Objects
- Economy: \(L\) (e.g. \(\mathbb{R}^m\)); households \(i\in I\) with \((X^i,u^i,e^i,\theta^i_j)\); firms \(j\in J\) with \(Y^j\); \(\sum_i\theta^i_j=1\).
- Signs: \(x>0\) purchase / \(x<0\) sale; \(y>0\) output / \(y<0\) input.
- Feasible: \(x^i\in X^i\), \(y^j\in Y^j\), \(\sum_ix^i=\sum_jy^j+\sum_ie^i\).
- CE \((p,\bar x,\bar y)\): \(\bar y^j\in\arg\max_{Y^j}p\,y\); \(\bar x^i\in\arg\max_{X^i}u^i\) s.t. \(p\,x\le p\,e^i+\sum_j\theta^i_j\pi^j\); feasible. L2
- PO: feasible, no feasible \(x\) with \(u^i(x^i)\ge u^i(\bar x^i)\ \forall i\), \(>\) for some. L3
- Walras: binding budgets ⇒ \(p\cdot z(p)=0\) for all \(p\).
Assumptions
| Name | Statement | Used by |
|---|
| LNS | \(\forall x\in X^i,\ \varepsilon>0\ \exists\hat x\in B_\varepsilon(x)\cap X^i:\ u^i(\hat x)>u^i(x)\). Implied by strict monotonicity on \(\mathbb{R}^m_+\); continuity is irrelevant. | 1WT |
| HH | \(X^i\) convex; \(u^i\) continuous; \(u^i(x')>u^i(x)\Rightarrow u^i(\theta x+(1-\theta)x')>u^i(x)\ \forall\theta\in(0,1)\) | 2WT |
| FF | Aggregate \(Y=\sum_jY^j\) convex | 2WT |
| Non-satiated \(h\) | \(\exists h,\tilde x^h:\ u^h(\tilde x^h)>u^h(\bar x^h)\) | 2WT Step 3 |
| Arrow | \(\forall i\ \exists\tilde x^i\in X^i:\ p\,\tilde x^i\lt p\,\bar x^i\) (not a cheapest point) | QE ⇒ CE |
1WT: CE ⇒ PO (under LNS) L4
Suppose a feasible \(\{x,y\}\) Pareto-dominates the CE allocation \(\{\bar x,\bar y\}\).
- \(u^i(x^i)\ge u^i(\bar x^i)\Rightarrow p\,x^i\ge p\,\bar x^i\). If cheaper, LNS gives a better affordable bundle nearby. (1)
- \(u^{i'}(x^{i'})>u^{i'}(\bar x^{i'})\Rightarrow p\,x^{i'}>p\,\bar x^{i'}\). Else it was affordable. (2)
- Sum: \(p\sum_ix^i>p\sum_i\bar x^i\). (3)
- Profit max: \(p\sum_j\bar y^j\ge p\sum_jy^j\). (4)
- Feasibility of both: \(p\sum x^i-p\sum\bar x^i=p\sum y^j-p\sum\bar y^j\). LHS \(>0\ge\) RHS. Contradiction. ∎
When the 1WT breaks L5
| Modification | 1WT? | Which step / why |
|---|
| No LNS (thick ICs, bliss) | Fails | (1): weakly preferred can be strictly cheaper |
| Convex \(u\), non-convex \(Y\), IRS | Holds | Not used; IRS threatens existence of CE |
| One price-taking firm | Holds | \(|J|\) never used. Monopoly pricing isn't a CE. |
| Consumption externality | Fails | (1)–(2): optimality holds others' \(\bar x\) fixed |
| Production externality | Fails | (4): \(Y^{j'}(y^j)\) differs from \(Y^{j'}(\bar y^j)\) |
| Incomplete markets | Fails (unconstrained) | (5): untraded goods have no price. Constrained PO in the fixed-endowment case. |
| Heterogeneous / wrong beliefs | Holds | PO w.r.t. agents' own beliefs (beliefs live in \(u^i\)) |
| Lump-sum taxes | Holds | Only endowment values change |
| Distorting taxes | Fails | Agents face different prices: MRS ≠ MRT |
| Infinitely many agents and goods (OLG) | Can fail | (3): \(p\sum\bar x^i=\infty\), so "∞ > ∞" L12 |
2WT: PO ⇒ quasi-equilibrium (under HH, FF, non-satiated \(h\)) L6
\(\exists p\ne0\): (i) \(p\,\bar y^j\ge p\,y\ \forall y\in Y^j\); (ii) \(\bar x^i\in\arg\min\{p\,x: x\in X^i,\ u^i(x)\ge u^i(\bar x^i)\}\).
- \(A=\sum_i\{x\in X^i:u^i(x)\ge u^i(\bar x^i)\}\); \(B=Y+\sum_ie^i\).
- Convex: HH (set-sum of convex sets) and FF.
- \(A\cap\operatorname{Int}B=\emptyset\): else mix \(h\)'s bundle toward \(\tilde x^h\). It stays in \(B\) (interior), \(h\) strictly gains, and that contradicts PO.
- Minkowski (needs \(\operatorname{Int}B\ne\emptyset\), e.g. free disposal): \(p\ne0\), \(p\,x\ge p\,z\ \forall x\in A,z\in B\). (6)
- \(\bar x=\sum\bar x^i\in A\), \(\bar z=\sum\bar y^j+\sum e^i\in B\), \(\bar x=\bar z\) ⇒ \(p\bar x=\min_Ap\,x=\max_Bp\,z\). (7)
- Firm \(j\) with \(p\hat y>p\bar y^j\): \(\hat z=\hat y+\sum_{k\ne j}\bar y^k+\sum e^i\in B\) has \(p\hat z>p\bar x\). Contradicts (6).
- Household \(i\) with \(u^i(\hat x)\ge u^i(\bar x^i)\), \(p\hat x\lt p\bar x^i\): \(\hat a=\hat x+\sum_{k\ne i}\bar x^k\in A\) has \(p\hat a\lt p\bar z\). Contradicts (6). ∎
Monotone \(u^i\) ⇒ \(p\ge0\): otherwise \(\bar x+te_k\in A\) and \(p\cdot(\bar x+te_k)\to-\infty\).
Arrow's remark: QE ⇒ CE L7
Suppose \(\hat x\) with \(p\hat x\le p\bar x^i\), \(u(\hat x)>u(\bar x^i)\). Let \(x^\theta=\theta\tilde x^i+(1-\theta)\hat x\).
- Continuity, \(\theta\) small: \(u(x^\theta)>u(\bar x^i)\).
- \(p\,x^\theta\lt \theta p\bar x^i+(1-\theta)p\bar x^i=p\bar x^i\).
- This contradicts expenditure minimization. ∎
Exceptional case
\(u=\sqrt{x_1}+x_2\), \(e=(0,1)\). The IC is vertical at \(e\), so \(p=(1,0)\) and \(e\) is the cheapest point. There is a quasi-equilibrium but no CE: \(x_1^d=(p_2/2p_1)^2>0\) whenever \(p_2>0\).
Rule it out: \(X^i=\mathbb{R}^m_+\), monotone (so \(p\ge0\)), \(\bar x^i\gg0\) ⇒ \(p\bar x^i>0=p\cdot0\).
Decentralization
- Need wealth \(=p\,\bar x^i\): \(p\,e^i+\sum_j\theta^i_j\pi^j=p\,\bar x^i\). One choice: any \(\theta\), with \(e^i=\bar x^i-\sum_j\theta^i_j\bar y^j\).
- Not unique. Equivalently, zero-sum lump-sum transfers \(T^i=p\bar x^i-p\hat e^i-\sum_j\hat\theta^i_j\pi^j\).
Non-convex failures of the 2WT
| Example | What breaks |
|---|
| \(u^A=x_1^2+x_2^2\), \(u^B=\min\{x^B_1,x^B_2\}\); \((t,t)\) is PO | A's demand on any line is an endpoint, never \((t,t)\) |
| Crusoe, \(c=n^\gamma\), \(\gamma>1\) | Firm earns \(\bar\pi=\bar n^\gamma(1-\gamma)<0\) at the tangency price |
| Fixed cost: \(c=(n-n_0)^\gamma\), \(u=\log c+b\log\ell\) | \(\bar n=\frac{\gamma+bn_0}{\gamma+b}\); fails iff \(n_0>\frac{1-\gamma}{1+b}\) (firm shuts down) |
Numbers worth remembering
- Cobb–Douglas box, \(p_2=1\): \(p_1^*=\frac{ae_2+b(1-e_2)}{1-ae_1-b(1-e_1)}\). Contract curve \(x_2=\frac{k_bx_1}{k_a(1-x_1)+k_bx_1}\) with \(k=\frac{a}{1-a}\).
- Crusoe \(c=n^\gamma\), \(u=\log c+b\log\ell\): \(\bar n=\gamma/(\gamma+b)\), \(w=\gamma\bar n^{\gamma-1}\). For \(\gamma=\tfrac12\), \(b=1\): \(\bar n=\tfrac13\), \(w=\tfrac{\sqrt3}2\).
- Pigou: \(u^1=\log s+m\), \(u^2=m-\gamma s\): \(s^{CE}=1\), \(s^{PO}=\frac1{1+\gamma}\), DWL \(=\gamma-\log(1+\gamma)\), tax \(t=\gamma\).