Lecture 21: Taxes

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In our analysis thus far, we have analyzed transactions from the perspective of the outcomes for individual agents. That is, we have been considering how each agent’s actions affect their own individual payoffs: for example, we think about how an individual consumer trades goods, from their perspective, by comparing their utility from their consumption bundle before the transaction with the utility from their consumption bundle after the transaction.

However, many actions taken by economic agents – individuals, firms, or governments – impact not only themselves but others as well. This week, we’ll introduce the field of public economics. This is a broad field, and is largely concerned with the role of government in a functioning economy. When are there situations in which the government can achieve an outcome which is better than the market can on its own – and when do governments make things worse, as much as they may be trying to help?

In this lecture, we will analyze what happens when a government imposes a tax (or tariff) on a good.
In the next lecture, we will examine situations with externalities, in which one agent’s actions affect those around them. These externalities may be negative (like pollution) or positive (like beautifying your front yard). We’ll see how appropriate taxes or subsidies could be used to solve the “missing markets” problem externalities pose.
On Friday, we’ll examine the case of goods which are not purely private goods. We’ll look at the provision of public goods, such as national defense or public radio, whose benefits accrue to a large group of people and can’t be excluded just to those who pay for them. And we’ll look at common resources, such as the environment, which may be overutilized by economic agents whose self-interest is left unconstrained.

In other words, we’re going to use all the tools we’ve been looking at to develop a coherent idea of the role of government in a market economy. Let’s start by looking at what happens when a government imposes a tax.

Effect of a tax

We’re all familiar with sales taxes: an amount that’s added on to the bill, which goes to one or more governing bodies. That is, if consumers pay $p_C$ and firms receive $p_F$, then in the presence of a tax $t$, $p_C = p_F + t$ For example, when you eat at a restaurant, the bill usually includes the price listed on the menu, plus a tax. If you buy gas from a gas station, generally the posted price includes all taxes. (In many states, but not all, you’ll see a little sticker on every gas pump saying how much of the list price is going to local, state, and federal taxes.) In each case, the amount the consumer pays goes to two recipients: part of it goes to the seller, and part goes to the government.

There are different kinds of taxes. Some are a fixed dollar amount per unit of the good: for example, since 1993, the federal government has imposed a tax of 18.4 cents a gallon regardless of the price of gas. Other taxes are a percentage of the listed price: for example, the State of California currently imposes a sales tax based on which county the sale takes place in; in Santa Clara County, it’s 9.125%. Mathematically, we would express this as $p_C = 1.09125p_F$ or more generally, if the tax rate is $\tau$, as $p_C = (1 + \tau)p_F$ where $\tau = 0.1$ would correspond to a 10% tax.

A lot of textbooks — perhaps even the one you had in Econ 1 or high school — treat the effect of a tax as a “shift” in a demand or supply curve. This can be a way of thinking about the effect, but it obfuscates as much as it clarifies. So let’s think about what’s really going on.

We’ve derived the demand and supply for a good as a function of the market price, $p$, and used this to arrive at our equilibrium condition $D(P) = S(P)$ However, this doesn’t hold in the presence of taxes. Rather, consumers base their buying decisions on the price they have to pay for a good, $P_C$, and firms base their selling decisions on the amount they receive for that good, $P_F$. Thus it’s better to say $D(P_C) = S(P_F)$ In the absence of a tax, $P_C$ and $P_F$ are both equal to the listed price $p$; so we can simplify the equation to just be $D(P) = S(P)$. This is what’s shown in the initial diagram below. But look at what happens when you impose a tax by dragging the tax slider to the right, and then adjust to the new equilibrium:

See interactive graph online here.

Note that neither of the curves “shift” — that is, the existence of the tax doesn’t change how firms and consumers respond to the prices they face.

However, let’s now look at the same graph, but this time let’s assume that the tax is entirely paid by consumers: firms receive the list price $(P_F = P)$, while consumers pay the price plus tax $(P_C = P + t)$. If what we show on the vertical axis is the list price (that is, $P$), then the two curves represent $S(P)$ and $D(P,t)$. Now the imposition of the tax “shifts” the demand curve down by the amount of the tax, since at every price $P$, the curve shows how much consumers would demand at that price plus the tax:

See interactive graph online here.

Likewise, if we impose the tax on firms, then if the vertical axis shows the list price, the two curves represent $D(P)$ and $S(P,t)$, and the imposition of the tax “shifts” ths supply curve up by the amount of the tax, since at every price $P$, the curve shows how much firms would supply at that price minus the tax:

See interactive graph online here.

Note that in each case, regardless of how we represent the imposition of the tax, the same amount of tax results in the same price paid by consumers, the same price received by firms, and the same equilibrium quantity.

Calculating the equilibrium with a tax

Let’s see how we solve this mathematically. Suppose we have $\begin{aligned} D(P_C) &= 100 - 3P_C\\ S(P_F) &= 2P_F \end{aligned}$ In the absence of any tax, we would set demand equal to supply at a common price $P$: $\begin{aligned} 2P &= 100 - 3P\\ 5P &= 100\\ P^\star &= 20 \end{aligned}$ If there is a tax of amount $t$ that’s paid by the buyer, then consumers pay $P_C = P + t$ and firms recieve $P_F = P$, we have $\begin{aligned} D(P,t) &= 100 - 3(P + t)\\ S(P) &= 2P \end{aligned}$ Supply will equal demand when $\begin{aligned} 2P &= 100 - 3(P + t)\\ 5P &= 100 - 3t\\ P^\star &= 20 - 0.6t \end{aligned}$ Remember, this is the “list price” which firms receive; consumers pay that amount, plus the tax: $P_C = P^\star + t = 20 - 0.6t + t = 20 + 0.4t$ To find the equilibrium quantity, we can plug either price back into its respective supply or demand function: $\begin{aligned} D(P_C) &= 100 - 3P_C = 100 - 3(20 + 0.4t) = 40 - 1.2t\\ S(P_F) &= 2P_F = 2(20 - 0.6t) = 40 - 1.2t \end{aligned}$ For example, if $t = 20$, we would have $\begin{aligned} P_F &= 20 - 0.6 \times 10 = 14\\ P_C &= 20 + 0.4 \times 10 = 24\\ Q &= 40 - 1.2 \times 10 = 28 \end{aligned}$

This is illustrated in the diagram below. Increase the tax to see how the prices and quantity respond:

See interactive graph online here.

Taxing buyers or sellers doesn’t matter

Sometimes, as in the price of a restaurant meal, the listed prices on the menu is the price excluding tax; the tax is added to the bill at the end. In such a case, if the listed price is $p$, then we could write $\begin{aligned} P_C &= P + t\\ P_F &= P\end{aligned}$In other cases, as in the price of gasoline, the price displayed to a consumer is the price including tax. In these cases, consumers pay the listed price $p$, but firms only get that minus the amount of the tax: $\begin{aligned} P_C &= P\\ P_F &= P - t\end{aligned}$ Does it matter which way the tax is imposed? This model would say absolutely not: the conditions $P_C = P_F + t$ and $P_F = P_C - t$ are identical. With the example above, if we had imposed the tax on firms, we would have had $\begin{aligned} D(P) &= 100 - 3P\\ S(P,t) &= 2(P - t) \end{aligned}$ Supply will equal demand when $\begin{aligned} 2(P - t) &= 100 - 3P\\ 5P &= 100 + 2t\\ P^\star &= 20 + 0.4t \end{aligned}$ This is exactly the amount consumers paid before, and firms would receive $P^\star - t = 20 - 0.6t$, same as before.

Psychologically, there may be a difference in perception. Payroll taxes, for example, fall evenly: half is paid by the employee, and half by the employer. This may seem “fairer” to employees than if they had to pay the entire tax, even if the equilibrium outcome in terms of what they actually do get paid is unaffected.

Relative elasiticies determine tax burden

If who you tax doesn’t affect the outcome, what does? The answer is elasticity: an in particular, the relative elasticities of demand and supply.

Before we dive into the math, let’s think about why this would be the case. Remember that elasticity is defined as the percentage change in quantity demanded or supplied per percentage change in price: $|\epsilon_{Q,P}| = \frac{\% \Delta Q}{\% \Delta P}$ If an increase in price causes you to dramatically change your quantity, you have very elastic demand or supply. This generally happens if you have many close substitutes for this good: for a consumer, it occurs if you can easily get the same utility from some other good as this one, while for a firm, it occurs if you can easily reallocate resources to producing some other more profitable good. By contrast, if agents very price insensitive — if consumers need to buy this particular good, or firms have a production process that’s set up only to produce this one good — then you’ll have very inelastic demand or supply.

Let’s think now about how this affects the response to a tax. If one side of the market is very elastic, and the other is very inelastic, the inelastic side of the market will bear the brunt of the tax, because they’re “locked in” to buying at or near the original quantity, regardless of the price. On the other hand, the elastic side of the market — who can easily go elsewhere if the price changes too much — will not see much of a change in the price they face.

Use the following graph to convince yourself of how this works, and then we’ll do the math. You can use the sliders to adjust the elasticities of demand and supply. Try making one very elastic, and the other very inelastic, then reverse them; and see how the prices after tax ($P_F$ and $P_C$) relate to the original price ($P$):

See interactive graph online here.

OK, now let’s do the math to see where this comes from. If we rearrange our elasticity equation, we can see that $\Delta Q = \Delta P \times |\epsilon_{Q,P}|$ In other words, the change in quantity due to a change in price is the percentage change in price times the magnitue of the elasticity. Now, this is true for both firms and consumers, so if we write $\epsilon_D$ as the magnitude of the price elasticity of demand, and $\epsilon_S$ as the magnitude of the price elasticity of supply, we have: $\begin{aligned} \Delta Q^D &= \Delta P_C \times \epsilon_D \Delta Q^S &= \Delta P_F \times \epsilon_S\\ \end{aligned}$ Now, in an after-tax equilibrium, the quantity demanded by consumers and firms must be the same, which means that the change in quantity demanded by consumers due to the tax must be equal to the change in quantity supplied by firms. Equating $\Delta Q^S$ and $\Delta Q^D$, therefore, gives us $\begin{aligned} \Delta Q^S &= \Delta Q^D\\ \Delta P_F \times \epsilon_S &= \Delta P_C \times \epsilon_D\\ \Delta P_F &= \Delta P_C \times \frac{\epsilon_D}{\epsilon_S} \end{aligned}$ Note that the fraction on the right-hand side is the ratio of the price elasticity of demand to the price elasticity of supply. So, if demand is more elastict than supply, the change in the price received by firms is greater than the change in the price paid by consumers; and vice versa. In other words, the burden of the tax falls more heavily on the less-elastic side of the market.

Note that the total change in price, $\Delta P_F + \Delta P_C$, must add up to the total amount of the tax. So if we substitute in $\Delta P_C = t - \Delta P_F$ and solve, a little bit of algebra gives us $\begin{aligned} \Delta P_C &= \frac{\epsilon_S}{\epsilon_S + \epsilon_D}t & \hspace{0.5in} & \Delta P_F &= \frac{\epsilon_D}{\epsilon_S + \epsilon_D}t \end{aligned}$ These are the formulas shown to the right of the graph above.

Finally, we might think about how the elasticity of demand and supply affect the equilibrium quantity of the good. If we plug either of these changes in price back into the first equation after the graph, we can see that $\Delta Q = \Delta P \times \epsilon = {\epsilon_S \times \epsilon_D \over \epsilon_S + \epsilon_D}$ From this equation we can derive some useful facts:

The more elastic demand or supply are (or both), the more effect a tax will have on the equilibrium quantity bought and sold.
If either demand or supply is perfectly inelastic, a tax will have no effect on the quantity demanded; and the perfectly inelastic side of the market will bear the entire burden of the tax.
If either demand or supply is perfectly elastic, price faced by that side of the market will not be affected by the tax at all; so the quantity will change based solely on the elasticity of the other side of the market, and that side of the market will bear the entire burden of the tax.

The deadweight loss of a tax

Recall our analysis of consumer and producer surplus: total surplus (total benefit to consumers minus total costs to firms) is maximized by equating marginal benefit to marginal cost. In a competitive market, this is achieved by having a single price to which all consumers set their marginal benefit equal, and to which all firms set their marginal cost equal. In other words, $MB = P = MC$ But when a tax is introduced, and consumers and firms face different prices, this condition is no longer met; rather, $MB = P_C > P_F = MC$ The effect of this is that it reduces the quantity from the efficient quantity to an inefficiently low quantity.

Drag the $t$ slider in the following graph to see the effect of the tax on welfare:

See interactive graph online here.

Now, you’ll note that a few things are happening here. The area of consumer surplus (the blue triangle) is shrinking: it’s the area above the price consumers pay $P_C$ and the marginal benefit/demand curve. Likewise, the area of producer surplus is shrinking: that’s the area below $P_F$ and above the marginal cost/supply curve. However, between these two areas are two new areas: a green rectangle and a black triangle.

The green rectangle is the government revenue from the tax; its height is the amount of the tax (dollars per unit) and its base is the quantity of units sold after the tax is introduced. So, $GR = t \times Q$ This is still welfare that accrues to society, and indeed may be spent on things like public goods that have a benefit even larger than it seems here. (We’ll talk about this more in a few lectures.)

The black triangle is the welfare loss (or “loss of surlus” or “deadweight loss”) that occurs because there are units of the good which could have been produced, and which would have yielded more benefit than their cost, but which are not because of the tax.

The following diagram helps you see how the welfare is affected by the tax. It starts out showing the consumer and producer surplus if there were no tax; then the loss of CS and PS because consumers pay a higher price than without a tax, and because producers receive less; then the gain of government revenue; the net welfare loss; and the combined view of the new situation with the tax:

See interactive graph online here.

Put the diagram in the “combined view” mode and you can see that how much of a welfare loss is generated depends on the elasticity of demand and supply: in particular, how much the tax affects the quantity in the market. If either demand or supply is perfectly inelastic, there is no deadweight loss, because the total quantity produced is the same before and after the tax. On the other hand, if both demand and supply are very elastic, there is also almost no welfare loss: this is because there are easy substitutes for this good (consumers can buy other things, and firms can produce other things with the same resources), so even if the government completely destroys the market for this good by imposing a relatively small tax, there isn’t much welfare loss.

The most welfare loss occurs, therefore, when demand and/or supply have a medium elasticity. This makes the market respond to the tax by producing much less quantity, and also represents a situation where there are no really great close substitutes for this good, either in consumption or production.

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