This whole course rests on one template. An economy is a short list of mathematical objects. A competitive equilibrium (Lesson 2) and a Pareto optimum (Lesson 3) are properties of allocations in that economy. Later models (OLG, Blanchard–Yaari, growth) are this same template with a particular commodity space and set of agents plugged in, so it pays to be fluent in it now.
The commodity space \(L\) is a vector space whose elements are bundles of commodities. Typically \(L=\mathbb{R}^m\): there are \(m\) goods, and a bundle \(x=(x_1,\dots,x_m)\) lists a quantity of each. Sometimes \(m=\infty\), e.g. \(L=\mathbb{R}^\infty\) in the overlapping-generations model (Lesson 8).
The key modelling idea is that a commodity is not just a physical object. Two units of the same physical good count as different commodities if they differ in any of the following S1 p.2:
| Distinguished by | Example | Use |
|---|---|---|
| characteristics | wheat vs. steel | classical static GE |
| location | wheat in Chicago vs. in Rotterdam | geography, trade |
| time | consumption in 2026 vs. 2027 | dynamics, saving, interest rates |
| state of nature | an umbrella if it rains vs. if it is dry | uncertainty, insurance |
Because "time" and "uncertainty" are just more coordinates, every theorem we prove for \(\mathbb{R}^m\) automatically covers dynamic and stochastic economies, as long as \(m\) stays finite. Whether it survives \(m=\infty\) is precisely the OLG question of Lesson 12.
| Space | What the coordinates are |
|---|---|
| \(m=2\) | static consumption and leisure (labour–leisure choice) |
| \(\mathbb{R}^m\) | \(m\) different goods |
| \(m=2\times 2\) | two inputs (capital, labour) and two final goods |
| \(m=k\times\#\text{locations}\) | basic trade with goods differentiated by origin (Armington) |
| \(\mathbb{R}^{2T}\) | consumption and investment in each of \(T\) periods |
| \(\mathbb{R}^{k\times\infty}\) | \(k\) goods every period, infinite horizon |
| \(\mathbb{R}^{k}\) | one period, \(k\) states of nature ("now and then") |
| \(\mathbb{R}^{k+1}\) | two periods: one good today, plus one good in each of \(k\) states tomorrow |
| \(\{x_t: Z^t\to\mathbb{R}^k\}_{t\ge0}\) | an event tree: \(x_t(z^t)\) is the \(k\)-vector consumed after history \(z^t=(z_0,\dots,z_t)\) |
The last row is the workhorse of stochastic macro. \(Z\) is a finite set of shocks, and a history grows recursively: \(z^{t+1}=(z^t,z_{t+1})\). Each node of the tree is a separate set of \(k\) commodities. Continuous time or a continuum of states can be handled the same way (sums become integrals); the slides use it "whenever is simpler" without dwelling on the measure theory S1 p.9.
Households are indexed by \(i=1,\dots,I\). Abusing notation, \(I\) also denotes the set of households. Each household has
The set of households may be infinite: countably many (\(i=1,2,3,\dots\), as in OLG) or a continuum (as in Blanchard–Yaari, Lesson 17).
\(X^i\) is where physical constraints live: non-negativity of consumption, or a cap on hours (with 24 hours a day, labour supply cannot exceed 24). It is not the budget set. \(X^i\) contains what is possible, the budget set contains what is affordable, and the budget only appears once we have prices.
Positive entries of \(x^i\) are quantities demanded (purchases); negative entries are quantities offered (sales). If labour is a commodity and the household supplies 8 hours, then \(x^i_{\text{labour}}=-8\). An equivalent and common alternative is to make leisure the commodity, endow the household with 24 hours of time, and keep \(x\ge 0\). Both conventions describe the same economy; just keep track of which one you are using.
Firms are indexed by \(j=1,\dots,J\) (and \(J\) also denotes the set). Firm \(j\)'s technology is a production possibility set \(Y^j\subset L\): the firm can carry out any plan \(y\in Y^j\). Positive entries of \(y\) are outputs (offered); negative entries are inputs (demanded).
A production function is a special case. A firm turning labour \(n\) into output with \(f\) has $$Y^j=\{(c,\,-n)\;:\; n\ge 0,\ c\le f(n)\}.$$ The inequality \(c\le f(n)\) allows waste (free disposal). With this sign convention, profit is simply the value of the plan, \(p\cdot y = p_c\,c + p_n\cdot(-n) = p_c c - p_n n\): revenue minus cost, with no special formula needed.
Ownership is what routes firm profits back to people, so that no value "leaks out" of the economy. The condition \(\sum_i\theta^i_j=1\) is used in exactly two places: in Walras' law (Lesson 2) and in the First Welfare Theorem (Lesson 4). The pair \((e^i,\theta^i)\) is a household's wealth endowment. In the Second Welfare Theorem it is the policy instrument: redistributing \(e\) and \(\theta\) lets us reach different Pareto optima (Lesson 7).
An allocation \(\{x^i,y^j\}\) is a consumption bundle for every household and a production plan for every firm. It is feasible if
Material balance is a vector equation, one equation per commodity S1 p.7: $$\begin{pmatrix}x^1_1\\ \vdots\\ x^1_m\end{pmatrix}+\cdots+\begin{pmatrix}x^I_1\\ \vdots\\ x^I_m\end{pmatrix} =\begin{pmatrix}y^1_1\\ \vdots\\ y^1_m\end{pmatrix}+\cdots+\begin{pmatrix}y^J_1\\ \vdots\\ y^J_m\end{pmatrix} +\begin{pmatrix}e^1_1\\ \vdots\\ e^1_m\end{pmatrix}+\cdots+\begin{pmatrix}e^I_1\\ \vdots\\ e^I_m\end{pmatrix}.$$ Read row \(s\): total use of good \(s\) by households = total net output of good \(s\) by firms + total endowment of good \(s\). Because inputs are negative, a firm's use of good \(s\) as an input is netted out on the right-hand side automatically.
Feasibility is a purely physical, planner's-eye notion: is this allocation possible with the resources and technologies available? Nothing about prices, budgets or ownership enters it. That is why Pareto optimality, which is defined over feasible allocations, is also price-free. The welfare theorems are bridges between this price-free world and the price-based world of competitive equilibrium.
An allocation can be feasible without each bundle being affordable at some prices, and every household can be on its budget without markets clearing. A CE requires both. Keep the words straight: feasible refers to the aggregate resource constraint, budget-feasible to an individual's budget set.
Alvarez's midterms use two formats: true/false statements with a short justification (about half the points on Midterm 1), and long derivations broken into small, hinted steps. The exercises below follow both. Lesson 1 is mostly vocabulary, so the exam questions that touch it test whether you can write an economy down correctly before solving anything.
For each statement, answer True or False and justify in at most four lines.
False. Commodities are distinguished by physical characteristics, location, date and state of nature. Each date gets its own coordinate in \(L\) and its own price; the ratio \(p_t/p_{t+1}=1+r_t\) is precisely the interest rate. Merging them would make intertemporal trade impossible to describe.
False. Feasibility is \(x^i\in X^i\), \(y^j\in Y^j\) and \(\sum_i x^i=\sum_j y^j+\sum_i e^i\): purely physical. Prices enter only the budget and profit problems, i.e. the definition of a competitive equilibrium.
False. The shares appear nowhere in feasibility. They only redistribute profit income across budget constraints, so they change which feasible allocation is a CE, never what is feasible. (This is the lever the Second Welfare Theorem uses, Lesson 7.)
True. For households, positive = purchases, negative = sales; for firms, positive = outputs, negative = inputs. Material balance then reads \(\sum_i x^i_{\text{lab}} = \sum_j y^j_{\text{lab}} + \sum_i e^i_{\text{lab}}\): both sides negative when there is no labour endowment entry, i.e. sales by households equal purchases by firms.
True. Convex combinations stay in each \(X^i\) and \(Y^j\), and material balance is linear, so a convex combination of two feasible allocations is feasible (Exercise 1.4 writes it out).
False. \(|Z|^T\) counts only the terminal nodes. Every node is a commodity: \(\sum_{t=0}^{T}|Z|^t=\frac{|Z|^{T+1}-1}{|Z|-1}\). (Exercise 1.1.)
Give \(\dim L\) as a formula, and say what a single coordinate means, in each case.
(a) \(k+n\). A household has positive entries for final goods and negative entries for inputs it sells; a firm the reverse.
(b) \(2T\); coordinate \((c_t, i_t)\) at each date.
(c) \(k(1+S)\): \(k\) spot goods today plus \(kS\) contingent claims \(x_{1\ell}(s)\), each delivering good \(\ell\) tomorrow only if state \(s\) occurs.
(d) There are \(|Z|^t\) histories \(z^t\) at date \(t\), so \(\dim L = k\sum_{t=0}^T|Z|^t = k\,\dfrac{|Z|^{T+1}-1}{|Z|-1}\).
(e) It becomes infinite. With infinitely many commodities, values such as \(p\cdot\sum_i\bar x^i\) can be infinite, and the First Welfare Theorem proof needs that value to be finite (Lesson 12).
A two-period economy has one consumption good per period. Household 1 is endowed with one unit of the good at \(t=0\) and nothing at \(t=1\); household 2 has nothing at \(t=0\) and one unit at \(t=1\). A single firm has a storage technology: storing \(s\ge0\) units at \(t=0\) yields up to \(Rs\) units at \(t=1\), with \(R>0\). Household 1 owns a share \(\theta\in[0,1]\) of the firm and household 2 owns the rest.
(a) \(L=\mathbb{R}^2\); \(X^i=\mathbb{R}^2_+\); \(e^1=(1,0)\), \(e^2=(0,1)\); \(\theta^1=\theta\), \(\theta^2=1-\theta\); \(Y=\{(y_0,y_1): y_0\le0,\ y_1\le -Ry_0\}\). The date-0 good is an input (negative) and the date-1 good an output.
(b) Date 0: \(x^1_0+x^2_0=y_0+1\). Date 1: \(x^1_1+x^2_1=y_1+1\).
(c) With \(y=(-s,y_1)\), \(y_1\le Rs\), and \(s\in[0,1]\) (non-negative consumption at date 0): aggregate consumption is \((C_0,C_1)\) with \(C_0=1-s\) and \(C_1\le 1+Rs\). The efficient frontier \(C_1 = 1+R(1-C_0)\), \(C_0\in[0,1]\), is a segment of slope \(-R\): giving up one unit today buys \(R\) tomorrow. Any split of \((C_0,C_1)\) between the two households with non-negative entries is feasible.
(d) Only in the budget constraints: household 1's income is \(p\cdot e^1+\theta\pi\) and household 2's is \(p\cdot e^2+(1-\theta)\pi\). (With constant returns, \(\pi=0\) in equilibrium, so \(\theta\) would not even matter there.)
A household has 1 unit of time, values consumption \(c\) and leisure \(\ell\), and works \(n=1-\ell\) hours. Describe its \(X^i\) and \(e^i\) in two ways: (i) with commodities (consumption, leisure) and a time endowment; (ii) with commodities (consumption, labour) and no time endowment. Show that the household's net trade with the market is identical in both.
(i) \(X=\{(c,\ell): c\ge0,\ 0\le\ell\le1\}\) and \(e=(0,1)\). Net trade: \(x-e=(c,\ell-1)=(c,-n)\).
(ii) \(X=\{(c,x_n): c\ge0,\ -1\le x_n\le 0\}\) with \(x_n=-n\), and \(e=(0,0)\). Net trade \(x-e=(c,-n)\).
Same net trade, so the same budget constraints (\(p\cdot(x-e)\le\) profits) and the same equilibria. Physical limits on hours appear in \(X^i\) either way (\(\ell\le 1\), or \(x_n\ge-1\)).
Let every \(X^i\) and \(Y^j\) be convex. Show that the set of feasible allocations is convex. (We will use convexity heavily in the Second Welfare Theorem, Lesson 6.)
Take feasible \(\{x^i,y^j\}\) and \(\{x'^i,y'^j\}\), and \(\lambda\in[0,1]\). Let \(x^i_\lambda=\lambda x^i+(1-\lambda)x'^i\) and similarly \(y^j_\lambda\). By convexity, \(x^i_\lambda\in X^i\) and \(y^j_\lambda\in Y^j\). Material balance is linear: \(\sum_i x^i_\lambda = \lambda\sum_i x^i+(1-\lambda)\sum_i x'^i = \lambda(\sum_j y^j+\sum_i e^i)+(1-\lambda)(\sum_j y'^j+\sum_i e^i) = \sum_j y^j_\lambda+\sum_i e^i\). ∎
This is the setup of the 2017/18 Midterm 1 ("OLG and trees"), here only written down; Lessons 8–12 solve economies like it. Time is \(t=1,2,\dots\). A measure one of agents is born each period and lives two periods. Generation \(t\ge1\) is endowed with \(e^t_t>0\) when young and nothing when old, and has strictly increasing utility \(u^t(c^t_t,c^t_{t+1})\). The initial old (generation 0) care only about \(c^0_1\). A tree produces \(d_t\ge0\) units of the good at each date; at \(t=1\) the initial old own the tree.
What would the initial old demand at date 1 if the value of their firm were infinite? Compare with what is physically available at date 1.
(a) \(L=\mathbb{R}^\infty\): one good per date.
(b) \(Y=\{y\in\mathbb{R}^\infty: 0\le y_t\le d_t \ \forall t\}\) (fruit can be discarded but not created). The initial old hold \(\theta^0=1\); everyone else holds 0. (In the sequential-markets version, the tree is sold each period; in Arrow–Debreu, ownership at date 0 is all that matters.)
(c) \(c^{t-1}_t+c^t_t = d_t+e^t_t\).
(d) With \(p_t>0\), profit maximization sets \(y_t=d_t\), so \(\pi=\sum_{t\ge1}p_td_t\). Initial old: \(p_1c^0_1\le \pi\).
(e) If \(\sum_tp_t=\infty\) then \(\pi=d\sum_tp_t=\infty\). Then the initial old's budget set contains every \(c^0_1\ge0\); with strictly increasing utility their problem has no solution (demand is unbounded), which contradicts the existence of an optimal \(\bar x^0\) with \(\bar c^0_1\le d_1+e^1_1\). Hence \(\sum_tp_t\lt \infty\). Interpretation: a positive-dividend asset forces the value of the aggregate endowment to be finite, i.e. interest rates that are positive on average. Lesson 12 shows that this is exactly the condition under which the First Welfare Theorem survives in OLG.