The big ideas of economics, with the fine print

Statement

First welfare theorem. If preferences are locally non-satiated, every Walrasian equilibrium allocation is Pareto efficient. Second welfare theorem. If preferences and technologies are convex (and continuous), every Pareto efficient allocation can be supported as a price equilibrium after suitable lump-sum transfers. In a two-person exchange economy the efficient allocations form the contract curve, where marginal rates of substitution agree:

Why it's true

First theorem: suppose $x'$ Pareto-dominates the equilibrium allocation $x^*$. Then whoever is strictly better off must find $x'_i$ unaffordable, $p\cdot x'_i>p\cdot\omega_i$, and by local non-satiation everyone else has $p\cdot x'_i\ge p\cdot\omega_i$. Summing gives $p\cdot\sum x'_i>p\cdot\sum\omega_i$, which contradicts feasibility. The proof uses no convexity at all. The second theorem is a separating-hyperplane argument: with convex preferences, the set of aggregate bundles that could make everyone at least as well off and someone better off is convex and doesn't contain the aggregate endowment. A hyperplane separating the two gives the supporting prices, and transfers put each agent on the right budget line.

Statement

Arrow (1951). With at least three alternatives, any social welfare function $F$ that maps profiles of strict rankings to a social ranking and satisfies unrestricted domain, weak Pareto, and independence of irrelevant alternatives (IIA) is dictatorial. Gibbard–Satterthwaite (1973, 1975). Any onto, non-dictatorial social choice function with three or more outcomes can be manipulated by some voter misreporting.

Why it's true

The Condorcet profile above already shows that pairwise majority isn't transitive. The Arrow proof (Geanakoplos gives three one-page versions) finds a pivotal voter. Move an alternative $b$ from the bottom to the top of each voter's ranking, one voter at a time. By Pareto and IIA, there is a voter whose move flips $b$ from socially last to socially first. IIA then forces that voter to dictate every pairwise comparison not involving $b$, and a second step extends the dictatorship to pairs involving $b$.

Statement

With inverse demand $P=a-Q$ and constant marginal cost $c$, the symmetric Cournot–Nash equilibrium with $n$ firms is

As $n\to\infty$, $P\to c$. Under Bertrand competition with homogeneous goods and no capacity limits, the unique equilibrium for $n\ge2$ is $P=c$.

Why it's true

Each Cournot firm maximizes $(a-q_i-Q_{-i}-c)\,q_i$, giving the best response $q_i=(a-c-Q_{-i})/2$. Imposing symmetry and solving gives the formula. Each firm ignores the price reduction it imposes on rivals' output, so total output exceeds the monopoly level, but each still restricts output a little, which leaves a markup that shrinks with its share. Under Bertrand, any price above cost invites undercutting by a penny to take the whole market, so only $P=c$ survives.

Statement

Nash (1950). Every finite game has an equilibrium in mixed strategies $\sigma\in\prod_i\Delta(S_i)$. In a symmetric population game with payoff matrix $A$, the replicator dynamics are

Every Nash equilibrium is a rest point of these dynamics. Every evolutionarily stable strategy (ESS) is an asymptotically stable rest point. Not every Nash equilibrium is stable.

Why it's true

Existence: the best-response correspondence on the compact convex set $\prod\Delta(S_i)$ is upper hemicontinuous with nonempty convex values, so Kakutani's fixed point theorem applies. For generalized rock–paper–scissors with win payoff $w$, loss payoff $-1$ and tie payoff 0, the interior equilibrium is $(\tfrac13,\tfrac13,\tfrac13)$. The function $V=\sum\log x_i$ satisfies $\dot V=(w-1)\big(3\textstyle\sum x_i^2-1\big)/2$. Since $\sum x_i^2\ge\tfrac13$, this has the sign of $w-1$ away from the centre, and $V$ is largest at the centre. So orbits spiral in when $w>1$, form closed cycles when $w=1$, and spiral out to the boundary when $w<1$.

Statement

Revenue equivalence (Vickrey 1961; Myerson 1981; Riley & Samuelson 1981). Suppose $n$ risk-neutral bidders have values drawn i.i.d. from $F$. Then any auction that allocates to the highest value and gives a bidder with the lowest possible value zero surplus yields the same expected revenue: the expected second-highest value. For $U[0,1]$ values, that is

Why it's true

By the envelope theorem, a bidder's equilibrium expected payoff is $U(v)=U(\underline v)+\int_{\underline v}^vQ(t)\,dt$, where $Q$ is the probability of winning. So payments depend only on the allocation rule and $U(\underline v)$. The same allocation gives the same payments. Myerson's optimal auction maximizes expected virtual value $v-\frac{1-F(v)}{f(v)}$, which for $U[0,1]$ means a reserve price of $\tfrac12$ regardless of $n$.

Statement

In a two-sided market with strict preferences, a matching is stable if no pair would both rather be with each other than with their assigned partners. Gale & Shapley (1962): a stable matching always exists, and the proposer-proposing deferred acceptance (DA) algorithm finds the stable matching that every proposer weakly prefers to all other stable ones, and every receiver likes least. Reporting true preferences is a dominant strategy for proposers (Dubins & Freedman, 1981; Roth, 1982).

Why it's true

Each proposer goes down their list. Each receiver holds the best offer so far and rejects the rest. Offers only move down proposers' lists and held offers only improve for receivers, so the algorithm ends after at most $n^2$ proposals. If $(m,w)$ blocked the result, $m$ would have proposed to $w$ before his final partner and been rejected for someone $w$ preferred, and her partner only improved after that. That's a contradiction. Proposer-optimality follows by showing that no proposer is ever rejected by a receiver who could be his partner in some stable matching.

Statement

The state is $\theta\in\{A,B\}$ with equal prior. Agent $t$ gets a private binary signal that is correct with probability $q\in(\tfrac12,1)$, observes $a_1,\ldots,a_{t-1}$, and chooses the action with the higher posterior. Once the difference between earlier $A$ and $B$ actions that revealed signals reaches 2, every later agent ignores their own signal: a cascade. Cascades start almost surely, and with positive probability on the wrong action.

Why it's true

Before a cascade, each action reveals its taker's signal (ties, where the posterior is exactly $\tfrac12$, are broken by following your own signal). Once revealed $A$ signals lead $B$ signals by 2, one contrary private signal only cuts the lead to 1, which still favours $A$. So everyone copies, actions stop carrying information, and public belief freezes. Agents effectively move in pairs: two matching signals start a cascade, and two different ones return the public belief to the prior. That gives the formula above. Two early wrong signals are enough to lock in error.

Statement

A household with assets $a\ge0$, income $y\in\{y_L,y_H\}$ drawn i.i.d., gross return $R$ and CRRA utility solves

The Bellman operator $T$ is a contraction with modulus $\beta$ in the sup norm (Blackwell's sufficient conditions: monotonicity and discounting). So $\|T^kV_0-V^*\|\le\beta^k\|V_0-V^*\|$.

Why it's true

Monotonicity: if $V\le W$ pointwise then $TV\le TW$. Discounting: $T(V+c)=TV+\beta c$. Together these give $\|TV-TW\|\le\beta\|V-W\|$, and the Banach fixed point theorem does the rest. With income risk and a borrowing limit, the policy shows precautionary saving: since $u'''>0$, uncertainty raises expected marginal utility tomorrow, and households hold a buffer stock even when $\beta R<1$, which wouldn't happen with certain income. Near $a=0$ the constraint binds and consumption tracks income.

Statement

Consider an accelerationist backward-looking model (a stripped-down version of Ball, 1999, and Svensson, 1997):

Inflation is stable iff $\phi_\pi>1$ (for small $\phi_y$). In the forward-looking New Keynesian model the analogous condition, $\phi_\pi+\frac{1-\beta}{\kappa}\phi_y>1$, gives determinacy: a unique bounded equilibrium.

Why it's true

The real-rate gap is $i-\pi-r^*=(\phi_\pi-1)(\pi-\pi^*)+\phi_y y$. If $\phi_\pi<1$, higher inflation lowers the real rate, which raises output, which raises inflation further. The deviation $\pi-\pi^*$ follows a linear recursion whose dominant root lies outside the unit circle. With $\phi_\pi>1$, policy leans against inflation and the root moves inside, as long as the response isn't so aggressive that it overshoots and oscillates.

Statement

Diamond & Dybvig (1983). A fraction $\lambda$ of depositors need to consume at $t=1$. The bank offers $r_1>1$ on early withdrawal, financed by a liquid reserve and a long asset that returns $R>1$ at $t=2$ but only $L<1$ if liquidated at $t=1$. If only impatient depositors withdraw, patient ones get more by waiting. But if patient depositors expect a large fraction $f$ to withdraw, liquidation leaves too little for $t=2$, and withdrawing early is a best response:

Why it's true

Withdrawal decisions are strategic complements: the more others withdraw, the more the bank liquidates at a loss, and the lower the payoff to waiting. Complementarity plus enough of it gives multiple fixed points of the best-response map. Deposit insurance, or suspending convertibility at $f=\lambda$, removes the bad equilibrium at no cost in equilibrium, because nobody runs on an insured bank.

Statement

The TWFE regression $y_{it}=\alpha_i+\lambda_t+\beta D_{it}+\varepsilon_{it}$ with staggered treatment timing gives (Goodman-Bacon, 2021)

a variance-weighted average of all two-group, two-period DiD comparisons, including "forbidden" ones where a cohort treated earlier serves as the control for one treated later. If effects vary over time, these comparisons subtract part of the earlier cohort's treatment effect. The implied weights on group-time effects $\mathrm{ATT}(g,t)$ can be negative (de Chaisemartin & D'Haultfœuille, 2020), so $\hat\beta$ can have the wrong sign even when every $\mathrm{ATT}(g,t)>0$.

Why it's true

By Frisch–Waugh–Lovell, $\hat\beta$ regresses $y$ on the two-way-demeaned treatment $\tilde D_{it}$. For early cohorts in late periods, $\tilde D_{it}$ can be negative: those observations act as controls. Their outcomes contain growing treatment effects, which get subtracted. Heterogeneity-robust estimators (Callaway & Sant'Anna, 2021; Sun & Abraham, 2021; Borusyak, Jaravel & Spiess, 2024) estimate each $\mathrm{ATT}(g,t)$ from clean comparisons with never-treated or not-yet-treated units, then aggregate with non-negative weights.

Statement

Top wealth and income are well described by $\Pr(W>w)\propto w^{-\zeta}$, with $\zeta$ around 1.5 for US wealth. If log wealth follows a random walk with drift $\mu$ and volatility $\sigma$, and each dynasty is reset with probability $p$ per period, the stationary distribution has a Pareto upper tail with

Why it's true

This is Reed's (2001) double-Pareto result. The age of a dynasty is geometric, and conditional on age, log wealth is approximately normal. Mixing normals over an exponential distribution of ages gives exponential tails in log wealth, which are Pareto tails in wealth. More generally, Kesten's (1973) theorem gives $\zeta$ as the root of $\mathbb E[G^\zeta]=1$ for the multiplicative factor $G$, provided there is a reflecting barrier or additive term. Higher return volatility or slower resets fatten the tail (lower $\zeta$).