13 Taxes

For more information on these topics, see Varian Chapter 16: Equilibrium.

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13.1 Classwork 13: Taxes

The Tax Wedge. One useful way to think about taxes is to think of them as putting a wedge between what the buyer pays and what the seller keeps. For example, consider a tax on gas of $2 per gallon, levied on sellers. That is, for each gallon of gas someone sells, they must send the government $2. Then the amount that the seller is actually able to keep is $2 less than what they receive for the gallon of gas. Letting $p^s$ refer to the price that the sellers receive and $p^d$ refer to the price that demanders pay:

\[p^s = p^d - 2\]

Notice if you solve for $p^d$, you get:

\[p^d = p^s + 2\]

Consider instead a gas tax of $2 per gallon, levied on demanders. For every gallon of gas a person buys, they must send the government $2. Then the price buyers pay for a gallon is $2 higher than the price sellers receive. Write down that equation in terms of $p^d$ and $p^s$ and compare to the equations above. This is the first tax result: theoretically, $p^s$ and $p^d$ (circle one: does/does not) depend on whether the tax is levied on suppliers or demanders.
Total surplus with no tax. Let demand be given by $p = 8 - \frac{1}{2} q$ and let supply be given by $p = 2 + \frac{1}{4} q$.
1. Use ggplot to draw the supply and demand curves on a plot. On a test, I might ask you to sketch both curves with a pencil on paper.
2. Calculate the equilibrium price and quantity exchanged.
3. Calculate the consumer surplus (area under the demand curve and above the price). Recall that the consumer surplus approximates how much consumers benefit from the existence of the market: it’s the difference between how much people would be willing to pay (demand curve) and how much they actually have to pay (the equilibrium price).
4. Calculate the producer surplus: it approximates how much producers benefit from the existence of the market: the difference between how much producers get (the equilibrium price) and how much they would be willing to sell their goods for (the supply curve).
5. Calculate the total surplus, which is the benefit to both sides of the market because the market exists. $TS = CS + PS$.
Total Surplus with a Tax. Take the same supply and demand curves as the previous problem, but now suppose there’s a $3 tax levied on one side of the market so that $p^d = p^s + 3$.
1. Show that in equilibrium, the quantity exchanged is 4 units, $p^*_d = 6$, and $p^*_s = 3$. Here’s a diagram of the $3 tax wedge, where CS stands for consumer surplus, TR stands for tax revenue, PS stands for producer surplus, and DWL stands for deadweight loss:
2. The total surplus is equal to the consumer surplus (CS), producer surplus (PS), and the tax revenue (TR), because the tax revenue can be put toward a socially useful purpose. Calculate the CS, PS, TR, and total surplus.
3. You should find that total surplus falls as a result of the tax. How much did it fall by? This change to total surplus is called the deadweight loss of taxation. When a tax is levied in a market, the tax wedge means that the price consumers pay increases and the price producers get decreases, which will decrease the quantity that is exchanged in equilibrium. The deadweight loss (DWL) is the loss to consumer and producer surplus because of those exchanges that no longer take place.
Tax Burden. Recall that elasticity describes how easy it is for market participants to escape the market when the price changes. Elastic demand means that consumers find it easy to escape into other markets when the price rises, and elastic supply means that producers find it easy to escape into other markets when the price falls.
1. Suppose producers have elastic suppy and consumers have inelastic demand. Sketch the curves and add a tax wedge. Which side of the market will bear the majority of the tax burden? That is, will the price demanders have to pay increase by more than the price suppliers get decreases, or is it the opposite?
2. Suppose consumers have elastic demand and producers have inelastic supply. Sketch the curves and add a tax wedge (although you don’t have to turn in your sketch). Which side of the market will bear the majority of the tax burden? That is, will the price demanders have to pay increase by more than the price suppliers get decreases, or is it the opposite?
3. If you want to minimize waste from deadweight loss as much as possible, should you levy a tax on a market where consumers and producers are both very elastic, or both very inelastic?
4. If you want to maximize tax revenue, should you levy a tax on a market where consumers and producers are both very elastic, or both very inelastic?
5. Fill in the blanks: As long as supply and demand curves have their normal shape (the demand curves have a negative slope while supply curves have a positive slope), if there is a tax, the equilibrium quantity must _______ and the price that buyers pay must _______.
  - Fall; rise
  - Fall; fall
  - Rise; fall
  - Rise; rise
6. Consider jewelry: consumers are certainly the more elastic side of the market. Is a luxury tax more likely to hurt the buyers of jewelry, or the sellers of jewelry?
  - The buyers
  - The sellers
7. Let’s apply the economics of taxation to romantic relationships. Sometimes relationships have taxes. Suppose that you and your boyfriend or girlfriend live one hour apart. Using the tools developed in the chapter, how can you predict which one of you will do most of the driving? That is, which one of you will bear the majority of the relationship tax?
  - The person with more inelastic demand for the relationship will bear the tax – he or she will do most of the driving.
  - The person with more elastic demand for the relationship will bear the tax – he or she will do most of the driving.