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Transcript
ECON 102 Tutorial: Week 3
Shane Murphy
www.lancaster.ac.uk/postgrad/murphys4/
[email protected]
Elasticity
Questions 1-3 deal with elasticity, specifically, we are looking at the price
elasticity of demand.
The price elasticity of demand is the percentage change in quantity
demanded given a percentage change in price. We can write this
mathematically as:
π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘Žπ‘”π‘’ π‘β„Žπ‘Žπ‘›π‘”π‘’ 𝑖𝑛 π‘žπ‘’π‘Žπ‘›π‘‘π‘–π‘‘π‘¦ π‘‘π‘’π‘šπ‘Žπ‘›π‘‘π‘’π‘‘
πœ€=
π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘Žπ‘”π‘’ π‘β„Žπ‘Žπ‘›π‘”π‘’ 𝑖𝑛 π‘π‘Ÿπ‘–π‘π‘’
In this course, we’ll use two methods for elasticity of demand. The Arc
Elasticity, and the Point Elasticity.
Arc Elasticity:
%βˆ† 𝑄
πœ€=
%βˆ† 𝑃
When we use this equation to find
the elasticity between two points,
we call this the arc elasticity.
Point Elasticity:
πœ€=
Where π‘ π‘™π‘œπ‘π‘’ =
1
π‘ π‘™π‘œπ‘π‘’
π‘Ÿπ‘–π‘ π‘’
π‘Ÿπ‘’π‘›
=
𝑃
𝑄
× ,
βˆ†π‘Œ
βˆ†π‘‹
=
βˆ†π‘ƒ
βˆ†π‘„
.
We can use this to find the elasticity at
any particular point.
Question 1(i)
1
2
The demand curve for potatoes is given by: P = 10 – Q
Where p is the price of a pound of potatoes.
Calculate the arc elasticity of demand if the price of potatoes
increases from £2 to £4
Let’s start with our equation for the arc elasticity:
%βˆ† 𝑄
πœ€=
%βˆ† 𝑃
To fill in our numerator, we need to know the quantity demanded for
each given price.
We find that by plugging in a price value into the demand equation,
and solving for Q.
So, At a price of £2,
1
2
2 = 10 – Q
1
2
-8 = – Q
16 = Q
quantity demanded is 16. At a price of £4, quantity demanded is 12.
Question 1(i)
The demand curve for potatoes is given by: P = 10 –
1
Q
2
Calculate the arc elasticity of demand if the price of potatoes
increases from £2 to £4
We found that At a price of £2, quantity demanded is 16. At a price of
£4, quantity demanded is 12.
We now have enough information to fill in our equation for the arc
elasticity:
πœ€=
πœ€=
%βˆ† 𝑄
%βˆ† 𝑃
=
(12 βˆ’16)/16
(4βˆ’2)/2
(𝑄2 βˆ’π‘„1 /𝑄1
(𝑃2 βˆ’π‘ƒ1 )/𝑃1
1
=
βˆ’4
1
= .25
So the arc elasticity of demand is |-1/4| = ¼ or .25.
Question 1(ii)
The demand curve for potatoes is given by:
1
P = 10 – Q
2
Calculate the point price elasticity of demand when p = £3.
Ok, let’s start with our equation for point price elasticity of demand:
1
𝑃
πœ€=
×
π‘ π‘™π‘œπ‘π‘’ 𝑄
There are three things we need to know to be able to solve this
equation: P, Q, and the slope of the demand curve.
We know that P = £3.
We can solve for Q by plugging P into the demand curve equation:
1
2
3 = 10 – Q
1
2
-7 = – Q
14 = Q
So, Q = 14.
Now, all we need to do is find the slope. How can we find the slope?
Question 1(ii)
The demand curve for potatoes is given by:
1
2
P = 10 – Q
Calculate the point price elasticity of demand when p = £3. When P = 3, we found Q = 14.
We have 3 different ways we can find the slope.
One is to utilize the fact that our demand curve is linear. So that means the equation of the
demand curve can be written in slope-intercept form.
Slope-intercept form is written as: Y = mX + c, where m is the slope, and c is the Y-intercept.
So, if we re-write our demand curve in Y = mX + c, format, we will have:
1
1
P = – 2 Q + 10, so our slope is – 2.
We now have enough information to fill in our equation for point price elasticity of demand:
πœ€=
1
π‘ π‘™π‘œπ‘π‘’
×
𝑃
𝑄
Plugging in, we get:
πœ€=
πœ€=
1
3
1×
14
βˆ’2
3
7
The two alternative methods of calculating slope, would be:
1) to use calculus, where the derivative of any function is it’s slope, or
2) to calculate slope using the equation: π‘ π‘™π‘œπ‘π‘’ =
π‘Ÿπ‘–π‘ π‘’
π‘Ÿπ‘’π‘›
=
βˆ†π‘Œ
βˆ†π‘‹
=
βˆ†π‘ƒ
.
βˆ†π‘„
Question 1(iii)
Suppose the price of potatoes is currently £2. If the price of potatoes
were to increase slightly, would total revenue increase or decrease?
How do you know?
1
2
First, let’s calculate elasticity when P = 2. We know that P = 10 – Q,
1
so Q = 16 when P = 2. From part (ii), we know our slope is – .
2
1
𝑃
1
2
1
πœ€=
×
=
×
=
1
π‘ π‘™π‘œπ‘π‘’ 𝑄
16
4
βˆ’
2
Elasticity and the Effect of a Price Change on Total Expenditure
– When price elasticity is greater than 1, changes in price and changes in total
expenditures always move in opposite directions.
– When price elasticity is less than 1, changes in price and changes in total
expenditures always move in the same direction.
– Another way to say this is: that a price increase will increase total revenue when the
% change in P (price) > than the % change in Q (quantity).
Question 2
The demand curve for some good is given by: P = a – bQ
Find the price (in terms of the parameters in the problem) where the
point price elasticity of demand is 1.
For what prices will demand be elastic? For what prices will demand
be inelastic?
Question 2
The demand curve for some good is given by:
P = a – bQ
Find the price (in terms of the parameters in the problem) where the
point price elasticity of demand is 1. For what prices will demand be
elastic? For what prices will demand be inelastic?
The point price elasticity at a point (P, Q) is (1/b)*(P/Q). Setting this
equal to 1 we obtain: P/Q = b, or equivalently, Q = P/b. Plugging this
quantity into the demand curve we obtain: P = a – b*(P/b). Which
means, P = a – P, and hence, P = a/2.
Question 2
The demand curve for some good is given by: P = a – bQ
Find the price (in terms of the parameters in the problem) where the
point price elasticity of demand is 1. For what prices will demand be
elastic? For what prices will demand be inelastic?
As P increases, the ratio P/Q will also increase.
This means that for P > a/2, elasticity will be
greater than 1 (ie demand is elastic), and for P
< a/2, elasticity will be less than 1 (demand is
inelastic).
Question 3
For each of the following pairs of goods, would you
expect the cross-price elasticities to be positive or
negative? Why?
First, let’s define the Cross-Price Elasticity of Demand:
The percentage change in quantity demanded of one good in response
to a 1 percent change in the price of another good.
If the other good is a Substitute Good:
Cross-Price Elasticity of demand is positive
If the other good is a Complement Good:
Cross-Price Elasticity of demand is negative
Question 3
For each of the following pairs of goods, would you expect the
cross-price elasticities to be positive or negative? Why?
a) Almonds and Peanuts
Almonds and peanuts are substitutes.
Therefore, the cross price elasticity of demand is positive.
b) Tortilla chips and Salsa
Tortilla chips and salsa are complements.
Therefore, the cross price elasticity of demand is negative.
c) iPhones and iPhone apps
iPhones and iPhone apps are complements.
Therefore, the cross price elasticity of demand is negative.
Question 4: Ch6 Q1(a)
Suppose the weekly demand and supply curves for used DVDs in Brussels,
are as shown in the graph below.
Consumer surplus is the triangular area
Calculate the weekly consumer surplus. between the demand curve and the
price line.
How do we measure Consumer Surplus?
We measure the area of that particular
triangular area, using the equation for
the area of a triangle.
1
𝐴 = ×𝑏×β„Ž
2
where b is the base of the triangle and h
is the height.
In this case, b = 6 units and h = 1.5 units,
measured in euros.
Therefore, consumer surplus is
1
𝐢𝑆 = × 6 𝑒𝑛𝑖𝑑𝑠 π‘€π‘’π‘’π‘˜ × 1.5 € 𝑒𝑛𝑖𝑑
2
𝐢𝑆 = €4.50 per week.
Question 4: Ch6 Q1(b)
Suppose the weekly demand and supply curves for used DVDs in
Brussels, are as shown in the graph below.
Calculate the weekly producer surplus.
Producer surplus is the triangular area
between the supply curve and the price line.
Using the base-height formula, it is
(0.5)(€4.50/unit)(6 units/wk), or €13.50 per
week.
Question 5: Ch6 Q1(c)
Calculate the maximum weekly amount that producers and consumers in
Brussels would be willing to pay to be able to buy and sell used DVDs in any
given week.
The maximum weekly amount that
consumers and producers together
would be willing to pay to trade in
used DVDs is the sum of gains from
trading in used DVDsβ€”namely, the
total economic surplus generated per
week.
The total economic surplus, in this
case is PS + CS, which is €18 per week.
Question 5: Ch 6 Q2(a)
Refer to Problem 1. Suppose a coalition of students from a Brussels secondary
school succeeds in persuading the local government to impose a price ceiling of €
7.50 on used DVDs, on the grounds that local suppliers are taking advantage of
teenagers by charging exorbitant prices.
Calculate the weekly shortage of used DVDs that will result from this policy.
Question 5: Ch 6 Q2(a)
Refer to Problem 1. Suppose a coalition of students from a Brussels
secondary school succeeds in persuading the local government to
impose a price ceiling of € 7.50 on used DVDs, on the grounds that
local suppliers are taking advantage of teenagers by charging
exorbitant prices.
Calculate the weekly shortage of used DVDs that will result from this
policy.
At a price of €7.50, the quantity supplied per week = 2. The quantity
demanded at this price is 18 per week, which implies a weekly
shortage of 16 used DVDs.
B
1
C64
Price
3
P=6+0.75Q
D
12
6
P=12-0.25Q
2
6
48
Quantity
A
Question 5: Ch 6 Q2(b)
Calculate the total economic surplus lost every week as a result of the
price ceiling.
Question 5: Ch 6 Q2(b)
Calculate the total economic surplus lost every week as a result of the
price ceiling.
The weekly economic surplus lost as a result of the price ceiling (also
called deadweight loss) is the area of the dark-shaded triangle in the
diagram, or the sum of the areas of the two triangles ABC and ACD.
Using the information given in the graph, this amount is calculated as
(0.5)(4)(1) + (0.5)(4)(3) = €8/wk.
B
1
C64
Price
3
P=6+0.75Q
D
12
6
P=12-0.25Q
2
6
48
Quantity
A
Textbook example of consumer surplus and producer
surplus when a price ceiling is imposed
The textbook uses an example of a market for home heating oil. With no price controls the
market equilibrium price is €1.40/litre and the equil. quantity is 3,000 litres/day. We can see
what happens when a price ceiling of €1 is imposed.
With no price controls:
With a price ceiling of €1.00 per litre:
Question 6: Ch 6 Q4(a)
Suppose the weekly demand for a certain good, in thousands of units,
is given by the equation P = 8 – Q, and the weekly supply of the good
is given by the equation P = 2 + Q, where P is the price in euros.
Calculate the total weekly economic surplus generated at the market
equilibrium.
Question 6: Ch 6 Q4(a)
Suppose the weekly demand for a certain good, in thousands of units, is given by the
equation P = 8 – Q, and the weekly supply of the good is given by the equation P = 2 + Q,
where P is the price in euros.
Calculate the total weekly economic surplus generated at the market equilibrium.
The total economic surplus = PS + CS
To find PS & CS, we need to find the
Equilibrium price and equilibrium quantity –
This is where S = D, where the two curves cross.
We can find this 2 ways: graphically, or algebraically.
If we graph, we should get the figure on the right:
Algebraically, we set S = D
2+Q=8–Q
2Q = 6
Q=3
Plugging Q = 3 in to either equation, we get P = 5
So, the equilibrium price is €5 and the equilibrium quantity is 3,000 units per week.
The consumer surplus is the area between the demand curve and the price lineβ€”
triangle ABC in the diagramβ€”which is €4,500/wk. The producer surplus generated is the
area of triangle ABD, which is €4,500/wk. We find these using the A = ½bh formula.
Adding PS + CS, we get the total economic surplus is €9,000/wk.
Question 6: Ch 6 Q4(b)
Demand: P = 8 – Q,
Supply: P = 2 + Q, where P is the price in euros.
Suppose a per-unit tax of €2, to be collected from
sellers, is imposed in this market.
Calculate the direct loss in economic surplus
experienced by participants in this market as a
result of the tax.
Question 6: Ch 6 Q4(b)
Demand: P = 8 – Q,
Supply: P = 2 + Q, where P is the price in euros.
Suppose a per-unit tax of €2, to be collected from sellers, is imposed in this market.
Calculate the direct loss in economic surplus experienced by participants in this
market as a result of the tax.
The tax shifts the vertical intercept of the supply
curve up by €2 to €4.
The new equilibrium price and quantity are €6 and
2,000 respectively.
The tax revenue is €2(2,000), or €4,000/wk.
Consumer surplus is now the area of the triangle
A’B’C, which is €2,000/wk.
Net of the €2 tax, sellers receive a price of €4 per
unit. Their surplus is the area of the triangle D’ED,
which is €2,000/wk.
The direct loss in economic surplus, or the
deadweight loss, is given by the triangle A’AE, and is
equal to €1,000/wk.
Question 6: Ch 6 Q4(c)
Demand: P = 8 – Q,
Supply: P = 2 + Q, where P is the price in euros.
How much government revenue will this tax generate each week? If the revenue is
used to offset other taxes paid by participants in this market, what will be their net
reduction in total economic surplus?
Question 6: Ch 6 Q4(c)
Demand: P = 8 – Q,
Supply: P = 2 + Q, where P is the price in euros.
How much government revenue will this tax generate each week? If the revenue is
used to offset other taxes paid by participants in this market, what will be their net
reduction in total economic surplus?
The tax revenue collected is (€2/unit)(2,000
units/wk) = €4,000/wk.
If we count the revenue from the tax as part of
total economic surplus, the new total
economic surplus is thus €2,000/wk +
€2,000/wk + €4,000/wk = €8,000/wk, or
€1,000/wk less than without the tax.
Textbook example of a market with a tax
The market for potatoes with tax
The market for potatoes without tax
Deadweight loss
tax revenue = (price paid by consumers – price kept by sellers ) *Q
Question 7(a)
As in prob. 4 in ch. 6, suppose the supply curve for some good is given by p = 2 + Q,
and the demand curve for the good is given by p = 8 – Q. Suppose the government
implements a €2 per-unit subsidy for suppliers.
What is the equation of the new supply curve under the subsidy?
p = 2 +Q
8
A
6
B
E
4
F
P=8-Q
2
4
Question 7(a)
As in prob. 4 in ch. 6, suppose the supply curve for some good is given by p = 2 + Q,
and the demand curve for the good is given by p = 8 – Q. Suppose the government
implements a €2 per-unit subsidy for suppliers.
What is the equation of the new supply curve under the subsidy?
The equation of the new supply curve
is p = Q.
The supply curve shifts to the right,
and the vertical distance between the
old and new supply curves is equal to
the amount of the subsidy, €2.
Question 7(b)
Supply Curve: p = Q
Demand curve: p = 8 – Q.
Find the market equilibrium price and quantity under the subsidy. In
this equilibrium what is the price paid by buyers, and what is the price
received by sellers?
p = 2 +Q
8
A
6
C
B
E
4
D
F
P=8-Q
2
4
Question 7(b)
Supply Curve with subsidy: p = Q
Demand curve: p = 8 – Q.
Find the market equilibrium price and quantity under the subsidy. In
this equilibrium what is the price paid by buyers, and what is the price
received by sellers?
p = 2 +Q
8
The market equilibrium price and
A
6
quantity are found by equating the
C
B
D
E
new supply curve with the demand curve.
4
F
2
P=8-Q
So, Q = 8 – Q
4
Q = 4,
which means p = 4.
Buyers pay the price of 4, and sellers receive a price of 4 + subsidy,
which is: 4 + 2 = 6.
Maths Questions
Simplify the following:
23
( i) 4
2
2 32 ο€­1
(ii) 4 ο€­2
22
2 -2 3ο€­1
(iii) -4 ο€­2
2 3
ο€­3
a
(i) ο€­4
a
3b ο€­1
a
(ii) 4 ο€­2
a b
-2 b ο€­1 3ο€­1
a
(iii) -4 ο€­2 ο€­2
a b 3
Maths Questions
Solve for x:
(i) y = x2
(ii) y2 ο€½ x4
(iii) ya ο€½ x b
(i) y = 9x2
(ii) y2 ο€½ 9x4
(iii) ya ο€½ 9x b
Next Week
 Please check Moodle for next week’s worksheet
from Prof. Rietzke (on Wednesdays), and for
maths questions from Prof. Peel (on Thursdays).