Difficulty: Intermediate-Advanced | Prerequisites: Consumer theory, market structures
This document covers the remaining exam topics. General equilibrium uses the Edgeworth box to show how two consumers trade two goods, with Pareto efficiency where no one can be made better off without making someone worse off. The welfare theorems link competitive equilibria to efficient outcomes. Antitrust uses the HHI to assess market concentration. The supply/demand/taxes section tests your ability to find equilibrium, impose a tax, identify buyer and seller prices, determine tax incidence, and calculate CS, PS, government revenue, and DWL.
Pareto efficiency
An allocation where no one can be made better off without making someone else worse off. In an Edgeworth box, this means the two consumers' indifference curves are tangent: MRS_A = MRS_B.
Edgeworth box
A diagram that shows all possible allocations of two goods between two consumers. Consumer A's origin is the bottom-left corner; Consumer B's origin is the top-right (the box is flipped for B). Any point in the box represents a complete allocation of both goods.
To check whether an allocation is Pareto efficient: compute MRS for each consumer at that allocation. If MRS_A = MRS_B, it is efficient. If not, there exist trades that make both better off.
Contract curve
The set of all Pareto efficient allocations inside the Edgeworth box. It connects all points where the two consumers' indifference curves are tangent. Any competitive equilibrium must lie on the contract curve.
First welfare theorem
A competitive equilibrium is Pareto efficient. In the Edgeworth box: MRS_A = MRT = MRS_B. The market gets you to an efficient outcome without any central planner.
Second welfare theorem
Any Pareto efficient allocation can be supported as a competitive equilibrium, given the right initial endowments and prices. In simple terms: pick any fair outcome on the contract curve, and there exists a set of prices and endowments that make the market reach it.
Competitive equilibrium in the Edgeworth box
Find relative prices (Px/Py) such that both consumers, maximising utility subject to their budget constraints (valued at initial endowments), choose demands that exactly clear the market for both goods. For Cobb-Douglas preferences, each consumer spends a fixed fraction of wealth on each good.
Special cases:
Perfect substitutes: indifference curves are straight lines; tangencies may not exist in the interior, so corner solutions or entire edges can be efficient
Perfect complements: L-shaped indifference curves; the contract curve connects the kinks
Cobb-Douglas: standard interior tangencies; the contract curve is a smooth path through the box
Herfindahl-Hirschman Index (HHI)
The sum of the squared market shares (in percentages) of all firms in an industry. Measures market concentration. A monopoly has HHI = 10,000. Perfect competition approaches HHI = 0.
FTC merger guidelines
Post-merger HHI < 1,000: merger is not challenged (unconcentrated market)
Post-merger HHI between 1,000 and 1,800: challenged only if the change in HHI is 100 or more
Post-merger HHI > 1,800: challenged only if the change in HHI is 50 or more
To calculate the change in HHI from a merger of two firms with shares s1 and s2: the increase is approximately 2 * s1 * s2 (because (s1 + s2)^2 - s1^2 - s2^2 = 2 * s1 * s2).
Equilibrium without tax
Set quantity supplied equal to quantity demanded: Qs = Qd. Solve for P* and Q*.
Per-unit tax
A tax of t per unit drives a wedge between the price buyers pay (Pb) and the price sellers receive (Ps): Pb = Ps + t.
To find the new equilibrium: substitute Pb = Ps + t into the demand equation (or equivalently shift the demand curve down by t), set equal to supply, and solve for Ps. Then Pb = Ps + t. The new quantity traded is lower than Q*.
Buyer's price (Pb)
The price consumers pay after the tax. Higher than the original equilibrium price.
Seller's price (Ps)
The price producers receive after the tax. Lower than the original equilibrium price.
Tax incidence
The proportion of the tax borne by buyers versus sellers depends on the relative elasticities of supply and demand. The more inelastic side bears more of the tax.
Buyer's share of tax: (Pb - P*) / t
Seller's share of tax: (P* - Ps) / t
Consumer surplus after tax
The triangle under the demand curve and above Pb, from 0 to Qt (the after-tax quantity).
Producer surplus after tax
The triangle above the supply curve and below Ps, from 0 to Qt.
Government revenue
t * Qt. The rectangle between Pb and Ps, from 0 to Qt.
Deadweight loss from tax
The triangle between the demand and supply curves, from Qt to Q*. This is surplus that existed before the tax but is captured by neither buyers, sellers, nor the government.
Students often think Pareto efficiency means "fair." It does not. An allocation where one person has everything and the other has nothing can be Pareto efficient. Efficiency only means no further trades can make both parties better off.
In the Edgeworth box, students sometimes forget that Consumer B's axes are flipped. B's quantity of good X increases to the left, and B's quantity of good Y increases downward.
When applying the HHI merger test, students sometimes compare the post-merger HHI to the threshold without checking the change in HHI. Both the level and the change matter.
In tax incidence problems, students often assume the tax falls entirely on the side it is legally imposed on. Economic incidence depends on elasticities, not on who writes the cheque.
⚠️ The general equilibrium/Pareto efficiency problem is one of the seven large final exam problems. You will need to: check whether an allocation is Pareto efficient, draw an Edgeworth box, find relative prices for competitive equilibrium, draw indifference curves through an initial allocation, and illustrate the contract curve. Know the special cases: perfect substitutes, perfect complements, and Cobb-Douglas.
⚠️ The supply/demand/elasticity/taxes problem is another of the seven large problems. Find P* and Q* before the tax, derive the new demand curve after the tax, find Pb and Ps, determine who bears more of the tax, find the after-tax quantity, and compute CS, PS, government revenue, and DWL.
⚠️ HHI calculations are a likely short-answer question. Know the thresholds and how to compute the change from a merger.
True or false: An allocation is Pareto efficient only if both consumers are equally well off. (False, efficiency says nothing about fairness.)
Fill in the blank: The First Welfare Theorem states that a competitive equilibrium is ______. (Pareto efficient)
True or false: If post-merger HHI is 1,500 and the change is 80, the merger will be challenged. (False, the change must be at least 100 for HHI between 1,000 and 1,800.)
Fill in the blank: Tax incidence depends on relative ______, not on who is legally responsible for the tax. (Elasticities)
True or false: DWL from a tax represents surplus captured by the government. (False, DWL is surplus lost entirely, captured by nobody.)
Q: Two consumers, A and B, have Cobb-Douglas utility. A has U = x^0.5 * y^0.5, B has U = x^0.5 * y^0.5. Initial endowments: A has (10, 0), B has (0, 10). Find the competitive equilibrium.
A: With symmetric Cobb-Douglas (alpha = 0.5), each consumer spends half their wealth on each good. A's wealth = 10Px. A demands: Xa = 0.5 * 10Px / Px = 5, Ya = 0.5 * 10Px / Py = 5Px/Py. B's wealth = 10Py. B demands: Xb = 0.5 * 10Py / Px = 5Py/Px, Yb = 5. Market clearing for X: 5 + 5Py/Px = 10, so Py/Px = 1, meaning Px = Py. Each gets (5, 5).
Q: Supply is Qs = 2P, demand is Qd = 100 - 2P. A tax of 10 per unit is imposed on buyers. Find the new equilibrium, Pb, Ps, and DWL.
A: Original equilibrium: 2P = 100 - 2P, P* = 25, Q* = 50. With tax on buyers, new demand: Qd = 100 - 2(Ps + 10) = 80 - 2Ps. Set equal to supply: 2Ps = 80 - 2Ps, 4Ps = 80, Ps = 20. Pb = 20 + 10 = 30. Qt = 2(20) = 40. DWL = 0.5 * 10 * (50 - 40) = 50.
Q: An industry has four firms with market shares of 40%, 30%, 20%, and 10%. Two firms with 20% and 10% plan to merge. Will the FTC challenge the merger?
A: Pre-merger HHI = 40^2 + 30^2 + 20^2 + 10^2 = 1600 + 900 + 400 + 100 = 3000. Post-merger HHI = 40^2 + 30^2 + 30^2 = 1600 + 900 + 900 = 3400. Change = 400. Since post-merger HHI > 1800 and change > 50, the merger will be challenged.
The welfare theorems tie together consumer theory (preferences, MRS) and market structures (competitive equilibrium). They provide the theoretical justification for why competitive markets are desirable.
Tax incidence connects to elasticity, which connects to the shape of supply and demand curves derived from the underlying consumer and producer theory.
The DWL from taxes parallels the DWL from monopoly: both arise because the quantity traded is below the efficient level.
Related Terms / Search Tags
Pareto efficiency, Pareto optimality, Edgeworth box, contract curve, welfare theorems, first welfare theorem, second welfare theorem, competitive equilibrium, general equilibrium, MRS equality, Herfindahl-Hirschman Index, HHI, antitrust, FTC merger guidelines, market concentration, supply and demand, tax incidence, per-unit tax, buyer's price, seller's price, consumer surplus, producer surplus, government revenue, deadweight loss, elasticity, tax burden