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supplied, and we will have excess demand
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INVERSE DEMAND AND SUPPLY FUNCTIONS
• In economics price is mostly of the vertical axis
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• Inverse demand function: price is subject of equation: 𝑝 =
• Inverse supply function: price is subject of equation: 𝑝 =
• See Figure 3
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17

𝑞𝑠
𝑐

𝑏
𝑎

1 𝑑
− 𝑞
𝑎

−

𝑑
𝑐

DEMAND AND SUPPLY ANALYSIS: EXAMPLES
1
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Solve for the equilibrium prices, 𝑝𝑖 , as well as the equilibrium quantities, 𝑞𝑖 , (recall
that in equilibrium, 𝑞𝑖𝐷 = 𝑞𝑖𝑆 )
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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
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EXAMPLE 1
• In equilibrium, 𝑞1𝐷 = 𝑞1𝑆 and 𝑞2𝐷 = 𝑞2𝑆
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t
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• 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
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(b)
• Now multiply (a) with 1 and (b) with 2
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(a)
• −10𝑝1 − 27
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5
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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
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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
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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
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Find the equilibrium price and equilibrium
quantity of widgets after the imposition of the per-unit tax
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e
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SOLUTION: SPECIFIC TAX
• Before tax: equilibrium is given by 𝑞 𝐷 = 𝑞 𝑆 = 𝑞 and 𝑝 = 𝑝
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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
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They pay 𝑅20 more
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After tax, they receive 𝑅200 − 𝑅60 =
𝑅140
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AD VALOREM TAX EXAMPLE
• The (inverse) demand for tobacco is 𝑝 = 80 − 5𝑞𝑑 , while the (inverse) supply of
tobacco is 𝑝 = 2𝑞 𝑠 + 10
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• Find equilibrium price and quantity of tobacco before and after the tax
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AD VALOREM TAX SOLUTION
• Equilibrium before tax: 80 − 5𝑞 = 2𝑞 + 10
• ∴ 𝑞 = 10; 𝑝 = 30

• After tax supply curve:

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

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

33
...
77
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23 less after the tax
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• Given 𝑝 = 69,56 − 4,348𝑞 𝑑 𝑎𝑛𝑑 𝑝 = 2𝑞 𝑠 + 10
• The price received is then R28
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08

NATIONAL INCOME
• Suppose that 𝑇𝐸 = 𝐶 + 𝐼 + 𝐺 + (𝑋 − 𝑀)
• W

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