Taxes take many forms in our economy, including income taxes, property tax, and sales tax. In this section, we will focus on a particular type of tax - an excise tax - and how the tax can be studied in the context of perfectly competitive markets. An excise tax is a tax that must be paid per unit of a good or service exchanged in a market. For example, the federal government places an 18.4-cent tax on each gallon of gasoline that is purchased in the U.S.. The state of Washington imposes an additional 49.4-cent tax on each gallon. Similar taxes exist on liquor (in the state of Washington), on sugary drinks in some cities, and on cigarettes. The federal government levies a $1.0066 tax per pack, while every state levies an additional tax per pack, ranging from $0.17 per pack in Missouri to $4.35 per pack in New York and Connecticut. 1
Figure6.2.1.Excise taxes on gasoline in the state of Washington. The state excise tax of $0.49.4 and federal excise tax of $0.18.4 combine to create a $0.67.8 tax per gallon.
Why would the government choose these goods to tax? In the case of gasoline, the government might have environmental concerns on its mind, and a tax is likely to reduce consumption (and, therefore, carbon emissions) by increasing the price consumers have to pay. Taxing cigarettes is a public health issue, so if higher prices lower consumption of cigarettes, reducing second-hand smoke and smoking-related illness, a tax may be warranted. The same could be said for taxes on sugary soft drinks. Often, with a tax, a reduction in quantity is the name of the game.
If you don’t remember writing a check to the government every time you stop for gas, that’s because you don’t need to. The tax is collected from sellers, who track how many gallons are sold and send an appropriately-sized check to the government. Does this mean that if the $1 per pack tax weren’t there, consumers would see a $1 reduction in the price of a pack of cigarettes?
The answer is no, but it isn’t immediately obvious why, so let’s examine with an example of the market for gasoline. Gasoline markets are fairly competitive in most areas, with numerous sellers without market power and identical products. In our example, let’s say the equilibrium outcome in the market is \((P^*, Q^*) = (\$3, 100)\text{.}\) 2 In a competitive market without any regulation, this would be the market outcome.
Now, suppose the government imposes a $1 per unit excise tax on every unit exchanged. This means the government needs to collect $1 from every transaction! To simplify the analysis, we’ll say that the policy gives sellers the responsibility to send one dollar from every transaction to the government. So, for example, if sellers still sold each gallon for $3 each, they would send one dollar off, and keep two for themselves.
This doesn’t sound like a good deal for sellers. So let’s hypothesize: what happens if sellers charge consumers a price of $4 instead of $3? Now, consumers will pay $4 for every gallon, and since $1 goes to the government, each gallon will generate $3 for the seller. In one way, then, there are now two prices in the market for gasoline: one price the consumer pays (\(P_C\)), and one price the seller receives (\(P_S\)). Of course only one price is apparent at the gas station, but the prices relevant to buyers’ and sellers’ decisions are now different.
This is the major distinction in markets with excise taxes. In any other market, the price sellers collect and the price buyers pay is the same. At the equilibrium outcome, buyers pay $3 and sellers get $3 for every unit. No conflict. But the tax drives a wedge between the price relevant to buyers (what they pay) and the price relevant to sellers (what they receive), and the difference arises. Any time there is an excess tax (or subsidy) in a market, \(P_C\) and \(P_S\) are not the same. 3 Moreover, the gap between the price consumers pay and the price sellers receive is exactly the size of the tax. If the excise tax is \(t = \$1\text{,}\) for example, then when buyers pay \(P_C = \$4\text{,}\) sellers receive \(P_C - t = P_S\) or \(4 - 1 = \$3\text{.}\)
In our example, the market price is $3 prior to the tax. But how will the market outcome be affected when the tax is introduced? If sellers attempt to pass on the $1 tax to consumers, then \(P_C = 4\) and \(P_S = 3\text{.}\) But what happens to the quantities supplied and demanded? \(P_S\) is the relevant price for determining \(Q_S\text{.}\) Sellers were willing to sell 100 units at a price of 3 at equilibrium, and under the tax, sellers still receive $3 per gallon. So the quantity supplied would remain unchanged:
\begin{equation*}
Q_S = 100
\end{equation*}
What about quantity demanded? Here, \(P_C\) is the relevant price for determining \(Q_D\text{,}\) since it is the price consumers ultimately pay. Consumers were willing to buy 100 units at a price of $3, but now their price has increased to \(P_C = 4\text{.}\) By the Law of Demand, consumers will not be willing to buy 100 gallons at a higher price:
\begin{equation*}
Q_D \lt 100
\end{equation*}
Since it must be true that \(Q_S > Q_D\text{,}\) there is a surplus in the market if the seller tries to charge a price of $4. The market doesn’t clear! Moreover, the surplus will put downward pressure on the price, per usual. The presence of the surplus shows that the seller cannot pass on the full dollar of the tax to consumers.
Figure6.2.2.If sellers raise price by the amount of the tax, there will be a surplus. As a result, the market will not clear.
If, on the other hand, sellers continued to sell each gallon for $3, sending $1 per gallon to the government, \(P_C = 3\) and \(P_S = 2\text{.}\) But consumers would demand 100 gallons at \(P_C = 3\) and sellers would not be willing to supply that many units. As a result, \(Q_D > Q_S\) and a shortage would occur, putting upward pressure on the price in the market.
Let’s combine: since there is upward pressure at a market price of 3, and downward pressure at a market price of 4, then under the tax, the market-clearing price must be somewhere between 3 and 4. 4 That is, the presence of the tax should 1) increase the price consumers pay, and 2) decrease the price sellers receive. This is often described by saying that both buyers and sellers bear some of the burden of the tax, since there is no mechanism by which 100% of the tax can be paid by either sellers or buyers.
Figure6.2.3.For the market to clear in the presence of an excise tax, the price sellers receive (\(P_S\)) must decrease and the price consumers pay (\(P_C\)) must increase. This means the burden of the tax must be felt by both buyers and sellers.
Let’s recap. When an excise tax is imposed on an competitive market, the tax drives a wedge between the price consumers pay (\(P_C\)) and the price sellers receive (\(P_S\)). Additionally,
the price consumers pay increases relative to \(P^*\text{,}\) leading to lower quantity demanded;
the price sellers receive decreases relative to \(P^*\text{,}\) leading to lower quantity supplied;
the market clears at a lower quantity than \(Q^*\text{.}\)
The reduction in number of transactions is, in many cases, a feature of the excise tax, not a bug. Recall that in many markets where excise taxes are imposed, such as markets for gasoline or cigarettes, policymakers’ objective is to reduce transactions. Since a transaction required both a buyer and a seller, reducing the number of transactions in a market requires reducing both the quantity demanded (by increasing \(P_C\)) and the quantity supplied (by decreasing \(P_S\)). The excise tax accomplishes exactly this! By driving a wedge between these two prices and spreading them apart, the tax discourages both buyers and sellers from participating, achieving the sought-after reduction in transactions.
The quantity reduction is critical in an analysis of the welfare impacts of the excise tax as well. First, notice that in the presence of an excise tax, both consumer surplus and producer surplus are reduced, relative to their size without the tax. On the consumer side:
each consumer who buys must pay a higher price (\(P_C > P^*\)); and,
fewer consumers are able to buy (\(Q_D\) decreases).
Both of these consequences drive consumer surplus lower. On the sellers’ side, we see something similar:
each seller who sells receives a lower price (\(P_S \lt P^*\)); and,
fewer sellers are able to sell (\(Q_S\) decreases).
Therefore, under the excise tax, consumers and sellers are both made worse off! Umm, why would we implement a policy that makes both parties worse off? Well, for a couple of reasons! First, once taxes are in play, there is now a third player in the model: the government. Someone benefits from the collection of taxes paid, and that someone is the government. In fact, if the government collects \(t\) dollars for each transaction that occurs, then it will collect tax revenue equal to
\begin{equation*}
\text{ tax revenue } = (\text{ excise tax } )\times(\text{ number of transactions } )
\end{equation*}
Graphically, the tax revenue is the rectangular area which is \(t\) dollars high and \(Q\) transactions wide. Therefore, much of the lost consumer surplus and producer surplus is transferred from those groups to the government. But, there is still some unaccounted for lost surplus. This is the deadweight loss under the excise tax!
Figure6.2.4.Left: Consumer surplus, producer surplus, and (non-existent) DWL prior to excise tax; Right: consumer surplus, producer surplus, and DWL after the excise tax. Tax revenue is the rectangular area in the middle, collecting \(t\) units per transaction.
Recall that deadweight loss is the result of denying mutually beneficial transactions, and represents a loss of total surplus. Here, it should be clear why the excise tax results in DWL, since the tax drives up \(P_C\) and lowers \(P_S\) precisely to reduce the number of transactions. The potential benefits to both buyers and sellers which do not materialize due to the tax are measured as the area of the typical DWL triangle. Moreover, notice that this surplus cannot be received by the government. Since tax revenue can only be collected from transactions which happen, mutually beneficial transactions that do not occur cannot generate tax revenue.
Figure6.2.5.Transferred surplus under an excise tax. Some of the lost consumer surplus (the red rectangular area to the left) becomes tax revenue, while the rest of the lost CS becomes DWL. Similarly, some of the lost producer surplus (the blue rectangular area to the left) is transferred to the government as tax revenue, while the rest becomes DWL.
Ultimately, the excise tax reduces consumer surplus and producer surplus. Much of the lost \(CS\) and \(PS\) is transferred to the government in the form of tax revenue, but some is not recovered and persists as deadweight loss. Therefore, an excise tax is, by definition, an inefficient outcome relative to competitive equilibrium. But this is sort of by design! If the objective of an excise tax is to reduce the number of transactions, then the market inefficiency - which is by definition about those same lost transactions - is likely to be more palatable to policymakers!
A subsidy is functionally similar to an excise tax, but works in the opposite direction. The objective of a subsidy is to encourage additional transactions in a market, by making government payments of \(s\) dollars per transaction that occurs. Subsidies are often implemented in markets where policymakers aim to encourage more transactions than would otherwise happen without the intervention. For example, some medical treatments, such as flu shots, and education, are often subsidized.
Just as we saw with the excise tax, it is irrelevant whether the subsidy is physically collected by buyers or sellers in the market. What matters is that both buyers and sellers are incentivized to participate more. This means the subsidy must reduce the price consumers pay and increase the price sellers receive.
Let’s take an example in the market for flu shots, which, with many buyers and sellers, can be modeled as a perfectly competitive market. Here, we assume the market equilibrium is \((P^*, Q^*) = (20, 500)\text{.}\) At a price of $20, 500 consumers are willing to purchase flu shots, and 500 providers are willing to sell them. But what if the local government wants to incentivize more flu shot transactions to promote public health?
We know that to increase the number of flu shots exchanged in the market, the government needs to incentivize buyers and sellers. So if the government is willing to subsidize each flu shot by \(s = \$10\text{,}\) it needs to make sure some of those dollars are received by consumers and the rest are received by sellers. Sending a $10 check to each consumer who buys a flu shot will increase \(Q_D\) by lowering \(P_C\text{,}\) but will do nothing to incentivize sellers to provide more flu shots to the market, and \(Q_S\) will remain at 500, leading to a shortage.
The difference between \(P_S\) and \(P_C\) must be the size of the subsidy here. For example, to incentivize more than 500 transactions from an initial equilibrium price of \(P^* = 20\) with a subsidy of \(s = \$10\text{,}\) the government will need make sure sellers receive, say, $23 per flu shot. Since 10 of those dollars are coming directly from the governmental subsidy, consumers only pay \(23 - 10 = P_C = \$13\text{.}\) Or, formally, \(P_S - s = P_C\text{.}\)
As we saw with the excise tax, the market-clearing condition here is
\begin{equation*}
Q_D\text{ at } P_C = Q_S\text{ at } P_S
\end{equation*}
Since the subsidy will lower the price consumers have to pay and increase the price sellers receive per transaction, the market will clear at a higher quantity than at the previous equilibrium, which is by design. To recap, under the per-unit subsidy:
the price consumers pay decreases relative to \(P^*\text{,}\) leading to higher quantity demanded;
the price sellers receive increases relative to \(P^*\text{,}\) leading to higher quantity supplied;
the market clears at a higher quantity than \(Q^*\text{.}\)
Figure6.2.6.For the market to clear in the presence of a per-unit subsidy, the price sellers receive (\(P_S\)) must increase and the price consumers pay (\(P_C\)) must decrease. Both buyers and sellers need to be incentivized into increasing the number of transactions.
Welfare analysis under a subsidy is similar, but most dimensions work in the opposite direction. For example, consumer surplus has to increase:
each consumer who buys gets to pay a lower price (\(P_C \lt P^*\)); and,
more consumers are able to buy (\(Q_D\) increases).
Both of these factors increase consumer surplus, since more consumers join the market (and receive benefits) and those who still participate earn higher individual surplus by paying the lower price. The same can be said for producer surplus:
each seller who sells receives a higher price (\(P_S > P^*\)); and,
more sellers are able to sell (\(Q_S\) increases).
The subsidy simultaneously increases both consumer surplus and producer surplus! So, who pays the cost to make this happen? The government, who pays \(s\) dollars for each transaction that happens. In other words, the government incurs the cost 5 here (measured by the cost of the subsidy), and consumers and producers benefit under the subsidy. Additionally, more transactions will occur, which is often the express purpose of the subsidy in the first place.
What about efficiency? Under the subsidy, both consumer surplus and producer surplus grow. In the graph below, notice that the larger \(CS\) and \(PS\) triangles now overlap. Normally, this can’t happen, since one surplus area cannot be simultaneously received by two separate groups. But the subsidy provides a second layer of available subsidy, allowing both groups to realize the added benefit. The rectangular area captures the cost of the subsidy, since it is measured as (subsidy per transaction)×(number of transactions). Most of what is spent goes to increasing the consumer and producer surplus, but a small portion - the rightmost triangle - does not contribute to any consumer or producer benefit.
This triangle is the deadweight loss (DWL) from the subsidy, and measures the degree of inefficiency which results from the subsidy. Normally, we describe deadweight loss as the result of missed opportunities for mutually beneficial transactions that otherwise would be allowed to occur without the policy intervention. But something different happens when there is a subsidy. The objective of the subsidy is to increase the number of flu shots, but in order to do this, the subsidy encourages transactions to happen that are not mutually beneficial! The additional transactions, which are to the right of the original equilibrium, occur between consumers with high WTP and sellers with low reservation prices. As we saw in the previous chapter, in the absence of the subsidy, these transactions would not happen. Even if forcing the transactions to occur achieves the objective of the subsidy, those resulting additional transactions are inefficient.
Figure6.2.7.Left: Consumer surplus, producer surplus, and (non-existent) DWL prior to per-unit subsidy; Right: consumer surplus, producer surplus, and DWL after the per-unit subsidy. The total cost to the government of the subsidy is the large rectangle which overlays across the middle of the graph, since the government must pay \(s\) dollars per transaction.
Both excise taxes and subsidies can be useful policy tools for increasing or decreasing the number of transactions in a market. This makes sense for markets where public interest might be best served by either fewer transactions (cigarettes, gasoline) or more transactions (flu shots, education). These markets share features which make them perfect candidates for these policies - features we will explore in the next section.
In Depth6.2.8.Taxes and subsidies with functions.
You surely noticed that in this section, instead of deriving an equilibrium from specific functions, we just assumed values for \(P^*\) and \(Q^*\text{.}\) Of course, it is possible to analyze with more precision using demand and supply functions, but they can be a bit challenging to work with. Let’s explore why.
Let’s again assume that the government wishes to impose a \(t = \$1\) excise tax. If the demand function in the market is \(Q_D = 180 - 30P\) and the supply function is \(Q_S = 30P\text{,}\) then the market equilibrium is \((P^*, Q^*) = (3, 90)\text{.}\)
We know that if an excise tax of $1 is imposed, the burden must be shared by both buyers and sellers, meaning \(P_C\) must increase and \(P_S\) must decrease. But previously, without specific functions, we could not identify where the new market outcome would be under the tax. With careful analysis now, we can.
Algebraically, this requires a bit of care. Mostly, the problem arises because of the two separate prices under the tax. Now, to be technical, we must write the demand and supply functions as
Again, without the tax, the price consumers pay and the price sellers receive is the same: \(P_C = P_S = P\text{.}\) But with a tax, \(P_C \neq P_S\text{.}\) This makes solving for equilibrium challenging: we have one equation - the market-clearing condition \(Q_D = Q_S\) - but two variables: \(P_C\) and \(P_S\text{.}\)
But! We do have an additional condition. We know how the tax interacts with the prices: \(P_S + t = P_C\text{.}\) This condition describes exactly how the tax drives a wedge between the price consumers pay and the price sellers receive. Therefore, in essence, finding the market outcome under a tax is about solving a system of two equations and two unknowns, where the two variables are \(P_C\) and \(P_S\) and the two equations are
\begin{equation*}
P_S + t = P_C\text{ (how the tax works) }
\end{equation*}
The easiest method for solving, therefore, is to substitute the tax condition into the market-clearing condition. In our example, the two conditions are
The tax condition, for example, should have some intuition behind it, since under a $1 tax, the consumers should pay a price exactly 1 dollar above what sellers receive. The first approach will be to substitute out \(P_C\text{.}\) If we plug the tax condition into the market-clearing condition, we get
Under the $1 tax, sellers receive $2.50, down from the previous equilibrium of $3. Since \(P_C = P_S + 1 = 2.50 + 1\text{,}\) we know the price buyers pay is \(P_C = \$3.50\text{.}\) Finally, the quantity in the market can be found from either the demand function of supply function:
The number of transactions exchanged under the tax is 75, down from the equilibrium quantity of 90. That is, the $1 tax reduces the quantity of transactions by 15! We can finally fully characterize the market outcome under the $1 excise tax. Consumers pay \(P_C = 3.50\text{,}\) sellers receive \(P_S = 2.50\text{,}\) and only 75 transactions happen. With these values, we can also conduct a full welfare analysis by calculating the areas for consumer surplus, producer surplus, tax revenue, and deadweight loss. A similar approach can be used to analyze a market with a per-unit subsidy.
Moreover, once we have the exact values for \(P_C\) and \(P_S\text{,}\) we can determine the tax incidence on each group. Tax incidence describes the share of the burden of the tax felt by either consumers or producers. Put simply, the tax incidence is the percentage of the tax that is paid by each group. In our example, the tax is $1 and the consumers’ price increases by $0.50, giving a tax incidence on consumers of \(\frac{0.50}{1} = \frac{1}{2}\text{.}\) Consumers pay for exactly one half of the excise tax. Similarly, then, since sellers’ price decreases by 0.50, they pay for the other half of the tax, and incur a tax incidence of \(\frac{1}{2}\text{.}\) Formally, the tax incidence on consumers can be calculated from the equation
One final aside: This problem has the same solution if we substitute out \(P_S\) when solving the equations above. We will find \(P_C = 3.50\) directly, and the rest of the solution is identical, since they are identical methods of solving the problem. On a technical level, the way you choose to substitute and solve is irrelevant.
But to some, there is a minor difference in interpretation in the two approaches. It is tempting to interpret the demand function \(Q_D = 180 - 30(P + 1)\) as a shifted demand curve as a result of consumers paying the tax. If you were to draw this shifted demand curve, which is a decrease in demand, the intersection would identify \(P_C = 3.50\) and a quantity exchanged of 75. If, instead, \(P_S\) were subbed out, the supply function \(Q_S = 30(P - 1)\text{,}\) you would be drawing a decrease in supply, and could interpret this as supply given sellers are the ones paying the tax. This intersection shows \(P_S = 2.50\) and a quantity of 75 as well.
There is no technical difference in using this virtual shifting curves approach. However, we do not use the shifting curve approach here for one primary reason: neither curve shifts under the tax. The demand curve is where it is because of demand parameters, such as consumer preferences, income, and the price of substitutes; a tax changes none of these things. In fact, the sole impact of the tax is to change \(P_C\text{,}\) but we know that a change in price does not shift the demand curve!
