Let \(\big(p,\{\bar x^i,\bar y^j\}\big)\) be a competitive equilibrium. Assume that the preferences of every agent satisfy local non-satiation. Then \(\{\bar x^i,\bar y^j\}\) is Pareto optimal.
In one sentence: if prices are such that everyone is optimizing and markets clear, there is no way to make someone better off without hurting someone else. This is Adam Smith's invisible hand made precise. It is remarkable mostly for how little it assumes. There is nothing about convexity, continuity, or the shape of technology; the CE just has to exist.
The heart of the proof is the Lemma from Lesson 3: at equilibrium prices, everything a household likes at least as much as \(\bar x^i\) costs at least as much as \(\bar x^i\). Geometrically, the upper contour set \(\{x: u^i(x)\ge u^i(\bar x^i)\}\) lies entirely on or above the budget hyperplane through \(\bar x^i\). A Pareto improvement would make every household's bundle weakly more expensive and one strictly so. Its total value would then exceed the value of what the economy can produce, which is impossible.
Strategy: contradiction. Suppose the CE allocation is not PO. Then some feasible allocation \(\{x^i,y^j\}\) Pareto dominates it: $$u^i(x^i)\ge u^i(\bar x^i)\ \ \forall i\in I,\qquad u^{i'}(x^{i'})>u^{i'}(\bar x^{i'})\ \text{ for some } i'\in I.$$ We will value both allocations at the equilibrium prices \(p\) and show that the dominating one would have to be worth more than the economy can supply.
Prices turn "better off" into "more expensive" (Steps 1–2). Profit maximization means no firm could have produced anything more valuable (Step 4). Feasibility says the value of what households consume equals the value of what firms produce plus endowments (Step 5). So a Pareto improvement would need households to consume more value than the economy can create at these prices. Every inequality is evaluated at the same \(p\); this is why everybody facing the same prices matters.
| Step | Inequality | Assumption used | Breaks if… |
|---|---|---|---|
| 1 | \(p x^i\ge p\bar x^i\) | LNS + utility maximization | thick indifference curves, satiation (Lesson 3) |
| 2 | \(p x^{i'}>p\bar x^{i'}\) | utility maximization; \(u^{i'}\) depends only on \(x^{i'}\) | consumption externalities (Lesson 5) |
| 3 | \(p\sum x^i>p\sum\bar x^i\) | sums are finite | infinitely many agents and goods with \(p\sum\bar x^i=\infty\) (OLG) |
| 4 | \(p\sum\bar y^j\ge p\sum y^j\) | profit maximization at \(p\); \(Y^j\) independent of others | market power, production externalities (Lesson 5) |
| 5 | contradiction | feasibility; a price for every good | missing markets (Lesson 5) |
Convexity of preferences or technology, continuity of \(u\), differentiability, \(p\gg 0\), and the ownership shares \(\theta\) (they cancel because feasibility is valued directly) are all absent. Increasing returns, non-convex preferences and so on do not by themselves invalidate the 1WT. They may destroy existence of a CE, but whenever a CE exists, it is PO. That is the point of slide questions p.19 (convex \(u\)) and p.20 (increasing returns), developed in Lesson 5.
The First Welfare Theorem is the single most tested result in this part of the course. On Alvarez's exams it appears as true/false statements that test its exact hypotheses (LNS, finitely many goods or a finite value of the aggregate endowment, and nothing about convexity), and as yes/no efficiency checks at the end of longer problems. The traps below come from past exams and from mistakes found in student review notes.
For each statement, answer True or False and justify in at most four lines.
True. These are exactly the hypotheses of the 1WT. Finitely many goods make every value \(p\cdot\sum_ix^i\) finite, so the inequalities from LNS and optimality can be summed across households and contradict feasibility.
False. The proof uses only LNS (equation (1)), optimality (2), profit maximization (4) and feasibility. Quasi-concavity is a hypothesis of the Second Welfare Theorem and of existence. Exercise 4.1 gives a CE with strictly convex utilities that is PO.
False. Profit maximization is used only as \(p\cdot\bar y^j\ge p\cdot y^j\) for feasible \(y^j\); no convexity is needed. With increasing returns a price-taking CE may fail to exist, but if one exists it is PO. (One set of student notes lists "convexity" as a 1WT requirement; it isn't.)
False. It needs LNS (Exercise 3.5 is a counterexample with satiation), price-taking with complete markets and no externalities (Lesson 5), and a finite value of the aggregate endowment. The OLG model violates the last one (Lesson 12).
False. With infinitely many agents and infinitely many goods, \(p\cdot\sum_i\bar x^i\) can be infinite, and the step "sum the inequalities across households" compares \(\infty\) with \(\infty\). The endowment OLG economy with \(\bar r<0\) is a CE that is not PO. Either finitely many goods (with a bounded aggregate endowment), or finitely many agents with strictly increasing utilities, is enough to keep that value finite.
True. Envy-free: all agents have the same budget set, so nobody can prefer a bundle she could have bought. Pareto optimal: by the 1WT. So efficiency and this notion of fairness are compatible.
True. \(r(0)=f'(k_0)-\delta\), and \(f'(k_0)\to0\) as \(k_0\to\infty\) (Inada), so \(r(0)<0\) once \(f'(k_0)\lt \delta\).
False. There is one agent. The equilibrium allocation maximizes her utility over everything that is feasible, so no feasible allocation makes her better off. A negative \(r\) is harmless here: it does not make the value of her endowment infinite, since \(r\) converges to \(\rho>0\). Negative rates signal inefficiency only in OLG-type economies (Exercise 4.6).
True. For strictly better-off households, revealed preference alone gives \(p\cdot x^i>p\cdot\bar x^i\). LNS is needed to rule out a weakly-preferred bundle being strictly cheaper. That is why the 1WT for weak Pareto optimality needs no LNS at all (Exercise 4.3).
"Suppose that all \(u\) are convex (instead of concave). Does the first welfare theorem hold?"
A convex function on a budget line is maximized at an endpoint. Check whether the two agents can choose different corners that together clear the market.
(a) Yes, whenever a CE exists and LNS holds. The proof never uses concavity or quasi-concavity. A strictly increasing convex \(u\) on \(\mathbb{R}^m_+\) satisfies LNS. The catch is existence: demands jump between corners, so a CE may fail to exist. But that is a separate question from efficiency.
(b) At \(p=(1,1)\), wealth is 2 and the budget is \(x_1+x_2\le2\). \(x_1^2+x_2^2\) is maximized at a corner, \((2,0)\) or \((0,2)\), each with utility 4. Let A choose \((2,0)\) and B choose \((0,2)\). Markets clear: \(2+0=2\) and \(0+2=2\). So this is a CE. By the 1WT it is PO. Directly: any feasible allocation has \(u^A+u^B=\|x^A\|^2+\|x^B\|^2\le\) its maximum on the box, which is 8 and is attained only at the corner allocations, so neither can gain without the other losing.
(c) The equal split gives each agent \(1+1=2<4\). With convex preferences, specializing is efficient and identical agents optimally consume different bundles. Equal treatment and efficiency can conflict. Convexity is what the Second Welfare Theorem will need (Lesson 6), not the first.
Review notes for this course present the 1WT through first-order conditions, and the Fall 2025 midterm tested its key step (Exercise 2.5). Take a pure exchange economy with finitely many goods, and \(u^i\) strictly concave, strictly increasing, differentiable, with Inada conditions. Let \((p,\bar x)\) be a CE with budget multipliers \(\mu^i\).
(a) CE FOC: \(u^i_\ell(\bar x^i)=\mu^ip_\ell\). Planner FOC: \(\lambda^iu^i_\ell(x^i)=\gamma_\ell\). With \(\lambda^i=1/\mu^i\) and \(\gamma_\ell=p_\ell\), the CE FOCs are the planner FOCs, and \(\bar x\) is feasible.
(b) The objective is a positively weighted sum of concave functions and the constraints are linear, so any point satisfying the FOCs (KKT) is a global maximum.
(c) If a feasible \(x'\) Pareto-dominated the solution \(\bar x\), then \(u^i(x'^i)\ge u^i(\bar x^i)\) for all \(i\) with one strict, so \(\sum_i\lambda^iu^i(x'^i)>\sum_i\lambda^iu^i(\bar x^i)\) because every \(\lambda^i>0\). That contradicts optimality of \(\bar x\).
(d) (a)–(c) give: every CE allocation is PO. This proof used differentiability, concavity of \(u^i\) (for sufficiency), interior solutions (Inada) and no production. The slides' proof needs only LNS and is therefore far more general; but this version is useful because it delivers the weights \(\lambda^i=1/\mu^i\) explicitly.
Show that without any assumption on preferences (no LNS), every CE allocation is weakly Pareto optimal: no feasible allocation makes every household strictly better off. Which step of the proof are you avoiding?
Suppose a feasible \(\{x^i,y^j\}\) has \(u^i(x^i)>u^i(\bar x^i)\) for all \(i\). Step 2's argument (which uses only optimality) now applies to every \(i\): \(p\cdot x^i>p\cdot\bar x^i\) for all \(i\). Summing gives (3). Steps 4–5 go through unchanged, giving a contradiction. We avoided Step 1, the only place LNS enters. LNS is needed only to handle the agents who are merely indifferent.
For each modification, say whether the 1WT can fail and identify the first equation of the proof, (1)–(4) or Step 5, that can no longer be established:
(a) Equation (1): the satiated household can have a weakly better, strictly cheaper bundle (Exercise 3.5).
(b) Equation (4): \(p\cdot\bar y^j\ge p\cdot y^j\) is no longer guaranteed. A dominating allocation can use a more valuable production plan.
(c) Equation (2) (and (1)): household 1's optimality was computed taking \(\bar x^2\) as given. In the dominating allocation, \(x^2\) changes too, so "\(u^1(x^1,x^2)>u^1(\bar x^1,\bar x^2)\)" no longer says \(x^1\) is better given \(\bar x^2\), and the revealed-preference argument says nothing about its cost.
(d) Equation (3): the strict inequality cannot be kept after summing infinite values. Lesson 12 shows this really happens in the OLG model.
Show that in any CE where some household satisfies LNS, \(p\ne 0\). Where is this implicitly needed in Step 1?
If \(p=0\), every bundle in \(X^i\) is affordable. LNS at \(\bar x^i\) gives a strictly better \(\hat x\in X^i\), which is affordable. This contradicts optimality, so \(p\ne0\). In the proof, \(p=0\) would make every valuation trivially zero, and (2) would read \(0>0\). The fact you just proved says this case never arises, so the price comparisons in Steps 1–5 are never vacuous. (Individual prices can still be zero, e.g. for a good nobody wants.)
Consider the continuous-time neoclassical growth model with fixed labour supply and an infinitely-lived representative agent. Utility and production functions are strictly increasing, strictly concave and satisfy Inada conditions; capital depreciates at rate \(\delta>0\), so \(r(t)=f'(k(t))-\delta\).
(a) Yes. By Inada, \(f'(k_0)\to0\) as \(k_0\to\infty\), so for \(k_0\) large, \(f'(k_0)\lt \delta\) and \(r(0)<0\). Capital is so abundant that it earns less than it depreciates.
(b) No. With a single (representative) agent and complete markets, the equilibrium allocation is the planner's solution: it maximizes her utility over the feasible set, so nothing feasible can make her better off. The high \(k_0\) is simply consumed or run down; \(r(t)\) rises toward \(\rho\) (the modified golden rule) as capital falls.
(c) In OLG there are infinitely many agents and infinitely many goods, and with \(r<0\) forever the value of the aggregate endowment \(\sum_tp_t\bar e_t\) is infinite: the 1WT proof breaks at the summation step. Here there is one agent, and \(r(t)\to\rho>0\), so the value of her lifetime endowment is finite and the 1WT applies.