Macro is all about vibes. If you can vibe the math, that’d be super cool.
Consider the following economy:
Denote as $L = \mathbb R^m$ the commodity space.
Denote as $i = 1, 2, \ldots I$ the $I$ agents. Each agent can consume any vector $X^i\subset L$, and has a utility function $u^i : X^i \mapsto \mathbb R$. Each agent has endowment $e^i\in L, \forall i$.
Denote as $j = 1, 2, \ldots, J$ the $J$ firms. Each firm can produce vectors $Y^j\subset L $.
Let consumer $i$ own a fraction $\theta^i_j \ge 0$ of firm $j$. Assume $\sum_{i\in [I]} \theta_j^i = 1, \forall j\in [J]$.
Definition A competitive equilibrium is a price vector $p\in \mathbb R^m$ and an allocation $\lbrace x^i, y^j\rbrace$ that satisfies
Each firm maximizes its profit: $$ y^j = \arg \max_{y^j \in Y^j} p\cdot y $$
Each consumer maximizes her utility subject to her budget constraint: $$ \begin{align*} & x^i = \arg\max_{x\in X^i}\ u^i(x)\ & \text{s.t. } p\cdot x\le p\cdot e^i + \sum_{j\in [J]}\theta_j^i(p \cdot y_j) \end{align*} $$
And the allocation is feasible (i.e. the market clears) $$ \sum_{i\in [I]}x^i = \sum_{j\in [J]}y^j + \sum_{i\in [I]}e^i. $$
Definition An allocation $\lbrace x^i, y^j\rbrace$ is said to be feasible if $x^i\in X^i, \forall i$, $y^j\in Y^j, \forall j$, and $$ \sum_{i\in [I]}x^i = \sum_{j\in [J]}y^j + \sum_{i\in [I]}e^i. $$
Definition A feasible allocation $\lbrace x^i, y^j\rbrace$ is Pareto Optimal if there is no other feasible allocation preferred by everybody, i.e. one such that $$ \begin{align*} u^i(x^i) \ge u^i(\bar x^i), \forall i\in [I],\ u^i(x^i) > u^i(\bar x^i), \exists i\in [I].\ \end{align*} $$
Definition (Local Non-Satiation): A preference relation $\succsim$ on $X^i$ satisfies local non-satiation if, for every bundle $x \in X^i$ and every $\epsilon > 0$, there exists some $x’ \in X^i$ with $\lVert x’ - x \rVert < \epsilon$ such that $x’ \succ x$.
Theorem (First Welfare Theorem).
If every all agent’s preferences are locally non-satiated, then every competitive equilibrium allocation is Pareto Optimal.
Proof (sketch).
Let $(p, {x^i}, {y^j})$ be a competitive equilibrium.
- Each firm $j$ maximizes profits given $p$.
- Each consumer $i$ maximizes utility subject to $p \cdot x^i \leq p \cdot e^i + \sum_j \theta_j^i (p \cdot y^j)$.
- Feasibility: $\sum_i x^i = \sum_j y^j + \sum_i e^i$.
Suppose, for contradiction, that this allocation is not Pareto Optimal. Then there exists another feasible allocation ${\bar x^i, \bar y^j}$ such that:
- $u^i(\bar x^i) \ge u^i(x^i)$ for all $i$,
- and $u^k(\bar x^k) > u^k(x^k)$ for some $k$.
Local non-satiation implies that each consumer $i$ must be spending their entire budget in equilibrium: $$ p \cdot x^i = p \cdot e^i + \sum_j \theta_j^i (p \cdot y^j). $$
Summing across all consumers: $$ \sum_i p \cdot x^i = \sum_i p \cdot e^i + \sum_j p \cdot y^j. $$ Which implies: $$ p \cdot \Big(\sum_i x^i - \sum_j y^j - \sum_i e^i\Big) = 0. $$ By feasibility, this holds.
Now consider the alternative allocation ${\bar x^i, \bar y^j}$. If it strictly improves some consumer’s utility without lowering anyone’s, then by local non-satiation, it must require strictly more expenditure at prices $p$: $$ \sum_i p \cdot \bar x^i > \sum_i p \cdot x^i. $$ But feasibility and firm profit maximization imply: $$ \sum_i p \cdot \bar x^i \le \sum_i p \cdot e^i + \sum_j p \cdot \bar y^j \le \sum_i p \cdot x^i, $$ a contradiction.
Hence, no such feasible Pareto improvement exists. The equilibrium allocation is Pareto Optimal.
Questions to Think About
The first welfare theorem is elegant, but its assumptions hide a lot of subtlety. Here are some questions (from class notes) that push on its limits:
Externalities in Consumption: Suppose the consumption of agent $i$ directly affects the utility of agent $i’$. In a competitive equilibrium, $i’$ takes $i$’s consumption as given. Does the theorem still hold?
Externalities in Production: Suppose the production of firm $j$ directly shifts the production possibilities of firm $j’$. In a competitive equilibrium, $j’$ takes $j$’s plan as given. Does the theorem still hold?
Incomplete Markets: If there are three goods but markets exist for only two, does the theorem still apply? What is the right notion of efficiency to compare equilibrium allocations with?
Disagreement about the world: In classical models, agents are assumed to “see the world” the same way. What happens if people disagree about probabilities, technologies, or even basic facts? Does efficiency require common beliefs?
Intertemporal trade-offs: How should we think about budget constraints that span across time? What does the equilibrium notion imply for saving and borrowing decisions?
Search frictions: In practice, finding jobs, housing, or trading partners takes effort and time. How would adding such frictions alter the efficiency conclusion?
Money and transactions: The model assumes goods-for-goods trade, but real economies use money. If we give money a role—as a unit of account, medium of exchange, or store of value—what changes in the welfare story?
Nominal contracts: What if contracts are written in money terms rather than real goods? Does this institutional detail affect the efficiency properties of equilibrium?