Governments at all levels (local, state, federal) tax all kinds of things, and they do it in different ways. Sometimes, suppliers are legally required to remit the tax. Stores in the United States collect sales taxes and send them to state revenue agencies, for example, just as producers in Canada and Europe collect and remit value-added tax (VAT). Sometimes the legal burden falls on consumers, like “use taxes” that states levy on purchases their residents make in other states. In still other cases, the legal burden is shared. For instance, half of the U.S. federal payroll tax (which funds the Social Security and Medicare programs) is paid by employers before workers get their wages, and the other half is paid by workers through a deduction from their wages.

In this section, we use the supply and demand model to show a striking finding of economics: In a competitive market, it doesn’t matter whether the buyer or the seller is required by law to actually sign the check and pay the tax to the government; the impact on consumers and producers is the same either way. We could change the law so that consumers paid sales tax instead of sellers, or employers paid the entire payroll tax, and market outcomes would not change. The total impact of a tax on consumers and sellers depends only on the steepness of the supply and demand curves, not on the identity of the payer. Before we can understand why this is true, however, we first need to look at how taxes affect a market.

We start with a no-tax market that is in equilibrium, the market for movie tickets in Boston, Massachusetts (Figure 3.11). The equilibrium is at point x, with a price of P₁, and the quantity of movie tickets sold is Q₁. In 2003 the then mayor of Boston, the late Tom Menino, proposed adding a 50-cent tax to movie tickets to help balance a budget deficit. Many thought Menino proposed the tax because a large number of moviegoers in Boston are college students who live in the Greater Boston area but are not Boston voters. Regardless of his motivation, the tax was defeated by the state legislature.

If it had been enacted, the tax would have required theater owners to pay 50 cents per ticket to the government. Let’s look at how such a change would affect the market for movie tickets. The tax is much like a 50-cent-per-ticket increase in the theaters’ costs. We know from Chapter 2 that increases in production costs cause suppliers to supply a smaller quantity at any given price. Therefore, in response to the tax, the supply curve shifts up by the amount of the tax (50 cents) to S₂, and the equilibrium quantity of movie tickets sold falls to Q₂.9

But taxes do something different from a typical supply shift: They drive a wedge between the price buyers pay (the market price) and the price that producers actually receive (the market price minus the tax). With a normal supply curve, the price at any point on the supply curve is the price a producer receives for selling its product. With a tax, the product sells for the price P_b (we denote the price the buyers pay with a “b”), but the sellers only receive P_s (we denote the price the seller receives with an “s”). This is the buyers’ price minus the tax: P_s = P_b – tax. In other words, the buyers have to pay 50 cents more for any quantity, but the movie theaters don’t get to keep the extra money—they receive only the higher price minus the tax.

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Because of this wedge, the new equilibrium in the Boston movie market involves two prices. The first, point y, is the price ($8.30) that the buyers pay at the theater that includes the 50-cent tax. Thus, $8.30 is the market price. The second, point z, is the price ($7.80) that the suppliers receive after taking the 50 cents out of the higher market price and sending it to the government.

There are two key characteristics to note about the post-tax market equilibrium. First, the price of movie tickets increases, but not by the full 50 cents of the tax. This can be seen in Figure 3.11. The size of the tax is reflected in the vertical distance between P_b ($8.30) and P_s ($7.80). The rise in the price of movie tickets, however, is the distance between $8.30 and the equilibrium price with no tax, $8.00. The reason for this discrepancy is that the tax wedge drives some of the highest marginal-cost theaters out of the market: Once the tax is added to the price, these theaters would have to sell their tickets at a price that is too high for buyers. The second characteristic to note is that the government generates revenue from the tax. The total revenue equals the 50-cent tax times Q₂, the new quantity of tickets sold.

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We can apply all the familiar concepts from consumer and producer surplus analysis to this new equilibrium. We just need to remember that the tax creates a second supply curve that we have to keep track of, rather than moving a single supply curve as before. The supply curve that the theater owner cares about is S₁, the initial supply curve. The number of tickets that theaters are willing to supply at any particular price is still given by this curve, even after the tax is imposed, because the level of S₁ reflects the after-tax dollars theater owners take home from selling tickets. But the supply curve that the buyers actually face is S₂. It has been shifted up by the amount of the tax, because that is the price moviegoers have to pay for a particular quantity supplied.

To make things clearer, let’s work through this example in more detail. In Figure 3.11, the demand curve for movie tickets in Boston is Q^D = 20 – 2P and the supply of movie tickets is Q^S = 3P – 20, where both quantities demanded and supplied in these curves are measured in hundreds of thousands of tickets. If the legislature passes the tax, all theater owners will be required to remit to the city 50 cents per ticket sold. We can analyze the tax’s effect on the market using graphs or equations.

Graphical Analysis With no taxes, solving the model our usual way gives a free-market equilibrium price P₁ and quantity Q₁ and the resulting consumer and producer surpluses (Figure 3.11).

The tax means buyers now face a new supply curve S₂, equal to S₁ shifted up by $0.50, the amount of the tax. This reduces the number of movie tickets bought in the market from 400,000 to 340,000. At that quantity, the price the buyers are paying rises from $8.00 to $8.30. Because the law requires suppliers (the theater owners) to pay the government a 50-cent tax for every ticket they sell, the suppliers don’t get to keep $8.30; they get to keep only $7.80 = $8.30 – 0.50.

What happens to consumer and producer surplus in the post-tax market? The new consumer surplus is smaller than before. In the no-tax market, consumer surplus was A + B + C. Now it is only A, the area below the demand curve but above the price that the buyers have to pay, $8.30.

The new producer surplus is also smaller than before. Before the tax, it was D + E + F. After the tax, it is only F, the area above the supply curve and below the price that the suppliers receive after they pay the tax, $7.80.

Imposing the tax reduces total producer and consumer surplus from area (A + B + C ) + (D + E + F ) to just area A + F. Where has the surplus in areas B, C, D, and E gone? The area B + D is no longer consumer or producer surplus; it is government tax revenue, the tax times the quantity sold after the tax is implemented. With a tax, there is no surplus transfer between producers and consumers, as we saw in earlier examples. Instead, both producers and consumers transfer some of their surpluses to the government. This tax revenue is then “returned” to consumers and producers in the form of government services, so it is not lost.

Areas C and E are the deadweight loss from the tax. They are surpluses that moviegoers and theater owners formerly got from buying and selling tickets at the competitive price. This surplus is gone now because consumers buy fewer tickets at the higher post-tax price and sellers supply fewer tickets at their lower post-tax price.

Just as in the price regulation cases, a natural way to look at the size of the deadweight loss is as a fraction of the surplus transfer. Before, that transfer occurred from producers to consumers (for a price ceiling) or from consumers to producers (for a price floor). Now, it’s from both to the government. The ratio in this case is the area C + E to the area B + D, which is the DWL as a share of revenue.

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Analysis Using Equations The no-tax market equilibrium equates quantity demanded and quantity supplied:

Q^D = Q^S

20 – 2P = 3P – 20

5P = 40

P₁ = 40/5 = $8 per ticket

Q^D = 20 – 2($8) = 4 or Q^S = 3($8) – 20 = 4

Therefore, before the tax, the equilibrium price is $8 and 400,000 tickets are sold.

The pre-tax consumer surplus is the triangle above the price and below the demand curve, as shown in Figure 3.11:

Again, the choke price is found by determining the price that pushes the quantity demanded to zero:

Q^D = 20 – 2P_DChoke = 0

P_DChoke = $10

In other words, this demand curve says that if tickets cost $10, no one will go to theaters in the city of Boston (perhaps because theaters in the suburbs are an attractive alternative).

Plugging the demand choke price into the CS formula gives a consumer surplus of

The producer surplus is the triangle above the supply curve and below the price:

The supply choke price is the price that moves quantity supplied to zero:

Q^S = 3P_SChoke – 20 = 0

P_SChoke = $6.67

That is, at any price below $6.67 a ticket, no theaters would operate in Boston. Plugging this supply choke price into the PS formula gives a producer surplus of

What happens to consumer and producer surplus if Mayor Menino applies his 50-cent tax? Theaters must pay the state for each ticket they sell. This creates a dual-supply-curve situation. The supply curve for the theater owners is the same as the initial supply curve. The theater is still willing to supply whatever number of tickets the supply curve says at the market price. But now, the supply curve buyers face is shifted up by the amount of the tax: At each price, the tickets supplied to consumers now cost $0.50 more. The difference between the supply curve that the buyers face and the supply curve that the sellers face is the amount of the tax. In words, the theaters’ supply curve says they would be willing to sell 400,000 tickets if they receive $8 per ticket (after the tax gets paid), but for theaters to get $8 per ticket, buyers would actually have to pay $8.50 per ticket because $0.50 of the tax needs to be paid out of the price received by the theaters. The prices that result for both the buyer and the seller are summed up in the equation P_b = P_s + $0.50.

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To solve for the post-tax quantity and prices, we substitute this expression, which links the two supply prices in our supply and demand equations:

Q^D = Q^S

20 – 2P_b = 3P_s – 20

20 – 2(P_s + 0.50) = 3P_s – 20

20 – 2P_s – 1 = 3P_s – 20

5P_s = 39

P_s = 39/5 = $7.80

Therefore, the buyers face the following price:

P_b = P_s + 0.50 = $7.80 + 0.50 = $8.30

Now if we plug the buyer price into the demand curve equation and the supplier price into the supply curve equation, they will both give the same after-tax market quantity:

Q₂ = 20 – 2(8.30) = 3.4
or

Q₂ = 3(7.80) – 20 = 3.4

Only 340,000 tickets will be sold once the tax is put into place.

The consumer surplus after a tax is the area below the demand curve but above the price that the buyers pay:

The producer surplus is the area above the supply curve and below the price that the suppliers receive:

So the tax makes consumer surplus fall by $111,000 and producer surplus fall by $74,000 from their values in the no-tax market equilibrium. Some of that $185,000 in lost surplus flows to the government in the form of revenue from the tax, however. That revenue is equal to $0.50 per ticket times the number of tickets sold after the tax, or

Revenue = $0.50Q₂

= $0.50(340,000) = $170,000

Notice that the total amount of the lost surplus, $185,000, is more than the amount of revenue that the government generated, $170,000. The difference of $15,000 is the deadweight loss of the tax.

A different way to calculate DWL is to compute the area of the triangle whose base is the change in quantity and whose height is the amount of the tax:

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That’s about 9% of the revenue generated by the tax. In other words, this tax burns up about $1 of surplus in DWL for every $11 of revenue it generates.

Just as we showed in the case of price and quantity regulations, the main determinant of the DWL from a tax as a share of revenue is how much the quantity changes when the tax is added. The size of that change depends, in turn, on how sensitive supply and demand are to prices. The deadweight loss from pizza price controls, for example, came about because there were consumers and suppliers who would like to trade at market prices and would have earned surplus from doing so, but were prevented from engaging in these transactions by the price ceiling. With taxes, there are no forbidden transactions. The source of the loss is the same, however. There are people who would have bought tickets at the market price without a tax and would have gained some surplus from doing so. Once the government adds a tax, the after-tax price rises enough so that these consumers no longer want to buy tickets. They get to keep their money, but they were previously able to buy something with it that gave them surplus. Likewise, movie theaters lose surplus because some would have shown movies at the pre-tax market price but find the after-tax price too low to justify operating. These lost surpluses are the DWL of the tax.

An interesting result of our analysis is that it implies the inefficiency represented by the size of the deadweight loss gets much bigger as the size of a tax becomes larger. In the movie ticket tax example (Figure 3.11), we saw that the DWL from the tax was area C + E and that the revenue generated was B + D. What would happen if Mayor Menino decided to increase the ticket tax? How much more revenue and how much more DWL would this tax increase create?

Figure 3.12 illustrates the outcome for a general case. The larger tax reduces the quantity even further, from Q₂ to Q₃. The DWL under the larger tax is area C + E + F + H (remember that it was only area F with the smaller tax). Government revenue, which was area D + E with a smaller tax, is now area B + D + G. That is, the government gains areas B and G because the people who still buy tickets are paying more in taxes. However, the government loses area E, because some people stop buying movie tickets after the tax is raised. If we look at the DWL as a share of the revenue generated, it is clear that the incremental revenue generated by increasing this tax causes more inefficiency than the smaller tax did. Initially, the DWL was F, with a revenue gain of D + E. But the incremental DWL here is C + E + H, while the revenue gain is only B + G – E. In fact, if taxes become high enough, the increase in revenue per ticket from the tax will be more than offset by the reduction in the quantity of tickets sold, and there will be no revenue gain at all!

A general rule of thumb is that the DWL of a tax rises with the square of the tax rate.10 That is, doubling the tax rate quadruples the DWL. That’s why economists tend to favor tax policies that exhibit what is called “low rates and broad bases.” That is just a way of saying that, all else equal, taxing ten things at a low rate is better than taxing five things at zero and five things at a high rate. Because DWL rises with the square of the tax rate, the overall DWL will be larger with the five high rates than with the ten low rates.

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An important thing to note about the movie ticket example is that although we supposed that it would be the theater owners who were legally obligated to remit the 50 cents per ticket tax to the City of Boston, they don’t bear the complete burden of the tax. Before the tax came in, the theater owners received $8 per ticket and moviegoers paid $8 a ticket. After the $0.50 tax, moviegoers pay $8.30 a ticket. After they send in the tax, however, the theaters only end up with $7.80 per ticket. Therefore, of the 50 cents going to the government, 30 cents (60%) of it is coming out of consumers’ pockets because their price went up by 30 cents. Movie theaters send the tax check to the government, but they are able to pass on much of the tax to consumers through higher prices. This means that the price realized by the suppliers goes down by only 20 cents. Who really bears the burden of a tax is called tax incidence. The incidence of this tax is 60% on the buyers and 40% on the suppliers.

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Now let’s say Boston changed the rule for who pays the tax to the government. Instead of the theater sending the tax payment to the government, moviegoers would pay the tax by, after buying their ticket at whatever price the theater charges, dropping two quarters in a “Menino Box” as they enter the theater (silly idea, yes, but this is just to make a point).

This does not alter the tax incidence. The equations below show that in the tax formula it doesn’t matter whether you subtract the tax from what the supplier receives or add the tax to what the buyer pays. P_s = P_b – tax is the same as P_s + tax = P_b.

This can also be seen graphically. The original case where the theater remits the tax is shown in Figure 3.13a. When the tax is instead paid by buyers, their quantity demanded depends on the price including the tax. But the price suppliers receive at this quantity demanded is only the price without the tax. To account for this difference, we shift down the demand curve by the amount of the tax, from D₁ to D₂ in Figure 3.13b. But the result hasn’t changed: Quantity demanded is still Q₂, and the difference between the buyer’s price and the seller’s price still equals the amount of the tax. Thus, the incidence of a tax does not depend on who is legally bound to pay it.

That’s why a helpful way to picture taxes on a graph is to just forget about whether the tax is moving the supply curve up or the demand curve down. Instead, start from the initial no-tax equilibrium point and move left until the vertical space between the supply and demand curves is the amount of the tax. This gives you the right answer regardless of whether the tax is being legally applied to suppliers or buyers.

This point about tax incidence is fundamental. If you have ever had a job, you probably know that the government takes many taxes out of your pay. Some of these are payroll taxes that appear on your pay stub as FICA (Federal Insurance Contributions Act). They are collected to pay for Social Security and Medicare. In the United States, payroll taxes are legally split evenly between workers and employers. In other words, if you earn wages of $1,000, you have to pay 7.65% of that in payroll taxes and the employer has to pay another 7.65% on its own.11 Would U.S. workers be better or worse off if the law changed so that the company paid 15.3% and workers paid nothing, or if it instead made the employee pay 15.3% and his or her employers paid nothing? The analyses we’ve just completed suggest that such changes wouldn’t make any difference. In a competitive market, the wage would adjust to the same level regardless of which side of the market is legally bound to pay the tax.

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It turns out that the only thing that matters about the economic effects of this tax is how elastic the supply and the demand for labor are. To see why, let’s consider the two extremes.

Relatively Elastic Demand with Relatively Inelastic Supply In a market characterized by relatively elastic demand and inelastic supply, buyers are very sensitive to price and the suppliers are not. Most labor economists tend to think of the labor market in this way, so that the 15.3% FICA tax (the combined tax rate on the two sides) applies to a market in which labor supply is fairly inelastic (people work a similar amount even if their wage goes up or down) and firms’ demand for labor is fairly elastic. This market is illustrated in Figure 3.14a.

Applying the methods we’ve used throughout this section, we see that the tax is borne almost entirely by the suppliers—here, workers supplying labor. With a tax, employers have to pay wages W_b that are a bit higher than wages without the tax, W₁. But after taxes, the workers receive a wage W_s that is much less than W₁. Therefore, workers are a lot worse off after the tax than employers. And we know from the discussion that we just had on tax incidence that even if the government switched the payroll tax rules so that employers paid the entire amount, workers would not do any better. Their wages would fall almost as much as the employers’ tax went up.

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Relatively Inelastic Demand with Relatively Elastic Supply Figure 3.14b shows a market characterized by relatively inelastic demand (buyers are not sensitive to price) and elastic supply (suppliers are very sensitive to price). In the market for cigarettes, for instance, many buyers are addicted and tend to buy a similar amount no matter how much the price goes up. Cigarette supply is more elastic. You can see in the figure that in this case, consumers bear the brunt of the tax. A tax on cigarettes causes the buyers’ price to rise from P₁ to P_b, almost the entire amount of the tax. Suppliers are only a bit worse off than they were before, because they can pass on the higher costs to the inelastic consumers.

We could do this entire analysis using equations, as we did in the movie ticket example. It turns out that there is a general formula that will approximate the share of the tax borne by the consumer and the share borne by the producer. Not surprisingly given what we’ve just discussed, these shares depend on elasticities:

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If the price elasticity of supply (E^S) is infinite, the consumers’ share is equal to 1; that is, consumers bear the whole burden when supply is perfectly elastic. If the absolute value of the price elasticity of demand is infinite, the consumers’ share of the tax burden is zero, and suppliers bear the whole burden of the tax.