Class 11: Equilibrium
Optional textbook reading: Varian, Chapter 16: “Equilibrium.”
Key Ideas
Classes 7 through 10 built the demand side of a market, first for one consumer and then for everyone together. Today the other side arrives. A supply curve \(S(p)\) records how much sellers offer at each price. Where it comes from is the subject of Classes 14 through 19; for now it is taken as given and slopes upward.
A market is in equilibrium at the price \(p^*\) where quantity demanded equals quantity supplied, \(D(p^*) = S(p^*)\). At any higher price some sellers are left holding goods, at any lower price some buyers leave empty handed, and in either case someone has a reason to change the price. The definition says nothing about whether the outcome is good for anyone; it says only that the price has no reason to move. Comparative statics asks how \(p^*\) and \(q^*\) respond when demand or supply shifts. Observed data are equilibrium points, and the first problem asks what such data can and cannot reveal about the curves behind them.
A tax of \(t\) per unit drives a wedge between the price buyers pay, \(p_d\), and the price sellers keep, \(p_s = p_d - t\). Buyers respond to \(p_d\) and sellers to \(p_s\), so the equilibrium condition becomes \(D(p_d) = S(p_d - t)\).
Consumer’s surplus from Class 9, the area under the demand curve and above the price, now has a mirror image: producer’s surplus is the area above the supply curve and below the price. Their sum, together with any tax revenue, is total surplus, a dollar measure of what the market delivers. The second and third problems ask who actually pays a tax, and what it costs.
Problems of the Day
Q34: The Case of the Unmoved Quantity
Every month, junior analyst Piper reports on the town coffee market. April: price $4, sales 1000 cups. May: price $5, sales 1000 cups. Piper refuses to file the report. “Impossible. A higher price means we slide up the demand curve, so sales must fall. The data are corrupted.”
Explain, with a diagram, what Piper has forgotten. Then check your story with algebra: April’s curves were \(D: q = 3000 - 500p\) and \(S: q = 500p - 1000\). Propose a shift of demand and a shift of supply that together produce May’s numbers exactly, and verify your proposal.
Extension: Your proposal slid each curve sideways. What if the curves did not move in parallel? Draw May’s market under each story and decide whether it can deliver $5 and 1000 cups:
Demand became less elastic, perhaps because the only other café in town closed.
Supply became less elastic, perhaps because the roasters hit capacity at 1000 cups a day.
Q35: Who Really Pays
Porthaven votes a $6 tax on every food truck meal, to be paid by the trucks. The truckers howl: “We will simply add $6 to every price. Customers will pay it all.” The customers’ league counters: “Fine, as long as the tax stays on you and not on us.”
Demand is \(q = 120 - 4p\) and supply is \(q = 2p - 30\). Find the market before the tax, then after it. How much of the $6 lands on customers, how much on the trucks, and what became of the truckers’ threat? What actually determines who pays: the law, or something else? Name the something else.
Q36: The Treasurer’s Ledger
Porthaven’s treasurer presents the food truck tax to the council as a single line: “Revenue: $72. Cost to the town: nothing. The money comes out of the trucks’ pockets and into ours.” A councillor asks to see the full ledger.
Using the market from Q35, compute consumer surplus, producer surplus, and total surplus before the tax and after it, counting the city’s revenue as part of total surplus after. Then build the ledger: who lost how much, who gained how much, and whether the column sums to zero. If it does not, say where the missing value went and why nobody can collect it.
Extension: Delighted by $72, the treasurer proposes doubling the tax to $12 “to double the revenue.” Will it? Compute the new ledger.
Answers
Q34: The Case of the Unmoved Quantity. Piper has forgotten that a market has two curves, and that both can move. “A higher price means we slide up the demand curve” is true only if the demand curve stayed put and the supply curve shifted. If supply alone had contracted, price would have risen and quantity would have fallen, as Piper expects. If demand alone had expanded, price and quantity would both have risen. May’s numbers, higher price and identical quantity, require both: demand shifted out, which pushes quantity up, and supply shifted in, which pushes it back down by exactly as much. In the diagram the new curves cross directly above the old equilibrium.
Check April first: \(3000 - 500p = 500p - 1000\) gives \(p = 4\) and \(q = 1000\). For May, shift each curve by 500 units: demand becomes \(q = 3500 - 500p\) and supply becomes \(q = 500p - 1500\). At \(p = 5\) the first gives \(3500 - 2500 = 1000\) and the second gives \(2500 - 1500 = 1000\), so the new curves meet at exactly May’s point. The data are not corrupted. Coffee drinkers wanted more coffee in May (demand out), and something made coffee costlier to supply (supply in).
Extension. Both curves must have moved. April’s demand curve gives \(q = 500\) at \(p = 5\), and April’s supply curve gives \(q = 1500\), so neither one passes through May’s point on its own. Beyond that, anything goes: any downward sloping demand curve through \((5, 1000)\) paired with any upward sloping supply curve through the same point produces May’s numbers. In particular the slopes need not survive. Story (a): if the rival café closed, coffee drinkers have fewer substitutes and demand is steeper than before, say \(D: q = 1500 - 100p\), which rose by 100 cups at the old price and now passes through May’s point; but April’s supply still delivers 1500 cups at $5, so supply must change too, say \(S: q = 200p\). A steeper demand curve alone cannot do it. Story (b): if the roasters are at capacity, supply is vertical at \(q = 1000\), and any outward shift of demand raises the price with no change in quantity; \(D: q = 3500 - 500p\) with \(S: q = 1000\) gives May exactly, and the change in supply is a change in shape rather than a sideways move. Whatever the slopes, both new curves must pass through \((5, 1000)\), and neither April curve does. If the slopes are unchanged, the shifts are pinned down, \(+500\) for demand and \(-500\) for supply, but Piper cannot know the slopes are unchanged from two price and quantity pairs. To sort out what happened she needs information about the shifters themselves: a cold snap, a rise in bean prices, a new office tower full of coffee drinkers. Price and quantity alone never reveal which curve moved. This is the identification problem, and it is why empirical economists spend so much effort hunting for things that shift one curve and not the other.
Q35: Who Really Pays. Before the tax, \(120 - 4p = 2p - 30\) gives \(p = 25\) and \(q = 20\) meals.
With a $6 tax paid by the trucks, a truck that charges customers \(p_d\) keeps \(p_s = p_d - 6\), and it is \(p_s\) that governs how many meals it supplies. Supply in terms of the customer price is \(q = 2(p_d - 6) - 30 = 2p_d - 42\). Setting \(120 - 4p_d = 2p_d - 42\) gives \(p_d = 27\), \(p_s = 21\), and \(q = 12\). Customers pay $2 more than before and the trucks keep $4 less. Of the $6, one third lands on customers and two thirds on the trucks. The truckers’ threat fails because it ignores the customers. At $31 quantity demanded is \(120 - 124 < 0\): nobody buys. Trucks with no customers cut prices, and they keep cutting until the market clears at $27.
Now charge the tax to customers. A customer who pays the truck \(p_s\) hands another $6 to the city, so demand in terms of the truck’s price is \(q = 120 - 4(p_s + 6) = 96 - 4p_s\). Setting this equal to \(2p_s - 30\) gives \(p_s = 21\), so the customer’s all in price is $27, and \(q = 12\). Every number is identical. Whose name is on the tax bill is irrelevant; what matters is a $6 wedge between what buyers pay and what sellers keep, and the market splits that wedge the same way whoever writes the check. The something else is the relative elasticity of the two sides. At the old equilibrium, \(\varepsilon_D = -4 \cdot 25/20 = -5\) and \(\varepsilon_S = 2 \cdot 25/20 = 2.5\). Customers can escape twice as easily as the trucks can, so the trucks bear twice as much of the tax. In general the buyers’ share is \(\varepsilon_S / (\varepsilon_S + |\varepsilon_D|) = 2.5/7.5 = 1/3\). The side that is less able to walk away pays more.
Q36: The Treasurer’s Ledger. Write the curves with price on the left: demand is \(p = 30 - q/4\) and supply is \(p = 15 + q/2\). Before the tax, \(p = 25\) and \(q = 20\). Consumer surplus is the triangle under demand and above $25: \(\frac{1}{2} \times (30 - 25) \times 20 = 50\). Producer surplus is the triangle above supply and below $25: \(\frac{1}{2} \times (25 - 15) \times 20 = 100\). Total surplus is $150.
After the tax, customers pay $27, trucks keep $21, and 12 meals are sold. Consumer surplus is \(\frac{1}{2} \times (30 - 27) \times 12 = 18\), producer surplus is \(\frac{1}{2} \times (21 - 15) \times 12 = 36\), and revenue is \(6 \times 12 = 72\). Total surplus, revenue included, is \(18 + 36 + 72 = 126\).
The ledger: customers lost \(50 - 18 = 32\), trucks lost \(100 - 36 = 64\), and the city gained 72. The losses come to 96 and the gain to 72, so the column does not sum to zero; $24 is missing. The treasurer was wrong twice. The money did not come only from the trucks: customers lost $32, a third of the total loss, which matches the one third of the tax they bore in Q35. And the cost to the town was not nothing: the two sides together gave up $24 more than the city collected. That $24 is meals 13 through 20, each worth more to its buyer than it cost its seller, with a gap of $6 at the 20th meal shrinking to zero at the 12th: the triangle \(\frac{1}{2} \times 6 \times 8 = 24\). Nobody collects it because the meals were never made. Revenue is a transfer; the $24 is deadweight loss, value that existed before the tax and now exists nowhere.
Extension. With a $12 wedge, \(120 - 4p_d = 2(p_d - 12) - 30\) gives \(p_d = 29\), \(p_s = 17\), and \(q = 4\). Revenue is \(12 \times 4 = 48\), less than before: doubling the rate cut the base from 12 meals to 4, and the base fell faster than the rate rose. Consumer surplus is \(\frac{1}{2} \times 1 \times 4 = 2\) and producer surplus is \(\frac{1}{2} \times 2 \times 4 = 4\), so total surplus is \(2 + 4 + 48 = 54\) and the deadweight loss is \(150 - 54 = 96\), the triangle \(\frac{1}{2} \times 12 \times 16\). Doubling the tax quadrupled the deadweight loss and lowered the revenue. The loss triangle grows with the square of the tax, because both its height (the wedge) and its base (the lost meals) scale with the rate, while revenue is the wedge times a shrinking base. Past some rate, a higher tax collects less.