Price Controls and Per-Unit Taxes
Rent in a big city feels too high, so the city council passes a law: no one can charge more than $\$900$ a month for a one-bedroom apartment. Everyone cheers. A year later, apartments are almost impossible to find. What went wrong? And if the government can set a price, why can't it just set every price?
In this lesson you will use supply and demand to predict what price controls and taxes actually do, and you will measure their costs with numbers.
Price ceilings
A price ceiling is a legal maximum price. Sellers cannot charge more than the ceiling. Rent control and caps on ticket resale prices are examples.
Whether a ceiling matters depends on where it sits compared with the equilibrium price:
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Ceiling above equilibrium: the market price is already below the cap, so the law changes nothing. It is not binding.
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Ceiling below equilibrium: buyers want more than sellers will supply at that price. It is binding.
When a ceiling binds, the quantity demanded is greater than the quantity supplied, so there is a shortage. Low prices make buyers want more and make sellers want to supply less.
In the graph, at the ceiling price sellers offer only $Q_s$, but buyers want $Q_d$. The gap is the shortage. Only $Q_s$ units are actually sold, because no one can force a seller to sell more than they want to at that price.
Back to the start: a rent ceiling of $\$900$ in a market where the equilibrium rent is $\$1,400$ binds. At $\$900$ more people want apartments and fewer landlords want to rent or build them, so there is a shortage. That is why apartments became hard to find.
A city government passes a law that says landlords cannot charge more than $\$900…
A city government passes a law that says landlords cannot charge more than $\$900$ per month for a one-bedroom apartment. The equilibrium rent in the city without the law is $\$1,400$ per month. Is this price control a price ceiling or a price floor? Is it binding? Explain in two or three sentences what you expect to happen to the quantity of apartments demanded and the quantity supplied.
The market for concert tickets in Chicago has the demand curve $Q_d = 800 - 10P$ and the supply curve $Q_s = 20P - 100$, where $P$ is the price in dollars and $Q$ is…
The market for concert tickets in Chicago has the demand curve $Q_d = 800 - 10P$ and the supply curve $Q_s = 20P - 100$, where $P$ is the price in dollars and $Q$ is tickets per night.
(a) Find the equilibrium price and quantity.
(b) The city imposes a price ceiling of $\$20$ per ticket. Find the quantity demanded, the quantity supplied, and the size of the shortage.
(c) Does the ceiling help or hurt consumers who are able to buy a ticket? Explain briefly.
Price floors
A price floor is a legal minimum price. Sellers must receive at least that much. Examples include the minimum wage (a price floor on labor) and farm price supports.
The logic mirrors a ceiling:
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Floor below equilibrium: not binding, no effect.
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Floor above equilibrium: binding. Buyers want less (higher price) and sellers want to supply more, so there is a surplus: $Q_s > Q_d$.
Notice that only $Q_d$ units are actually bought. The quantity traded is always the smaller of the two quantities, because a trade needs both a willing buyer and a willing seller. The surplus units go unsold (or the government buys them).
A state sets a minimum price of $\$5.00$ per gallon for milk to help dairy farmers. Without the policy, the equilibrium price is $\$3.50$ per gallon. Define a price…
A state sets a minimum price of $\$5.00$ per gallon for milk to help dairy farmers. Without the policy, the equilibrium price is $\$3.50$ per gallon. Define a price floor in your own words, and state whether this one creates a shortage or a surplus. Identify which side of the market (buyers or sellers) wants to trade more at the controlled price.
In a market for corn, the demand curve is $Q_d = 500 - 20P$ and the supply curve is $Q_s = 30P$, where $P$ is dollars per bushel and $Q$ is thousands of bushels. The…
In a market for corn, the demand curve is $Q_d = 500 - 20P$ and the supply curve is $Q_s = 30P$, where $P$ is dollars per bushel and $Q$ is thousands of bushels. The government sets a price floor of $\$12$ per bushel.
(a) Find the free-market equilibrium price and quantity.
(b) Find the quantity demanded and the quantity supplied at the floor.
(c) State the size of the surplus and explain why the quantity actually traded is the smaller of the two quantities.
Per-unit taxes
A per-unit tax (like an excise tax on cigarettes or gasoline) is a fixed dollar amount charged on every unit sold. Suppose it is collected from sellers.
For a seller, the tax is just an extra cost on every unit. To stay willing to sell the same quantity, a seller now needs a price that is higher by the amount of the tax. So the supply curve shifts up (left) by exactly the tax. Demand does not move, because nothing changed about what buyers want at each price.
Moving along the demand curve to the new equilibrium gives:
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a higher price paid by buyers ($P_b$),
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a lower price kept by sellers ($P_s = P_b - t$),
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a smaller quantity.
The difference $P_b - P_s = t$ is the tax wedge.
The green rectangle is tax revenue: tax per unit $\times$ units sold. The small purple triangle is the deadweight loss (DWL): trades that would have helped both buyer and seller but no longer happen because of the tax.
Drag the tax up and watch the quantity fall and the wedge grow.
Who really pays?
It does not matter whether the law says buyers or sellers must pay the tax to the government. The graph gives the same result either way. What matters is the tax incidence, meaning who bears the burden:
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Buyers' burden per unit $= P_b - P^*$ (how much more they pay than before).
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Sellers' burden per unit $= P^* - P_s$ (how much less they keep than before).
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The two burdens add up to the tax: $(P_b - P^*) + (P^* - P_s) = t$.
The federal government places a per-unit tax of $\$2…
The federal government places a per-unit tax of $\$2$ on every pack of cigarettes sold, collected from sellers. On a standard supply and demand graph, state which curve shifts, in which direction, and whether the equilibrium quantity rises or falls. Also say whether the price buyers pay rises or falls.
The market for pizza slices in a college town has an equilibrium price of $\$4.00…
The market for pizza slices in a college town has an equilibrium price of $\$4.00$ and an equilibrium quantity of 1,000 slices per day. The town imposes a per-unit tax of $\$1.00$ per slice, collected from sellers. After the tax, buyers pay $\$4.60$ per slice and the quantity falls to 800 slices per day.
(a) How much of the tax does each slice's buyer bear? How much does the seller bear (the price the seller keeps is the buyer's price minus the tax)?
(b) Calculate the tax revenue the government collects per day.
(c) Calculate the deadweight loss, treating the lost-trade region as a triangle.
(d) Which side of the market bears the larger share of the tax, and what does that suggest about relative elasticities?
Surplus, deadweight loss, and linear models
To put numbers on the cost of a control, use consumer surplus (CS) and producer surplus (PS).
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CS is the triangle between the demand curve and the price, up to the quantity traded.
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PS is the triangle between the price and the supply curve, up to the quantity traded.
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Each triangle has area $\tfrac{1}{2} \times \text{base} \times \text{height}$. The base is the quantity, and the height is the vertical distance on the price axis.
Deadweight loss is the surplus that disappears. It is the triangle between demand and supply from the new quantity out to the old equilibrium quantity:
$$\text{DWL} = \tfrac{1}{2} \times (\text{lost quantity}) \times (\text{gap between demand price and supply price at the new quantity}).$$For a tax, that gap is the tax itself. For a ceiling or floor, it is the vertical distance between the demand and supply curves at the quantity actually traded.
Working with price-form curves
When curves are written as $P = \dots$ (price in terms of $Q$), the process is:
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Equilibrium: set demand price = supply price and solve for $Q$, then find $P$.
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Tax $t$: the supply price rises by $t$. Add $t$ to the supply equation, then solve again.
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Buyers' price $P_b$ comes from the demand curve at the new $Q$. Sellers' price is $P_s = P_b - t$ (this equals the original supply curve at the new $Q$).
Surplus under a ceiling
The market for rental apartments in a city has the linear demand curve $P = 2{,}000 - 2Q$ and the linear supply curve $P = 200 + Q$, where $P$ is monthly rent in dollars…
The market for rental apartments in a city has the linear demand curve $P = 2{,}000 - 2Q$ and the linear supply curve $P = 200 + Q$, where $P$ is monthly rent in dollars and $Q$ is thousands of apartments.
(a) Find the equilibrium rent and quantity, and compute consumer surplus and producer surplus.
(b) The city imposes a rent ceiling of $\$800$. Find the quantity supplied and the quantity demanded, and state the quantity actually rented.
(c) Compute consumer surplus, producer surplus, and the deadweight loss under the ceiling. Assume that the apartments that are rented go to the consumers with the highest willingness to pay.
(d) Explain why this assumption is probably unrealistic, and how relaxing it would change the size of the deadweight loss.
The market for gasoline in a state has the demand curve $P = 8 - 0.02Q$ and the supply curve $P = 2 + 0.01Q$, where $P$ is dollars per gallon and $Q$ is millions of…
The market for gasoline in a state has the demand curve $P = 8 - 0.02Q$ and the supply curve $P = 2 + 0.01Q$, where $P$ is dollars per gallon and $Q$ is millions of gallons per week. The state imposes a per-unit tax of $\$0.60$ per gallon collected from sellers.
(a) Find the pre-tax equilibrium price and quantity.
(b) Write the new supply curve including the tax and find the new quantity, the price paid by buyers, and the price received by sellers.
(c) Compute the tax incidence on buyers and on sellers per gallon, and the government's weekly tax revenue.
(d) Compute the deadweight loss.
(e) The state is considering a second proposal: a tax of $\$1.20$ per gallon. Without recalculating everything, explain what happens to the deadweight loss relative to the $\$0.60$ tax, and why economists describe the relationship the way they do.