General equilibrium and the limits of the model

Every market in this course has been analysed on its own, with everything outside it held fixed, and that is not a safe thing to do.

Raise the price of petrol and the market for cars moves, which moves the market for steel, which moves the market for coking coal, which moves the market for rail freight, some of which comes back to petrol. Partial equilibrium ignores those loops on the argument that they are small, which is often true and is never guaranteed. General equilibrium insists that every market clear at once, and it is where the strongest claims in economics are made and where their limits are clearest. This last lesson states both, then asks how much of the whole framework the evidence supports.

Pure exchange and the box

The essentials appear with no production at all. Take two people, A and B, two goods, and fixed total quantities of each. Every possible allocation can be drawn in a rectangle whose width is the total of good $x$ and whose height is the total of good $y$ , with A's holdings measured from the bottom-left corner and B's from the top-right. Every point in the rectangle is a complete description of who has what. This is the Edgeworth box, drawn in its modern form by Pareto in 1906.

Put each person's indifference curves in, A's bowed towards the bottom-left and B's towards the top-right, and the interesting structure appears immediately. Through most points the two families of curves cross, and where they cross there is a lens-shaped region between them containing allocations that both prefer. Any point with such a lens is one where a mutually beneficial trade remains available.

An allocation is Pareto efficient when no such lens exists: nobody can be made better off without making someone worse off. That happens exactly where the two indifference curves are tangent, so the marginal rates of substitution are equal:

{MRS}_{A} = {MRS}_{B}

If they differed, the two people would value the goods at different relative rates and a trade between those rates would help both, exactly as the arbitrage argument in the consumer's problem said. The set of all tangency points is the contract curve, and it runs from one corner of the box to the other. Note what Pareto efficiency does not do: giving one person everything is Pareto efficient, since the other cannot be helped without hurting them. The criterion is deliberately silent about distribution, and that silence is the whole reason it can command wide agreement.

Example. A box holds 120 units of $x$ and 60 of $y$ , and both people have $u = \sqrt{x y}$ , so each marginal rate of substitution is $y / x$ . A holds $(40, 40)$ and B holds the rest, $(80, 20)$ . Is the allocation efficient, and if not, find a trade that helps both.

A's marginal rate of substitution is $40 / 40 = 1$ and B's is $20 / 80 = 0.25$ . They differ, so the allocation is not efficient. A will give up one unit of $y$ for one unit of $x$ ; B will part with a unit of $x$ for only a quarter of a unit of $y$ . Any exchange rate between 0.25 and 1 units of $y$ per unit of $x$ makes both better off. At half a unit of $y$ per unit of $x$ , A gives 8 units of $y$ for 16 of $x$ , reaching $(56, 32)$ with utility $\sqrt{1792} = 42.3$ against 40 before, while B reaches $(64, 28)$ with the same 42.3 against 40. Both gained, and the ratios are now $32 / 56 = 0.571$ and $28 / 64 = 0.438$ , still unequal, so more trade remains available.

Now you. In the same box, A holds $(20, 40)$ and B holds $(100, 20)$ . Check efficiency and find a trade that helps both.

Answer

A's marginal rate of substitution is $40 / 20 = 2$ , B's is $20 / 100 = 0.2$ , so the allocation is far from efficient and the gains from trade are large. Any rate between 0.2 and 2 works. Trading one for one, A gives 10 units of $y$ for 10 of $x$ : A moves to $(30, 30)$ with utility 30 against 28.3, and B moves to $(90, 30)$ with utility 52.0 against 44.7.

Competitive equilibrium in exchange

Now let the two trade at prices instead of bargaining. Each starts with an endowment, values it at the going prices, and buys their preferred bundle from that budget. A competitive equilibrium is a price at which the two demands exactly exhaust the totals.

Only relative prices matter, since demand is homogeneous of degree zero, so one good can be used as the unit of account. Léon Walras set this system out in 1874 and noticed a useful fact about it: because everyone spends exactly their income, the values of all the excess demands sum to zero, so if every market but one clears, the last one clears automatically. That is Walras's law, and it is why a two-good exchange problem needs only one equation solved.

Example. A holds 90 units of $x$ and 10 of $y$ ; B holds 30 of $x$ and 50 of $y$ . Both have $u = \sqrt{x y}$ . Find the equilibrium price ratio and the final allocation.

Set the price of $y$ to 1 and let $p$ be the price of $x$ . A's income is $90 p + 10$ and B's is $30 p + 50$ . Cobb-Douglas with equal exponents spends half of income on each good, so total demand for $x$ is $0.5 (120 p + 60) / p$ . Setting that equal to the total supply of 120 gives $60 p + 30 = 120 p$ , so $p = 0.5$ : one unit of $x$ trades for half a unit of $y$ .

A's income is then £55, so A buys $x_{A} = 55$ and $y_{A} = 27.5$ . B's income is £65, giving $x_{B} = 65$ and $y_{B} = 32.5$ . The totals check: $55 + 65 = 120$ and $27.5 + 32.5 = 60$ . A has sold 35 units of $x$ for 17.5 of $y$ , and A's utility rose from $\sqrt{900} = 30$ to $\sqrt{1512.5} = 38.9$ while B's rose from 38.7 to 46.0. Both gained, from trade alone, with nothing produced.

Now you. A holds 60 of $x$ and 30 of $y$ ; B holds 30 of $x$ and 150 of $y$ . Both have $u = \sqrt{x y}$ . Find the price and the allocation.

Answer

Total $x$ is 90 and total $y$ is 180. Total demand for $x$ is $0.5 (90 p + 180) / p = 90$ , so $45 p + 90 = 90 p$ and $p = 2$ . A's income is £150, so $x_{A} = 37.5$ and $y_{A} = 75$ ; B's is £210, so $x_{B} = 52.5$ and $y_{B} = 105$ . Good $x$ is scarcer here, and its price is four times what it was in the previous case.

Check the efficiency condition on that first answer. A's marginal rate of substitution is $y_{A} / x_{A} = 27.5 / 55 = 0.5$ , and B's is $32.5 / 65 = 0.5$ . They are equal, so the competitive allocation lies on the contract curve. That is not a coincidence, and generalising it is the first welfare theorem.

The First Welfare Theorem

Every competitive equilibrium is Pareto efficient.

The proof in this setting is one line of the reasoning already used. Every consumer chooses where their marginal rate of substitution equals the price ratio, and everyone faces the same prices, so all marginal rates of substitution are equal and the allocation is on the contract curve. With production included, the same argument runs through firms: each sets marginal cost equal to price, so every firm's marginal cost equals every consumer's marginal valuation, and no reallocation can help anyone.

This is the formal version of Adam Smith's invisible hand, and it is a genuinely remarkable result. Nobody in the system knows the technology, the preferences or the endowments of anyone else. Each agent solves a small private problem using one number, the price, and the outcome cannot be improved on by any planner with complete information.

Its assumptions are exactly the four failures this course has already worked through. Everybody is a price taker, so no market power. No externalities, so all costs and benefits are internal to the trades. Complete information, so nothing is hidden. And a complete set of markets, so everything anyone cares about, including states of the world and future dates, can be traded. Remove any one and the theorem fails.

The theorem also claims much less than it is usually made to claim. Efficiency is compatible with any distribution at all, including one person holding everything, and it is silent about whether the resulting allocation is acceptable. Kenneth Arrow and Gérard Debreu proved in 1954 that an equilibrium exists under stated conditions; existence, efficiency and desirability are three different questions and only the first two have theorems.

The Second Welfare Theorem

The converse is the more politically loaded result. Any Pareto efficient allocation can be reached as a competitive equilibrium, given a suitable redistribution of endowments, provided preferences are convex.

Read carefully, it says something specific: the market mechanism does not fix the distribution. Choose the distribution you want by moving endowments around, and then let the market run, and it will reach an efficient outcome consistent with your choice. Efficiency and equity are separable questions, and the price system is a tool that can serve any distribution.

Example. In the first example above, suppose society decides B has too little. Transfer 20 units of $x$ from A's endowment to B's, so A starts with $(70, 10)$ and B with $(50, 50)$ . Find the new equilibrium.

The totals are unchanged, so the price is still $p = 0.5$ . A's income falls to £45, giving $x_{A} = 45$ and $y_{A} = 22.5$ ; B's rises to £75, giving $x_{B} = 75$ and $y_{B} = 37.5$ . A's utility falls from 38.9 to 31.8 and B's rises from 46.0 to 53.0. Both allocations are on the contract curve, so both are efficient: the transfer changed who gained, not whether anything was wasted.

Now you. Instead transfer 10 units of $x$ from B to A, so A starts with $(100, 10)$ and B with $(20, 50)$ . Find the equilibrium.

Answer

The price is still 0.5. A's income is £60 and B's is £60, so each ends with $x = 60$ and $y = 30$ and both reach utility 42.4. Equal incomes give an equal split, and this allocation is efficient as well.

The catch is in the word "suitable". The redistribution has to be lump sum, meaning it cannot depend on anything the recipient can change, or it distorts behaviour and the efficiency is lost. A transfer based on income changes the incentive to earn; one based on wealth changes the incentive to save. Genuinely lump-sum transfers would have to be based on unalterable characteristics, which is both informationally impossible and ethically unattractive. So the second welfare theorem is a clean statement about a policy instrument that does not exist, and every real redistribution trades some efficiency for the distribution it buys. That tradeoff, rather than the theorem, is what public finance is about.

Second best

One more theoretical result deserves to be widely known and is not. Richard Lipsey and Kelvin Lancaster showed in 1956 that if one of the conditions for the first welfare theorem cannot be met, it does not follow that satisfying the others as far as possible is the next best thing. In an economy with one uncorrectable distortion, the optimal policy elsewhere generally involves deliberately introducing further departures from the competitive conditions.

The consequence is uncomfortable and important. A recommendation of the form "this industry is not competitive, so make it more competitive" is not supported by the first welfare theorem when other distortions remain, and piecemeal liberalisation can make things worse. It does not say that markets should be interfered with, only that arguments of the form "closer to the ideal must be better" are invalid. Each case has to be worked out.

What the evidence does to the axioms

The framework rests on the preference axioms of the second lesson, and those have been tested extensively since. The results are consistent and unflattering.

Maurice Allais showed in 1953 that people reliably violate the independence axiom underlying expected utility, choosing certainty over a slightly better gamble in one framing and reversing when the same difference is embedded in two risky options. Sarah Lichtenstein and Paul Slovic found in 1971 that people asked to choose between two gambles rank them one way and asked to price them rank them the other way, which contradicts the existence of a single underlying preference. Daniel Kahneman and Amos Tversky's prospect theory of 1979 assembled the pattern: people evaluate changes from a reference point rather than final positions, weigh losses roughly twice as heavily as equivalent gains, and distort small probabilities. The endowment effect follows directly and has been measured many times: in the standard 1990 experiment, students given a mug demanded around twice as much to part with it as students not given one were willing to pay for it, though both groups were assigned at random.

Set against this is an equally real finding about where the framework works. John List showed in 2003 that the endowment effect shrinks substantially among experienced traders at sports card conventions and largely disappears for dealers. Experience, repetition and stakes push behaviour towards the model, which is precisely the domain the model was built for. That is the sensible reading of the evidence: the axioms describe experienced participants in repeated markets far better than they describe someone making an unfamiliar decision once.

There is a purely theoretical limit too. Hugo Sonnenschein, Rolf Mantel and Gérard Debreu proved in the early 1970s that aggregate excess demand functions inherit almost none of the structure of individual demand, so general equilibrium models can have multiple equilibria and need not converge to any of them. The theory guarantees an equilibrium exists; it does not guarantee it is unique, stable, or the one you get.

What the course leaves you with

The apparatus of this course is a way of turning a question about scarcity into a constrained optimisation and reading the answer. The specific results are worth having: opportunity cost, the tangency condition, the Slutsky decomposition, elasticity and revenue, marginal cost cutting average cost at its minimum, price equals marginal cost, the markup rule, the deadweight loss triangle, the Samuelson condition, the two welfare theorems.

What is worth having more is the habit those results are built out of. Ask what is being held fixed. Ask what the marginal unit costs and what it is worth. Ask who bears a cost and whether they chose it. Ask what the counterfactual is. Those questions survive every one of the failures in this lesson, because they are what generated the theory rather than what the theory concluded.

And the honest summary of the model's status is the one this lesson has been building towards. It is a benchmark rather than a description. Its assumptions are stated precisely enough that each can be checked and each can be shown to fail, which is a strength and not a weakness: a framework whose failures can be located and named is more useful than one whose cannot. The four failures identified here, market power, externalities, hidden information and missing markets, are not embarrassments to the theory. They are its most useful output, because they say exactly where to look when a market is not doing what it should.