Glossary & Notation

Part I: The economy and equilibrium

\(L\), commodity space

Vector space of bundles; usually \(\mathbb{R}^m\), with \(m=\infty\) in OLG. Commodities are distinguished by physical characteristics, location, date and state of nature. L1

\(X^i\), \(u^i\)

Consumption possibility set and utility function of household \(i\in I\). In \(x\in X^i\), positive entries are purchases and negative entries are sales (e.g. labour). L1

\(Y^j\)

Technology (production possibility set) of firm \(j\in J\). In \(y\in Y^j\), positive entries are outputs and negative entries are inputs.

\(e^i\), \(\theta^i_j\)

Endowment of \(i\); share of firm \(j\) owned by \(i\), with \(\theta^i_j\ge0\) and \(\sum_i\theta^i_j=1\).

Feasible allocation

\(\{x^i,y^j\}\) with \(x^i\in X^i\), \(y^j\in Y^j\), and material balance \(\sum_i x^i=\sum_j y^j+\sum_i e^i\) (every good, every date and state).

\(p\), \(p\cdot x\)

Price vector and the value of bundle \(x\) (inner product). Only relative prices matter, so we normalize one. L2

Competitive equilibrium (CE)

\((p,\{\bar x^i,\bar y^j\})\): each \(\bar y^j\) maximizes \(p\cdot y\) on \(Y^j\), each \(\bar x^i\) maximizes \(u^i\) subject to \(p\cdot x\le p\cdot e^i+\sum_j\theta^i_j\pi^j\), and the allocation is feasible. L2

Walras' law

With binding budgets, \(p\cdot z(p)=0\) at every \(p\), where \(z\) is aggregate excess demand. So \(m-1\) clearing conditions suffice.

Pareto optimal (PO)

A feasible allocation that no other feasible allocation Pareto-dominates, i.e. makes everyone weakly better off and someone strictly better off. L3

Local non-satiation (LNS)

For every \(x\in X^i\) and \(\varepsilon>0\), there is \(\hat x\in B_\varepsilon(x)\cap X^i\) with \(u^i(\hat x)>u^i(x)\). No bliss points and no thick indifference curves. L3

First Welfare Theorem (1WT)

Under LNS, every CE allocation is PO. It needs no convexity. In infinite economies it also needs \(p\cdot\sum_i\bar x^i\lt \infty\). L4, L12

Upper contour set

\(\{x\in X^i: u^i(x)\ge u^i(\bar x)\}\). It is strictly convex under strict quasi-concavity (Assumption HH).

Aggregate production set \(Y\)

\(Y=\sum_j Y^j\) (set sum). Assumption FF: \(Y\) is convex.

Second Welfare Theorem (2WT)

Under HH and FF, plus one non-satiated household, any PO allocation is supported by some \(p\ne0\): firms maximize profits and households minimize expenditure (a quasi-equilibrium). L6

Separating hyperplane theorem

Two disjoint convex sets (here \(A\) and \(\operatorname{Int}B\)) can be separated by a hyperplane \(\{x: p\cdot x = c\}\), \(p\ne0\).

Arrow's remark / cheapest point

If \(\bar x^i\) is not the cheapest point in \(X^i\) (some \(\tilde x^i\) has \(p\cdot\tilde x^i\lt p\cdot\bar x^i\)), expenditure minimization implies utility maximization, so the quasi-equilibrium is a CE. L7

Part II–III: Overlapping generations

Generation \(i\)

Agents born at date \(i\ge1\) live for dates \(i,i+1\) with utility \(v^i(x_i^i,x_{i+1}^i)\). The initial old (\(i=0\)) live only at \(t=1\), with \(u^0=x_1^0\). L8

\(\bar e_t\)

Aggregate endowment at \(t\): \(e_t^{t-1}+e_t^t\). Feasibility: \(x_t^{t-1}+x_t^t=\bar e_t\).

\(r_t\)

Net interest rate, \(1/(1+r_t)=p_{t+1}/p_t\), so \(p_t=1/\prod_{s\lt t}(1+r_s)\) with \(p_1=1\). L9

\(s(r;\alpha,\beta)\)

Savings of the young: \(\beta(1-\alpha)-(1-\beta)\alpha/(1+r)\) in the log example. Equilibrium sets \(s=0\), so \(\bar r=(\alpha-\beta)/(\beta(1-\alpha))\).

Symmetric allocation

\((c_y,c_o)\), the same for every generation; feasible iff \(c_y+c_o=1\) (no growth). The best one for the young is \((1-\beta,\beta)\). L10

Dynamic inefficiency

A CE that is Pareto-dominated, e.g. by intergenerational transfers. It arises when \(\bar r<0\), or with growth when \(1+\bar r<(1+g)(1+n)\). L13

Pay-as-you-go (PAYG) social security

Tax each young agent \(\tau\) and transfer the proceeds to the current old. It is balanced every period. With population growth each old agent receives \((1+n)\tau\). L11

Value of aggregate consumption

\(p\cdot\sum_i\bar x^i\). The 1WT proof needs it finite. In the OLG example it equals \(\sum_t p_t\), which diverges iff \(\bar r\le0\). L12

Ricardian equivalence

Changing the timing of taxes while preserving each generation's present value leaves consumption and prices unchanged. L14, L20

\(B_t\)

Government assets in Class Note III (\(B_t<0\) is debt). Asset market clearing: \(B_t+s_t=0\) (endowment) or \(B_t+s_t=K_{t+1}\) (with capital).

Golden rule

The steady-state capital stock that maximizes steady-state consumption: \(F'(K)=\delta\) gross, or \(r=n\) with growth (\(r=0\) without). L15

Part IV: Perpetual youth (Blanchard–Yaari)

\(p\)

Death rate (hazard), not a price; expected lifetime \(1/p\). Cohort born at \(s\) has size \(N(s,t)=pe^{-p(t-s)}\) at \(t\); total population is 1. L17

Actuarially fair annuity

Wealth held with an insurer earns \(r+p\) while alive and is forfeited at death. The insurer breaks even.

\(\theta\)

Pure rate of time preference. The effective discount rate is \(\theta+p\).

\(R(t,z)\)

Price at \(t\) of a good at \(z\) for a surviving agent: \(\exp(-\int_t^z[r(\mu)+p]d\mu)\).

\(h\), \(v\); \(H\), \(V\)

Individual human and non-human wealth; aggregates over cohorts. With log utility, \(c=(\theta+p)(h+v)\) and \(C=(\theta+p)(H+V)\).

\(\alpha\) (BY)

Rate at which labour income declines with age: \(y(s,t)=aY(t)e^{-\alpha(t-s)}\) with \(a=(p+\alpha)/p\). L18

Aggregate system

\(\dot C=(r+\alpha-\theta)C-(p+\alpha)(p+\theta)V\), \(\dot V=rV+Y-C\). Endowment economy: \(r=\theta-\alpha\). L19

\(F(K)\) (BY)

Net output \(F(K,1)-\delta K\), so \(r=F'(K)\) and \(Y=F(K)-F'(K)K\).

Part V: Aggregation and the social planner (Class Note 3)

Cardinal vs ordinal property

An ordinal property survives every strictly increasing transformation of \(u^i\), so it depends only on preferences (and demand). Quasi-concavity is ordinal; concavity is cardinal: \(e^{u}\) of a concave \(u\) need not be concave. L21

Utility possibility set \(U\)

\(\{u\in\mathbb{R}^I: u_i\le u^i(x^i)\ \forall i\) for some feasible allocation\(\}\). It is convex under HH, FF and concave \(u^i\) (assumption CC). PO utility vectors lie on its upper boundary. L21

Downward comprehensive

\(u\in U\) and \(u'\le u\) (component by component) imply \(u'\in U\). This is what forces supporting weights to be non-negative.

Pareto (welfare) weights \(\lambda\)

Non-negative, nonzero \(\lambda\in\mathbb{R}^I_+\) in the planner objective \(\sum_i\lambda_iu^i\). Every PO allocation solves the planner problem for some \(\lambda\) (under HH, CC, FF). With \(\lambda\gg0\), every solution is PO; with zero weights, a solution need not be PO. L21

Planner problem \(W\)

\(\max_{\{x^i,y^j\}}\sum_i\lambda_iu^i(x^i)\) subject to feasibility.

Representative agent / representative utility \(u(x;\lambda)\)

\(\max_{\{x^i\in X^i\}}\sum_i\lambda_iu^i(x^i)\) s.t. \(\sum_ix^i=x\). It is concave in \(x\) when the \(X^i\) are convex and the \(u^i\) concave, and HD1 in \(\lambda\). The weights that reproduce a CE are the Negishi weights. L22

Pure exchange economy

No production: \(Y=-\mathbb{R}^m_+\) (free disposal only), so feasibility is \(\sum_ix^i\le\sum_ie^i\equiv\bar e\).

Shadow value \(\gamma_l\)

The multiplier on feasibility of good \(l\) in the planner problem. It equals the marginal value of one more unit of \(\bar e_l\) to the planner, \(\partial u(\bar e;\lambda)/\partial x_l\). Planner FOC: \(\lambda_iu^i_l=\gamma_l\). It depends on \(\lambda\) in general.

Marginal utility of income \(\mu_i\)

The multiplier on \(i\)'s budget constraint. CE FOC: \(u^i_l=\mu_ip_l\). By the envelope theorem it is the utility gain from one more unit of numeraire income. Richer agents have lower \(\mu_i\). L23

Negishi weights / Negishi method

The planner weights that reproduce a CE: \(\lambda_i=\kappa/\mu_i\), with \(\gamma=\kappa p\). Matching the planner and household FOCs this way proves both welfare theorems (interior, differentiable case). Not \(\lambda_i=\mu_i\). L23

Envy-free allocation

Feasible \(\{x^i\}\) with \(u^i(x^i)\ge u^i(x^j)\) for all \(i,j\). The CE from equal endowments \(e^i=\bar e/I\) is envy-free (identical budget sets); equal endowments are sufficient, not necessary.

Homogeneous of degree \(k\) (HD\(k\))

\(h(\eta x)=\eta^kh(x)\) for all \(\eta>0\). If \(h\) is HD1, its partial derivatives are HD0: \(h_l(\eta x)=h_l(x)\). L24

Euler's theorem

If \(h\) is differentiable and HD1, then \(h(x)=\sum_lx_l\,h_l(x)=\nabla h(x)\cdot x\).

Homothetic

Preferences representable as \(g(h(x))\) with \(g\) strictly increasing and \(h\) HD1. Indifference curves are radial blow-ups of each other, MRS is constant along rays, and Engel curves are rays through the origin.

Gorman aggregation / Gorman form

Equilibrium relative prices (equivalently \(\gamma_l/\gamma_k\)) do not depend on the distribution of wealth or Pareto weights, so a representative agent evaluated at \(\bar e\) prices the economy. It holds for \(u^i=g^i(h(x))\) with common HD1 \(h\): \(p\propto\nabla h(\bar e)\). More generally (Gorman polar form) Engel curves are linear with a slope common to all agents. L24, L25

Subsistence / translation vector \(\theta^i\)

An agent-specific shift of the origin: \(u^i(x)=g(H(x-\theta^i))\). If \(\theta^i_l>0\), good \(l\) is a necessity for \(i\). Aggregation then works on net aggregate endowment \(\bar e-\bar\theta\), with \(\bar\theta=\sum_i\theta^i\). L25

Stone–Geary

\(u^i=\sum_l\beta_l\log(x_l-\theta^i_l)\). It is non-homothetic and non-identical when the \(\theta^i\) differ, yet it aggregates if \(\beta\) is common: \(\gamma_l/\gamma_k=\frac{\beta_l(\bar e_k-\bar\theta_k)}{\beta_k(\bar e_l-\bar\theta_l)}\). It is the \(\sigma=1\) case of translated CES.

Satiation (bliss) point

A bundle beyond which more is not better, e.g. \(\theta^i\) in \(-\sum_l\pi_l(\theta^i_l-x_l)^{1+\sigma}/(1+\sigma)\) on \(x\le\theta^i\). It violates LNS at \(\theta^i\). That family still aggregates (needs \(\bar e\lt\bar\theta\)).

Notation traps across slide decks