Theory of Income · Reference · Class Note 3

Aggregation & the Social Planner: Cheatsheet

Planner weights, the representative agent, the welfare theorems via first-order conditions, Gorman aggregation and its limits. Lessons 21–25 · Slides · Glossary · Index

Utility possibility set L21 · slides 3–4

$U=\{u\in\mathbb{R}^I:\ u_i\le u^i(x^i)\ \forall i\ \text{for some feasible}\ \{x^i,y^j\}\}$.
It is downward comprehensive by construction (the "$\le$"): $u\in U,\ u'\le u\Rightarrow u'\in U$.
PO allocation ⇒ $\bar u=(u^i(\bar x^i))_i$ is on the (upper) boundary of $U$.

$X^i$ and $Y$ convex ⇒ the set of feasible allocations is convex (the constraint is linear).
Add CC (every $u^i$ concave) ⇒ $U$ convex: $u^i(tx^i+(1-t)x'^i)\ge tu^i(x^i)+(1-t)u^i(x'^i)$.
Concavity is cardinal ($e^{u}$ can destroy it). Quasi-concavity is ordinal, and it is not enough: one good, $u^i=(x^i)^2$, $\bar e=1$ gives the frontier $\sqrt{u_1}+\sqrt{u_2}=1$, which bows inward.

Planner-weight theorem L21 · slides 5–10

Assume $u^i$ strictly increasing, HH, CC, FF, and $\operatorname{Int}U\ne\emptyset$. Let $W(\lambda)$: $\max_{\{x^i,y^j\}}\sum_i\lambda_iu^i(x^i)$ s.t. feasibility.

Sufficiency. $\lambda\in\mathbb{R}^I_{++}$ and $\{\bar x,\bar y\}$ solves $W(\lambda)$ ⇒ PO.
Necessity. $\{\bar x,\bar y\}$ PO ⇒ it solves $W(\lambda)$ for some $\lambda\in\mathbb{R}^I_+$, $\lambda\ne0$.

Sufficiency. A Pareto improvement $\hat x$ has $u^i(\hat x^i)\ge u^i(\bar x^i)$ for all $i$, strictly for some. With every $\lambda_i>0$, $\sum\lambda_iu^i(\hat x^i)>\sum\lambda_iu^i(\bar x^i)$. This contradicts optimality. No convexity is used.

Necessity. $U$ is convex and $\bar u\in\partial U$, so a supporting hyperplane exists: $\lambda^*\ne0$ with $\lambda^*\bar u\ge\lambda^*u$ for all $u\in U$. Sign: if $\lambda^*_i\lt 0$, then $\bar u-\varepsilon e_i\in U$ (comprehensive) and $\lambda^*(\bar u-\varepsilon e_i)>\lambda^*\bar u$. Contradiction, so $\lambda^*\ge0$.

Zero weights

A PO point may need some $\lambda_i=0$ (e.g. at a corner of the frontier). Then $W(\lambda)$ can have other maximizers that are not PO: they give agent $i$ less without changing anyone else's utility. "Solves $W$ for some $\lambda\ge0$" ⇏ PO. Picture: $\lambda$ is the normal to a supporting line of $U$ at $\bar u$; if $U$ is non-convex, inward-bowed frontier points are PO but are supported by no $\lambda$.

Representative agent L22 · slides 11–14

$u(x;\lambda)=\max_{\{x^i\in X^i\}}\sum_i\lambda_iu^i(x^i)$ s.t. $\sum_ix^i=x$, defined on $\sum_iX^i\times\mathbb{R}^I_+$. It involves no technology, so the production side can be handled by one "Crusoe" facing $Y$.
Concave in $x$ if every $X^i$ is convex and every $u^i$ concave: mix the two optimal allocations, which is feasible at the mixed aggregate. It is HD1 in $\lambda$.
CE ⇒ PO (1WT) ⇒ some $\lambda$ whose single agent reproduces the CE. PO ⇒ CE for some ownership (2WT) ⇒ the same.
Exchange economy: $Y=-\mathbb{R}^m_+$ (free disposal only); feasibility is $\sum_ix^i\le\sum_ie^i\equiv\bar e$.
Envelope theorem: $\partial u(\bar e;\lambda)/\partial x_l=\gamma_l$, the shadow value of good $l$.

Planner FOCs vs CE FOCs (interior, exchange economy) L22, L23 · slides 16–19

	Planner $W(\lambda)$	Competitive equilibrium
FOC for $x^i_l$	$\lambda_i\,u^i_l(\bar x^i)=\gamma_l$	$u^i_l(\bar x^i)=\mu_i\,p_l$
Multiplier	$\gamma_l\ge0$: shadow value of one more unit of $\bar e_l$ (depends on $\lambda$ and all $u^i$)	$\mu_i\ge0$: marginal utility of income of $i$ (envelope theorem)
MRS (allocation is efficient)	$u^i_l/u^i_k=\gamma_l/\gamma_k$, the same for every $i$; no $\lambda$	$u^i_l/u^i_k=p_l/p_k$, the same for every $i$; no $\mu$
Levels across agents	$u^i_l/u^j_l=\lambda_j/\lambda_i$	$u^i_l/u^j_l=\mu_i/\mu_j$
Necessary and sufficient when	HH + CC, interior, $\lambda\gg0$	HH + CC, interior (linear budget)

Welfare theorems via FOCs: Negishi weights L23 · slides 20–25

$$\lambda_i=\frac{\kappa}{\mu_i},\qquad \gamma_l=\kappa\,p_l,\qquad \kappa>0\ \text{arbitrary (normalization).}$$

1WT (CE ⇒ PO). Use necessity of the household FOC at the CE. Divide by $\mu_i$: $\frac1{\mu_i}u^i_l=p_l$, which is the planner FOC with $\lambda_i=1/\mu_i$, $\gamma=p$. Then use sufficiency of the planner FOC, plus feasibility, ⇒ solves $W$ ⇒ PO.

2WT (PO ⇒ CE). Use necessity of the planner FOC at $\bar x$ with $\lambda\gg0$. Set $p=\gamma$, $\mu_i=1/\lambda_i$, and $e^i=\bar x^i$ (so $\bar x^i$ is affordable). Then use sufficiency of the household FOC ⇒ $\bar x^i$ is optimal; it is feasible ⇒ CE.

Distribution: rich agents consume a lot ⇒ low $\mu_i$ (concavity) ⇒ high $\lambda_i$. Goods with high $p_l$ have high social value $\gamma_l$.
Prices are a weighted average of marginal utilities with good-independent weights: $p_l\propto\sum_i\frac1{\mu_i}u^i_l(\bar x^i)$ (Fall 2025 MT1 T/F: True, needs interiority).
In general $p$ and $\bar x$ depend on the endowment distribution $\{e^i\}$, and $\gamma$ and $\bar x$ depend on $\lambda$.

Gorman aggregation L24 · slides 26–30

Assume $u^i(x)=g^i(h(x))$, with $g^i$ strictly increasing and concave and $h$ common and HD1. Endowments are proportional, $e^i=\delta^i\bar e$ with $\sum_i\delta^i=1$. Then $\bar x^i=\delta^i\bar e$ (no trade), and

$$p_l=\kappa\,\frac{\partial h(\bar e)}{\partial x_l},\qquad \mu_i=\frac{g^{i\prime}\big(h(\delta^i\bar e)\big)}{\kappa},\qquad \lambda_i=\frac{\kappa}{g^{i\prime}\big(h(\delta^i\bar e)\big)},\qquad \gamma_l=\kappa\,\frac{\partial h(\bar e)}{\partial x_l}.$$

Key step: $h$ HD1 ⇒ $h_l$ HD0 ⇒ $h_l(\delta^i\bar e)=h_l(\bar e)$. The household FOC $g^{i\prime}h_l(\delta^i\bar e)=\mu_ip_l$ then holds with the $\mu_i$ above.
Relative prices are independent of $\{\delta^i\}$, and $\gamma_l/\gamma_k$ is independent of $\lambda$. The level of $\gamma$ depends on the normalization; $\lambda$ depends on $\delta$.
With common $g$: larger $\delta^i$ ⇒ larger $\lambda_i$ ($g'$ decreasing, $h$ increasing). With different $g^i$, this need not hold.
Arbitrary endowments: all that is needed is $p\cdot e^i=\delta^i\,p\cdot\bar e$, i.e. define $\delta^i=p\cdot e^i/p\cdot\bar e$ at $p\propto\nabla h(\bar e)$. Then $\bar x^i=\delta^i\bar e$, and there is trade.

Which preference families aggregate? L25 · slides 31–37

Method: planner FOC $\lambda_iu^i_l=\gamma_l$ → solve for $x^i_l$ → sum using $\sum_ix^i_l=\bar e_l$ → solve for $\gamma_l(\lambda)$ → test whether $\gamma_l/\gamma_k$ depends on $\lambda$. Notation: $\bar\theta=\sum_i\theta^i$.

Family	$\gamma_l$	Aggregation?
Identical homothetic $g^i(h(x))$	$\kappa\,h_l(\bar e)$	Yes
Non-identical Cobb–Douglas $\sum_l\beta^i_l\log x_l$	$\dfrac{\sum_i\lambda_i\beta^i_l}{\bar e_l}$	No, unless $\beta^i=\beta$ for all $i$. Homotheticity alone is not enough.
Identical non-homothetic $\frac{x_1^{1-\rho}}{1-\rho}+\frac{x_2^{1-\beta}}{1-\beta}$	$\gamma_1=\dfrac{(\sum_i\lambda_i^{1/\rho})^{\rho}}{\bar e_1^{\rho}}$, $\gamma_2=\dfrac{(\sum_i\lambda_i^{1/\beta})^{\beta}}{\bar e_2^{\beta}}$	No if $\rho\ne\beta$. The ratio is HD0 in $\lambda$ but depends on its distribution.
Stone–Geary, common $\beta$: $\sum_l\beta_l\log(x_l-\theta^i_l)$	$\dfrac{\beta_l\sum_i\lambda_i}{\bar e_l-\bar\theta_l}$	Yes; needs $\bar e>\bar\theta$; $\theta^i$ may differ
Translated CES: $\sum_l\beta_l\frac{(x_l-\theta^i_l)^{1-\sigma}}{1-\sigma}$	$\beta_l\Big(\dfrac{\sum_i\lambda_i^{1/\sigma}}{\bar e_l-\bar\theta_l}\Big)^{\sigma}$	Yes; $\frac{\gamma_l}{\gamma_k}=\frac{\beta_l}{\beta_k}\big(\frac{\bar e_k-\bar\theta_k}{\bar e_l-\bar\theta_l}\big)^{\sigma}$. Stone–Geary is $\sigma=1$.
Satiation power: $-\sum_l\pi_l\frac{(\theta^i_l-x_l)^{1+\sigma}}{1+\sigma}$, $x\le\theta^i$	$\pi_l\dfrac{(\bar\theta_l-\bar e_l)^{\sigma}}{\big(\sum_j\lambda_j^{-1/\sigma}\big)^{\sigma}}$	Yes; needs $\bar e\lt\bar\theta$. $\sigma=1$ is quadratic. $x^i_l=\theta^i_l+\delta^i(\bar e_l-\bar\theta_l)$ with $\delta^i=\frac{\lambda_i^{-1/\sigma}}{\sum_j\lambda_j^{-1/\sigma}}$, so $x^i_l$ rises with $\lambda_i$.
Quasi-linear: $x_1+v(x_2,\dots,x_m)$, $v$ strictly concave	CE: $\mu_i=1/p_1$ for all $i$; $\frac{\gamma_l}{\gamma_1}=v_l(\bar e_{-1}/I)$	Yes in CE, while every $x^i_1$ is interior (wealth effects fall only on $x_1$). The planner with unequal $\lambda$ puts all $x_1$ at a corner.

Unifying idea

Every family marked "Yes" is $u^i=g\big(H(x-\theta^i)\big)$, with $g$ and HD1 $H$ common across agents: identical homothetic preferences after an agent-specific shift of the origin (slide 37). Let $\hat x^i=x^i-\theta^i$, so feasibility becomes $\sum_i\hat x^i=\bar e-\bar\theta$, and $\gamma\propto\nabla H(\bar e-\bar\theta)$. Equivalently, Engel curves are linear with a common slope (Gorman polar form).

Representative agent with translations (Fall 2025 Final Q2) L25

Problem: $U(\bar e;\lambda)=\max\sum_i\lambda_iH(c^i-\theta^i)$ s.t. $\sum_ic^i=\bar e$, with $H$ HD1, increasing and quasi-concave (hence concave and superadditive on its positive cone).
Equal weights (the CE weights, since $\mu_i=1/b(p)$ is common under HD1 $H$): $U(\bar e;\lambda)=\lambda\,H(\bar e-\bar\theta)$.
Unequal weights with $c^i\ge\theta^i$: $U=(\max_i\lambda_i)\,H(\bar e-\bar\theta)$, with all the surplus going to the top-weight agent. In every case $\nabla_{\bar e}U\propto\nabla H(\bar e-\bar\theta)$, independent of $\lambda$.

Circulating answer is wrong

$U(\bar e;\lambda)=H\big(\bar e-\sum_i\lambda_i\theta^i\big)$ fails two checks. With $I=1$ it should be $\lambda_1H(\bar e-\theta^1)$. And any such value function is HD1 in $\lambda$, which this formula is not. The right transformation is the sum $\bar\theta=\sum_i\theta^i$, not a $\lambda$-weighted sum.

Exam traps

Statement	Verdict	Why
Setting $\mu_i=\lambda_i$, $p=\gamma$ makes the CE and planner allocations coincide (Fall 2005).	False	The match is $\mu_i=1/\lambda_i$. High weight means low $\mu_i$ and high consumption.
The CE with equal endowments $e^i=\bar e/I$ is envy-free (Fall 2025 MT1).	True	Every $x^j$ costs $\le p\bar e/I$, so it is in $i$'s budget set. No convexity or identical preferences needed.
Envy-freeness requires equal endowments.	False	$u^1=x_1$, $u^2=x_2$, $e^1=(1,0)$, $e^2=(0,1)$: no trade is a CE, envy-free at any $p\gg0$.
If all agents have the same preferences, aggregation holds.	False	Identical CRRA with $\rho\ne\beta$ (slide 33).
If all agents have homothetic preferences, aggregation holds.	False	Non-identical Cobb–Douglas (slides 31–32): the same $h$ is needed.
Translated CRRA / Stone–Geary is homothetic.	False	It is homothetic around $\theta^i$, not the origin (quasi-homothetic). Aggregation still holds.
Satiation power utility $-\sum_l\frac{(\theta^i_l-x_l)^{1+\sigma}}{1+\sigma}$ gives Gorman aggregation (Fall 2025 MT1).	True	$\gamma_l/\gamma_k=\big(\frac{\bar\theta_l-\bar e_l}{\bar\theta_k-\bar e_k}\big)^{\sigma}$.
Different subjective probabilities, all risk averse ⇒ consumption depends only on $\bar e_s$ (Fall 2005).	False	$\lambda_i\pi^i_sv^{i\prime}(x^i_s)=\gamma_s$: the belief ratio $\pi^2_s/\pi^1_s$ enters.
Planner table: $u^i_l/u^j_l=\lambda_i/\lambda_j$ (a circulating review).	False	Inverted: it is $\lambda_j/\lambda_i$.

Notation clashes in Class Note 3

The footer says "Class Notes III", but this is the folder-3 deck (aggregation), not Class Note III on OLG extensions (privatization, capital).
$\theta^i$: translation vector (subsistence if $x\ge\theta^i$, satiation if $x\le\theta^i$). Elsewhere $\theta$ is the ownership share $\theta^i_j$ (Slides 1), the PAYG tax (Note III), or the discount rate (BY).
$\delta^i$: agent $i$'s share of the aggregate endowment. In Note III, $\delta$ is depreciation.
$\lambda_i$: Pareto weight. $\gamma_l$: shadow value of good $l$. $\mu_i$: marginal utility of income. $\kappa$: free normalization.
$\beta$ on slide 33 is a curvature parameter, not a discount factor. $h$ is an HD1 aggregator here; in BY it is human wealth. In the Fall 2025 MT1 statement, "$p$" is an exponent, not a price.

Course index Welfare theorems cheatsheet Glossary

	Planner \(W(\lambda)\)	Competitive equilibrium
FOC for \(x^i_l\)	\(\lambda_i\,u^i_l(\bar x^i)=\gamma_l\)	\(u^i_l(\bar x^i)=\mu_i\,p_l\)
Multiplier	\(\gamma_l\ge0\): shadow value of one more unit of \(\bar e_l\) (depends on \(\lambda\) and all \(u^i\))	\(\mu_i\ge0\): marginal utility of income of \(i\) (envelope theorem)
MRS (allocation is efficient)	\(u^i_l/u^i_k=\gamma_l/\gamma_k\), the same for every \(i\); no \(\lambda\)	\(u^i_l/u^i_k=p_l/p_k\), the same for every \(i\); no \(\mu\)
Levels across agents	\(u^i_l/u^j_l=\lambda_j/\lambda_i\)	\(u^i_l/u^j_l=\mu_i/\mu_j\)
Necessary and sufficient when	HH + CC, interior, \(\lambda\gg0\)	HH + CC, interior (linear budget)

Family	\(\gamma_l\)	Aggregation?
Identical homothetic \(g^i(h(x))\)	\(\kappa\,h_l(\bar e)\)	Yes
Non-identical Cobb–Douglas \(\sum_l\beta^i_l\log x_l\)	\(\dfrac{\sum_i\lambda_i\beta^i_l}{\bar e_l}\)	No, unless \(\beta^i=\beta\) for all \(i\). Homotheticity alone is not enough.
Identical non-homothetic \(\frac{x_1^{1-\rho}}{1-\rho}+\frac{x_2^{1-\beta}}{1-\beta}\)	\(\gamma_1=\dfrac{(\sum_i\lambda_i^{1/\rho})^{\rho}}{\bar e_1^{\rho}}\), \(\gamma_2=\dfrac{(\sum_i\lambda_i^{1/\beta})^{\beta}}{\bar e_2^{\beta}}\)	No if \(\rho\ne\beta\). The ratio is HD0 in \(\lambda\) but depends on its distribution.
Stone–Geary, common \(\beta\): \(\sum_l\beta_l\log(x_l-\theta^i_l)\)	\(\dfrac{\beta_l\sum_i\lambda_i}{\bar e_l-\bar\theta_l}\)	Yes; needs \(\bar e>\bar\theta\); \(\theta^i\) may differ
Translated CES: \(\sum_l\beta_l\frac{(x_l-\theta^i_l)^{1-\sigma}}{1-\sigma}\)	\(\beta_l\Big(\dfrac{\sum_i\lambda_i^{1/\sigma}}{\bar e_l-\bar\theta_l}\Big)^{\sigma}\)	Yes; \(\frac{\gamma_l}{\gamma_k}=\frac{\beta_l}{\beta_k}\big(\frac{\bar e_k-\bar\theta_k}{\bar e_l-\bar\theta_l}\big)^{\sigma}\). Stone–Geary is \(\sigma=1\).
Satiation power: \(-\sum_l\pi_l\frac{(\theta^i_l-x_l)^{1+\sigma}}{1+\sigma}\), \(x\le\theta^i\)	\(\pi_l\dfrac{(\bar\theta_l-\bar e_l)^{\sigma}}{\big(\sum_j\lambda_j^{-1/\sigma}\big)^{\sigma}}\)	Yes; needs \(\bar e\lt\bar\theta\). \(\sigma=1\) is quadratic. \(x^i_l=\theta^i_l+\delta^i(\bar e_l-\bar\theta_l)\) with \(\delta^i=\frac{\lambda_i^{-1/\sigma}}{\sum_j\lambda_j^{-1/\sigma}}\), so \(x^i_l\) rises with \(\lambda_i\).
Quasi-linear: \(x_1+v(x_2,\dots,x_m)\), \(v\) strictly concave	CE: \(\mu_i=1/p_1\) for all \(i\); \(\frac{\gamma_l}{\gamma_1}=v_l(\bar e_{-1}/I)\)	Yes in CE, while every \(x^i_1\) is interior (wealth effects fall only on \(x_1\)). The planner with unequal \(\lambda\) puts all \(x_1\) at a corner.

Statement	Verdict	Why
Setting \(\mu_i=\lambda_i\), \(p=\gamma\) makes the CE and planner allocations coincide (Fall 2005).	False	The match is \(\mu_i=1/\lambda_i\). High weight means low \(\mu_i\) and high consumption.
The CE with equal endowments \(e^i=\bar e/I\) is envy-free (Fall 2025 MT1).	True	Every \(x^j\) costs \(\le p\bar e/I\), so it is in \(i\)'s budget set. No convexity or identical preferences needed.
Envy-freeness requires equal endowments.	False	\(u^1=x_1\), \(u^2=x_2\), \(e^1=(1,0)\), \(e^2=(0,1)\): no trade is a CE, envy-free at any \(p\gg0\).
If all agents have the same preferences, aggregation holds.	False	Identical CRRA with \(\rho\ne\beta\) (slide 33).
If all agents have homothetic preferences, aggregation holds.	False	Non-identical Cobb–Douglas (slides 31–32): the same \(h\) is needed.
Translated CRRA / Stone–Geary is homothetic.	False	It is homothetic around \(\theta^i\), not the origin (quasi-homothetic). Aggregation still holds.
Satiation power utility \(-\sum_l\frac{(\theta^i_l-x_l)^{1+\sigma}}{1+\sigma}\) gives Gorman aggregation (Fall 2025 MT1).	True	\(\gamma_l/\gamma_k=\big(\frac{\bar\theta_l-\bar e_l}{\bar\theta_k-\bar e_k}\big)^{\sigma}\).
Different subjective probabilities, all risk averse ⇒ consumption depends only on \(\bar e_s\) (Fall 2005).	False	\(\lambda_i\pi^i_sv^{i\prime}(x^i_s)=\gamma_s\): the belief ratio \(\pi^2_s/\pi^1_s\) enters.
Planner table: \(u^i_l/u^j_l=\lambda_i/\lambda_j\) (a circulating review).	False	Inverted: it is \(\lambda_j/\lambda_i\).