Lesson 21 showed that, with concave utilities, every Pareto optimum solves a weighted planner problem $$W:\quad \max_{\{x^i\}}\ \sum_{i\in I}\lambda_i\,u^i(x^i)\quad\text{s.t.}\quad \sum_{i\in I}x^i\le\bar e,$$ and Lesson 22 wrote its first-order conditions for an interior solution with \(\lambda\gg0\) CN3 p.16: $$\lambda_i\,\frac{\partial u^i(\bar x^i)}{\partial x_l}=\gamma_l\qquad\text{for all goods }l\text{ and agents }i. \tag{2}$$ Two consequences: every agent's MRS equals \(\gamma_l/\gamma_k\) (the weights drop out), and the weights fix levels: \(u^i_l/u^j_l=\lambda_j/\lambda_i\). This lesson does the same exercise for the market and lines the two up. The punchline is one sentence: a competitive equilibrium is a planner's optimum in which each agent's weight is the inverse of her marginal utility of income, and the shadow values are the prices.
In a pure exchange economy, household \(i\) solves $$\max_{x\in X^i}\ u^i(x)\quad\text{s.t.}\quad p\cdot x\le p\cdot e^i .$$ Under assumptions HH and CC (convex \(X^i\), concave \(u^i\)), an interior solution \(\bar x^i\) is characterized by CN3 p.18 $$\frac{\partial u^i(\bar x^i)}{\partial x_l}=\mu_i\,p_l\qquad\text{for all }l, \tag{3}$$ together with \(\mu_i\ge0\) and complementary slackness \(\mu_i\,(p\cdot e^i-p\cdot\bar x^i)=0\).
Let \(V^i(p,w)=\max\{u^i(x): p\cdot x\le w\}\) be the indirect utility function and \(x^*(p,w)\) the (differentiable, interior) demand. Then \(\displaystyle\frac{\partial V^i(p,w)}{\partial w}=\mu_i\). Moreover, if \(u^i\) is concave then \(V^i\) is concave in \(w\), so \(\mu_i\) is non-increasing in wealth.
Divide (3) by \(p_l\): \(u^i_l/p_l=\mu_i\) for every good. At an optimum, the last dollar spent on any good buys the same utility, and that common "utility per dollar" is \(\mu_i\). It is agent-specific but not good-specific, because each agent faces one budget. A rich agent with concave utility is already consuming a lot, so her last dollar buys little extra utility: low \(\mu_i\).
Replace \(u^i\) by \(2u^i\): preferences, demands and the equilibrium are unchanged, but every \(u^i_l\) doubles, so \(\mu_i\) doubles and the weight \(1/\mu_i\) halves. Pareto weights are only meaningful relative to a chosen utility representation. Comparing \(\mu_i\) across agents ("the rich have low \(\mu\)") is unambiguous only when agents share the same \(u\); otherwise the comparison depends on how each \(u^i\) is scaled. (Lesson 21 showed that concavity itself is a cardinal property.)
Dividing (3) for goods \(l\) and \(k\), and comparing two agents for the same good CN3 p.19:
| Planner problem \(W\) | Competitive equilibrium | |
|---|---|---|
| FOC | \(\lambda_i\,u^i_l=\gamma_l\) | \(u^i_l=\mu_i\,p_l\) |
| Efficiency (MRS) | \(\dfrac{u^i_l}{u^i_k}=\dfrac{\gamma_l}{\gamma_k}\) for all \(i\) | \(\dfrac{u^i_l}{u^i_k}=\dfrac{p_l}{p_k}\) for all \(i\) |
| Distribution (levels) | \(\dfrac{u^i_l}{u^j_l}=\dfrac{\lambda_j}{\lambda_i}\) | \(\dfrac{u^i_l}{u^j_l}=\dfrac{\mu_i}{\mu_j}\) |
| What sets the levels | the weights \(\lambda\) | the wealth distribution \(\{p\cdot e^i\}\), through \(\mu\) |
The MRS rows contain neither \(\lambda\) nor \(\mu\): both problems equalize marginal rates of substitution, which is the "efficiency" part. The level rows say the two problems pick the same point on the contract curve exactly when \(\lambda_j/\lambda_i=\mu_i/\mu_j\), i.e. \(\lambda_i\propto1/\mu_i\). (A circulating review sheet prints the planner's distribution row as \(\lambda_i/\lambda_j\); that is upside down. From (2), \(u^i_l=\gamma_l/\lambda_i\), so a higher weight means a lower marginal utility.)
In a pure exchange economy with HH and CC, let \((p,\{\bar x^i\})\) be a CE in which each \(\bar x^i\) is interior and each \(u^i\) is differentiable and strictly increasing. Then \(\{\bar x^i\}\) solves the planner problem \(W\) with weights and multipliers $$\lambda_i=\frac{\kappa}{\mu_i},\qquad \gamma_l=\kappa\,p_l,\qquad\text{for any }\kappa>0,$$ and is therefore Pareto optimal.
The constant \(\kappa\) reflects two harmless normalizations: prices are only defined up to scale (Lesson 2), and so is the planner's objective. A convenient choice is \(\kappa\) such that \(\sum_i\lambda_i=1\).
| General 1WT (Lesson 4) | 1WT via FOCs (this lesson) | |
|---|---|---|
| Preferences | local non-satiation only | differentiable, strictly increasing, concave (CC) |
| Consumption sets | anything | convex (HH); solution interior |
| Production | allowed, any \(Y^j\) | none (pure exchange) |
| Value of the endowment | finite (finitely many goods, or \(\sum p_t\bar e_t\lt\infty\)) | finitely many goods |
| What you get | CE is PO | CE is PO and the weights \(\lambda_i=\kappa/\mu_i\) |
So the FOC proof is a weaker theorem. If utility is not concave, the planner's FOCs are no longer sufficient and Step 5 breaks; but the CE is still PO, by Lesson 4. What breaks is this proof, not the result. The payoff of the FOC route is the explicit dictionary between market objects \((p,\mu)\) and planner objects \((\gamma,\lambda)\).
In a pure exchange economy with HH and CC, let \(\{\bar x^i\}\) be an interior Pareto optimum that solves \(W\) for weights \(\lambda\gg0\), with differentiable, strictly increasing \(u^i\). Set $$p=\gamma,\qquad \mu_i=\frac{1}{\lambda_i},\qquad e^i=\bar x^i .$$ Then \((p,\{\bar x^i\})\) is a competitive equilibrium for the endowments \(\{e^i\}\).
Lesson 6 needed only convexity (HH, FF) and LNS, allowed production and non-differentiable utility, and found the price as the normal of a hyperplane separating "better" bundles from feasible ones. It delivered only a quasi-equilibrium, and Arrow's cheapest-point problem had to be ruled out separately (Lesson 7). Here the gradient \(\gamma\) of the planner's problem is that normal, and interiority with \(p\gg0\) means no agent sits at a cheapest point, so the quasi-equilibrium caveat disappears. The price paid: differentiability, concavity, interiority, \(\lambda\gg0\), and no production.
In general, the equilibrium price \(p\) and the allocation \(\{\bar x^i\}\) depend on the distribution of endowments \(\{e^i\}\); equivalently, the shadow values \(\gamma\) and the PO allocation depend on the weights \(\lambda\) CN3 p.25. The reason is simple. Moving wealth from one agent to another changes aggregate demand unless both spend the marginal dollar in the same way. If A likes good 1 more than B does, giving A more wealth raises the demand for good 1 and its relative price.
The figure makes this concrete. Two agents, two goods, aggregate endowment \(\bar e=(1,1)\), \(u^A=a\log x_1+(1-a)\log x_2\), \(u^B=b\log x_1+(1-b)\log x_2\). Normalize prices so that \(p_1+p_2=1\), i.e. \(p\cdot\bar e=1\). With Cobb–Douglas utilities the FOC gives \(p_lx^i_l=a^i_l/\mu_i\); summing over goods, \(1/\mu_i=p\cdot e^i\). With this normalization the Negishi weight of each agent is her share of aggregate wealth, and the weights sum to one automatically. (Exercise 23.4 derives all of this in general.)
The two theorems say that the set of CE allocations (for all endowment distributions) and the set of PO allocations (for all weights) are the same set, indexed differently. Negishi (1960) turned this into a way to find the CE for a given distribution of endowments: search over weights, not prices.
What it buys you:
Why the update works here: with Cobb–Douglas utilities, the value of the planner's bundle for A at shadow prices is exactly \(\gamma\cdot x^A(\lambda)=\lambda_A\) (Exercise 23.4). So the budget condition \(T^A=0\) reads \(\lambda_A=\gamma(\lambda)\cdot e^A\), a fixed-point equation in one unknown. The map \(\lambda_A\mapsto\gamma(\lambda)\cdot e^A\) has slope \((a-b)(e_1-e_2)\), which is less than one in absolute value, so iterating it converges.
On past Alvarez exams this material shows up in two ways: true/false statements on the mapping \(\lambda_i=1/\mu_i\), \(\gamma=p\) (Fall 2005, Fall 2025), and envy-freeness of equal-division equilibria (Fall 2025). The derivations below are in exam format, with general parameters.
Answer True or False and justify in at most four lines.
False. The CE FOC is \(u^i_s=\mu_ip_s\); the planner's is \(\lambda_iu^i_s=\gamma_s\). With \(\gamma=p\) they coincide iff \(\lambda_i=1/\mu_i\). Under \(\lambda_i=\mu_i\), the planner would favour agents with a high marginal utility of income, i.e. the poor, the opposite of a CE. (The statement holds only in the knife-edge case \(\mu_i=1\) for all \(i\).)
True (for interior allocations with concave, differentiable utility). Multiplying \(u^i_s=\mu_ip_s\) by \(\kappa/\mu_i\) gives \(\lambda_iu^i_s=\gamma_s\). The planner problem is invariant to scaling all weights by \(\kappa\), and \(\gamma\) scales with it CN3 p.21.
True. Inada makes demands interior, so \(p_l=\frac1{\mu_i}u^i_l\) for every \(i\). Averaging over \(i\) with any fixed weights \(\omega_i\) gives \(p_l=\sum_i\omega_i\frac1{\mu_i}u^i_l\): weights \(\omega_i/\mu_i\), proportional to the Negishi weights, the same for every good because each agent has a single budget multiplier. Equivalently, \(p_l=\gamma_l\), the representative agent's marginal utility of good \(l\).
False in general. At a corner, \(u^i_l\le\mu_ip_l\) with strict inequality for goods \(i\) does not consume. Then \(p_l=\max_iu^i_l/\mu_i\), and a positively weighted average of \(u^i_l/\mu_i\) undershoots \(p_l\) for some goods but not others, so it is not proportional to \(p\). Exercise 23.2 works this out.
False. From \(\lambda_iu^i_l=\gamma_l\), \(u^i_l/u^j_l=\lambda_j/\lambda_i\). A larger weight means a lower marginal utility, i.e. more consumption CN3 p.16. A commonly circulated summary table has this ratio upside down.
False. The conclusion holds under LNS (Lesson 4), but the FOC proof needs differentiability, concavity (so the planner's FOCs are sufficient), interior allocations and pure exchange. LNS alone gives none of these.
False. The proof sets \(e^i=\bar x^i\). Relative to original endowments \(\omega^i\), this is a transfer \(T^i=p\cdot(\bar x^i-\omega^i)\), and \(\sum_iT^i=0\). No transfers are needed only if \(p\cdot\omega^i=p\cdot\bar x^i\) for all \(i\) already, i.e. \(\lambda\) happens to be the CE weight of the original endowments.
False in general. With \(u^A=a\log x_1+(1-a)\log x_2\), \(u^B=b\log x_1+(1-b)\log x_2\), \(\bar e=(1,1)\) and \(p_1+p_2=1\), we get \(p_1=b+\lambda_A(a-b)\) where \(\lambda_A\) is A's wealth share; it moves with the distribution unless \(a=b\) (see the first figure). Independence requires special preferences: Lesson 24.
True. Every agent has wealth \(p\cdot\bar e/I\), so all budget sets are the same set. Each \(\bar x^j\) lies in it, hence in \(i\)'s budget set; since \(\bar x^i\) is optimal for \(i\) there, \(u^i(\bar x^i)\ge u^i(\bar x^j)\). Nothing else (LNS, convexity, identical preferences) is needed.
False. Equal endowments are sufficient (item 9), not necessary. Example: \(u^1=x_1\), \(u^2=x_2\), \(e^1=(1,0)\), \(e^2=(0,1)\). At any \(p\gg0\), each agent keeps her endowment, and with \(p=(1,10)\) agent 2 is ten times richer. Neither envies the other: agent 1 values agent 2's bundle at 0. (A commonly circulated answer key justifies a related statement with "envy-freeness requires equal endowments"; that justification is wrong.)
Agent \(i\) has a strictly increasing, strictly concave, differentiable \(u^i\) on \(\mathbb R^m_+\) satisfying Inada conditions, faces prices \(p\gg0\), and has wealth \(w>0\).
For (c), differentiate the identity \(p\cdot x^*(p,w)=w\) with respect to \(w\). For (e), recall that \(\lambda_i\propto1/\mu_i\) and that \(\mu_i\) scales with the utility representation.
(a) \(\mathcal L=u^i(x)+\mu_i(w-p\cdot x)+\sum_l\nu_lx_l\). Conditions: \(u^i_l-\mu_ip_l+\nu_l=0\); \(\mu_i\ge0\), \(\mu_i(w-p\cdot x)=0\); \(\nu_l\ge0\), \(\nu_lx_l=0\). Inada gives \(x_l>0\), so \(\nu_l=0\) and \(u^i_l=\mu_ip_l\).
(b) \(u^i_l>0\) (strictly increasing; Inada gives \(u_l\to\infty\) at zero, and differentiable strictly concave increasing functions have \(u_l>0\) in the interior), so \(\mu_i=u^i_l/p_l>0\) and complementary slackness makes the budget bind. This is local non-satiation in action.
(c) \(V^i=u^i(x^*(p,w))\), so \(V^i_w=\sum_lu^i_l\,x^*_{l,w}=\mu_i\sum_lp_lx^*_{l,w}=\mu_i\), using \(\sum_lp_lx^*_{l,w}=1\) from the binding budget.
(d) For \(w,w'\) with optimal \(x,x'\) and \(t\in(0,1)\): \(tx+(1-t)x'\) costs \(tw+(1-t)w'\) and is in \(\mathbb R^m_+\), so \(V^i(p,tw+(1-t)w')\ge u^i(tx+(1-t)x')\ge tV^i(p,w)+(1-t)V^i(p,w')\). A concave function's derivative \(\mu_i=V^i_w\) is non-increasing. (Strict concavity of \(u\) makes \(V\) strictly concave whenever \(x\neq x'\), so \(\mu\) is strictly decreasing.)
(e) Same \(u\), same \(p\): \(\mu^1=V_w(p,w^1)\lt V_w(p,w^2)=\mu^2\), so \(\lambda_1>\lambda_2\): the richer agent gets the larger weight. With different \(u^i\), no: multiplying \(u^1\) by a constant changes \(\mu_1\) and \(\lambda_1\) without changing anything real, so cross-agent comparisons of \(\mu\) are not meaningful.
Pure exchange, \(I\) agents, \(m\) goods, each \(u^i\) strictly increasing, strictly concave, differentiable. Let \((p,\{\bar x^i\})\) be a CE with budget multipliers \(\mu_i\).
(a) Interior demands: \(u^i_l=\mu_ip_l\), so \(u^i_l/\mu_i=p_l\) for each \(i\). Multiply by \(\omega_i\) and sum: \(\sum_i\omega_iu^i_l/\mu_i=p_l\sum_i\omega_i=p_l\). The weight \(\omega_i/\mu_i\) does not depend on \(l\) because \(\mu_i\) is the multiplier on \(i\)'s single budget constraint.
(b) By the 1WT via FOCs, \(\bar x\) solves the planner's allocation problem at \(\bar e\) with \(\lambda_i=1/\mu_i\) and multipliers \(\gamma=p\). By the envelope theorem, \(\partial u(\bar e;\lambda)/\partial x_l=\gamma_l=p_l\) (Lesson 22).
(c) \(u^i_l(\bar x^i)\le\mu_ip_l\), with equality if \(\bar x^i_l>0\). So \(u^i_l/\mu_i\le p_l\) for all \(i\), with equality for every agent who consumes good \(l\). Some agent consumes each good (markets clear with \(\bar e_l>0\)), so \(p_l=\max_iu^i_l/\mu_i\).
(d) For good 2, both agents consume it, so \(u^1_2/\mu_1=u^2_2/\mu_2=p_2\) and the average is \(p_2\). For good 1, \(u^1_1/\mu_1=p_1\) but \(u^2_1/\mu_2\lt p_1\), so the average is \(\omega_1p_1+\omega_2u^2_1/\mu_2\lt p_1\) when \(\omega_2>0\). The ratio of averages is below \(p_1/p_2\): not proportional. The statement survives only by putting zero weight on agent 2, so the Inada conditions in the exam statement do real work.
Pure exchange; HH and CC; differentiable, strictly increasing utilities; all allocations interior.
For (d), take one good, two agents, \(\bar e=1\), and \(u^i(x)=x^2\). Look at the equal split with equal weights.
(a) See §3, Steps 1–6. Concavity is used in Step 5 (the planner's Lagrangian is concave, so its FOCs are sufficient); \(\lambda\gg0\) is used in Step 6 (a Pareto improvement would strictly raise \(\sum\lambda_iu^i\)). Household FOCs are used as necessary conditions in Step 1.
(b) See §4, Steps 1–6. The planner's FOCs are used as necessary (Step 1: \(\bar x\) is an interior maximizer with linear constraints); the household's FOCs as sufficient (Step 5: concave \(u^i\), linear budget). This is the slide's warning: "keep in mind when we use the necessity of the FOC and when we use its sufficiency" CN3 p.15.
(c) \(T^i=p\cdot(\bar x^i-\omega^i)\); then \(p\cdot\bar x^i=p\cdot\omega^i+T^i\), and \(\sum_iT^i=p\cdot(\sum_i\bar x^i-\sum_i\omega^i)=p\cdot(\bar e-\bar e)=0\).
(d) With \(u^i=x^2\), \(\lambda_1=\lambda_2=1\), the FOC \(2x^1=\gamma=2x^2\) holds at \(x^1=x^2=\tfrac12\), where \(\sum\lambda_iu^i=\tfrac12\). But \(x=(1,0)\) gives \(1\): the FOC point is a minimum of \(W\) on the feasible line. Is it PO? Every split of one good with increasing utilities is PO (you cannot raise one agent without lowering the other), so the equal split is PO, yet it solves \(W\) for no weights at all (\(\lambda_1x^2+\lambda_2(1-x)^2\) is convex in \(x\), so its maximum is always at a corner). No contradiction with the 1WT: with one good, the only CE is the endowment allocation, and it is PO, as Lesson 4 guarantees. What fails is the FOC method: without concavity, FOCs neither identify planner optima nor support PO allocations as CE (Lesson 21's non-convex utility possibility set).
\(I\) agents, \(m\) goods, \(u^i(x)=\sum_la^i_l\log x_l\) with \(a^i_l>0\), \(\sum_la^i_l=1\). Endowments \(e^i\gg0\), aggregate \(\bar e=\sum_ie^i\). Write \(s^i_l=e^i_l/\bar e_l\) for agent \(i\)'s share of the endowment of good \(l\).
(a) The planner FOC is \(\lambda_ia^i_l/x^i_l=\gamma_l\). Solve for \(x^i_l\) and sum over \(i\). (c) By the 2WT argument, \(\lambda\) is a CE weight iff \(\gamma(\lambda)\cdot x^i(\lambda)=\gamma(\lambda)\cdot e^i\) for every \(i\).
(a) \(x^i_l=\lambda_ia^i_l/\gamma_l\). Feasibility \(\sum_ix^i_l=\bar e_l\) gives \(\gamma_l=\sum_j\lambda_ja^j_l/\bar e_l\), hence \(x^i_l=\dfrac{\lambda_ia^i_l}{\sum_j\lambda_ja^j_l}\,\bar e_l\): each agent gets a share of each good proportional to \(\lambda_ia^i_l\).
(b) \(\sum_l\gamma_l\bar e_l=\sum_j\lambda_j\sum_la^j_l=\sum_j\lambda_j=1\). And \(\gamma\cdot x^i=\sum_l\gamma_l\lambda_ia^i_l/\gamma_l=\lambda_i\sum_la^i_l=\lambda_i\).
(c) CE weight iff budgets balance: \(\lambda_i=\gamma(\lambda)\cdot e^i=\sum_le^i_l\sum_j\lambda_ja^j_l/\bar e_l=\sum_j\lambda_j\sum_la^j_ls^i_l\). Column sums: \(\sum_iM_{ij}=\sum_la^j_l\sum_is^i_l=\sum_la^j_l=1\). So \(M\) is column-stochastic with positive entries and \(\lambda\) is its Perron eigenvector with eigenvalue 1, unique on the simplex. With the normalization \(\gamma\cdot\bar e=1\), \(\lambda_i=\gamma\cdot e^i\) is agent \(i\)'s share of aggregate wealth: here, the Negishi weight is wealth.
(d) \(\gamma_1=\lambda_Aa+(1-\lambda_A)b=b+\lambda_A(a-b)\) and \(\gamma_2=1-\gamma_1\). Budget balance for A: \(\lambda_A=\gamma_1e_1+\gamma_2e_2=be_1+(1-b)e_2+\lambda_A(a-b)(e_1-e_2)\). Solve for \(\lambda_A^*\). Prices are \(p=\gamma(\lambda^*)\).
(e) If \(a=b\), \(\gamma_1=a\) for every \(\lambda\): \(p_1/p_2=a/(1-a)\), independent of the distribution. If \(a>b\), \(\gamma_1\) rises with \(\lambda_A\): richer A, who values good 1 more, raises its relative price. Identical homothetic preferences are exactly what kills the dependence (Lesson 24).
(f) The map is \(\phi(\lambda_A)=be_1+(1-b)e_2+(a-b)(e_1-e_2)\lambda_A\). Its slope satisfies \(|(a-b)(e_1-e_2)|\lt1\) because \(a,b,e_1,e_2\in(0,1)\). So \(\phi\) is a contraction, the iterates converge to its unique fixed point, and that fixed point is \(\lambda_A^*\) from (d). (This is what the "Negishi step" button computes.)
Pure exchange, \(I\) agents, \(m\) goods. A feasible allocation is envy-free if \(u^i(x^i)\ge u^i(x^j)\) for all \(i,j\).
(a) All agents have wealth \(p\cdot\bar e/I\) and face the identical budget set \(B=\{x\in X: p\cdot x\le p\cdot\bar e/I\}\) (assuming a common consumption set). Each \(\bar x^j\in B\). If \(u^i(\bar x^j)>u^i(\bar x^i)\), agent \(i\) could have chosen \(\bar x^j\), contradicting optimality. Assumptions used: only that each agent maximizes over the same budget set. No LNS, convexity, differentiability or identical preferences.
(b) \(u^1=x_1\), \(u^2=x_2\), \(e^1=(1,0)\), \(e^2=(0,1)\). For any \(p\gg0\), agent 1 spends everything on good 1 and agent 2 on good 2, so the endowment is the CE allocation (for example with \(p=(1,10)\), so wealths are 1 and 10). Each agent values the other's bundle at 0 \(\lt1\): envy-free.
(c) \(u^i(\bar x^j)=g^i(\delta^jh(\bar e))\). Envy-free requires \(g^i(\delta^ih(\bar e))\ge g^i(\delta^jh(\bar e))\) for all \(i,j\), i.e. \(\delta^i\ge\delta^j\) for all \(i,j\), i.e. all \(\delta^i\) equal, \(=1/I\). Conversely, equal shares give equal bundles, and nobody envies an identical bundle.
(d) The equal-division CE exists under standard assumptions (continuity, convexity and monotonicity of preferences) and is PO by the 1WT, so a fair (envy-free and efficient) allocation exists. This is Varian's (1974) observation in "Equity, envy, and efficiency," Journal of Economic Theory.
There are \(S\) states and one good per state. Agent \(i\) has endowment \(e^i\in\mathbb R^S_{++}\) and strictly concave, increasing, differentiable \(u^i(x)\). There are \(K\) securities; security \(k\) pays \(d_{ks}\) in state \(s\) and costs \(q_k\). Agent \(i\) buys a portfolio \(a^i\in\mathbb R^K\) with \(\sum_kq_ka^i_k\le0\) and consumes \(x^i_s=e^i_s+\sum_kd_{ks}a^i_k\). Let \(\mu^i\) be the multiplier on the portfolio budget and define \(\eta^i_s\equiv u^i_s(x^i)/\mu^i\).
(a) \(\partial/\partial a^i_k\): \(\sum_su^i_sd_{ks}=\mu^iq_k\); divide by \(\mu^i\). \(\eta^i_s\) is agent \(i\)'s marginal valuation, in units of account, of one unit of consumption in state \(s\): her personal state price.
(b) PO and interiority imply equal MRS: \(u^i_s/u^i_1=g_s\equiv\gamma_s/\gamma_1\) for every \(i\). Then \(\eta^i_s=g_s\,\eta^i_1\). Plug into (a): \(q_k=\eta^i_1\sum_sd_{ks}g_s\), so \(\eta^i_1=q_k/\sum_sd_{ks}g_s\) for any \(k\) with \(q_k\neq0\). The right side does not depend on \(i\), so \(\eta^i_1\) and therefore every \(\eta^i_s\) is common.
(c) It assumes the conclusion. \(\mu^i\) is determined by agent \(i\)'s own budget, not chosen freely. PO gives some weights \(\lambda\) with \(\lambda_iu^i_s=\gamma_s\); that \(\lambda_i\) can be taken equal to \(1/\mu^i\) (up to a common \(\kappa\)) is what has to be shown, and it is shown by the argument in (b) using the securities' pricing equations.
(d) No. If \(\eta^i\neq\eta^j\), they cannot be proportional: \(\eta^i=c\,\eta^j\) would give \(q=c\,q\), so \(c=1\) (for \(q\neq0\)). Non-proportional \(\eta\) vectors mean \(u^i_s/u^i_{s'}\neq u^j_s/u^j_{s'}\) for some \(s,s'\): unequal MRS, which violates the planner's FOCs, so a Pareto improvement exists. With complete markets (the payoff matrix has rank \(S\)), \(q=D\eta^i\) pins down \(\eta^i\) uniquely, so it is common and the allocation is PO.
One good in each of \(S\) states, common probabilities \(\pi_s>0\), aggregate endowment \(\bar e_s\). Agent \(i\) has expected utility \(\sum_s\pi_sv^i(x^i_s)\) with CARA \(v^i(x)=-e^{-\rho_ix}/\rho_i\), \(\rho_i>0\). Let \(R=\sum_j1/\rho_j\).
(a) \(\lambda_i\pi_se^{-\rho_ix^i_s}=\gamma_s\), so \(x^i_s=\frac{1}{\rho_i}\big[\log\lambda_i+\log\pi_s-\log\gamma_s\big]\).
(b) Sum over \(i\): \(\bar e_s=\sum_i\rho_i^{-1}\log\lambda_i+R(\log\pi_s-\log\gamma_s)=R(\log\pi_s-\log\gamma_s)\). So \(\log\pi_s-\log\gamma_s=\bar e_s/R\), and substituting gives the result.
(c) Each agent bears a fixed fraction \((1/\rho_i)/R\) of aggregate risk, her share of total risk tolerance. The weights only shift each agent's consumption by a state-independent constant \(\rho_i^{-1}\log\lambda_i\): higher weight, more consumption in every state. In a CE this constant is pinned down by the agent's wealth, exactly as \(\lambda_i\propto1/\mu_i\) says.
(d) From (b), \(\gamma_s=\pi_se^{-\bar e_s/R}\), so \(\gamma_s/\gamma_{s'}=(\pi_s/\pi_{s'})e^{-(\bar e_s-\bar e_{s'})/R}\), free of \(\lambda\). CARA demands shift one-for-one with wealth in a way that is common across agents, so redistribution does not change aggregate demand. This is a preview of when aggregation holds (Lessons 24–25), and a contrast with Exercise 23.4, where different Cobb–Douglas shares break it.