Equilibrium in an Exchange Economy
Microeconomics for Finance
Juan F. Imbet
Master in Finance, 1st Year - Paris Dauphine University - PSL
Overview and Notation
- We study an exchange economy with \(N \ge 2\) agents (individuals) and \(L \ge 2\) goods.
- Each agent \(i\) has:
- A consumption bundle: \(x_i = (x_{i1},\dots,x_{iL}) \in \mathbb{R}^L_{+}\).
- An initial endowment: \(\omega_i = (\omega_{i1},\dots,\omega_{iL}) \in \mathbb{R}^L_{+}\).
- The total resources in the economy: \(\Omega = \sum_{i=1}^N \omega_i\).
- Goods have prices \(p = (p_1,\dots,p_L) \in \mathbb{R}^L_{+}\) (later we will normalize).
Consumption Sets and Budgets
- Each agent can only consume non-negative quantities: \(X_i = \mathbb{R}^L_{+}\).
- At given prices \(p\), the budget set is: \[
B_i(p) = \{x_i \in X_i : p \cdot x_i \le p \cdot \omega_i\}.
\]
- The value of agent \(i\)’s endowment is \(w_i(p) = p \cdot \omega_i\) (wealth).
- A feasible allocation: \((x_1,\dots,x_N)\) such that \(\sum_i x_i \le \Omega\).
- At equilibrium we require market clearing: \(\sum_i x_i = \Omega\). (Inequalities are componentwise.)
Preferences and Utility
- Each agent has preferences \(\succsim_i\) over \(X_i\).
- We assume:
- Completeness: for any two bundles \(x,y \in X_i\), the agent can compare them: \(x \succsim_i y\) or \(y \succsim_i x\) (or both, i.e., indifference). No incomparable pairs.
- Transitivity: rankings are consistent over chains: if \(x \succsim_i y\) and \(y \succsim_i z\), then \(x \succsim_i z\). Prevents preference cycles.
- Continuity: small changes in bundles do not cause jumps in ranking. Formally, the upper and lower contour sets \(\{x: x \succsim_i y\}\) and \(\{x: y \succsim_i x\}\) are closed for each \(y\).
- Local non-satiation (LNS): around every bundle \(x\) and for every \(\varepsilon>0\), there exists \(x'\) with \(\|x'-x\|<\varepsilon\) such that \(x' \succ_i x\). Intuition: “more is (locally) better,” which implies budgets bind at optima.
- Convexity: if \(x \succsim_i y\), then for any \(\theta\in[0,1]\), \(\theta x + (1-\theta)y \succsim_i y\); strictly convex if the mixture is strictly preferred when \(x\ne y\). Intuition: preference for diversification (“averages at least as good as extremes”).
- Under these assumptions (plus mild technical conditions), preferences admit a utility representation \(u_i: X_i \to \mathbb{R}\) with \(x \succsim_i y\) iff \(u_i(x)\ge u_i(y)\).
Utility Examples
- Cobb–Douglas:
\(u_i(x_i) = \prod_{\ell=1}^L x_{i\ell}^{\alpha_{i\ell}}, \quad \alpha_{i\ell} > 0.\)
- CES:
\(u_i(x_i) = \Big(\sum_{\ell=1}^L a_{i\ell} x_{i\ell}^\rho\Big)^{1/\rho}, \quad a_{i\ell}>0.\)
- Quasilinear:
\(u_i(x_i) = x_{i1} + v_i(x_{i2},\dots,x_{iL}).\)
- If utilities are strictly concave and monotone, demand is unique (often interior; corners can still occur with non-negativity constraints).
Allocations and Feasibility
- An allocation: \(x = (x_1,\dots,x_N)\) with \(x_i \in X_i\).
- Feasible if \(\sum_i x_i \le \Omega\) (componentwise).
- With monotone preferences, efficient allocations satisfy \(\sum_i x_i = \Omega\) (no waste).
- We only study pure exchange (no production).
Pareto Efficiency
- Let \(x = (x_1,\dots,x_N)\) and \(y = (y_1,\dots,y_N)\) be two feasible allocations.
- \(x\) Pareto dominates \(y\) if:
- \(u_i(x_i) \ge u_i(y_i)\) for all \(i\),
- and strict inequality holds for at least one agent.
- An allocation is Pareto efficient if no feasible allocation dominates it.
- Efficient allocations usually form a “frontier.”
Planner’s Problem
- One way to find Pareto efficient allocations is to solve:
\[
\max \sum_{i=1}^N \lambda_i u_i(x_i) \quad \text{s.t. } \sum_i x_i \le \Omega \; (\text{componentwise}),
\] with weights \(\lambda_i > 0\).
- First-order conditions imply marginal rates of substitution (MRS) are equal across agents.
Marginal Rates of Substitution (MRS)
- Definition:
\[
\text{MRS}_{i,\ell k} = \frac{\partial u_i/\partial x_{i\ell}}{\partial u_i/\partial x_{ik}}.
\]
- Individual problem: maximize \(u_i(x_i)\) s.t. \(p\cdot x_i \le w_i(p) = p\cdot \omega_i\)
(with monotonicity/LNS, the budget binds at the optimum: \(p\cdot x_i = w_i(p)\)).
- Lagrangian (using the binding constraint):
\[
\mathcal{L}(x_i,\lambda) = u_i(x_i) - \lambda\,(p\cdot x_i - w_i(p)).
\]
- FOCs give: \(\partial u_i/\partial x_{im} = \lambda p_m\).
So for goods \(\ell\) and \(k\):
\[
\frac{\partial u_i/\partial x_{i\ell}}{\partial u_i/\partial x_{ik}} = \frac{p_\ell}{p_k}.
\]
- At an interior optimum: \(\text{MRS}_{i,\ell k} = p_\ell/p_k\).
Individual Choice Problem
- At prices \(p\), agent \(i\) solves:
\[
\max_{x_i \in B_i(p)} u_i(x_i).
\]
- If preferences are monotone:
\[
p \cdot x_i(p) = p \cdot \omega_i.
\]
- Interior optimum condition:
\[
\frac{\partial u_i/\partial x_{i\ell}}{\partial u_i/\partial x_{ik}} = \frac{p_\ell}{p_k}.
\]
Marshallian (Uncompensated) Demand: What and Why
- Definition: The Marshallian demand of agent i, \(x_i(p,w_i)\), gives the utility-maximizing bundle as a function of prices \(p\) and wealth (income) \(w_i\) subject to the budget constraint \(p\cdot x_i \le w_i\).
- “Uncompensated” means income is held fixed; changes in prices affect both relative prices and real purchasing power.
- Name: “Marshallian” honors Alfred Marshall, who formalized demand as a function of prices and income in the partial-equilibrium tradition.
Marshallian Demand Properties
- Homogeneous of degree zero: scaling all prices and wealth doesn’t affect demand.
- Budget exhaustion (with LNS/monotonicity): \(p \cdot x_i(p) = p \cdot \omega_i\).
- Individual Walras’ law (net demand): \(p \cdot z_i(p) = p \cdot (x_i(p) - \omega_i) = 0\).
- If utilities are strictly concave and monotone: demand is unique.
- Corner solutions possible if some prices are very high or endowments zero.
Excess Demand and Market Clearing
- Individual net demand: \(z_i(p) = x_i(p) - \omega_i\).
- Aggregate excess demand: \(z(p) = \sum_i z_i(p) = \sum_i x_i(p) - \Omega\).
- Market clearing requires \(z(p^*) = 0\).
- An equilibrium is a pair \((p^*, x^*)\) such that:
- Each \(x_i^*\) solves the individual problem,
- \(\sum_i x_i^* = \Omega\).
Equilibrium Properties
- Walras’ Law: for any \(p\),
\[
p \cdot z(p) = 0.
\]
- Prices are defined up to scale (normalize: \(\sum_\ell p_\ell=1\)).
First Welfare Theorem
- Any competitive equilibrium is Pareto efficient.
- Idea: if an allocation could be improved, someone would demand it, contradicting optimal choice and market clearing.
Second Welfare Theorem
- Any Pareto efficient allocation can be supported by some prices and transfers.
- Consider you want to implement a specific Pareto efficient allocation \(x^*\).
- You can find prices \(p\) and transfers \(T_i\) such that: \[
p \cdot x_i^* + T_i = w_i(p) \quad \forall i.
\]
Cobb–Douglas Preferences: Setup
- Agent i’s utility:
\[
u_i(x_i) = \prod_{\ell=1}^L x_{i\ell}^{\alpha_{i\ell}}, \qquad \alpha_{i\ell} > 0,\; \sum_{\ell} \alpha_{i\ell} = 1\; (\text{without loss of generality}).
\]
- Equivalent formulation (monotone transform):
\[
\max \sum_{\ell=1}^L \alpha_{i\ell} \log x_{i\ell} \quad \text{s.t.} \quad p\cdot x_i \le w_i(p) = p\cdot \omega_i,\; x_i \ge 0.
\]
- With LNS/monotonicity the budget binds: \(p\cdot x_i = w_i(p)\).
Cobb–Douglas: Lagrangian and FOCs
- Lagrangian (binding budget):
\[
\mathcal{L}(x_i,\lambda_i) = \sum_{\ell} \alpha_{i\ell}\, \log x_{i\ell} - \lambda_i\, (p\cdot x_i - w_i(p)).
\]
- First-order conditions (interior solution since \(\alpha_{i\ell}>0\)):
\[
\frac{\partial \mathcal{L}}{\partial x_{i\ell}} = \frac{\alpha_{i\ell}}{x_{i\ell}} - \lambda_i p_\ell = 0
\quad\Rightarrow\quad
x_{i\ell} = \frac{\alpha_{i\ell}}{\lambda_i p_\ell}.
\]
- Budget:
\[
p\cdot x_i = \sum_{\ell} p_\ell\, x_{i\ell} = \sum_{\ell} p_\ell\, \frac{\alpha_{i\ell}}{\lambda_i p_\ell} = \frac{1}{\lambda_i} \sum_{\ell} \alpha_{i\ell} = \frac{1}{\lambda_i} = w_i(p).
\]
- Hence \(\lambda_i = 1 / w_i(p)\).
Cobb–Douglas: Marshallian Demand
- Substituting \(\lambda_i\) back:
\[
\boxed{\; x_{i\ell}(p) = \alpha_{i\ell}\, \frac{w_i(p)}{p_\ell} = \alpha_{i\ell}\, \frac{p\cdot \omega_i}{p_\ell} \;}
\]
- Properties:
- Homothetic: proportional to wealth \(w_i\).
- Constant expenditure shares: \(\tfrac{p_\ell x_{i\ell}}{w_i} = \alpha_{i\ell}\).
- Interior for \(\alpha_{i\ell}>0\), \(p_\ell>0\), \(w_i>0\).
The Hicksian (Compensated) Demand.
- Every convex optimization problem has a dual problem.
- Original: Choose consumption to maximize utility subject to a budget constraint (Marshallian demand).
- Dual: Choose consumption to minimize expenditure subject to achieving a target utility level (Hicksian demand).
- General case
\[
\min_{x_i \ge 0} p \cdot x_i \quad \text{s.t.} \quad u_i(x_i) \ge \bar u.
\]
Lagrangian:
\[
\mathcal{L}(x_i, \mu) = p \cdot x_i + \mu (\bar u - u_i(x_i))
\]
F.O.C.
\[
\frac{\partial \mathcal{L}}{\partial x_{i\ell}} = p_\ell - \mu \frac{\partial u_i}{\partial x_{i\ell}} = 0 \quad \Rightarrow \quad \frac{\partial u_i}{\partial x_{i\ell}} = \frac{p_\ell}{\mu}.
\]
Cobb–Douglas: Hicksian Demand and Expenditure
- Expenditure minimization for target utility \(\bar u\):
\[
\min_{x_i\ge 0} \; p\cdot x_i \quad \text{s.t.} \quad \prod_{\ell=1}^L x_{i\ell}^{\alpha_{i\ell}} \ge \bar u,\; \sum_{\ell} \alpha_{i\ell}=1.
\]
- Equivalent log form constraint: \(\sum_{\ell} \alpha_{i\ell} \log x_{i\ell} \ge \log \bar u\).
- Lagrangian (with multiplier \(\mu\)):
\[
\mathcal{L}(x_i,\mu) = p\cdot x_i + \mu\, \Big(\log \bar u - \sum_{\ell} \alpha_{i\ell} \log x_{i\ell}\Big).
\]
- FOCs: \(\dfrac{\partial \mathcal{L}}{\partial x_{i\ell}} = p_\ell - \mu\, \dfrac{\alpha_{i\ell}}{x_{i\ell}} = 0 \Rightarrow x_{i\ell} = \dfrac{\mu\, \alpha_{i\ell}}{p_\ell}\).
Cobb–Douglas: Hicksian Demand and Expenditure (cont.)
- Plug into the constraint to determine \(\mu\): \[
\sum_{\ell} \alpha_{i\ell} \log\left(\frac{\mu\, \alpha_{i\ell}}{p_\ell}\right) = \log \bar u
\; \Rightarrow \;
\log \mu + \sum_{\ell} \alpha_{i\ell} (\log \alpha_{i\ell} - \log p_\ell) = \log \bar u.
\]
- Hence
\[
\mu = \bar u\; \exp\!\left( -\sum_{\ell} \alpha_{i\ell} (\log \alpha_{i\ell} - \log p_\ell) \right)
= \bar u\; \frac{\prod_{\ell} p_\ell^{\alpha_{i\ell}}}{\prod_{\ell} \alpha_{i\ell}^{\alpha_{i\ell}}}.
\]
- Hicksian demand:
\[
\boxed{\; h_{i\ell}(p,\bar u) = \alpha_{i\ell}\, \frac{\mu}{p_\ell} = \alpha_{i\ell}\, \bar u\, \frac{\prod_{m} p_m^{\alpha_{im}}}{\alpha_{i\ell}\, p_\ell\, \prod_{m} \alpha_{im}^{\alpha_{im}}} = \bar u\; \frac{\prod_{m} p_m^{\alpha_{im}}}{p_\ell\, \prod_{m} \alpha_{im}^{\alpha_{im}}}\; \alpha_{i\ell}.\;}
\] A cleaner standard form groups constants:
\[
h_{i\ell}(p,\bar u) = \alpha_{i\ell}\, \bar u\, \frac{\prod_m p_m^{\alpha_{im}}}{p_\ell} \cdot C_i, \quad \text{with } C_i = \left(\prod_m \alpha_{im}^{\alpha_{im}}\right)^{-1}.
\]
Continuation: Hicksian Demand and Expenditure
- Expenditure function (Shephard’s lemma): \[
e_i(p,\bar u) = p\cdot h_i(p,\bar u) = \bar u\, C_i\, \prod_{m} p_m^{\alpha_{im}}.
\]
- Check (duality): Marshallian demand satisfies \(x_{i\ell}(p,w) = h_{i\ell}\big(p, u_i(x_i(p,w))\big)\) and \(e_i(p, u_i(x_i)) = p\cdot x_i\).
Cobb–Douglas: Aggregation and Clearing
Aggregate demand for good \(\ell\):
\[
X_\ell(p) = \sum_i x_{i\ell}(p) = \frac{1}{p_\ell} \sum_i \alpha_{i\ell}\, w_i(p), \qquad w_i(p) = p\cdot \omega_i.
\]
Market clearing for each \(\ell\): \[
\boxed{\; p_\ell\, \Omega_\ell = \sum_i \alpha_{i\ell}\, w_i(p) \;}
\] Note: RHS depends on prices via \(w_i(p)\).
- The equilibrium is then defined as
- a price vector \(p^*\) satisfying the above for all \(\ell\),
- and the associated demands \(x_i(p^*)\) clearing the markets.
- How to find \(p^*\)? System of \(L\) equations in \(L\) unknowns (up to scale).
Equilibrium with 2 agents and two goods.
Assume two agents with Cobb-Douglas preferences with parameters \(\alpha_1\) and \(\alpha_2\) for good 1 and \(1-\alpha_1\) and \(1-\alpha_2\) for good 2. Initial endowments are \(\omega_1=(\omega_{11},\omega_{12})\) and \(\omega_2=(\omega_{21},\omega_{22})\).
- Individual demands:
\[
x_{11}(p) = \alpha_1 \frac{w_1(p)}{p_1}, \quad x_{12}(p) = (1-\alpha_1) \frac{w_1(p)}{p_2},
\] \[
x_{21}(p) = \alpha_2 \frac{w_2(p)}{p_1}, \quad x_{22}(p) = (1-\alpha_2) \frac{w_2(p)}{p_2}.
\]
- Market clearing:
\[
\begin{align}
\omega_{11} + \omega_{21} &= x_{11}(p) + x_{21}(p) = \alpha_1 \frac{w_1(p)}{p_1} + \alpha_2 \frac{w_2(p)}{p_1}, \\[6pt]
\omega_{12} + \omega_{22} &= x_{12}(p) + x_{22}(p) = (1-\alpha_1) \frac{w_1(p)}{p_2} + (1-\alpha_2) \frac{w_2(p)}{p_2}.
\end{align}
\]
Continuation
- Normalize prices \(p_1 + p_2 = 1\) to solve for \(p^*\). Only one equation is independent (Walras’ law).
- Substitute \(w_i(p) = p_1 \omega_{i1} + p_2 \omega_{i2}\) and solve for \(p_1^*\):
\[
\begin{align}
\omega_{11} + \omega_{21} &= \alpha_1 \frac{p_1\omega_{11} + p_2\omega_{12}}{p_1} + \alpha_2 \frac{p_1\omega_{21} + p_2\omega_{22}}{p_1},\\
&= \frac{\alpha_1\big(p_1\omega_{11} + (1-p_1)\omega_{12}\big) + \alpha_2\big(p_1\omega_{21} + (1-p_1)\omega_{22}\big)}{p_1},\\
p_1(\omega_{11} + \omega_{21}) &= p_1\Big(\alpha_1(\omega_{11}-\omega_{12}) + \alpha_2(\omega_{21}-\omega_{22})\Big) + (\alpha_1 \omega_{12} + \alpha_2 \omega_{22}),\\
p_1(\omega_{11} + \omega_{21}) - p_1\Big(\alpha_1(\omega_{11}-\omega_{12}) + \alpha_2(\omega_{21}-\omega_{22})\Big) &= (\alpha_1 \omega_{12} + \alpha_2 \omega_{22}),\\
p_1 &= \frac{\alpha_1 \omega_{12} + \alpha_2 \omega_{22}}{(\omega_{11} + \omega_{21}) - \Big(\alpha_1(\omega_{11}-\omega_{12}) + \alpha_2(\omega_{21}-\omega_{22})\Big)}\\
p_2 &= 1 - p_1.
\end{align}
\]