14.8 Taxes and Subsidies

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:

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:

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:

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:

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 ($P$):

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.

(Note that the graph above doesn’t yet handle the edge cases of perfectly elastic or inelastic demand…I fought with it for many hours today…)

Subsidies

A subsidy is just a negative tax: so instead of the price paid by consumers being the price received by firms plus a tax, it’s the price received by firms minus a subsidy. As a result, the quantity is greater than the equilibrium quantity without a subsidy.

I’ll add more descriptions of that later. For now, we’ll focus our energies on taxes. :)