UNIVERSITY OF PERADENIYA
Mathematical Models
Algebra Review and Applications
Exponential Functions
Name Thevuni Kotigala
University University of Peradeniya, Sri Lanka
Faculty Faculty of Arts
Programme Masters in Development Practice (MDP)
Course Year 2020/21
Academic Year Pre-MDP
Lecturer Dr. S.J.S. de Mel
Department Department of Economics & Statistics
Subject Code DPR 402
Subject Name Basic Mathematics
Medium English
Mathematical Models, Algebra Review & Applications, and
Topic
Exponential Functions
Word Count 1,136 words
, 1. Assume that the demand for and supply of cigarettes are given by q + 2 p = 40 and
q − 6 p = −12 respectively, where p is the price per cigarette and q is quantity (in
thousands).
(a) Find the supply function q S ( p) , the inverse supply function p S (q) , the demand
function q D ( p ) , and the inverse demand function p D (q ) .
Demand → 𝑞 + 2𝑝 = 40 Supply → 𝑞 − 6𝑝 = −12
Demand function: Supply function:
𝑞 𝐷 (𝑝) = 𝒒 = 𝟒𝟎 − 𝟐𝒑 𝑞 𝑆 (𝑝) = 𝒒 = 𝟔𝒑 − 𝟏𝟐
Inverse demand function: Inverse supply function:
𝑝𝐷 (𝑞) = 2𝑝 = 40 − 𝑞 𝑝 𝑆 (𝑞) = 6𝑝 = 𝑞 + 12
40−𝑞 𝑞+12
𝑝= 2 𝑝= 6
𝒒 𝒒
𝒑 = 𝟐𝟎 − 𝒑= +𝟐
𝟐 𝟔
(b) Find the equilibrium price and quantity
Market Equilibrium → Quantity Supplied = Quantity Demanded
𝑞 𝑆 (𝑝) = 𝑞 𝐷 (𝑝) When 𝑝 = 6.5,
6𝑝 − 12 = 40 − 2𝑝 𝑞 = 6𝑝 − 12
6𝑝 + 2𝑝 = 40 + 12 𝑞 = 6(6.5) − 12
8𝑝 = 52 𝑞 = 39 − 12
𝒑 = 𝟔. 𝟓 𝒒 = 𝟐𝟕
At the market equilibrium, the price per cigarette is Rs. 6.50, and the quantity demanded is
27,000 cigarettes.
(c) Suppose that the government imposes an excise tax of Rs T per unit of each good.
Find the new equilibrium price and quantity (in terms of T).
𝑞 𝐷 (𝑝) = 40 − 2𝑝 → (1)
𝑞 𝑆 (𝑝) = 6(𝑝 − 𝑇) − 12 → (2)
Market Equilibrium → 𝑞 𝐷 (𝑝) = 𝑞 𝑆 (𝑝) When 𝑝𝑇 = 6.5 + 0.75𝑇
40 − 2𝑝 = 6(𝑝 − 𝑇) − 12 𝑞 𝑇 = 40 − 2(6.5 + 0.75𝑇)
40 − 2𝑝 = 6𝑝 − 6𝑇 − 12 𝑞 𝑇 = 40 − 13 − 1.5𝑇
8𝑝 = 52 + 6𝑇 𝒒𝑻 = 𝟐𝟕 − 𝟏. 𝟓𝑻
52+6𝑇
𝑝𝑇 = 8
𝒑𝑻 = 𝟔. 𝟓 + 𝟎. 𝟕𝟓𝑻
At the new market equilibrium, where the excise tax is imposed, the price per cigarette is
increased to 6.5 + 0.75T rupees, and the quantity demanded is decreased to 27 – 1.5T cigarettes.
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Mathematical Models
Algebra Review and Applications
Exponential Functions
Name Thevuni Kotigala
University University of Peradeniya, Sri Lanka
Faculty Faculty of Arts
Programme Masters in Development Practice (MDP)
Course Year 2020/21
Academic Year Pre-MDP
Lecturer Dr. S.J.S. de Mel
Department Department of Economics & Statistics
Subject Code DPR 402
Subject Name Basic Mathematics
Medium English
Mathematical Models, Algebra Review & Applications, and
Topic
Exponential Functions
Word Count 1,136 words
, 1. Assume that the demand for and supply of cigarettes are given by q + 2 p = 40 and
q − 6 p = −12 respectively, where p is the price per cigarette and q is quantity (in
thousands).
(a) Find the supply function q S ( p) , the inverse supply function p S (q) , the demand
function q D ( p ) , and the inverse demand function p D (q ) .
Demand → 𝑞 + 2𝑝 = 40 Supply → 𝑞 − 6𝑝 = −12
Demand function: Supply function:
𝑞 𝐷 (𝑝) = 𝒒 = 𝟒𝟎 − 𝟐𝒑 𝑞 𝑆 (𝑝) = 𝒒 = 𝟔𝒑 − 𝟏𝟐
Inverse demand function: Inverse supply function:
𝑝𝐷 (𝑞) = 2𝑝 = 40 − 𝑞 𝑝 𝑆 (𝑞) = 6𝑝 = 𝑞 + 12
40−𝑞 𝑞+12
𝑝= 2 𝑝= 6
𝒒 𝒒
𝒑 = 𝟐𝟎 − 𝒑= +𝟐
𝟐 𝟔
(b) Find the equilibrium price and quantity
Market Equilibrium → Quantity Supplied = Quantity Demanded
𝑞 𝑆 (𝑝) = 𝑞 𝐷 (𝑝) When 𝑝 = 6.5,
6𝑝 − 12 = 40 − 2𝑝 𝑞 = 6𝑝 − 12
6𝑝 + 2𝑝 = 40 + 12 𝑞 = 6(6.5) − 12
8𝑝 = 52 𝑞 = 39 − 12
𝒑 = 𝟔. 𝟓 𝒒 = 𝟐𝟕
At the market equilibrium, the price per cigarette is Rs. 6.50, and the quantity demanded is
27,000 cigarettes.
(c) Suppose that the government imposes an excise tax of Rs T per unit of each good.
Find the new equilibrium price and quantity (in terms of T).
𝑞 𝐷 (𝑝) = 40 − 2𝑝 → (1)
𝑞 𝑆 (𝑝) = 6(𝑝 − 𝑇) − 12 → (2)
Market Equilibrium → 𝑞 𝐷 (𝑝) = 𝑞 𝑆 (𝑝) When 𝑝𝑇 = 6.5 + 0.75𝑇
40 − 2𝑝 = 6(𝑝 − 𝑇) − 12 𝑞 𝑇 = 40 − 2(6.5 + 0.75𝑇)
40 − 2𝑝 = 6𝑝 − 6𝑇 − 12 𝑞 𝑇 = 40 − 13 − 1.5𝑇
8𝑝 = 52 + 6𝑇 𝒒𝑻 = 𝟐𝟕 − 𝟏. 𝟓𝑻
52+6𝑇
𝑝𝑇 = 8
𝒑𝑻 = 𝟔. 𝟓 + 𝟎. 𝟕𝟓𝑻
At the new market equilibrium, where the excise tax is imposed, the price per cigarette is
increased to 6.5 + 0.75T rupees, and the quantity demanded is decreased to 27 – 1.5T cigarettes.
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