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ECONOMICS I – ACADEMIC YEAR 2017-2018
TUTORIAL 3 – Government intervention, taxation, economic welfare
TA: ANDREA VENEGONI
EXERCISE 1. Consumer and Producer Surplus, Price Floors and Ceilings (14 pts)
Suppose that in the Malaysian market of coconuts the demand function is Qd= 110-P and the supply
function is Qs= -90+3P
a. Calculate equilibrium price and quantity, consumer surplus, producer surplus and total economic
welfare.
Now suppose the government imposes a floor price of $60.
b. Under the price floor how many coconuts will be sold? What will happen to consumer, producer
surplus and overall economic welfare compared to the levels previously computed?
Solutions:
a) Recalling the equilibrium condition that wants Qd=Qs we can equate the two functions
deriving the equilibrium price.
110-P=-90+3P
200=4P
P*=50
By substituting the equilibrium price in either the demand or supply function we can
compute the equilibrium quantity. For sake of computation simplicity we choose the
demand function obtaining:
Q*=110-50=60
To graph the outcome we have to compute the intercepts between the two curves and the
axes. We start with the demand equation:
To compute the intercept between the curves and the vertical (Y) axis we have to set Q=0
In this way we obtain:
Demand: 0=110-P, and so, by taking P on the right side: P=110
Supply: 0=-90+3P , and so, by taking -90 on the right side: 90=3P, P=90/3=30
To compute the intercept between the curves and the vertical (Y) axis we have to set P=0
In this way we obtain:
Demand: Q=110-0, and so, Q=110
Supply: Q=-90+0 , and so, Q=-90
b) To compute the new quantities demanded and supplied after the introduction of the price
floor of 90, it is sufficient to substitute, both in the demand and supply equation, the new
price level.
DEMAND: Qd=110-P Qd=110-60=50
SUPPLY: Qs=-90+3P Qs=-90+3*60=-90+180=90
So, to summarize, by recalling that the equilibrium quantity before the implementation of
the price floor was equal to 60, we have that such an intervention makes the quantity
demanded drop by 10 and the quantity supply increase by 30. This generates a surplus
(calculated as Qs-Qd) equal to 90-50= 40.
Y
DA
S
D
110
30
60
50
CS
PS
Y
DA
S
D
110
30
50
60
PRICE FLOOR
50 90
SURPLUS
CS
PS DWL
46.6
60
EXERCISE 2.
The US government is concerned about the high incidence of heart diseases among the
population.
Soft drinks demand in US is given by: Qd = 60 - 3P.
Soft drinks supply in US is given by: Qs =-20 + 5P.
To curtail junk food consumption, it is considering imposing a 2 dollars tax on soft drinks, the so
called sugar-tax.
a. (4 pts.) Calculate the market clearing price per pack and packs sold prior to the imposition
of the tax.
b. (6 pts.) Calculate the market clearing price and quantity after the imposition of the 10 euros
tax. Graph the outcome with and without the tax. What is the price that buyers pay and the
price that sellers receive with the tax?
c. (4 pts.) From your answers to (a) – (b) above, what is the incidence of the tax on consumers
versus producers, i.e. what proportion of the tax will consumers vs. producers pay?
d. (2 pts.) From your answer to (c ) above, what do you deduct about the relative elasticities of
buyers versus producers of liquors in US?
e. (4 pts.) Now calculate the incidence of the tax on buyers and sellers using the “pass
through” formula. (Hint: First calculate the elasticity of demand and supply at the market
equilibrium without the tax).
f. (4 pts.) Calculate total tax revenues to the US government from the tax and Dead Weight
Loss (DWL) associated with the tax and show them on the graph.
g. (4 pts.) Is the DWL larger or smaller than the tax? Discuss the implications of your findings.
Is this an effective tax? Why or why not? What considerations you believe are excluded
from this type of analysis.
Answer:
a. The market equilibrium price absent the tax is given by finding the P that equates Qd and Qs:
40 – P = -10 + 4P
50= 5P
P* = 10
The equilibrium quantity is given by plugging the equilibrium price into the Qd or Qs equation:
Qd = 40 –10 = 30
b. The tax of 2$ creates a wedge between the price buyers pay (Pb) and the price sellers
receive (Ps). Note that it does not matter to the market equilibrium whether the tax is
imposed on buyers or sellers of cigarettes.
Pb = Ps + 10
The demand and supply equations become:
Qd = 40 – Pb = 40 - (Ps + 2) = 40 – Ps - 2
Qs = -10 + 4Ps
The new market equilibrium will occur where Qd = Qs
40 - Ps - 2= -10 + 4Ps
48 = 5Ps
Ps = 9.6. This is the price sellers receive.
Pb = Ps + 2 = 9.6 + 2 = 11.6. This is the price buyers pay.
The new equilibrium quantity Q* is determined by substituting the new price paid by buyers Pb of
11.6 in the Qd equation or the new price received by sellers Ps= 9.6 in the Qs equation:
Qd = 40 – 11.6 = 28.4
Or alternatively,
Qs = -10 + 4(9.6) = 28.4
P
S
11.6
10 Old equilibrium without tax
9.6
D
28.4 30 Q
c. To calculate the proportion of the tax borne by buyers, we have to compare the price paid by
buyers with and without the tax. As a result of the tax, the market price went up by 1.6, from 10 to
11.6. Therefore consumers bear 1.6 of the 2 dollars of the tax or 80% of the tax. To calculate the
proportion of the tax paid by sellers, we have to compare the price received by sellers before and
after the tax. As a result of the tax, the price received by sellers fell from 10 to 9.6. Therefore,
sellers bear the remaining 0.4 dollars of the tax or 20% of it.
d. From (c ) above, buyers bear a significantly larger proportion of the tax than sellers. This implies
that the demand of buyers is less elastic (with regard to price) than the supply of sellers.
e. Pass-through formula: Es/(Es – Ed).
where Es = Elasticity of supply and Ed = elasticity of demand
It measures the incidence of the tax borne by buyers.
Ed = Q/ΔP * P/Q = -1 (10/30) = -10/30 = -1/3
Es = Q/ΔP * P/Q = 4 (10/30) = 40/30 = 4/3
Therefore Es/(Es-Ed) = (4/3)/(4/3 – (-1/3)) = (4/3)/(5/3) = 4/3 *3/5 = = 4/5 = .8
Thus, 80% of the tax is borne by buyers and 20% of the tax is borne by sellers.
f. Tax revenues are 28.4 * 2 = 56.8. Dead weight loss associated with the tax = 1.5 given by
[(30 – 28.4) * 2]/2= 1.5 million pesos/month.
P
S
11.6
10 Old equilibrium without tax
9.6
D
28.4 30 Q
g. DWL is somewhat less than tax revenue from the tax. This would indicate that the tax is
effective in raising tax revenue and with causing only a little distortion in the market (in the form of
DWL). .
However, excluded from these calculations are the potential benefits to society from reduced soft
drink consumption which are not measurable with this type of static analysis. For example: the tax
revenues could be used for information campaigns, health care costs which may have benefits that
are greater over time than the size of the tax revenues themselves. Also, we may argue that the loss
in consumer surplus and producer surplus associated with reduced consumption/sale of an
unhealthy good is not an appropriate way of evaluating this type of market.
Tax
Revenue
DWL
EXERCISE 3
In the market for apples, the demand curve is Q = 50-3P and the supply curve
is Q = 2P. The Government decides to raise revenue by taxing consumers 2 for
every apple purchased.
a) Graph the supply and demand curves. Calculate the pre-tax and after-tax equilibrium
quantities and prices.
b) Calculate the change in consumer and producer surplus from the tax.
c) Calculate the burden of the tax borne by each party.
d) Calculate elasticity of demand and supply at the equilibrium and use the
elasticity formula to check your calculations at point b).
e) Calculate the amount of revenues raised by the Government and the loss of
efficiency for the society (DWL)
f) Intuitively, why is there dead-weight loss from a tax? That is, what exactly
does deadweight loss represent?
SOLUTIONS:
a) Graph the supply and demand curves. Calculate the pre-tax and after-tax
equilibrium quantities and prices.
b) Calculate the change in consumer and producer surplus from the tax.
To compute the pre-tax equilibrium lets recall the equilibrium condition Qd=Qs.
In this way we can equate the demand and supply equations obtaining: 50-3P=2P
Solving for P we have: 50=5P P*=10
To compute the equilibrium quantity (Q*) we have to substitute P* in the demand or the
supply equation. Choosing the supply equation we have Q*=2*10=20
So, to summarize: P*=10, Q*=20
To compute the after tax equilibrium we have to remind, besides the equilibrium
condition (Qd=Qs) that still holds, that now we are going to have two prices: a price of
the buyers and a price of the sellers, that are related by the following equations:
Pb=Ps+tax
Ps=Pb-tax
Knowing this we can rewrite the demand and supply equations as:
Qd=50-3Pb
Qs=2Ps
Using now one of the two equations (free to choose which) that describe the relation
between Pb and Ps we obtain:
Qd=50-3Pb
Qs=2(Pb-tax)=2(Pb-2)
Equating as usual this two expressions: 50-3Pb=2(Pb-2)
Solving for Pb: 50-3Pb=2Pb-4
54=5Pb
Pb=10.8
Ps=10.8-2=8.8
To obtain quantity we can substitute Pb in demand function or Ps in the supply
function. Beware that now we have two prices, so we have to carefully aassociate
the price to the right equation!
Choosing for sake of simplicity Ps, we substitute its value in the supply equation:
Qs=2*8.8=17.6
So the introduction of the tax (recalling that in the pre-tax equilibrium P*=10, Q*=20)
makes quantity decrease, price of buyers increase and price of suppliers decrease.
To graph the equilibrium we need to compute the intercepts between the axes and the
curves.
We start with the demand equation:
To compute the intercept between the curves and the vertical (Y) axis we have to set
Q=0
In this way we obtain:
Demand: 0=50-3P, and so, by taking P on the right side: 3P=50 P=50/3=16.6
Supply: 0=2P , and so P=0
To compute the intercept between the curves and the vertical (Y) axis we have to set P=0
In this way we obtain:
Demand: Q=50-0, and so, Q=50
Supply: Q=2*0 , and so, Q=0
Y
DA
S
D
16.6
10
20
CS
PS
As is possible to observe we are able to spot two areas of economic significance. The
upper triangle indicates the consumer surplus, while the lower the producer surplus.
To calculate the consumer surplus (CS) the procedure is the following. You have to
multiplicate the difference between the highest price possibly paid by customers (the
intercept between the demand curve and the vertical axis) and the equilibrium price
times the equilibrium quantity. All divided by 2: CS= (16.6−10)∗20
2=
6.6∗20
2 =132/2= 66
For the producer surplus the reasoning is analogous: you have to multiply the difference
between the equilibrium price and the lowest price at which suppliers can sell the good
(the intercept between the supply curve and the vertical axis) times the equilibrium
quantity. All divided by 2: 𝑃𝑆 = (10−0)∗20
2=
200
2= 100.
To compute again consumer and producer surpluses after the introduction of the tax the
procedure is the same, but now when considering consumer surplus we have to take into
account Pb and for producer surplus Ps.
𝐶𝑆 =(16.6 − 10.8) ∗ 17.6
2= 51,04
𝑃𝑆 =(8,8 − 0) ∗ 17.6
2= 77,44
Now we are able to compute the variation in CS and PS after the introduction of the tax
𝛥𝐶𝑆 = 𝐶𝑆𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 − 𝐶𝑆𝑝𝑟𝑒−𝑡𝑎𝑥 = 51,04 − 66 = −14,96
𝛥𝑃𝑆 = 𝑃𝑆𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 − 𝑃𝑆𝑝𝑟𝑒−𝑡𝑎𝑥 = 77,44 − 100 = −12,56
c) Calculate the burden of the tax borne by each party.
Y
DA
S
D
16.6
10
10.8
10
8.8
10
20
10
17.6
10
CS
PS
DWL Tax
Revenues
The burden of the tax borne by each party (consumers and producers) is defined as the
portion of the tax paid by either of the two in the form of price variation compared to the
pre-tax equilibrium price. Given that the tax introduction causes a price increase for the
consumers and a price decrease for the suppliers, the burden of the tax results in the amount
of such price variation compared to the level of the tax.
In formula:
TAX BURDEN BORNE BY CONSUMERS: 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑟𝑠 𝑎𝑓𝑡𝑒𝑟 𝑡ℎ𝑒 𝑡𝑎𝑥−𝑝𝑟𝑒 𝑡𝑎𝑥 𝑝𝑟𝑖𝑐𝑒
𝑇𝐴𝑋
TAX BURDEN BORNE BY PRODUCERS: |𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟𝑠 𝑎𝑓𝑡𝑒𝑟 𝑡ℎ𝑒 𝑡𝑎𝑥−𝑝𝑟𝑒 𝑡𝑎𝑥 𝑝𝑟𝑖𝑐𝑒|
𝑇𝐴𝑋
To compute them is now easy, we have just to substitute data in the above formulas:
TAX BURDEN BORNE BY CONSUMERS: 10.8−10
2=
0.8
2= 0.4 = 40%
TAX BURDEN BORNE BY PRODUCERS: |8.8−10|
2=
1.2
2= 0.6 = 60%
Please bear in mind that the sum of the two must be equal to 1 (or, if you prefer, 100%)!!
(in the second formula, the one used to calculate the tax burden borne by producers, we consider, at
the numerator, the absolute value of the difference between the post-tax and the pre-tax prices, as
we are concerned about the amount of tax burden borne by suppliers, that surely is a positive value)
d) Calculate elasticity of demand and supply at the equilibrium and use the
elasticity formula to check your calculations at point b
To compute the tax burden borne by consumers can be employed another formula, the so called
pass-through formula:
PASS-THROUGH FORMULA: 𝜀𝑠/(𝜀𝑠 − 𝜀𝑑)
To use it is necessary to compute the elasticity of demand (𝜀𝑑) and the elasticity of supply(𝜀𝑠):
𝜀𝑑 =𝛥𝑄
𝛥𝑃∗
𝑃
𝑄=
17.6 − 20
10.8 − 10∗
10
20=
−2.4
0.8∗
1
2= −3 ∗
1
2= −
3
2
𝜀𝑠 =𝛥𝑄
𝛥𝑃∗
𝑃
𝑄=
17.6 − 20
8.8 − 10∗
10
20=
−2.4
−1.2∗
1
2= 2 ∗
1
2=
2
2= 1
PASS-THROUGH FORMULA: 𝜀𝑠/(𝜀𝑠 − 𝜀𝑑)= 1
1−(−3
2)
=15
2
=2
5= 0.4 = 40%
This is confirms that the bove calculations were good. The pass-through formula states that the tax
burden borne by customers is the 40%, as we should expect given that the elasticity of demand is
higher than the elasticity of supply. The amount of tax born is higher on the party that displays the
lowest elasticity to price.
e) Calculate the amount of revenues raised by the Government and the loss of efficiency for
the society (DWL)
To compute tax revenues we have to calculate the area of the rectangle that goes from PB to PS in
height and from 0 to the after-tax quantity in length.
TAX REVENUES= (10.8 − 8.8) ∗ (20 − 17.6) = 2 ∗ 2.4 = 4.8
The dead-weight loss (DWL), i.e. the loss of economic welfare caused by the distortionary effect of
the tax introduction, is described by the area of the triangle at the right of the tax revenues’
rectangle.
To compute it, hence, is necessary to calculate the area of that triangle:
DWL= 𝑇𝑎𝑥∗(𝑝𝑟𝑒 𝑡𝑎𝑥 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦−𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦)
2=
2∗(20−17.6)
2=
2∗2.4
2= 2.4
So we have that Tax revenues are equal to 4.8 while the DWL amounts at 2.4.
So DWL is the 50% of tax revenues: from this we ca argue that the amount of the tax is a little bit
high, as it introduces an appreciable degree of distortion in the market functioning.
EXERCISE 4
Suppose that the market for green tea can be described by the following demand and supply
curves (prices are per kg):
Qd = 260 − 5P
QS = 8P
a) Find the market equilibrium in the absence of taxes. Draw the demand and
supply curves, labelling all intercepts and the market equilibrium
b) Draw the curves as in the last item, showing clearly the areas representing the
consumer surplus (CS) and the producer surplus (PS). Calculate their values
and the value of the total surplus (TS).
c) Suppose now that the government decides to tax green tea by $13 per kg.
Calculate the tax equilibrium.
Calculate and show on the graph the values of CS, PS, TAXR, DWL
SOLUTIONS:
a) To compute the pre-tax equilibrium lets recall the equilibrium condition Qd=Qs.
In this way we can equate the demand and supply equations obtaining: 260-5P=8P
Solving for P we have: 260=13P P*=20
To compute the equilibrium quantity (Q*) we have to substitute P* in the demand or the
supply equation. Choosing the supply equation we have Q*=8*20=160
So, to summarize: P*=20, Q*=160
To graph the equilibrium we need to compute the intercepts between the axes and the
curves.
We start with the demand equation:
To compute the intercept between the curves and the vertical (Y) axis we have to set
Q=0
In this way we obtain:
Demand: 0=260-5P, and so, by taking P on the right side: 5P=260 P=260/5=52
Supply: 0=8P , and so P=0
To compute the intercept between the curves and the vertical (Y) axis we have to set P=0
In this way we obtain:
Demand: Q=260-0, and so, Q=260
Supply: Q=8*0 , and so, Q=0
b) As is possible to observe we are able to spot two areas of economic significance. The
upper triangle indicates the consumer surplus, while the lower the producer surplus.
To calculate the consumer surplus (CS) the procedure is the following. You have to
multiplicate the difference between the highest price possibly paid by customers (the
intercept between the demand curve and the vertical axis) and the equilibrium price
times the equilibrium quantity. All divided by 2: CS= (52−20)∗160
2=
32∗160
2 =5120/2= 2560
For the producer surplus the reasoning is analogous: you have to multiply the difference
between the equilibrium price and the lowest price at which suppliers can sell the good
(the intercept between the supply curve and the vertical axis) times the equilibrium
quantity. All divided by 2: 𝑃𝑆 = (20−0)∗160
2=
3200
2= 1600.
c) To compute the after tax equilibrium we have to remind, besides the equilibrium
condition (Qd=Qs) that still holds, that now we are going to have two prices: a price of
the buyers and a price of the sellers, that are related by the following equations:
Pb=Ps+tax
Ps=Pb-tax
Knowing this we can rewrite the demand and supply equations as:
Qd=260-5Pb
Qs=8Ps
Using now one of the two equations (free to choose which) that describe the relation
between Pb and Ps we obtain:
Qd=260-5Pb
Qs=8(Pb-tax)=8(Pb-13)
Equating as usual this two expressions: 260-5Pb=8(Pb-13)
Solving for Pb: 260-5Pb=8Pb-104
364=13Pb
Pb=28
Ps=28-13=15
Y
DA
S
D
52
20
0
260 160
CS
PS
To obtain quantity we can substitute Pb in demand function or Ps in the supply
function. Beware that now we have two prices, so we have to carefully associate the
price to the right equation!
Choosing for sake of simplicity Ps, we substitute its value in the supply equation:
Qs=8*15=120
So the introduction of the tax (recalling that in the pre-tax equilibrium P*=20, Q*=160)
makes quantity decrease, price of buyers increase and price of suppliers decrease.
To compute again consumer and producer surpluses after the introduction of the tax the
procedure is the same, but now when considering consumer surplus we have to take into
account Pb and for producer surplus Ps.
𝐶𝑆 =(52 − 28) ∗ 120
2= 1440
𝑃𝑆 =(15 − 0) ∗ 120
2= 900
Now we are able to compute the variation in CS and PS after the introduction of the tax
𝛥𝐶𝑆 = 𝐶𝑆𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 − 𝐶𝑆𝑝𝑟𝑒−𝑡𝑎𝑥 = 1600 − 2560 = −960
𝛥𝑃𝑆 = 𝑃𝑆𝑎𝑓𝑡𝑒𝑟−𝑡𝑎𝑥 − 𝑃𝑆𝑝𝑟𝑒−𝑡𝑎𝑥 = 900 − 1600 = −700
To compute tax revenues we have to calculate the area of the rectangle that goes from PB to PS in
height and from 0 to the after-tax quantity in length.
TAX REVENUES= (28 − 20) ∗ (160 − 120) = 8 ∗ 40 = 320
The dead-weight loss (DWL), i.e. the loss of economic welfare caused by the distortionary effect of
the tax introduction, is described by the area of the triangle at the right of the tax revenues’
rectangle.
Y
DA
S
D
52
20
28
15
CS
PS
TAX
REVENUES DWL
160 120
To compute it, hence, is necessary to calculate the area of that triangle:
DWL= 𝑇𝑎𝑥∗(𝑝𝑟𝑒 𝑡𝑎𝑥 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦−𝑎𝑓𝑡𝑒𝑟 𝑡𝑎𝑥 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦)
2=
13∗(160−120)
2=
13∗40
2= 260
So we have that Tax revenues are equal to 320 while the DWL amounts at 260.
So DWL is the 81.25% of tax revenues (260/320): from this we can argue that the amount of the tax
is excessive, as it introduces a very high degree of distortion in the market functioning.
