With \(m\) commodities, a price vector is \(p = (p_1,\dots,p_m)\). It turns any bundle \(x \in \mathbb{R}^m\) into a number, its value in units of account: $$p\cdot x \;=\; \sum_{s=1}^m p_s x_s .$$ Alvarez usually drops the dot and writes \(px\) S1 p.10. Two facts about this valuation get used in every proof in Part I:
The commodity space decides what a price is. If commodities are dated, then \(p_t\) is the price today of a good delivered at \(t\), and \(p_{t+1}/p_t = 1/(1+r_t)\) is a gross interest rate. If commodities are state-contingent, \(p(z^t)\) is an Arrow–Debreu price for delivery after history \(z^t\). Whatever the space, the value of a bundle is always an inner product: \(\sum_t\sum_{z^t} p_t(z^t)x_t(z^t)\), or \(\int p(s)x(s)\,ds\) S1 p.11. Real wages, terms of trade, factor prices and interest rates are all ratios of entries of the same vector \(p\).
A competitive equilibrium is a price vector \(p\in\mathbb{R}^m\) and an allocation \(\{\bar x^i,\bar y^j\}\) such that
Read the budget constraint as uses of funds ≤ sources of funds. On the left is the value of net purchases. On the right are the value of the endowment, which the household can sell, plus its share of every firm's profits, which it owns via the \(\theta^i_j\).
Multiply every price by \(\lambda>0\). Each firm's objective \(\lambda p\cdot y\) has the same maximizers, so \(\bar y^j\) is unchanged and \(\pi^j\) scales by \(\lambda\). Each household's budget set $$\{x: \lambda p\cdot x \le \lambda p\cdot e^i + \textstyle\sum_j\theta^i_j\lambda\pi^j\}$$ is identical to the original one. So if \((p,\bar x,\bar y)\) is a CE, then so is \((\lambda p,\bar x,\bar y)\). That is why we can always normalize one price (e.g. \(p_1 = 1\), or \(\sum_s p_s = 1\)): with \(m\) goods, there are only \(m-1\) prices to find.
If every household's budget constraint binds, then for any prices \(p\) and any profit-maximizing and utility-maximizing plans, $$p\cdot\Big(\underbrace{\textstyle\sum_i x^i - \sum_j y^j - \sum_i e^i}_{\text{aggregate excess demand } z(p)}\Big) = 0 .$$ Hence if \(m-1\) markets clear and \(p_m\neq 0\), the \(m\)-th clears too.
Why do budgets bind? Under local non-satiation (the key assumption of Lesson 3), a household with money left over could buy a slightly better bundle nearby. Notice that Walras' law used exactly the two ingredients that reappear in the First Welfare Theorem proof: binding budgets and \(\sum_i\theta^i_j=1\).
Two consumers, two goods, no firms, and a total endowment of one unit of each good. Consumer A's origin is bottom-left; consumer B's origin is top-right, so a point in the box is an allocation of both goods. Both have Cobb–Douglas utility, \(u^A = a\log x_1 + (1-a)\log x_2\) and \(u^B = b\log x_1 + (1-b)\log x_2\). Normalize \(p_2 = 1\) and let \(p\equiv p_1\).
Take endowments \(e^A = (e_1,e_2)\) and \(e^B=(1-e_1,1-e_2)\), with \(p_2=1\).
With the default values \(a=0.5\), \(b=0.3\), \(e=(0.8,0.2)\), this gives \(p^* = (0.1+0.24)/(1-0.4-0.06) = 0.34/0.54 \approx 0.63\).
Alvarez's Midterm 1 opens with true/false statements worth about half the points; two of the Fall 2025 statements are about this lesson. The longer exercises below are derivations in the exam's format: small steps, each worth a few points, with general parameters rather than numbers.
For each statement, answer True or False and justify in at most four lines.
True. Inada conditions make every optimum interior, so each FOC holds with equality: \(u^i_\ell(\bar x^i)=\mu^ip_\ell\), where \(\mu^i>0\) is \(i\)'s budget multiplier. Hence \(p_\ell=\frac1{\mu^i}u^i_\ell\) for every \(i\), and averaging, \(p_\ell=\frac1I\sum_i\frac{1}{\mu^i}u^i_\ell\). The weight \(1/\mu^i\) is agent-specific but the same for all goods, because each agent has one budget constraint. (Exercise 2.5.)
True. All agents face the same budget set \(\{x: p\cdot x\le p\cdot\bar e/I\}\), and every equilibrium bundle \(\bar x^j\) is in it. If \(u^i(\bar x^j)>u^i(\bar x^i)\), agent \(i\) could have afforded \(\bar x^j\), contradicting optimality of \(\bar x^i\).
False. The revealed-preference argument uses only that budget sets are identical. Each agent compares bundles with her own utility function; preferences can be arbitrary.
False. It holds at every price vector at which budgets bind: sum the binding budgets and use \(\sum_i\theta^i_j=1\). That is why it can be used to drop one market-clearing condition when searching for equilibrium prices.
True. If \(p\cdot\bar x\lt M\), the budget set contains a ball around \(\bar x\) (intersected with \(X\)). LNS gives a strictly better \(\hat x\) in that ball, and it is affordable: contradiction.
False in general. Walras' law then gives \(p_mz_m=0\), which forces \(z_m=0\) only if \(p_m\neq0\). A free good (\(p_m=0\)) can be in excess supply in equilibrium. The statement is true when \(p_m>0\).
False. Profits double, but so do the values of all bundles and endowments. Each budget set \(\{x: 2p\cdot x\le 2p\cdot e^i+\sum_j\theta^i_j2\pi^j\}\) is unchanged, as is each firm's argmax. Only relative prices matter.
For each commodity space from Lesson 1, interpret a ratio of two prices as a familiar economic object:
(a) \(p_{\text{leisure}}/p_{c}\) is the real wage: how much consumption one hour of leisure costs. (b) \(p_t/p_{t+1} = 1+r_t\) is the gross real interest rate between \(t\) and \(t+1\). (c) \(p_s/\sum_{s'}p_{s'}\) is the price of an Arrow security paying one unit in state \(s\), relative to a riskless bond. Ratios \(p_s/p_{s'}\) combine probabilities and marginal-utility weights. (d) The ratio of the price of the export bundle to the import bundle is the terms of trade; for the same good at two locations it reflects transport cost.
Two consumers, two goods. \(u^A = a\log x_1 + (1-a)\log x_2\) with \(e^A=(1,0)\), and \(u^B = b\log x_1+(1-b)\log x_2\) with \(e^B=(0,1)\), where \(a,b\in(0,1)\). Normalize \(p_2=1\) and write \(p\equiv p_1\).
Cobb–Douglas spends a constant share of wealth on each good. A's wealth is \(p\), B's is \(1\).
(a) \(m^A=p\), \(m^B=1\). \(x_1^A=a\,m^A/p=a\) and \(x_1^B=b/p\).
(b) \(a+b/p=1\), so \(p=\dfrac{b}{1-a}\).
(c) \(x_2^A=(1-a)m^A=(1-a)\frac{b}{1-a}=b\); \(x_1^B=b/p=1-a\); \(x_2^B=(1-b)m^B=1-b\).
(d) \(p\) rises with \(b\): B, who owns good 2, wants more good 1 and bids its price up. \(p\) rises with \(a\): A, the owner of good 1, wants to keep more of it, so less is supplied to B.
(e) \(x_2^A+x_2^B=b+(1-b)=1\) ✓.
Later lessons use the special case \(a=\tfrac13\), \(b=\tfrac12\): \(p=(\tfrac34,1)\), \(\bar x^A=(\tfrac13,\tfrac12)\), \(\bar x^B=(\tfrac23,\tfrac12)\).
One household with endowment \(e=(0,1)\) (no consumption good, one unit of time) and utility \(u(c,\ell)=\log c+b\log\ell\), \(b>0\). One firm with technology \(Y=\{(c,-n): 0\le c\le An^\gamma,\ n\ge0\}\), \(A>0\), \(\gamma\in(0,1)\), owned entirely by the household. Normalize the price of consumption to 1 and let \(w\) be the price of time.
The household's bundle is \(x=(c,\ell)\) with \(\ell\in[0,1]\). The time market clears when \(\ell+n=1\). The firm's \(y=(An^\gamma,-n)\) uses time as an input. For (c), use \(wn=\gamma Y\) from (a).
(a) \(\max_n An^\gamma-wn\). FOC: \(\gamma An^{\gamma-1}=w\), so \(wn=\gamma Y\) and \(\pi=Y-wn=(1-\gamma)Y\).
(b) \(c+w\ell\le w\cdot1+\pi\). Log utility with weights \(1\) and \(b\) spends shares \(\frac1{1+b}\) and \(\frac{b}{1+b}\) of \(M\): \(c=\frac{M}{1+b}\), \(\ell=\frac{bM}{(1+b)w}\).
(c) Goods market: \(c=Y\), so \((1+b)Y=w+(1-\gamma)Y\), i.e. \(w=(b+\gamma)Y\). With \(wn=\gamma Y\): \(n=\frac{\gamma}{\gamma+b}\). The time market then clears by Walras' law: \(\ell=\frac{bc}{w}=\frac{b}{b+\gamma}=1-n\) ✓.
(d) FOC: \(\frac{\gamma}{n}=\frac{b}{1-n}\), so \(n=\frac{\gamma}{\gamma+b}\): the CE allocation equals the planner's, as the welfare theorems predict.
(e) Under log utility, the income effect of higher \(A\) (richer, so more leisure) exactly offsets the substitution effect (a higher wage, so less leisure).
Special case used in Lesson 6: \(A=1\), \(\gamma=\tfrac12\), \(b=1\) gives \(n=\tfrac13\) and \(w=\tfrac32\cdot\tfrac{1}{\sqrt3}=\tfrac{\sqrt3}{2}\).
Show that if some household has a slack budget constraint at its optimum, Walras' law can fail, i.e. \(p\cdot z(p) < 0\). Construct a one-line example and say which assumption it violates.
Let one household have a bliss point \(x^b\) with \(p\cdot x^b < p\cdot e\), and let every bundle be weakly worse than \(x^b\). For example, \(u(x) = -\|x - x^b\|^2\). It chooses \(x^b\) and leaves money unspent, so \(p\cdot z = p\cdot(x^b - e) < 0\). This violates local non-satiation: at \(x^b\) there is no better bundle nearby. In Lesson 3 we'll see that LNS is exactly what forces budgets to bind.
Pure exchange, \(I\) agents, \(m\) goods. Each \(u^i\) is strictly concave, strictly increasing, differentiable and satisfies Inada conditions. Let \((p,\bar x)\) be a competitive equilibrium.
(a) \(u^i_\ell(\bar x^i)=\mu^ip_\ell\). Inada conditions rule out corners (\(u^i_\ell\to\infty\) as \(x_\ell\to0\)), so the non-negativity constraints are slack and the Kuhn–Tucker condition holds with equality. \(\mu^i>0\) because \(u^i\) is strictly increasing.
(b) From (a), \(p_\ell=\frac{1}{\mu^i}u^i_\ell\) for each \(i\). Sum over \(i\) and divide by \(I\): \(p_\ell=\frac1I\sum_i\frac1{\mu^i}u^i_\ell\).
(c) Each agent has a single budget constraint, so a single multiplier \(\mu^i\) prices all of her goods.
(d) FOC: \(a^i_\ell/x^i_\ell=\mu^ip_\ell\), so \(p_\ell x^i_\ell=a^i_\ell/\mu^i\). Summing over \(\ell\): \(p\cdot\bar x^i=1/\mu^i\). The budget binds, so \(1/\mu^i=p\cdot e^i\).
(e) Planner's FOC: \(\lambda^iu^i_\ell(x^i)=\gamma_\ell\). Setting \(\lambda^i=1/\mu^i\) and \(\gamma_\ell=p_\ell\) turns it into (a). Agents with a low marginal utility of income, i.e. rich agents, get a high weight. Because the planner's problem is concave, these FOCs are also sufficient; Exercise 4.2 uses this to prove the First Welfare Theorem a second way.
Generation \(t\ge1\) lives two periods; the initial old (generation 0) live only at \(t=1\). A unit mass is born each period. Taxes are \(\tau^t_t\) (when young) and \(\tau^t_{t+1}\) (when old); \(s_t\) is the saving of the young; \(B_t\) is the government's assets (\(B_t<0\) is debt); \(g_t\) is government spending. The constraints are:
$$\text{(1) } s_t+c^t_t=e^t_t-\tau^t_t,\qquad c^t_{t+1}=s_t(1+r_t)+e^t_{t+1}-\tau^t_{t+1},$$ $$\text{(2) } c^0_1=s_0(1+r_0)+e^0_1-\tau^0_1,\qquad \text{(3) } B_t+g_t=(1+r_{t-1})B_{t-1}+\tau^t_t+\tau^{t-1}_t,$$ $$\text{(4) } g_t+c^t_t+c^{t-1}_t=e^t_t+e^{t-1}_t .$$
(a) Young: \(s_t+c^t_t=e^t_t-\tau^t_t\). Old: \(c^{t-1}_t=(1+r_{t-1})s_{t-1}+e^{t-1}_t-\tau^{t-1}_t\). Adding and using (4) to replace \(c^t_t+c^{t-1}_t\) by \(e^t_t+e^{t-1}_t-g_t\):
$$s_t+\tau^t_t+\tau^{t-1}_t=(1+r_{t-1})s_{t-1}+g_t .$$
(At \(t=1\) the old constraint is (2).)
(b) Add this to (3): the taxes and \(g_t\) appear on both sides and cancel, leaving \(B_t+s_t=(1+r_{t-1})(B_{t-1}+s_{t-1})\). (Class Note III p.4 says "subtracting"; with the equations as written, you add them.)
(c) Induction: if \(B_{t-1}+s_{t-1}=0\) then \(B_t+s_t=0\); start from \(B_0+s_0=0\).
(d) If all budgets hold and every goods market clears, the one remaining market, for assets, clears automatically, just as \(p\cdot z(p)=0\) lets you drop one market in the static economy. Lesson 14 uses this result.