Download Chapter 27: Taxation

Chapter 27: Taxation
27.1: Introduction
We consider the effect of taxation on some good on the market for that good. We ask the questions:
who pays the tax? what effect does it have on the equilibrium price and on the equilibrium quantity?
what effect does it have on the surpluses? We shall see that taxation reduces the total surplus
generated by the market – this loss caused by the tax is called the ‘deadweight loss of the tax’. It
reduces the efficiency of the market – it reduces the surplus generated by the market – and for that
reason might be considered ‘a bad thing’. Though it is important to remember that there are
obvious offsetting advantages – the government can use the tax generated elsewhere in the
economy. This ‘good thing’ might well be worth the ‘bad thing’ caused by the reduction of the total
surplus generated by the market in which the tax is collected.
In practice there are different kinds of taxes. The two most common are flat rate taxes and
proportional taxes. The first of these are fixed taxes – taxes independent of the price of the good.
The second are taxes such as value added tax – which are levied at a rate proportional to the price of
the good. The most common form of a tax is the proportional form – in many countries
governments levy a value added tax. In the UK at present this is equal to 17.5% of the (pre-tax)
price of the good. However, there also flat rate taxes – such as was the case until recently with the
vehicle registration licence. An even more famous example is the ‘Poll Tax’ that Mrs Thatcher
introduced on property ownership in the UK – the owner of any house, however big or small, had to
pay the same amount of Poll Tax to the government. We shall examine both of these taxes in this
chapter. However our methods can be applied to any kind of tax.
27.2: The Two Prices with a Tax
Whatever kind of tax we are examining, the crucial point is that with a tax there are two prices in
the market: the price that buyers pay; and the price that sellers receive. The difference between
these two prices is the tax – which the government takes. Only when there is no tax are these prices
equal. Accordingly when we use ‘the price’ as one of the variables in our analysis we must specify
which price is it that we mean. As we will see, we can do our analysis with either price – we just
have to be careful about which price it is.
27.2: The Pre-Tax Position
Let us start with a situation in which there is no tax. We are going to work with a specific example
initially and later we will generalise. In this specific example we take simple linear demand and
supply schedules so we can see exactly what is going on. This initial position is pictured in figure
27.1.
The demand curve is given by
qd = 100 – pb
(27.1)
qs = ps – 10
(27.2)
and the supply curve is given by
In these equations, qd indicates the quantity demanded and qs the quantity supplied. The price pb is
the price paid by the buyers and ps the price received by the sellers. Here these two prices are the
same as there is no tax, but when there is a tax they will be different.
We note that the original equilibrium price is 55 and the original equilibrium quantity is 45.
27.3: A Flat Rate Tax
Let us now suppose that the government imposes a flat rate tax on the good. Let us suppose that this
is at the rate 10 on every unit bought and sold. For every unit exchanged the government takes 10.
What effect does this have? We look at the effect on the demand and supply schedules and hence on
the equilibrium.
Let us first do the analysis with the price paid by the buyers on the vertical axis. Since this is the
relevant variable in the demand curve, the demand curve does not change. It is pictured in figure
27.2. What about the supply curve? Well, we cannot use equation (27.2) directly as the variable
there is the price received by the sellers - which is not the price variable that we are using in this
analysis. Let us work out the new supply curve - first using a bit of algebra and then using some
numbers. Algebraically it is simple. Since the price paid by buyers differs from the price received
by the sellers by the tax we have that
pb = ps +10
(27.3)
If we use this in (27.2) to find the relationship between the quantity supplied and the price paid by
the buyers we start with the supply curve:
qs = ps – 10
and then insert into it equation (27.3) to get
qs = (pb –10) – 10
hence getting the supply curve expressed as a relationship between the quantity supplied and the
price paid by buyers:
qs = pb – 20
If we plot this in figure 27.2 we get the new supply curve pictured there.
In this figure the downward sloping line is the (new and old) demand curve; the thick upward
sloping curve is the new supply curve and the thin upward sloping line is the old supply curve. So
the supply curve shifts upwards by the amount of the tax. Notice that the vertical distance between
the two supply curves is equal to 10 – the amount of the tax – everywhere.
An alternative way of seeing this is by working out some numbers. Consider the table below.
Price received by the sellers
0
10
20
30
40
50
60
70
80
90
100
Supply
0
0
10
20
30
40
50
60
70
80
90
Price paid by the buyers1
10
20
30
40
50
60
70
80
90
100
110
The supply curve is illustrated in the first two columns – which show the relationship between the
price received by the sellers and the quantity supplied. Recall it is this relationship that defines the
supply curve. We now derive the final column from the first – taking into the account that the tax is
10 which makes the price paid by the buyers always 10 more than the price received by the sellers.
Now in figure 27.2 we graph the relationship between the second and the third columns – because
1
Always 10 more than the price received by the sellers.
the second column is the variable on the horizontal axis and the third column (not the first column)
is the variable on the vertical axis. This is the thick upward sloping line in the figure.
The effect of the tax on the equilibrium can now be seen – it is at the intersection of the demand
curve and the new supply curve. It is indicated in the figure. The new equilibrium quantity is 40 and
the new equilibrium price is 60. But let us be precise – the new equilibrium price paid by the buyers
is 60. It follows that the new equilibrium price received by the sellers is 50.
So before the tax, 45 units are exchanged, with the buyers paying 55 for each unit and the sellers
receiving 55 for each unit. With the tax, only 40 units are exchanged, with the buyers paying 60 for
each unit, the sellers receiving 50 for each unit and the government taking 10 in tax on each unit.
Let us now repeat the analysis with the other price variable – the price received by the sellers. If this
is the variable on the vertical axis then the supply curve is the same as it was originally, as the price
received by sellers is the variable which determines the supply. This is shown in figure 27.3. What
about the demand curve? Well, we cannot use equation (27.1) directly as the variable there is the
price paid by the buyers - which is not the price variable that we are using in this analysis. Let us
work out the new demand curve - first using a bit of algebra and then using some numbers.
Algebraically it is simple. Since the price paid by buyers differs from the price received by the
sellers by the tax we have equation (27.3) as before. If we use this in (27.1) to find the relationship
between the quantity demanded and the price received by the sellers we get that
qd = 90 - ps
If we plot this in figure 27.3 we get the new demand curve pictured there.
In this figure the upward sloping line is the (new and old) supply curve; the thick downward sloping
curve is the new demand curve and the thin downward sloping line is the old demand curve. So the
demand curve shifts downwards by the amount of the tax. Notice that the vertical distance between
the two demand curves is equal to 10 – the amount of the tax – everywhere.
An alternative way of seeing this is by working out some numbers. Consider the table below.
Price paid by the buyers
0
10
20
30
40
50
60
70
80
90
100
Demand
100
90
80
70
60
50
40
30
20
10
0
Price received by the sellers2
03
0
10
20
30
40
50
60
70
80
90
The demand curve is illustrated in the first two columns – which show the relationship between the
price paid by the buyers and the quantity demanded. Recall it is this relationship that defines the
demand curve. We now derive the final column from the first – taking into the account that the tax
is 10 which makes the price paid by the buyers always 10 more than the price received by the
sellers. Now in figure 27.3 we graph the relationship between the second and the third columns –
because the second column is the variable on the horizontal axis and the third column (not the first
column) is the variable on the vertical axis. This is the thick downward sloping line in the figure.
The effect of the tax on the equilibrium can now be seen – it is at the intersection of the new
demand curve and the supply curve. It is indicated in the figure. The new equilibrium quantity is 40
and the new equilibrium price is 50. But let us be precise – the new equilibrium price received by
the sellers is 50. It follows that the new equilibrium price paid by the buyers is 60.
So we get exactly the same conclusions as before. Before the tax, 45 units are exchanged, with the
buyers paying 55 for each unit and the sellers receiving 55 for each unit. With the tax only 40 units
are exchanged, with the buyers paying 60 for each unit, the sellers receiving 50 for each unit and the
government taking 10 in tax on each unit.
27.4: The Effect on the Surpluses
We have seen that the tax has the effect of reducing the quantity exchanged in the market. In this
section we investigate the effect that the tax has on the surpluses. First we note the surpluses before
the tax. They are shown in figure 27.5.
2
Always (but see the next footnote) 10 less than the price paid by the buyers.
Strictly speaking the price should be –10 which implies that the sellers pay 10 for every unit sold. But clearly this
makes no sense.
3
The buyer surplus is the area between the price (paid) and the demand curve – in this case 0.5 x 45 x
45 = 1012.5. The seller surplus is the area between the price (received) and the supply curve – in
this case 0.5 x 45 x 45 = 1012.5.
To analyse the position with the tax we will find it useful to use a slightly different diagram – one
that contains the original demand and supply curves and the new equilibrium. We can do this by
noting the properties of the new equilibrium: that the tax drives a wedge between the price that
buyers pay and the price that sellers receive – and this wedge is exactly equal to the tax. We can
think of the equilibrium quantity as the quantity at which the vertical distance between the demand
and supply curves is exactly equal to the tax. Figure 27.8 illustrates – though you should be careful
if you are using this kind of diagram to explain exactly what it is that you are illustrating.
The quantity indicated is the new equilibrium quantity. It is so because the vertical gap between the
demand curve and the supply curve at that quantity (note that it is unique) is exactly equal to the tax
(10). The price given by the demand curve at the quantity of 40 is the new equilibrium price paid
by the buyers. The price given by the supply curve at the quantity of 40 is the new equilibrium price
received by the sellers. The new buyer surplus is the area indicated – between the new price paid by
the buyers and the demand curve. The new surplus is 0.5 x 40 x 40 = 800, giving a loss of surplus of
212.5. The new seller surplus is the area indicated – between the new price received by the sellers
and the supply curve. The new surplus is 0.5 x 40 x 40 = 800, giving a loss of surplus of 212.5.
The government raises money through the tax – this is the area bounded by the two prices (the price
paid by the buyers and the price received by the sellers and the quantity exchanged. In figure 27.8
the tax yield is the rectangle with area 40 x 10 = 400. The government takes most of the reduction
in the surpluses of the buyers and sellers.
But there is a bit of the original surpluses that no-one gets – not the buyers, not the sellers, not the
government – it just disappears from the market, because the quantity exchanged has fallen. This
lost surplus is the triangular area illustrated in the figure below. In this example, its magnitude is 0.5
x 5 x 10 = 25. This is called the deadweight loss of the tax.
If we do the accounts we find that
Original surpluses: buyers
Sellers
Total
1012.5
1012.5
2025
New surpluses:
800
800
400
2000
buyers
Sellers
Government
Total
Deadweight loss of tax
25
So the tax causes a loss of surplus. In this example the magnitude is quite small but this depends on
a number of things including the reduction in the quantity traded. It is this reduction that causes the
loss of surplus – as there were trades being consummated before the tax that are no longer being
consummated. The tax causes fewer trades and hence a lower surplus.
27.5: A Proportional Tax
The crucial point about all off the above is that the tax causes the demand curve to shift downwards
(when the price variable on the vertical axis is the price received by sellers) and causes the supply
curve to shift upwards (when the price variable on the vertical axis is the price paid by the buyers) –
with the magnitude of the vertical shift exactly equalling the amount of the tax. We showed that this
was the case with a flat rate tax but it is true with any kind of tax – including a proportional tax. Let
us consider this case now.
We take a 20% tax. So the price paid by the buyers is 20% more than the price received by sellers
and the difference is the tax. Let us consider the same case as before – so our starting position is
figure 27.1. Let us again consider the analysis with the two different price variables – first starting
where the price is the price paid by the buyers. In this case, as we know, the supply curve shifts. We
can found out to where algebraically or numerically. Algebraically we have
pb = 1.2 ps
because the buyers pay 20% more than the sellers. Substituting this in the supply curve (27.2) gives
us the new supply curve as a function of the price paid by buyers:
qs = pb /1.2 – 10
If we graph this we get figure 27.104.
Alternatively we can use numbers. Proceeding as before we have the following table and when we
graph the final two columns we get figure 27.10.
Price received by the sellers
0
10
20
30
40
50
60
70
80
90
100
Supply
0
0
10
20
30
40
50
60
70
80
90
Price paid by the buyers5
0
12
24
36
48
60
72
84
96
108
12 0
In figure 27.10 the downward-sloping line is the (new and old) demand curve. The thin upwardsloping line is the original supply curve and the thick upward-sloping line is the new supply curve.
Notice that the vertical gap between the old and the new curves is exactly the tax – the new curve is
20% higher than the old – even at the intercept. Why? Because to produce a given quantity the
sellers need the buyers to pay 20% more than they did originally – the 20% which goes to the
government in tax.
The new equilibrium is where the (new and old) demand curve intersects the new supply curve. As
it happens this is at a quantity 40 and a price paid by the buyers of 60 – this, in turn, implies a new
price received by the sellers of 50. Note that 50 plus 20% equals 60.
It is pure coincidence that the tax of 20% has exactly the same effect as a tax of 10 per unit. In
general, rather obviously, different taxes will have different effects.
4
5
Notice that the intercept of the supply curve is at a price of 24 – which is 20% higher than the original intercept.
Always 20% more than the price received by the sellers.
We can, once again, do all the analysis with the price received by sellers on the vertical axis. If we
do, the supply curve is the original supply curve – as the price relevant to the sellers is the price that
they receive. However, the demand curve moves as the price relevant to the buyers is the price that
they pay. To find the demand as a function of the price received by sellers we need to substitute
(27.4) into the demand curve (27.1). This gives us
qd = 100 –1.2 ps
as the relevant schedule. Plotted we get figure 27.11.
Alternatively we can use numbers. Proceeding as before we have the following table and when we
graph the final two columns we get figure 27.11. We note that the first column is always 20% more
than the final column.
Price paid by the buyers
0
10
20
30
40
50
60
70
80
90
100
Demand
100
90
80
70
60
50
40
30
20
10
0
Price received by the sellers6
0
8⅓
16⅔
25
33⅓
41⅔
50
58⅓
66⅔
75
83⅓
In figure 27.11 the upward-sloping line is the (new and old) supply curve. The thin downwardsloping line is the original demand curve and the thick downward-sloping line is the new demand
curve. Notice that the vertical gap between the old and the new curves is exactly the tax – the old
curve is 20% higher than the new – even at the intercept. Why? Because to buy a given quantity the
buyers need the sellers to sell for 20% less than they did originally – the 20% which goes to the
government in tax.
The new equilibrium is where the (new and old) supply curve intersects the new demand curve. As
before, obviously, this is at a quantity 40 and a price received by the sellers of 50 – this, in turn,
implies a new price paid by the buyers of 60. Note that 50 plus 20% equals 60.
6
Always such that the price paid by the buyers is20% more than this.
We notice that once again the tax drives a wedge between the price paid by the buyers and the price
received by the sellers. In this instance, because the 20% tax has exactly the same effect at the
fixed-rate tax of 10, which we analysed earlier, and because we started in exactly the same position
the effect on the surpluses is exactly as it was before. In particular there is a deadweight loss of
surplus caused by the tax.
27.6: Who Pays the Tax?
In the example above all the effects were nicely symmetrical: the price paid by the buyers went up
by 5 and the price received by the sellers went down by 5. So the buyers paid half the tax of 10 and
the sellers paid half the tax of 10. Moreover the effect on the surpluses was symmetrical – both
buyers and sellers lost the same part of their surplus to the government and the same part to the
deadweight loss. The reason for this symmetry is that the demand and supply curves were nicely
symmetrical. If they were not there is no reason why the effects should be symmetrical.
This section looks at asymmetrical cases and answers the question ‘who pays the tax?’ in these
cases. As may be apparent the answer depends upon the slope of the demand and supply curves. We
consider four cases, in which we consider three possibilities – flat, average and steep – where by
‘average’ we mean a case in between flat and steep:
1)
demand average, supply flat
2)
demand average, supply steep
3)
demand flat, supply average
4)
demand steep, supply average
You may like to consider other cases yourself. In each case we take a flat-rate tax equal to 10.
If the demand curve is of average steepness but the supply curve is very flat, we have the position
pictured in figure 27.17. The original equilibrium has a quantity of 45 and a price of just under 55.
The fact that the supply curve is very flat indicates that it is very sensitive to the price: small
increases in the price lead to large increases in the quantity supplied. You will see that originally the
buyer surplus is very high and the seller surplus very low. With the tax the quantity exchanged falls
to a little under 37. The price paid by the buyers increases from just under 55 to almost 64 – a rise
of almost 9. In contrast the price received by the sellers falls from just under 55 to just under 54 – a
fall of just over 1. In this case, the tax of 10 is almost all paid for by the buyers – as they were less
sensitive to the price than the sellers. Note that there is a large fall in the buyer surplus but only a
modest fall in the seller surplus.
If the demand curve is of average steepness but the supply curve is very steep, we have the position
pictured in figure 27.18. The original equilibrium has a quantity of just over 54 and a price of
around 45.5. The steepness of the supply curve indicates that it is not very responsive to the price.
Notice that there are large buyer and seller surpluses.
In the new equilibrium the quantity exchanged falls very little – to just under 54 (because the supply
curve is steep). The price paid by the buyers rises from around 45.5 to around 46.5 – a rise of just 1.
The price received by sellers, however, falls from around 45.5 to almost 36 – a fall of about 9 – the
reason being that the supply is not very sensitive to the price. The sellers pay the tax. There is a
small fall in the buyer surplus but a large fall in the seller surplus. In this case the deadweight loss
of the tax is very small – because the quantity exchanged falls very little.
If the demand curve is very flat but the supply curve is of average slope, we have the position
pictured in figure 27.19. The original equilibrium has a quantity of 36.5 and a price of just over 46.
It will be seen that the buyer surplus is very small – because the demand curve is flat, indicating that
the demand is very sensitive to the price.
The tax causes a big reduction in the quantity exchanged because the demand is very sensitive to
the price. For the same reason the price paid by the buyers rises very little – from just over 46 to
around 47. In contrast the price received by the sellers falls a lot – from just over 46 to around 37
and there is a big fall in the seller surplus. In this case the sellers bear most of the burden of the tax.
There is a large deadweight loss – because the quantity exchanged falls considerably.
Finally, if the demand curve is very steep but the supply curve is of average slope, we have the
position pictured in figure 27.20. The original equilibrium has a quantity of just under 45 and a
price of just over 54. There are large buyer and seller surpluses.
In the new equilibrium the quantity exchanged falls just a little - from just under 45 to a little under
44. As a consequence the deadweight loss is rather small. The price paid by the buyers rises sharply
– from just over 54 to almost 64 – the reason being that the demand is rather insensitive to the price.
In contrast there is just a small fall in the price received by the sellers – from just over 54 to just
under 54. In this case the tax burden is almost all taken by the buyers – because the demand is
rather insensitive to the price.
The conclusion to be drawn is that the effects of the tax depend upon the form of the demand and
supply schedules. If one of the two is relatively insensitive to the price then that side of the market
bears the burden of the tax. If one of the two is relatively sensitive to the price then the other side of
the market bears the burden of the tax.
27.7: Summary
The crucial point when analysing the effect of a tax on a market is that there are two prices when
there is a tax: the price that buyers pay and the price that sellers receive.
If the price variable is the price paid by the buyers then the demand curve does not move but the
supply curve moves up vertically by the amount of the tax.
If the price variable is the price received by the sellers then the supply curve does not move but the
demand curve moves down vertically by the amount of the tax.
A tax causes a reduction in the surpluses going to the buyers and sellers and also causes an overall
loss of surplus generated by the market – the deadweight loss of the tax.
The shape of the demand and supply schedules determines the tax burden and the size of the
deadweight loss. Generally the less sensitive is the schedule the greater the burden of the tax.
27.8: Indirect or direct taxation?
Direct taxation is the term that economists use for taxes on income (in any form, including profits).
Indirect taxation is the term used for taxes on expenditure – the kind of taxes that we have been
considering in this chapter. One clear message from this chapter is that this form of taxation causes
a deadweight loss – some of the surplus previously generated by the market disappears when
taxation is introduced. The inference from this message might be that this form of taxation is
therefore necessarily bad. Here we consider whether this is, in fact, the case.
We use a very simple story – with linear demand and supply curves – like the one we have told in
this chapter. Consider a market without tax where the demand and supply curves are as in the
figure below.
The equilibrium price is 50, the equilibrium quantity is 50, the buyer surplus is 1250 and the seller
surplus is 1250, giving a total surplus of 2500.
Now suppose the government introduces a flat-rate tax of 20 on the good. Then new equilibrium
price paid by the buyers becomes 60, the equilibrium price received by the sellers is 40 (the
difference between the two being the tax of 20, of course), the surplus of the buyers becomes 800,
the surplus of the sellers becomes 800, the government gets 800 in tax – and there is a deadweight
loss of 100 – as shown in the figure below.
The total surplus is now just 2400 – the deadweight loss of 100 represents the loss of surplus
generated in the market as a consequence of the introduction of the tax.
Critics might say that this is inefficient. A more efficient situation might be to levy the tax directly
on the incomes of the agents involved. Let us see what happens if the government, instead of
levying the tax indirectly, does so directly.
The analysis of one side of the market might be simple. If the suppliers consist of firms whose
objective is to maximise profits, and if the direct tax introduced by the government is a tax on
profits – then there is no shift in the supply curve. The quantity of output at which pre-tax profits is
maximised must be the same quantity at which post-tax profits are maximised – if the tax simply
takes a proportion of the profits. You should be clear about this. If we denote profits by π and
output by q, and if we suppose that the government takes a proportion t of the profits in tax, then the
without-tax optimisation problem is to choose that q which maximises π, while the with-tax
optimisation problems is to choose that q which maximises tπ. Clearly the solution is the same.
Multiplying the objective by a constant changes nothing7.
So the supply curve might not shift. But the demand curve might, if the tax is on income and
income affects demand. In general (that is, if the preferences are not quasi-linear), we would expect
income to affect demand and we would therefore expect an increase in income tax to lower incomes
and thus affect demand. We would expect therefore that the demand curve would shift down. In
general, it is not clear how much it would shift down – as this would depend on the amount of the
tax and the responsiveness of demand to income. One possibility is illustrated in the graph below, in
which we have drawn the supply curve unchanged, and the demand curve shifted down by 10.
In this we have no indirect taxation. Here the equilibrium price is 45, the equilibrium quantity is 45,
the surplus of the buyers 1012.5 and the surplus of the sellers is 1012.5. The government takes no
indirect tax but has the revenue from the direct taxation. Note here the total surplus is 2025. This is
less than the original total surplus of 2500 before any tax was introduced. This last situation is
efficient in terms of the market, but less surplus is generated than initially. There is, of course,
however, the tax revenue raised by the direct taxation.
7
We should make one proviso. If the profits become negative then the firm may prefer to go out of business. If that
happens we lose a bit of the supply. But it the tax is proportional and the pre-tax profit is positive, then so must the posttax profit.
We cannot conclude from this analysis whether one method of taxation is better than another,
though we have shown the tools that economists can use to analyse the situation. But note: we have
analysed only a part of the problem. The taxation will also have implications in other markets.
There will be spillover effects elsewhere – other prices will be affected and other quantities. To take
into account all such effects we would need to do what is called General Equilibrium analysis,
which is way beyond the scope of this book.