Lab 6: Practice with Equilibrium
Q37: The Island Landlords
Cormorant Island has exactly 100 cottages, and no one may build more. Across the strait, Gull Point builds freely: its landlords supply \(q = 2p - 100\) cottages. Renters in both towns have monthly demand \(q = 300 - 2p\). Both councils, short of funds, vote the same $20 monthly tax per rented cottage, charged to owners. The Landlords’ Association, which speaks for both shores, shrugs: “Rents rise $20 tomorrow. Tenants will pay every penny.”
Find the rent in each town before the tax. Then test the Association’s claim: if every landlord posts $20 more, what happens to their cottages? Where does the rent settle in each town, who truly pays, and how much does each council collect? Which tax destroys trades, and which destroys none? Use the elasticity formula from Q35 to explain why the two shores differ.
Extension: Both councils double the tax to $40 “to double the revenue.” Which council succeeds?
Q37: The Island Landlords. Start with the supply curves. On the island no one may build, and an empty cottage earns nothing, so all 100 cottages are offered whatever the rent: supply is a vertical line at \(q = 100\). At Gull Point supply slopes up, \(q = 2p - 100\), because a higher rent draws cottages into the rental market and a lower one sends them out of it. Before the tax the island’s rent solves \(300 - 2p = 100\), so \(p = 100\); Gull Point’s solves \(300 - 2p = 2p - 100\), so \(p = 100\) and \(q = 100\). The two markets are identical on the eve of the tax, which is what makes the comparison clean.
Now every landlord posts $120. On both shores tenants demand \(300 - 240 = 60\) cottages, and 40 stand empty. What happens next depends on the shore. On the island an empty cottage brings in nothing, so each landlord with one shades the rent to fill it, and since all 100 cottages stay on the market the shading continues until all 100 are let, which happens at exactly $100. Tenants pay what they paid before; landlords keep $80. The tax falls entirely on the landlords, and the Association’s shrug was wrong in every particular. At Gull Point the shading starts the same way, but as the rent falls landlords withdraw cottages, so the market clears before the rent gets back to $100. A landlord who collects \(p_d\) keeps \(p_d - 20\) and supplies \(q = 2(p_d - 20) - 100 = 2p_d - 140\) cottages. Setting this equal to \(300 - 2p_d\) gives \(p_d = 110\), so landlords keep $90 and 80 cottages are let. Tenants pay $10 more and landlords keep $10 less: half each.
The island council collects \(20 \times 100 = 2000\); Gull Point’s collects \(20 \times 80 = 1600\). The island tax also destroys nothing: the same 100 cottages are rented to the same 100 tenants at the same rent, and the $2000 is a pure transfer from landlords to the council. At Gull Point 20 cottages leave the rental market. Each was worth more to its tenant than to its landlord, by a gap that runs from $20 at the 100th cottage down to zero at the 80th, so the lost value is the triangle \(\frac{1}{2} \times 20 \times 20 = 200\). (Check it with surplus: before the tax, consumer and producer surplus are each \(\frac{1}{2} \times 50 \times 100 = 2500\); after it, each is \(\frac{1}{2} \times 40 \times 80 = 1600\), and with revenue of 1600 the total is 4800, down from 5000.) The tax that raises more money is the one that destroys nothing.
The formula from Q35 explains the split. At \((100, 100)\) demand has \(\varepsilon_D = -2 \times 100/100 = -2\) on both shores. The island’s supply elasticity is zero, so the tenants’ share \(\varepsilon_S / (\varepsilon_S + |\varepsilon_D|)\) is \(0/2 = 0\); Gull Point’s is \(\varepsilon_S = 2 \times 100/100 = 2\), so the share is \(2/4 = 1/2\). The side that cannot walk away pays, and on the island the landlords cannot walk away at all. Land is not produced, so taxing it cannot discourage anyone from producing it; that is why a tax on pure land is sometimes called the perfect tax, and why Henry George built a political movement on it. The practical trouble is separating the land from the improvements on it: a tax on cottages, as opposed to the ground beneath them, would discourage building wherever building is allowed, which is exactly Gull Point.
Extension. On the island the $40 tax changes nothing but the landlords’ take: the rent stays at $100, landlords keep $60, and revenue doubles to exactly $4000 with no loss at all. A tax on a fixed supply can be raised all the way to the rent itself without the ledger growing a missing line. At Gull Point, \(300 - 2p_d = 2(p_d - 40) - 100\) gives \(p_d = 120\), \(p_s = 80\), and \(q = 60\). Revenue is \(40 \times 60 = 2400\), up by half rather than doubled, because the base fell from 80 cottages to 60. The deadweight loss is \(\frac{1}{2} \times 40 \times 40 = 800\), four times what it was: both the height of the triangle and its base scale with the tax, as in Q36. The island council succeeds; Gull Point’s does not.