Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The PreTax Position


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1 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 (pretax) 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 PreTax 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.
2 The demand curve is given by q d = 100 p b (27.1) and the supply curve is given by q s = p s 10 (27.2) In these equations, q d indicates the quantity demanded and q s the quantity supplied. The price p b is the price paid by the buyers and p s 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 : 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 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 p b = p s +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: and then insert into it equation (27.3) to get q s = p s 10
3 q s = (p b 10) 10 hence getting the supply curve expressed as a relationship between the quantity supplied and the price paid by buyers: q s = p b 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 Supply Price paid by the buyers 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.
4 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 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 q d = 90  p s 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.
5 Price paid by the buyers Demand Price received by the sellers 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 Always (but see the next footnote) 10 less than the price paid by the buyers. 3 Strictly speaking the price should be 10 which implies that the sellers pay 10 for every unit sold. But clearly this makes no sense.
6 The buyer surplus is the area between the price (paid) and the demand curve in this case 0.5 x 45 x 45 = The seller surplus is the area between the price (received) and the supply curve in this case 0.5 x 45 x 45 = 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 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 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 noone gets not the buyers, not the sellers, not the government it just disappears from the market, because the quantity exchanged has fallen. This
7 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 2025 New surpluses: buyers 800 Sellers 800 Government 400 Total 2000 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 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 p b = 1.2 p s
8 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: If we graph this we get figure q s = p b / Alternatively we can use numbers. Proceeding as before we have the following table and when we graph the final two columns we get figure Price received by the sellers Supply Price paid by the buyers In figure the downwardsloping line is the (new and old) demand curve. The thin upwardsloping line is the original supply curve and the thick upwardsloping 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 Notice that the intercept of the supply curve is at a price of 24 which is 20% higher than the original intercept. 5 Always 20% more than the price received by the sellers.
9 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 q d = p s as the relevant schedule. Plotted we get figure Alternatively we can use numbers. Proceeding as before we have the following table and when we graph the final two columns we get figure We note that the first column is always 20% more than the final column. Price paid by the buyers Demand Price received by the sellers ⅓ ⅔ ⅓ ⅔ ⅓ ⅔ ⅓ In figure the upwardsloping line is the (new and old) supply curve. The thin downwardsloping line is the original demand curve and the thick downwardsloping 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 Always such that the price paid by the buyers is20% more than this.
10 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 fixedrate 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 flatrate 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 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.
11 If the demand curve is of average steepness but the supply curve is very steep, we have the position pictured in figure The original equilibrium has a quantity of just over 54 and a price of around 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 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 The original equilibrium has a quantity of just under 45 and a price of just over 54. There are large buyer and seller surpluses.
12 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
13 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 Now suppose the government introduces a flatrate 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.
14 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 pretax profits is maximised must be the same quantity at which posttax 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 withouttax optimisation problem is to choose that q which maximises π, while the withtax optimisation problems is to choose that q which maximises tπ. Clearly the solution is the same. Multiplying the objective by a constant changes nothing 7. 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 quasilinear), 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 and the surplus of the sellers is The government takes no indirect tax but has the revenue from the direct taxation. Note here the total surplus is 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 pretax profit is positive, then so must the posttax profit.
15 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.
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