Why one market is never the whole story: the Edgeworth box, the two welfare theorems, efficiency with production, and what partial equilibrium gets wrong about a tax.
Week 1 steps back from one market and looks at the whole economy at once. Everything you did last year was partial equilibrium: draw supply and demand for one good, hold everything else fixed, read off the answer. That is a useful lie. Taxing cider changes what people spend on pasties; the pasty price then feeds back into the cider market. General equilibrium asks whether there is a set of prices that clears every market simultaneously — and then uses that machinery to say something genuinely surprising about efficiency, about what markets achieve, and about what redistribution costs.
Partial equilibrium analysis finds the price and quantity in one market while treating every other price as a fixed constant. General equilibrium analysis lets all prices move together and insists that all markets clear at once.
The distinction matters whenever markets are linked — and they are linked whenever goods are substitutes or complements in demand, or compete for the same inputs in supply. A tax on good X does not simply vanish: the money not spent on X reappears somewhere, and the price that reappears there bounces back into the market for X.
General equilibrium theory asks: do prices exist that clear all markets? It does not ask how a market finds them. That second question needs a theory of price adjustment out of equilibrium, and economics does not have a settled one — it would require knowing not just supply and demand but how each reacts, in real time, to every other variable moving at once.
We work in the short run, where production capacity is roughly fixed. Long-run general-equilibrium effects — capital accumulating, technology shifting — belong to growth and development models.
A tariff raises import prices at home, but can lower prices in unrelated third countries as supply is diverted there. Co-movement between equity prices and commodity prices is largely a general-equilibrium phenomenon.
First an economy with no production at all — just endowments to be traded. Then add production. Efficiency, then equilibrium, then welfare, in each case.
Strip the economy to its bones. Two people, two goods, no production. Each person simply starts holding some of each good — their endowment — and the only economic activity available is trade.
Our two traders are Morwenna, who works a boat and lands pilchards, and Tegan, who keeps an orchard. Good 1 is pilchards, good 2 is apples. Write ωkj for person j's endowment of good k, and xkj for what they end up consuming. Preferences are well behaved:
with α > 0 and β > 0. Nothing hangs on the logarithms — they just make the algebra clean, because with log utility each person spends a fixed share of income on each good.
An allocation is a list of what each person consumes: (x₁M, x₁T, x₂M, x₂T). It is feasible if it does not use more than exists:
Set ownership aside for a moment and ask only which allocations are good ones.
If x₁M + x₁T < ω₁M + ω₁T, pilchards are being left to rot. Both goods are normal, so handing the surplus to anyone makes them better off and nobody worse off. Efficiency needs every available unit consumed — the feasibility condition holds with equality.
Suppose everything is consumed but MU₁M/MU₂M > MU₁T/MU₂T. Pilchards are worth more to Morwenna, measured in apples, than they are to Tegan. Move a pilchard from Tegan to Morwenna and an apple the other way: both gain. Efficiency needs equal MRS across all consumers.
With the log utilities above, the marginal rates of substitution are
Setting them equal, and using condition 1 to substitute xkT = ωkM + ωkT − xkM, gives the whole efficient set as a curve in Morwenna's consumption:
as x₁M runs from 0 to the total stock of pilchards. That curve is the contract curve, and the next section draws it.
The Edgeworth box is the single most useful picture in this unit, and it is built on a trick. Draw a rectangle whose width is the total stock of pilchards and whose height is the total stock of apples. Measure Morwenna's consumption from the bottom-left corner in the usual way. Then measure Tegan's consumption from the top-right corner, with both axes reversed.
Every single point in the box is now a complete feasible allocation that uses up exactly all of both goods — because whatever Morwenna does not consume, Tegan does. Two people's consumption bundles, one point.
Morwenna's indifference curves have their usual convex shape. Tegan's are drawn from the opposite corner, so they look convex "upside down" from where you are standing. At a point where the two are tangent, the MRSs are equal — condition 2 — and since every point in the box already satisfies condition 1, tangency is exactly Pareto efficiency.
Mark the endowment point E: the allocation people start from. Draw the indifference curve through E for each person. They cross at E, carving out a lens-shaped region between them.
Every allocation strictly inside that lens makes both people better off than the endowment does. Those are the mutually beneficial trades. The part of the contract curve inside the lens is the set of allocations that are both efficient and acceptable to both parties — no-one has to be forced. Notice that the lens is generally much bigger than that segment: most Pareto improvements are not efficient, and most efficient points are not Pareto improvements on this particular E.
Saying people "own" their endowments is already assuming property rights: everything belongs to somebody, and everybody knows what is theirs and what is not. Competitive markets are then a mechanism for reallocating resources that respects those rights.
A market works by making traders anonymous. Instead of Morwenna negotiating with Tegan, both face the same posted exchange rate between goods — the price ratio p₁/p₂. Each takes it as given and chooses their best affordable bundle. Geometrically, the price ratio is a straight line through E with slope −p₁/p₂: the set of bundles Morwenna can reach by trading at those prices.
p₁/p₂ = (£ per pilchard) ÷ (£ per apple) = apples per pilchardThe line itself drops straight out of the budget constraint. Morwenna can afford any bundle worth no more than what she started with:
Divide through by p₂ and solve for x₂:
which is a straight line in (x₁, x₂) space with slope −p₁/p₂. The prices only ever appear as a ratio, so no money axis is needed to draw it.
x₂ = 40 − 2x₁, and it passes through E, as it must: you can always afford the bundle you already hold. Give up one pilchard and you move one step left and two steps up — slope −2.MRS = p₁/p₂ sets your own rate of exchange against the market's, like against like.
Take the specific case α = 1 and β = ½, so that
Morwenna starts with 12 pilchards and 16 apples; Tegan starts with 6 pilchards and 36 apples. So the box is 18 wide and 52 tall.
With log utility, each person spends fixed shares of income on each good — the share on good k is that good's exponent divided by the sum of exponents. Morwenna spends ½ on each; Tegan spends ⅓ on pilchards and ⅔ on apples. Normalise p₂ = 1 and write p for p₁.
| Income at prices (p, 1) | Demand for pilchards | Demand for apples | |
|---|---|---|---|
| Morwenna | 12p + 16 | ½(12p+16)/p | ½(12p+16) |
| Tegan | 6p + 36 | ⅓(6p+36)/p | ⅔(6p+36) |
Market clearing for pilchards requires total demand to equal the total stock of 18:
At p = 2, one pilchard trades for two apples. Substituting back:
| Endowment | Equilibrium bundle | Net trade | |
|---|---|---|---|
| Morwenna | (12, 16) | (10, 20) | sells 2 pilchards, buys 4 apples |
| Tegan | (6, 36) | (8, 32) | buys 2 pilchards, sells 4 apples |
| Total | (18, 52) | (18, 52) | both markets clear |
Check the trades are consistent: Morwenna gives up 2 pilchards and receives 4 apples, which at a price ratio of 2 is exactly fair value, and Tegan's trade is the mirror image. Both markets clear — as they must, since clearing one market in a two-good economy forces the other to clear too (Walras' Law).
Suppose prices are not equilibrium prices. Then in some market, quantity demanded differs from quantity supplied. Either goods pile up unsold, or demand goes unmet. Neither state can persist: sooner or later somebody stops taking the price as given and offers a different one. Prices must change.
But notice what we have not done. We have shown that equilibrium prices exist; we have not shown how the economy gets to them. That gap is real, it is well known, and a great many able people have worked on it without settling it.
The argument is almost embarrassingly short. In equilibrium every consumer sets their own MRS equal to the same price ratio. So all MRSs are equal to each other — which is precisely condition 2. And market clearing is condition 1. Both efficiency conditions hold, so the allocation is on the contract curve.
Read economically: if everyone can trade freely at common prices, every mutually beneficial trade gets made, because any unexploited gain from trade would show up as somebody wanting to trade more at the going prices — which would mean the market had not cleared. This is Adam Smith's invisible hand stated precisely: the economy reaches a Pareto efficient allocation with nobody organising it.
The theorem is only as strong as its assumptions, and they are demanding:
Everything is owned by somebody, and everyone knows what they own and what they do not.
Your consumption affects only you — no externalities, no public goods.
No search costs, no bargaining costs, no frictions in making the trade happen.
Everything relevant is known and observable to everyone. Nobody has private information about quality or effort.
Every one of these will fail somewhere in the rest of the unit, and each failure is a reason a market might not deliver efficiency.
The First Welfare Theorem is a statement about efficiency and nothing else. Every point on the contract curve is efficient, including the ones at the extreme corners where one person has almost everything. Which efficient point the market delivers depends entirely on who started with what.
The contract curve lives in goods space. The same information can be redrawn in utility space: plot Morwenna's utility on one axis and Tegan's on the other, and trace out every efficient allocation. The result is the utility possibilities frontier (UPF).
The endowment E sits strictly inside the frontier — there are gains from trade left on the table. Competitive markets move the economy from E out to a point on the frontier. Which point depends on E. Moving along the frontier always means one person gaining at the other's expense, which is exactly why efficiency alone cannot settle distributional questions.
To choose a point on the frontier you need a criterion that ranks people's utilities against each other — a social welfare function, which maps the whole profile of individual utilities into a single number.
W = Σk Uk(xk)
Maximise the total. Indifferent to who gets what, so long as the sum is as large as possible.
W = mink Uk(xk)
Maximise the worst-off person's position. Society is only as well off as its least fortunate member.
Each embodies a different view of equity, and each picks a different point on the frontier. Choosing between them is not a technical question and economics has no privileged answer to it. In a pure endowment economy the only decision is consumption, so these definitions are enough; once production enters, redistribution can blunt incentives and the analysis gets harder.
Suppose you look at the market outcome and find it unacceptable — efficient, but unfair. Must you interfere with the market itself?
This is the more remarkable of the two theorems, and it runs the first one backwards. The First says markets land you somewhere on the contract curve. The Second says you can land anywhere on the contract curve you like — you just have to move the starting point first.
The mechanism is a lump-sum transfer. Pick the allocation D you want. Draw the common tangent to the two indifference curves at D — that is the price line that would support D. Now move the endowment from E to any point Ẽ on that line. Reopen the market at those prices, let people trade freely, and they will trade their way to D of their own accord.
Now let the goods be made rather than merely found. Production combines inputs to produce outputs, which adds two further layers of efficiency on top of the consumption condition already established.
The condition is the exact analogue of the consumption one, with marginal products in place of marginal utilities. For any two outputs A and B and any two inputs x and y:
Give input prices wx and wy and this stops being abstract. If input markets are competitive, every industry faces the same input prices, so efficiency requires each industry to set
which is exactly what a firm does anyway when it minimises the cost of hitting a given output target. Cost minimisation by price-taking firms delivers input efficiency for free.
The MRT is the rate at which the economy — not a single firm — can turn one good into another. Take an economy whose only input is labour, with total supply L = 100, producing boats and nets:
Inverting, L₁ = q₁² and L₂ = 4q₂. Since all labour is used, L₁ + L₂ = 100, which gives the production possibilities frontier directly:
Differentiating, the MRT is the magnitude of the PPF's slope:
The PPF bows outward and the MRT rises as you move along it: the first boat is cheap in forgone nets, the tenth is expensive. Every point on the frontier is input-efficient; points inside it waste labour.
Being on the PPF is not enough. You can be on it at the wrong point — producing things nobody particularly wants.
Why: the MRT is the cost of one more boat in nets forgone; the MRS is the benefit of one more boat in nets consumers would willingly give up. If these differ, there is a gain available.
A fully efficient economy satisfies all three at once. Learn them as a set — they are the backbone of the whole topic and they recur every time we diagnose a market failure.
For every pair of goods x, y, the MRS is the same for every consumer.
MUx¹/MUy¹ = ⋯ = MUxⁿ/MUyⁿ
For every pair of inputs a, b, the relative marginal product is the same in every industry.
MPa¹/MPb¹ = ⋯ = MPak/MPbk
Every consumer's MRS equals the economy's MRT between the same two goods.
MUx¹/MUy¹ = ⋯ = MCx/MCy
In a competitive economy with production, equilibrium is a set of prices, an allocation of goods and an allocation of inputs such that consumers maximise utility, firms maximise profits, both take prices as given, and no market has excess demand or supply. Prices are the signals that carry all three conditions at once: consumers set MRS = price ratio, firms set MRT = price ratio, so MRS = MRT without anyone intending it.
This is where general equilibrium earns its keep, and it is the calculation the seminar asks you to do. A specific tax is imposed on one good. Partial equilibrium answers the question in one market. General equilibrium answers it in both — and the two answers differ.
An economy has two goods: cider (C) and pasties (P). They are complements — each market's demand falls when the other good gets dearer. Let PC and PP be the prices producers receive.
| Market | Demand | Supply |
|---|---|---|
| Cider | DC = 58 − PCcons − PP | SC = 18 + 2PC |
| Pasties | DP = 38 − 2PP − PCcons | SP = 14 + PP |
A specific tax t is levied on cider. Consumers pay PCcons = PC + t while producers keep PC. The tax enters both demand functions, because the consumer price of cider appears in the pasty demand too — that is the whole point.
SC = 18 + 2(P − t) with demand written in the market price P, and clearing that market gives 3P + PP = 40 + 2t — which is exactly 3PC + PP = 40 − t once you substitute P = PC + t. Identical prices, identical quantities, identical revenue.DC = 58 − (PC + t) − PP DP = 38 − 2PP − (PC + t)58 − (PC+t) − PP = 18 + 2PC ⟹ 3PC + PP = 40 − t38 − 2PP − (PC+t) = 14 + PP ⟹ PC + 3PP = 24 − tStep 3 — solve the system. From the second equation, PC = 24 − t − 3PP. Substituting into the first:
Step 4 — quantities and revenue. Read them straight off the supply curves:
| t | PC (producer) | PC + t (consumer) | PP | QC | QP | Tax revenue T |
|---|---|---|---|---|---|---|
| 0 | 12 | 12 | 4 | 42 | 18 | 0 |
| 2 | 11.5 | 13.5 | 3.5 | 41 | 17.5 | 82 |
| 4 | 11 | 15 | 3 | 40 | 17 | 160 |
| 8 | 10 | 18 | 2 | 38 | 16 | 304 |
Now do it the naive way. Analyse the cider market alone, holding the pasty price at its pre-tax level of 4 as though it were a constant:
Compare the two. Because the goods are complements, taxing cider depresses pasty demand and pulls PP down. Cheaper pasties then make cider more attractive, which partly offsets the tax. General equilibrium captures that cushion; partial equilibrium misses it and so overstates the damage.
| t | QC — general | QC — partial | T — general | T — partial | Revenue error |
|---|---|---|---|---|---|
| 2 | 41 | 40.67 | 82 | 81.33 | −0.8% |
| 4 | 40 | 39.33 | 160 | 157.33 | −1.7% |
| 6 | 39 | 38.00 | 234 | 228.00 | −2.6% |
| 8 | 38 | 36.67 | 304 | 293.33 | −3.5% |
The error is small for a small tax and grows with the tax. That is the general lesson: partial equilibrium is a good approximation precisely when the intervention is small. For marginal changes the feedback is second-order and safely ignored; for large interventions it is not, and the bigger the shock the worse the single-market answer gets.
Drag the tax rate. The cider market is drawn in producer-price space, so the supply curve never moves. Both demand curves shift left as the tax rises — but the general-equilibrium curve shifts less, because the falling pasty price is pulling cider demand back up. Toggle to the pasty market to see the spillover in the market nobody taxed.
We emphasise competitive markets for three honest reasons: they give a clean, workable definition of efficiency; a good many real markets are close to competitive, with easy entry and price-taking behaviour; and they explain much of the specialisation we observe, which is the basis of the gains from trade in international economics.
But every assumption behind the First Welfare Theorem can fail, and each failure is a topic later in this unit.
Price-taking is equivalent to an absence of market power. A firm with power sets MRT ≠ the price ratio; a union does the same on the input side, breaking MPx/MPy = wx/wy. Analysing it needs game theory.
Not everything is known or observable. Once some parties know more than others, the efficiency argument collapses and you need the economics of information.
Not all benefits and costs show up in private decisions. What unites all three is a price that no longer measures everything at stake — and once it stops carrying the full story, it stops enforcing the three efficiency conditions that rested on it.
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