrium, 𝑞1𝐷 = 𝑞1𝑆 and 𝑞2𝐷 = 𝑞2𝑆
...
t
...

• 𝑝3 → 𝑝3 = 77 − 2𝑝1 − 4𝑝2

EXAMPLE 2 - CONTINUED
• Plug this expression into the second and third equations:
• 3𝑝2 = 114 − 4𝑝1 − 7 77 − 2𝑝1 − 4𝑝2

• Simplify: −10𝑝1 − 25𝑝2 = −425
...
(b)
• Now multiply (a) with 1 and (b) with 2
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(a)
• −10𝑝1 − 27
...
5
...
Like multiply (a)
with 4 and (b) with 10 to eliminate 𝑝1
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5 ⟹ 𝑝2 = 13
• Plug this into (a):

• −10𝑝1 − 25 13 = −425 ⟹ −10𝑝1 = −100, ∴ 𝑝1 = 10
• Plug these two results into the expression for 𝑝3 :

• 𝑝3 = 77 − 2 10 − 4 13 = 5
• Note that you can use both substitution and elimination in the same problem (as

was done here)

TAXES, DEMAND AND SUPPLY
• Two types of taxes levied on buying/selling of goods and services:
• Specific tax (Per unit tax): Fixed amount per unit bought/sold
...
g
...
g
...
If this is the case, then the supply curve
will change after the tax is imposed
• Quantity supplied depends on the price received by sellers/producers, NOT on
the price that they charge
• For a specific tax: Price charged = Price received + tax or 𝑝 = 𝑝′ + 𝑇
...
Then the price received is:

𝑝′

=

𝑝
1+𝑡

TAXES 2
• For a specific tax, the supply curve becomes 𝑞 𝑠 = 𝑑 + 𝑐(𝑝 − 𝑇), while the
1
𝑐

1
𝑐

1
𝑐

1
𝑐

inverse supply curve becomes 𝑝 − 𝑇 = 𝑞 𝑠 − 𝑑 ⟹ 𝑝 = 𝑞 𝑠 − 𝑑 + 𝑇

• For an ad valorem tax, the supply curve becomes 𝑞 𝑠 = 𝑑 + 𝑐
inverse supply curve becomes

𝑝
1+𝑡

=

𝑞𝑠
𝑐

𝑑
−
𝑐

⟹𝑝 = 1+𝑡

𝑞𝑠
𝑐

−

𝑝
1+𝑡
𝑑
𝑐

, while the

TAXES 3
• Suppose that buyers have to pay the tax
...
Find the equilibrium price and equilibrium
quantity of widgets after the imposition of the per-unit tax
...
e
...


SOLUTION: SPECIFIC TAX
• Before tax: equilibrium is given by 𝑞 𝐷 = 𝑞 𝑆 = 𝑞 and 𝑝 = 𝑝
...
The price received would be 140, while price paid
would be 200
• Note: after the tax: higher price, lower quantity
• Regarding the tax burden/incidence:
• Before tax, consumers paid 𝑅180
...
They pay 𝑅20 more
...
After tax, they receive 𝑅200 − 𝑅60 =
𝑅140
...


AD VALOREM TAX EXAMPLE
• The (inverse) demand for tobacco is 𝑝 = 80 − 5𝑞𝑑 , while the (inverse) supply of
tobacco is 𝑝 = 2𝑞 𝑠 + 10
...

• Find equilibrium price and quantity of tobacco before and after the tax
...


AD VALOREM TAX SOLUTION
• Equilibrium before tax: 80 − 5𝑞 = 2𝑞 + 10
• ∴ 𝑞 = 10; 𝑝 = 30

• After tax supply curve:

𝑝
1+0
...
15 2𝑞 𝑠 + 10 = 2
...
5
• Equilibrium after tax: 80 − 5𝑞 = 2
...
5
• ∴ 𝑞 = 9
...
08

AD VALOREM TAX SOLUTION 2
• Incidence:
• Before the tax, consumers paid 𝑅30
...
08
...
08 more
after tax
...
After the tax, they receive

33
...
77
...
23 less after the tax
...


• Given 𝑝 = 69,56 − 4,348𝑞 𝑑 𝑎𝑛𝑑 𝑝 = 2𝑞 𝑠 + 10
• The price received is then R28
...
08

NATIONAL INCOME
• Suppose that 𝑇𝐸 = 𝐶 + 𝐼 + 𝐺 + (𝑋 − 𝑀)
• Where: 𝐶 = 𝐶0 + 𝑐(𝑌 − 𝑇), 𝐼 = 𝐼0 , 𝐺 = 𝐺0 , 𝑋 = 𝑋0 , 𝑀 = 𝑀0 + 𝑚𝑌 and 𝑇 = 𝑇0 + 𝑡𝑌

• In macroeconomic equilibrium, 𝑌 = 𝑇𝐸 → 𝑌 = 𝐶 + 𝐼 + 𝐺 + (𝑋 − 𝑀)
• So, in macroeconomic equilibrium: 𝑌 = 𝐶0 + 𝑐 𝑌 − 𝑇0 + 𝑡𝑌

(𝑀0 + 𝑚𝑌))
• To solve, collect all 𝑌𝑠 on the LHS

+ 𝐼0 + 𝐺0 + (𝑋0 −

NATIONAL INCOME
• Note that 𝑌 − 𝑇 = 𝐶 + 𝑆
...
9(𝑌 − 𝑇), 𝐼 =

900; 𝐺 = 1200; 𝑋 = 700, 𝑀 = 200 + 0
...
3𝑌
• Solve for equilibrium national income
• Write down the savings function
...
Is the budget in a
surplus or deficit?

• Calculate the value of the trade balance at the equilibrium income level
...
9(𝑌 − 0
...
1𝑌)

• → 𝑌 = 2820 + 0
...
47𝑌 = 2820

• ∴ 𝑌 = 6000

SOLUTION: NATIONAL INCOME 2
• Saving: 𝑆 = 𝑌 − 𝑇 − 𝐶
• 𝑆 = 𝑌 − 𝑇 − (220 + 0
...
1 𝑌 − 𝑇 − 220 = 0
...
3𝑌 − 220
• 𝑆 = 0
...
07 6000 − 220 = 200 > 0

SOLUTION: NATIONAL INCOME 2
• Budget deficit = 𝐺 − 𝑇
• 𝐺 = 1200; 𝑇 = 0
...
1 6000 = 800
• ⟹ 𝑋 − 𝑀 = −100 < 0
• Therefore, trade deficit

IS-LM ANALYSIS
• IS = shows combinations of Y and r for which the goods market is in equilibrium
• Equilibrium condition: 𝑌 = 𝑇𝐸; 𝑇𝐸 = 𝐶 + 𝐼 + 𝐺 ⟹ 𝑌 = 𝐶 + 𝐼 + 𝐺

• LM = shows combinations of Y and r for which the money market is in equilibrium
• Equilibrium condition: 𝑀𝑑 = 𝑀𝑠

• Macroeconomic equilibrium (simultaneous equilibrium in goods and money
markets) = intersection of IS-LM (set IS equal to LM)

IS-LM ANALYSIS EXAMPLE
• The consumption and investment functions of Free state are 𝐶 = 102 + 0
...
Furthermore, suppose that the demand for money and the
supply of money in Free State are 𝑀𝑑 = 124 − 200𝑟 + 0
...
7𝑌 + 150 − 100𝑟
...
3𝑌 = 252 − 100𝑟 (IS)
• For LM → Equilibrium condition is 𝑀𝑑 = 𝑀𝑠

• This means that 124 − 200𝑟 + 0
...
25𝑌 = 176 + 200𝑟 (LM)

IS-LM SOLUTION
• Now, multiply LM by 4: 𝑌 = 704 + 800𝑟
• Plug this solution into IS: 0
...
2 + 240𝑟 = 252 − 100𝑟
• 340𝑟 = 40
...
12,

• ∴ 𝑌 = 704 + (800 × 0

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