The 1WT proof from Lesson 4 has four moves. Each one uses a specific feature of the environment. To decide whether some modification breaks the theorem, ask which move it attacks.
| Move | What it says | What it uses |
|---|---|---|
| (1) | \(u^i(x^i)\ge u^i(\bar x^i)\Rightarrow p x^i\ge p\bar x^i\) | LNS; \(\bar x^i\) maximizes \(u^i\) on the budget set; \(u^i\) depends only on \(x^i\) |
| (2) | \(u^{i'}(x^{i'}) > u^{i'}(\bar x^{i'})\Rightarrow p x^{i'} > p\bar x^{i'}\) | Utility maximization; \(u^{i'}\) depends only on \(x^{i'}\) |
| (3) | Sum (1)–(2): \(p\sum_i x^i > p\sum_i\bar x^i\) | Linearity of \(p\cdot\); finite sums (Lesson 12!) |
| (4) | \(p\sum_j\bar y^j\ge p\sum_j y^j\) | Each firm maximizes profit taking \(p\) as given; \(Y^j\) doesn't depend on other firms |
| (5) | Feasibility of both allocations, valued at \(p\), contradicts (3)–(4) | Every commodity that matters has a price, i.e. complete markets |
The proof never uses convexity, continuity, differentiability, or the number of firms. It uses one fact about individuals: anything you'd like at least as much as what you have costs at least as much (moves 1–2). It also uses one fact about the system: every effect of anyone's action on anyone else goes through a price (moves 4–5). Externalities, market power and missing markets all break the second fact. Satiation breaks the first.
Recall LNS: for every \(x\in X^i\) and \(\varepsilon>0\), there is \(\hat x\in B_\varepsilon(x)\cap X^i\) with \(u^i(\hat x)>u^i(x)\) (Lesson 3). Throughout, take \(X^i=\mathbb{R}^m_+\). Slide 19 asks four questions. The first three test whether you know what LNS is implied by; the fourth tests whether you know what the 1WT doesn't need.
| If \(u\) is… | LNS? | Why |
|---|---|---|
| continuous, strictly increasing | Yes | \(\hat x = x + \tfrac{\varepsilon}{2\sqrt m}\mathbf 1\) is within \(\varepsilon\) of \(x\), is in \(\mathbb{R}^m_+\), and is strictly larger, so \(u(\hat x)>u(x)\). |
| continuous, (weakly) increasing | Not necessarily | A constant \(u\) is continuous and weakly increasing but satiated everywhere. So is \(u(x)=\min\{x_1+x_2,\,1\}\) on the region \(x_1+x_2\ge 1\). |
| strictly increasing, not continuous | Yes | The same \(\hat x\) works. The argument never used continuity. Continuity is irrelevant to LNS. |
| convex (instead of concave) | Irrelevant | The 1WT uses only LNS. A convex \(u\) such as \(x_1^2+x_2^2\) is strictly increasing on \(\mathbb{R}^2_+\), so LNS holds and any CE is PO. Convexity affects existence of a CE (demand can jump to corners) and the 2WT, not the 1WT. |
What goes wrong without LNS? Move (1) fails: a household at a "thick" part of its indifference map can have a weakly preferred bundle that is strictly cheaper. The figure shows the simplest case.
The 1WT still holds: if a CE exists, it is PO. Move (4) uses only that \(\bar y^j\) maximizes \(p\cdot y\) over \(Y^j\). It never asks \(Y^j\) to be convex. The bite of increasing returns is elsewhere: a CE may not exist. With \(Y=\{(-n,c): c\le n^\gamma\}\), \(\gamma>1\), the profit \(c - wn = n^\gamma - wn\) is unbounded in \(n\) at any \(w>0\) (if \(n\) can grow without bound), so there is no profit-maximizing plan. The only candidate CE has no production at all, and that one usually fails market clearing. Use the slider below to see how the profit function changes shape.
If the single firm takes prices as given, the 1WT holds. Nothing in the proof counts firms: \(J=\{1\}\) is a perfectly good index set. The theorem fails when the single firm exercises market power, for instance a monopolist choosing output knowing that its price falls as it sells more. Then the firm sets marginal revenue = marginal cost, so price exceeds marginal cost. Consumers equate their MRS to the price, the technology's MRT equals marginal cost, and MRS ≠ MRT. A small increase in output is a Pareto improvement. But that outcome is not a competitive equilibrium in the sense of Lesson 2, because the firm is not solving \(\max_{y\in Y}p\cdot y\) at given \(p\). The real lesson: price-taking is a behavioural assumption, not a statement about the number of firms. Large numbers are one justification for it; contestability and Bertrand competition are others.
Suppose \(u^{i'}(x^{i'},x^i)\): agent \(i'\) cares about \(i\)'s consumption, but in a CE takes \(\bar x^i\) as given. Then utility maximization only tells us $$u^{i'}(x,\bar x^i) > u^{i'}(\bar x^{i'},\bar x^i)\ \Rightarrow\ p x > p\bar x^{i'} .$$ A Pareto-dominating allocation changes both \(x^{i'}\) and \(x^i\). Then \(u^{i'}(x^{i'},x^{i})\ge u^{i'}(\bar x^{i'},\bar x^i)\) says nothing about \(u^{i'}(x^{i'},\bar x^{i})\), which is what the budget argument needs. Agent \(i'\) may be happier only because \(i\) changed behaviour, even though \(x^{i'}\) itself is cheap. So move (1) fails and the 1WT generally fails.
Suppose firm \(j'\)'s technology is \(Y^{j'}(y^j)\): an upstream factory's output pollutes a downstream fishery. In a CE, firm \(j'\) maximizes over \(Y^{j'}(\bar y^j)\). In the dominating allocation, \(y^{j'}\in Y^{j'}(y^j)\), a different set, so \(p\bar y^{j'}\ge p y^{j'}\) need not hold. Move (4) fails, and the 1WT generally fails. In both cases the root problem is the same: there is no market, and hence no price, for the side effect. Read this way, an externality is a missing market (see §5).
Agent 1 smokes: \(u^1 = \log s + m^1\), where \(s\) is cigarettes and \(m\) is money (the numeraire). Agent 2 doesn't smoke and dislikes smoke: \(u^2 = m^2 - \gamma s\). Cigarettes are produced at constant cost: one unit of \(m\) makes one unit of \(s\), so the CE price of \(s\) is \(1\).
Three goods but markets for only two. Each agent consumes its endowment of the untraded good 3 and trades goods 1–2. Relative to the unconstrained notion of PO, the 1WT fails. Move (5) values feasibility using a price for every good, but good 3 has none, so agents' marginal rates of substitution involving good 3 are never equalized. Relative to constrained efficiency, it holds. Constrained efficiency is the planner who must also leave good 3 where it is. Fix \(x_3^i = e_3^i\) and define \(\tilde u^i(x_1,x_2)=u^i(x_1,x_2,e^i_3)\). The CE is then a CE of a complete-markets two-good economy, and the 1WT applies to it. So the answer to "what is the notion of efficiency?" is: the benchmark must respect the same trading restrictions the agents face. (Warning for later: with several goods per date or state, even constrained efficiency typically fails, because prices on the open markets change how valuable the incomplete trades are. That is Geanakoplos–Polemarchakis 1986.)
The 1WT does not require agents to agree, nor to be "right". With commodities indexed by states of nature, \(u^i(x)=\sum_s\pi^i_s\,v^i(x_s)\) can embed any subjective probabilities \(\pi^i\). The proof never looks inside \(u^i\). So a CE is PO with respect to the agents' own beliefs, which is ex ante efficiency. If beliefs differ, the CE involves betting (Exercise 5.6), and at most one person's beliefs can be correct. A planner who uses "true" probabilities might call the outcome excessively risky, but that is a different welfare criterion.
They must agree on the commodity space and the prices. Everyone has to recognise the same list of commodities: the same dates, the same set of possible states, and states that are publicly verifiable, so that a contract "deliver 1 unit if state \(s\)" can be enforced. And everyone faces the same price for each commodity. Agents need not agree on how likely the states are.
With dated commodities there is a single budget constraint. All trades for all dates (and states) are contracted at date 0 at prices \(p_t\), and then simply executed. That is the Arrow–Debreu interpretation. It is equivalent to a sequence of budget constraints with borrowing and lending at interest rates \(1+r_t = p_t/p_{t+1}\), plus a limit on borrowing (a no-Ponzi condition). Lesson 9 makes this concrete in the OLG model, and we'll see that what is "summable" turns out to matter.
These follow the format of Alvarez's exams. True/false items like these were about half the points on recent first midterms, and the derivations are broken into short sub-steps. Answer each sub-part in a few lines before opening the solution.
For each statement, say true or false and justify it in one to four lines. The quickest justification names the step of the 1WT proof (§1) that does or does not go through.
False. Increasing returns make the production set non-convex. The 1WT never uses convexity: its moves need only LNS, profit maximization at \(p\), and feasibility. So if a CE exists, it is PO. Non-convexity breaks the 2WT (Lesson 7), and it can make a price-taking CE fail to exist, but neither of those is inefficiency of a CE. This is the classic Midterm 1 trap.
False. The only assumption on preferences is local non-satiation (moves 1–2). Quasi-concavity (convex upper contour sets) is a hypothesis of the 2WT and of existence theorems.
False as a list of requirements: convexity does not belong on it (see item 2). The correct list is: a CE with complete markets (a price for every commodity), no externalities, LNS, and a finite value of aggregate consumption \(p\cdot\sum_i\bar x^i\lt \infty\). The last condition is automatic with finitely many goods and agents. It is exactly what fails in OLG (Lesson 12).
False. Incomplete markets make inefficiency possible, not necessary. Counterexample: endowments that are already PO (e.g. no aggregate risk, constant individual endowments, common beliefs). No trade is needed, so the no-trade equilibrium is PO even with zero securities. More generally, the outcome is efficient whenever the available securities happen to span the trades needed.
False. Efficiency requires \(\dfrac{u_i'(c_i(s))}{u_j'(c_j(s))}=\dfrac{\alpha_j\pi_j(s)}{\alpha_i\pi_i(s)}\). With common beliefs the right side is constant across states, so \(c_i\) depends on \(s\) only through \(Y(s)\). With different beliefs the likelihood ratio varies across states, so two states with the same \(Y\) give different \(c_i\): agents bet on their beliefs. Exercise 5.6 derives this.
False. With a consumption externality, move 1 compares \(u^{i'}(x^{i'},x^i)\) with \(u^{i'}(\bar x^{i'},\bar x^i)\). The household optimized only over its own \(x^{i'}\) holding \(\bar x^i\) fixed. A dominating allocation can make \(i'\) happier by changing \(x^i\) while \(x^{i'}\) stays cheap, so "weakly better ⇒ weakly more expensive" fails.
False. LNS is sufficient, not necessary. For example, two agents who are both at their bliss points at the endowment form a no-trade CE that is trivially PO.
False. A lump-sum transfer changes the right-hand side \(p\cdot e^i\) of the budget but not the prices anyone faces. Moves 1–2 use only that each agent optimizes on its own budget set at the common \(p\). Summing still gives the contradiction, because the transfers sum to zero. That is why the 2WT pairs prices with lump-sum transfers.
True (generically). Households face the wage \((1-t)w\) while firms face \(w\). So there is no single price vector at which everyone optimizes, and move 5 has no common valuation. Concretely, the MRS between leisure and consumption equals \((1-t)w\) while the MRT equals \(w\), and shifting a little time toward work is a Pareto improvement.
False as stated. The proof sums over \(j\in J\) and never uses \(|J|\). What it needs is that the firm takes \(p\) as given and maximizes \(p\cdot y\). A monopolist that internalizes its price impact is not behaving as in a CE, so the theorem says nothing about that outcome. But a single price-taking firm is fine.
Let \(X=\mathbb{R}^m_+\). Prove or give a counterexample.
For (a) and (c), try \(\hat x = x+\delta\mathbf{1}\). What is \(\|\hat x - x\|\)? For (b), look for the laziest possible weakly increasing function. On an exam, state which definition of "increasing" you are using.
(a) True. Fix \(x\) and \(\varepsilon>0\), and let \(\hat x = x + \delta\mathbf 1\) with \(\delta=\varepsilon/(2\sqrt m)\). Then \(\|\hat x - x\| = \delta\sqrt m = \varepsilon/2\lt \varepsilon\), \(\hat x\in\mathbb{R}^m_+\), and \(\hat x\gg x\), so strict monotonicity gives \(u(\hat x)>u(x)\).
(b) False. \(u\equiv 0\) is continuous and weakly increasing, and no bundle is ever strictly better. Less trivially, \(u=\min\{x_1+x_2,1\}\) is satiated on \(\{x_1+x_2\ge1\}\). (If "increasing" means \(x\gg x'\Rightarrow u(x)>u(x')\), the argument of (a) applies and the answer is true.)
(c) True. The proof of (a) never used continuity.
(d) True, provided the convex \(u^i\) satisfy LNS (e.g. \(u=x_1^2+x_2^2\), which is strictly increasing on \(\mathbb{R}^2_+\)), because the 1WT uses only LNS. Caveat: with convex \(u\), demands jump to corners, so a CE is less likely to exist, and the 2WT fails (Lesson 7).
This is the derivation behind statement 1 of Exercise 5.1. One consumption good \(c\) and labour \(n\) (price of \(c\) normalized to 1, wage \(w>0\)). A single price-taking firm has \(Y=\{(-n,c): 0\le c\le n^\gamma,\ n\ge0\}\) with \(\gamma>1\).
(a) \((-1,1)\in Y\) and \((0,0)\in Y\), but their midpoint \((-\tfrac12,\tfrac12)\) requires \(\tfrac12\le(\tfrac12)^\gamma\), which fails for \(\gamma>1\).
(b) \(\pi(n)=n^\gamma-wn\to\infty\) as \(n\to\infty\) because \(\gamma>1\). There is no maximizer.
(c) \(\pi''(n)=\gamma(\gamma-1)n^{\gamma-2}>0\), so \(\pi\) is strictly convex on \([0,1]\). Its maximum is at an endpoint: \(\pi(0)=0\) or \(\pi(1)=1-w\). Supply is \(n=1\) if \(w<1\), \(n\in\{0,1\}\) if \(w=1\), and \(n=0\) if \(w>1\). The interior FOC point \(n=(w/\gamma)^{1/(\gamma-1)}\) is a profit minimum.
(d) Labour demand jumps between 0 and 1. If households want to supply some interior amount, no wage clears the market.
(e) None. Moves 4–5 use only \(p\cdot\bar y^j\ge p\cdot y^j\) for all \(y^j\in Y^j\) and feasibility. So any CE that exists is PO. IRS is a problem for existence and for the 2WT, not for the 1WT.
Use the smoking economy of §4 with a general \(\gamma>0\): \(u^1=\log s+m^1\), \(u^2=m^2-\gamma s\), and one unit of money makes one cigarette.
(a) Agent 1 maximizes \(\log s-s\), so \(s^{CE}=1\). With quasi-linear utility, transferring money moves utility one-for-one between agents. So the utility possibility frontier is a line, and every PO allocation (with interior money) maximizes the sum \(\log s - s-\gamma s\). The FOC \(1/s=1+\gamma\) gives \(s^{PO}=1/(1+\gamma)\).
(b) Surplus \(W(s)=\log s-(1+\gamma)s\). Then \(W(s^{PO})-W(1)=-\log(1+\gamma)-1+(1+\gamma)=\gamma-\log(1+\gamma)\). Since \(\Delta(0)=0\) and \(\Delta'(\gamma)=1-\tfrac{1}{1+\gamma}>0\) for \(\gamma>0\), we get \(\Delta>0\).
(c) Agent 1's private surplus \(\log s-s\) falls by \(L=(0-1)-\big(-\log(1+\gamma)-\tfrac{1}{1+\gamma}\big)=\log(1+\gamma)+\tfrac{1}{1+\gamma}-1\ge0\). Agent 2 gains \(G=\gamma(1-\tfrac1{1+\gamma})=\tfrac{\gamma^2}{1+\gamma}\). Both strictly gain iff \(L\lt T\lt G\). The interval is non-empty because \(G-L=\Delta(\gamma)>0\).
(d) With tax \(t\), agent 1 solves \(\max\log s-(1+t)s\), so \(s=1/(1+t)\). This equals \(s^{PO}\) iff \(t=\gamma\). The tax is the price of smoke that the market was missing: agent 1 now pays the marginal damage \(\gamma\).
(e) The factory maximizes \(q-q^2/2\), so \(q^{CE}=1\). Efficiency maximizes \(q-q^2/2-\gamma q\), so \(q^{PO}=1-\gamma\). Move 4 fails: the fishery's production set depends on \(q\). At the CE it optimized over \(Y^{\text{fish}}(\bar q)\), but in the dominating allocation it operates on \(Y^{\text{fish}}(q)\), a different set, so \(p\cdot\bar y^{\text{fish}}\ge p\cdot y^{\text{fish}}\) need not hold.
Two agents with \(u^i=\log x_1+\log x_2+\log x_3\). Endowments are \(e^A=(1,1,a)\) and \(e^B=(1,1,b)\) with \(a\ne b\), \(a,b>0\). Only goods 1 and 2 can be traded.
For (d), use the weighted planner with weights \(\lambda^i=1/\mu^i\) and multipliers \(\gamma_s\) on each resource constraint.
(a) Goods 1 and 2 enter symmetrically, so \(p_1=p_2\). Each agent spends half its tradeable wealth \(2p_1\) on each, giving \(x_1=x_2=1\). No trade: \(x^A=(1,1,a)\) and \(x^B=(1,1,b)\).
(b) \(MRS^i_{31}=x^i_1/x^i_3\), so \(MRS^A_{31}=1/a\) and \(MRS^B_{31}=1/b\). Say \(a>b\): then B values good 3 more. Let A give B \(\delta\) units of good 3 for \(\kappa\delta\) units of good 1, with \(1/a\lt \kappa<1/b\). To first order, A's utility changes by \(\delta(\kappa-1/a)>0\) and B's by \(\delta(1/b-\kappa)>0\). For small \(\delta\) both gain.
(c) Fix \(x^i_3=e^i_3\). Each agent's utility over goods 1–2 is \(\log x_1+\log x_2+\text{const}\), which satisfies LNS, and the CE is a CE of this complete two-good economy. By the 1WT it is PO in that economy, i.e. constrained PO.
(d) An interior PO allocation solves \(\max\sum_i\lambda^iu^i\) subject to resource constraints for some weights. FOC: \(\lambda^i\,\partial u^i/\partial x_s=\gamma_s\) for every \(i\). With \(\lambda^i=1/\mu^i\), this says \(\eta^i_s=\gamma_s\), the same for all \(i\). If the \(\eta^i\) differ, two agents value state-\(s\) goods differently at the margin. Moving goods from the low-\(\eta\) to the high-\(\eta\) agent and compensating with another good makes both better off. In (b), \(\eta\) for good 3 differs across A and B exactly because no market equalizes it.
One good, \(S\) states, aggregate endowment \(Y(s)>0\). Agent \(i\) has expected utility \(\sum_s\pi_i(s)\log c_i(s)\) with subjective probabilities \(\pi_i\). Markets are complete.
(a) \(\max\sum_i\alpha_i\sum_s\pi_i(s)\log c_i(s)\) subject to \(\sum_ic_i(s)\le Y(s)\) for each \(s\). FOC: \(\alpha_i\pi_i(s)/c_i(s)=\mu(s)\).
(b) \(c_i(s)=\alpha_i\pi_i(s)/\mu(s)\). Summing over \(i\) gives \(\mu(s)=\sum_j\alpha_j\pi_j(s)/Y(s)\), so
$$c_i(s)=\frac{\alpha_i\pi_i(s)}{\sum_j\alpha_j\pi_j(s)}\,Y(s).$$
(c) With common \(\pi\), the share \(\alpha_i/\sum_j\alpha_j\) is the same in every state, so \(c_i(s)=\text{share}_i\cdot Y(s)\). With two agents, \(\alpha_A=\alpha_B\), \(Y(1)=Y(2)=2\), \(\pi_A(1)=\pi_A\) and \(\pi_B(1)=1-\pi_A\ne\pi_A\), agent A gets \(c_A(1)=2\pi_A\ne c_A(2)=2(1-\pi_A)\) even though \(Y\) is the same in both states. The agents bet.
(d) Yes. Each \(u^i\) satisfies LNS and markets are complete, so the 1WT applies. Efficiency is with respect to each agent's own \(u^i\), which embeds \(\pi_i\).
(e) Yes, it can be. In the example of (c) with \(\pi^*(1)=\tfrac12\), the no-trade allocation \((1,1)\) gives both agents \(\tfrac12\log1+\tfrac12\log1=0\) under \(\pi^*\). The CE gives \(\tfrac12[\log2\pi_A+\log2(1-\pi_A)]<0\) whenever \(\pi_A\ne\tfrac12\), since \(4\pi_A(1-\pi_A)<1\). So under \(\pi^*\) the CE is dominated. The 1WT does not require beliefs to be correct; the efficiency it delivers is relative to the agents' own beliefs. Agents need only agree on the commodity space (which states exist and are verifiable) and face the same prices.
Answer each in at most three lines. (a) What must agents agree about in a classical CE? (b) Interpret the single intertemporal budget constraint. (c) Why can't intrinsically useless money have positive value in a finite Arrow–Debreu economy? (d) Give one friction that leads to inefficiency and one that need not.
(a) The list of commodities (dates, locations, verifiable states) and the price of each. They need not agree on probabilities.
(b) All trades are contracted at date 0 at present-value prices \(p_t\). This is equivalent to sequential budget constraints with borrowing and lending at \(1+r_t=p_t/p_{t+1}\), plus a no-Ponzi limit.
(c) At the final date, money buys nothing in the future, so nobody gives up goods for it and its price is 0. At the date before, it can only buy something worth 0 later, so it is worth 0 then too. By induction it is worth 0 at every date.
(d) Random matching with bargaining generates congestion externalities and is inefficient except under the Hosios condition. Competitive (directed) search, with a price for every margin, can be efficient.