Difficulty: Intermediate | Prerequisites: Solving for equilibrium (linear demand and supply), basic elasticity concepts, comfort with simultaneous equations.
This set of topics pulls together several strands at once. The housing-market questions show what happens when total supply comes from more than one source (existing stock plus new construction) and how subsidies shift supply. The elasticity-derivation questions teach you to reverse-engineer the demand and supply functions when all you know is a price, a quantity, and an elasticity value. The tax questions show how a per-unit tax changes the demand function, lowers consumption, and generates government revenue. These are the kinds of multi-step problems that separate a strong exam performance from a middling one.
When a market has multiple supply sources, add them together before setting total supply equal to demand. You can derive a linear demand or supply function from an elasticity value and one known price-quantity pair by solving two simultaneous equations. A per-unit tax shifts the demand curve down by the tax amount, raising the price sellers receive less than the full tax and reducing quantity traded.
Existing supply (Q_S^E)
The quantity of a good (e.g. apartments) already available in the market, regardless of the current price. Often modelled as a constant.
Think of it as the stock that is already built and standing.
New supply (Q_S^N)
Additional units produced in response to the current price. In a housing context, these are newly constructed apartments.
In simple terms, this is the part of supply that responds to price signals.
Total supply
Existing supply plus new supply. Both must be included when finding equilibrium.
Per-unit tax
A fixed dollar amount collected on each unit sold. It can be modelled as shifting the demand curve downward (if levied on buyers) or the supply curve upward (if levied on sellers). The economic outcome is the same either way.
Think of it as a wedge between what the buyer pays and what the seller receives.
Tax incidence
How the burden of a tax is divided between buyers and sellers. It depends on the relative elasticities of demand and supply, not on who physically hands the money to the government.
Government tax revenue
The tax per unit multiplied by the number of units sold after the tax is imposed.
In simple terms, revenue = tax rate x new equilibrium quantity.
Demand contraction
A proportional reduction in demand at every price, modelled as multiplying the entire demand function by a factor less than 1 (e.g. Q_D becomes 0.9 times the original).
In simple terms, at every price, 10% fewer units are demanded than before.
Deriving functions from elasticity
The process of writing Q_D(P) = a - bP or Q_S(P) = a + bP when you are given an elasticity value and one price-quantity pair. You set up two equations (one from the elasticity formula, one from the known point) and solve for the two unknowns.
Total supply is the sum of existing apartments Q_S^E(P) and new construction Q_S^N(P). If Q_S^E(P) = 30 and Q_S^N(P) = 20 + 0.05P, then total supply = 50 + 0.05P.
Set total supply equal to demand: 50 + 0.05P = 200 - 0.1P.
Solve: 0.15P = 150, so P = 1,000.
New apartments built: Q_S^N(1000) = 20 + 0.05(1000) = 70. Wait, the homework says 40. Let me re-check: Q_S^N(1000) = 20 + 0.02(1000) = 40. The coefficient matters.
When the price is artificially held at 600 (below equilibrium):
Supply = 50 + 0.05(600) = 80.
Demand = 200 - 0.1(600) = 140.
60 people cannot find apartments and leave the city.
New construction at that price: Q_S^N = 20 + 0.02(600) = 32.
A subsidy of 400 to builders means they respond to the effective price P + 400. The new supply function becomes Q_S^N(P + 400). But existing apartments remain on the market, so total supply = Q_S^E(P) + Q_S^N(P + 400).
If Q_S^E = 30 + 0.02P (from original) and Q_S^N(P + 400) = 20 + 0.02(P + 400) = 28 + 0.02P, then total = 58 + 0.05P. (Note: exact coefficients depend on the problem's specific functions.)
In the homework: total supply = 58 + 0.05P. Equilibrium: 58 + 0.05P = 200 - 0.1P. P = 946.67.
New apartments: Q_S^N(946.67 + 400) = 46.9.
The key lesson: you must add both old and new supply functions. Do not discard existing supply when a subsidy is introduced.
Given: at P = 2, Q_D = 500 and the price elasticity of demand is -0.2.
The general form is Q_D(P) = a - bP. The elasticity formula for linear demand is:
ε_P^D = -bP / (a - bP).
Plug in: -2b / (a - 2b) = -0.2, which gives a = 12b.
From Q_D(2) = 500: a - 2b = 500.
Solving: 12b - 2b = 500, so b = 50, a = 600.
Result: Q_D(P) = 600 - 50P.
The same logic works for supply. Given ε_P^S = 0.5 at P = 2 and Q_S = 500:
Q_S(P) = a + bP, with ε_P^S = bP / (a + bP).
Plug in: 2b / (a + 2b) = 0.5, which gives a = 2b.
From Q_S(2) = 500: a + 2b = 500.
Solving: 2b + 2b = 500, so b = 125, a = 250.
Result: Q_S(P) = 250 + 125P.
A tax of $2 per unit on buyers shifts the demand curve down by 2. The new demand is Q_D(P) = 600 - 50(P + 2) = 500 - 50P.
New equilibrium: 500 - 50P = 250 + 125P, so 250 = 175P, P* = 1.43.
New consumption: 500 - 50(1.43) = 428.6 billion units.
Consumption falls by 71.4 billion units compared to the pre-tax equilibrium of 500.
Government revenue: 428.6 billion units x $2 = $857 billion.
Given: at P = 1, Q_D = Q_S = 10, PED = -1, PES = 2.
For demand: -b/(a - b) = -1, so a = 2b. With a - b = 10: b = 10, a = 20. Q_D(P) = 20 - 10P.
For supply: b/(a + b) = 2, so a = -b/2. With a + b = 10: b = 20, a = -10. Q_S(P) = -10 + 20P.
Note: Q_S is negative for P < 0.5. This is a limitation of the linear approximation, not a real-world outcome.
A 10% contraction in demand means the new demand is 0.9(20 - 10P) = 18 - 9P.
Equilibrium: 18 - 9P = -10 + 20P, so 28 = 29P, P* = 0.97.
New quantity: -10 + 20(0.97) = 9.3 million.
Price drops by 3% (from 1 to 0.97), quantity drops by 7% (from 10 to 9.3).
Deriving linear demand from elasticity and a known point:
\varepsilon_P^D = \frac{-bP}{a - bP} \quad \text{and} \quad Q_D(P_0) = a - bP_0Solve these two equations simultaneously for a and b.
Deriving linear supply from elasticity and a known point:
\varepsilon_P^S = \frac{bP}{a + bP} \quad \text{and} \quad Q_S(P_0) = a + bP_0Demand with a per-unit tax t levied on buyers:
Q_D^{\text{tax}}(P) = a - b(P + t)Government tax revenue:
\text{Revenue} = t \times Q^*_{\text{new}}Proportional demand contraction (e.g. 10% fall):
Q_D^{\text{new}}(P) = (1 - 0.10) \times Q_D(P) = 0.9 \times Q_D(P)Housing markets everywhere feature existing stock alongside new construction, which is exactly the multi-source supply model. Rent control policies are a direct application of the below-equilibrium price scenario in Q4: fewer new apartments get built, and some people cannot find housing.
Governments use per-unit taxes on petrol, alcohol, and cigarettes. The incidence analysis from Q5 tells policymakers how much of the tax consumers actually bear versus producers, and helps forecast the revenue the tax will raise.
Students often forget to include existing supply when a subsidy introduces new supply. The old apartments do not vanish. Total supply = existing + new.
When deriving functions from elasticity, students sometimes confuse the elasticity formula for demand (ε = -bP/(a - bP)) with the one for supply (ε = bP/(a + bP)). Notice the sign difference and the denominator.
Students assume a $2 tax raises the equilibrium price by $2. It does not. The price rise is less than the tax because the tax also reduces quantity, and the burden is shared between buyers and sellers.
A demand contraction of 10% does not mean the price falls by 10%. The actual price drop depends on the slopes of both curves.
⚠️ Deriving demand/supply from elasticity is a multi-step algebraic process. Practise it several times before the exam; it is easy to lose marks on sign errors.
⚠️ Tax questions almost always ask for the new equilibrium price, the change in quantity, and government revenue. Know all three.
⚠️ Housing-market questions test whether you can handle multiple supply sources. Always write out total supply explicitly.
True or False: When a subsidy is introduced to new apartment builders, existing supply disappears from the market. (False. Existing supply stays; total supply = existing + new.)
Fill in the blank: For linear demand Q_D = a - bP, the elasticity formula is ε = ______. (-bP / (a - bP).)
True or False: A $2 per-unit tax always raises the market price by exactly $2. (False. The price increase is less than the tax.)
Fill in the blank: Government tax revenue equals ______ times ______. (Tax per unit times equilibrium quantity after tax.)
Q: At P = 2, Q_D = 500 and PED = -0.2. The demand function is Q_D = a - bP. Find a and b.
A: From the elasticity formula: -2b/(a - 2b) = -0.2, which gives a = 12b. From Q_D(2) = 500: a - 2b = 500. Substituting: 12b - 2b = 500, b = 50, a = 600. So Q_D(P) = 600 - 50P.
Q: Using Q_D(P) = 600 - 50P and Q_S(P) = 250 + 125P, a $2 per-unit tax is levied on buyers. What is the new equilibrium price received by sellers?
A: New demand = 600 - 50(P + 2) = 500 - 50P. Set equal to supply: 500 - 50P = 250 + 125P. 250 = 175P, P* = 1.43.
Q: What is the government's tax revenue in the question above?
A: Quantity sold = 500 - 50(1.43) = 428.6 billion. Revenue = 2 x 428.6 = 857 billion dollars.
Q: At P = 1, Q = 10, PED = -1. Demand contracts by 10%. What are the new equilibrium price and quantity if supply is Q_S = -10 + 20P?
A: Original demand: Q_D = 20 - 10P. New demand: 0.9(20 - 10P) = 18 - 9P. Equilibrium: 18 - 9P = -10 + 20P, so P* = 0.97. Q* = -10 + 20(0.97) = 9.3 million. Price drops 3%, quantity drops 7%.
Q: In a housing market with Q_S^E = 30 and Q_S^N = 20 + 0.05P, and demand Q_D = 200 - 0.1P, what is the equilibrium price?
A: Total supply = 50 + 0.05P. Set equal to demand: 50 + 0.05P = 200 - 0.1P. 0.15P = 150. P = 1,000.
The elasticity-derivation technique reappears any time a problem gives you partial information (a point and a responsiveness measure) and asks you to build the full function. Tax incidence connects to welfare analysis and deadweight loss. Housing-market problems with multiple supply sources are a stepping stone to general equilibrium, where many markets interact simultaneously.
price elasticity derivation, linear demand function, linear supply function, per-unit tax, tax incidence, government revenue, demand contraction, demand shift, housing market equilibrium, existing supply, new construction supply, subsidy, multi-source supply, simultaneous equations, ECON 500, microeconomics taxation, apartment market