The slide footer says "Class Notes III", but this is folder 3 (Aggregation, Social Planner), not Class Note III on OLG extensions (Lessons 14–16). In this lesson:
Lesson 24 proved that if every household has \(u^i=g^i(h(x))\) with a common HD1 function \(h\), then relative prices are \(p_l/p_k=h_l(\bar e)/h_k(\bar e)\), whatever the distribution of wealth. Both hypotheses in that sentence, identical and homothetic, do real work. This lesson shows what happens when each one is dropped, and how a change of origin rescues a large class of non-homothetic, non-identical preferences.
All examples use the same recipe. It is the skeleton of every past exam question on this note.
The logic of step 4 deserves a sentence. Every CE is a planner optimum for some weights, and redistributing endowments moves the supporting weights. If the planner's relative shadow values are the same for all weights, redistribution cannot move relative prices. If they differ across weights, it generically does. Section 2 shows this directly from the CE in one case.
Let household \(i\) have Cobb–Douglas preferences with its own taste weights N3 p.31:
$$u^i(x^i)=\sum_{l=1}^m\beta^i_l\log x^i_l,\qquad \beta^i_l>0,\quad\sum_l\beta^i_l=1 .$$Each \(u^i\) is homothetic: it is a strictly increasing transformation of the HD1 function \(\prod_l x_l^{\beta^i_l}\). But the HD1 function differs across households.
Let agent \(i\) have wealth \(w^i=p\cdot e^i\). Cobb–Douglas demand is \(x^i_l=\beta^i_lw^i/p_l\). Market clearing for good \(l\) gives
$$p_l\bar e_l=\sum_i\beta^i_l\,w^i .$$The value of good \(l\) equals the total spending on it, and that is a wealth-weighted average of the taste weights. Take even proportional endowments \(e^i=\delta^i\bar e\), so that \(w^i=\delta^i\,p\cdot\bar e\). Then
$$\frac{p_l\bar e_l}{p_k\bar e_k}=\frac{\sum_i\delta^i\beta^i_l}{\sum_i\delta^i\beta^i_k},$$which moves with the distribution \(\delta\). Moving wealth toward households that like good \(l\) raises its relative price. The two formulas match exactly. With log utility and \(\sum_l\beta^i_l=1\), the marginal utility of income is \(\mu_i=1/w^i\), so the Negishi weight is \(\lambda_i=1/\mu_i=w^i\propto\delta^i\).
Each household's Engel curves are rays from the origin, so each spends a fixed share of a marginal dollar on each good. But the shares differ across households. A transfer from \(j\) to \(i\) therefore changes the composition of aggregate demand, and relative prices must adjust. What exact aggregation needs is a single preference ordering \(h\) shared by everyone, up to individual increasing transformations \(g^i\).
Now give everyone the same preferences, \(I=m=2\) N3 p.33:
$$u^i(x_1,x_2)=\frac{x_1^{1-\rho}}{1-\rho}+\frac{x_2^{1-\beta}}{1-\beta},\qquad\rho,\beta>0,\quad\rho\ne\beta .$$(Read \(\frac{x^{1-1}}{1-1}\) as \(\log x\).) Good 1 has curvature \(\rho\) and good 2 has curvature \(\beta\). These preferences are not homothetic: along a ray \(x=t\bar x\), the MRS \(\frac{x_1^{-\rho}}{x_2^{-\beta}}\) scales like \(t^{\beta-\rho}\), so it changes with \(t\) unless \(\rho=\beta\).
With identical non-homothetic preferences, rich and poor households spend a marginal dollar differently: Engel curves bend. Suppose \(\rho>\beta\). Good 1 is then more of a necessity, because its marginal utility falls off faster. A rich agent's extra dollar goes disproportionately to good 2. Shifting wealth from poor to rich raises aggregate demand for good 2 and lowers it for good 1, so \(p_1/p_2\) falls. Identical preferences do not help when the slope of the Engel curve depends on the level of wealth.
Two agents, two goods, \(\bar e=(1,1)\). Let \(s\equiv\lambda_1/(\lambda_1+\lambda_2)\) be agent 1's share of the Pareto weight. Since every \(\gamma_1/\gamma_2\) above is HD0 in \(\lambda\), it is a function of \(s\) alone. A family has aggregation exactly when its curve is flat.
Now let preferences be neither identical nor homothetic N3 p.34:
$$u^i(x^i)=\sum_{l=1}^m\beta^i_l\log\big(x^i_l-\theta^i_l\big),\qquad x^i_l>\theta^i_l .$$\(\theta^i_l\) is a subsistence level. Agent \(i\) needs at least \(\theta^i_l\) of good \(l\) before it gets any utility from the good, and marginal utility explodes as \(x^i_l\downarrow\theta^i_l\). The quantity \(x^i_l-\theta^i_l\) is supernumerary consumption, the consumption above subsistence.
Define translated consumption \(\hat x^i\equiv x^i-\theta^i\). In these coordinates, \(u^i=\sum_l\beta_l\log\hat x^i_l\) is the same homothetic function for everyone. The feasibility constraint becomes
$$\sum_i\hat x^i=\bar e-\bar\theta,$$an exchange economy with identical homothetic preferences and aggregate endowment \(\bar e-\bar\theta\). Lesson 24 applies verbatim with \(\bar e\) replaced by \(\bar e-\bar\theta\). In the original coordinates the preferences are not homothetic (Engel curves start at \(\theta^i\), not at 0) and not identical (the \(\theta^i\) differ). They are homothetic around a household-specific point, which is what economists call quasi-homothetic.
Replace the logs by power functions N3 p.36:
$$u^i(x^i)=\sum_{l=1}^m\beta_l\,\frac{\big(x^i_l-\theta^i_l\big)^{1-\sigma}}{1-\sigma},\qquad\sigma>0 .$$Compare §3: there the exponent was good-specific (\(\rho\) versus \(\beta\)), so the shares \(\lambda_i^{1/\rho}/\sum\lambda^{1/\rho}\) and \(\lambda_i^{1/\beta}/\sum\lambda^{1/\beta}\) differed across goods, and the \(\lambda\) terms did not cancel. A common curvature \(\sigma\) across goods is what makes \(u\) homothetic in \(\hat x\).
Slide 37 sets up the general case and stops. Let \(\theta^i\in\mathbb R^m\) for every \(i\), and N3 p.37
$$u^i(x)=g^i\big(h(x-\theta^i)\big),$$with \(h\) common, HD1, increasing and concave, and \(g^i\) strictly increasing and concave. If \(\theta^i_l>0\), good \(l\) is a necessity for household \(i\). Redefine consumption as \(\hat x^i=x^i-\theta^i\), so feasibility reads \(\sum_i\hat x^i=\bar e-\bar\theta\). Here is the conclusion the slide leaves implicit.
Suppose \(\bar e-\bar\theta\gg0\). In any interior competitive equilibrium, whatever the endowments \(\{e^i\}\) (provided each agent can afford its translation point, \(p\cdot e^i>p\cdot\theta^i\)), $$p=\kappa\,\nabla h\big(\bar e-\bar\theta\big)\quad\text{for some }\kappa>0,\qquad x^i=\theta^i+\delta^i(\bar e-\bar\theta),\quad\delta^i=\frac{p\cdot(e^i-\theta^i)}{p\cdot(\bar e-\bar\theta)} .$$ On the planner side (with strictly concave \(g^i\)), \(\gamma_l/\gamma_k=h_l(\bar e-\bar\theta)/h_k(\bar e-\bar\theta)\) for every \(\lambda\gg0\).
The slide notes that one can take \(g^i\) to be the identity. For the CE that is fine (step 2 above never used \(g^i\)). For the planner it changes the problem: with \(u^i=h(x-\theta^i)\) linear along rays, utility is transferable, and with unequal weights the planner hands the whole surplus \(Z\) to a top-weight agent. With equal weights every split on the ray is optimal. This is the source of a wrong answer to a recent final (Exercise 25.3).
Two past exams use a translation in the other direction. Let
$$u^i(x^i)=-\sum_{l=1}^m\pi_l\,\frac{\big(\theta^i_l-x^i_l\big)^{1+\sigma}}{1+\sigma},\qquad x^i_l\le\theta^i_l,\quad\pi_l,\sigma>0 .$$Now \(\theta^i_l\) is a satiation (bliss) point, and \(\theta^i_l-x^i_l\ge0\) is the agent's shortfall. Utility is a common homothetic function of the shortfall, decreasing in each shortfall. With \(\sigma=1\) it is quadratic utility.
The condition \(\bar e\lt\bar\theta\) is essential, and the exam leaves it unstated. Since \(x^i\le\theta^i\), feasibility with equality needs \(\bar e_l\le\bar\theta_l\). If \(\bar e_l=\bar\theta_l\), everyone is satiated in good \(l\) and \(\gamma_l=0\). With free disposal and \(\bar e_l>\bar\theta_l\), good \(l\) is free. Interior, positive prices need \(\bar e\ll\bar\theta\). Relative prices \(p_l/p_k\) then fall with \(\bar e_l\) and rise with \(\bar e_k\): the less scarce good \(l\) is relative to satiation, the cheaper it is.
Why do translations work while §2 and §3 fail? Look at demand. Gorman (1953) showed that aggregate demand depends only on aggregate wealth, for every wealth distribution, iff each household's indirect utility has the form (MWG Prop. 4.B.1)
$$v^i(p,w)=a^i(p)+b(p)\,w ,$$with \(b(p)\) common to all households. By Roy's identity, demand is then \(x^i_l(p,w)=-\frac{a^i_l(p)}{b(p)}-\frac{b_l(p)}{b(p)}\,w\): linear Engel curves with the same slope for everyone. A dollar moved from \(j\) to \(i\) changes \(i\)'s demand by exactly as much as it changes \(j\)'s, so aggregate demand cannot move.
| Family | Identical? | Homothetic? | Engel curves | Aggregation? |
|---|---|---|---|---|
| \(g^i(h(x))\), common \(h\) | ordinally yes | yes | rays from 0, common slope | yes (Lesson 24) |
| \(\sum_l\beta^i_l\log x_l\) | no | yes | rays, slopes differ by \(i\) | no (§2) |
| \(\frac{x_1^{1-\rho}}{1-\rho}+\frac{x_2^{1-\beta}}{1-\beta}\), \(\rho\ne\beta\) | yes | no | curved | no (§3) |
| \(\sum_l\beta_l\log(x_l-\theta^i_l)\) | no | no (quasi-homothetic) | straight, from \(\theta^i\), common slope | yes (§5) |
| \(\sum_l\beta_l\frac{(x_l-\theta^i_l)^{1-\sigma}}{1-\sigma}\) | no | no (quasi-homothetic) | straight, common slope | yes (§6) |
| \(-\sum_l\pi_l\frac{(\theta^i_l-x_l)^{1+\sigma}}{1+\sigma}\) | no | no | straight, toward \(\theta^i\), common slope | yes (§8) |
For \(u^i=h(x-\theta^i)\) with \(h\) HD1, let \(P(p)=\min\{p\cdot z:h(z)\ge1\}\) be the unit cost of \(h\)-utility. Then \(v^i(p,w)=\frac{w-p\cdot\theta^i}{P(p)}\), which is the Gorman form with \(a^i(p)=-p\cdot\theta^i/P(p)\) and \(b(p)=1/P(p)\). Translation is therefore one example of the Gorman polar form, not the whole of it. Exercise 25.7 shows another: CARA utilities with different risk aversions, which aggregate even though preferences are not identical.
Gorman's condition delivers aggregation for every wealth distribution (with interior demands). The course's version (slides 26–30) assumed identical homothetic preferences and proportional endowments, then noted that proportionality is not needed: sufficient is \(p\cdot e^i=\delta^ip\cdot\bar e\), and with identical homothetic preferences any distribution satisfies this for \(\delta^i\equiv p\cdot e^i/p\cdot\bar e\). One widely shared review sheet calls this condition "necessary". It is sufficient, not necessary: Stone–Geary aggregates without it.
Put heterogeneous types inside each generation of a two-period OLG economy (Lesson 16). Each cohort's "goods" are consumption at \(t\) and at \(t+1\). Two facts combine:
With the satiation utility of §8 and weights \((\pi_y,\pi_o)\), this gives the Spring 2023 answer:
$$\frac{1}{1+r}=\frac{p_{t+1}}{p_t}=\frac{\pi_o}{\pi_y}\left(\frac{\bar\theta_o-\bar\alpha_o}{\bar\theta_y-\bar\alpha_y}\right)^{\sigma}.$$The interest rate depends only on cohort aggregates. Whether \(r\) is positive then decides efficiency, exactly as in the homogeneous model (Lesson 10). With log utility and heterogeneous endowments (the 2025 problem set), the same logic gave \(\bar r=\frac{\bar\alpha-\beta}{\beta(1-\bar\alpha)}\): only the mean \(\bar\alpha\) matters.
Aggregation questions on Alvarez's exams have two shapes. One is a long guided derivation on translated utilities (FOC, invert, aggregate, test, representative agent, prices). The other is a set of one-line true/false items. Exercises 25.2, 25.3 and 25.5 are reworded past exam questions. Attempt each part before opening the solution.
For each statement, say whether it is true or false and justify in one to four lines.
True. This is §8 with \(\pi_l=1\). The planner FOC \(\lambda_i(\theta^i_l-x^i_l)^\sigma=\gamma_l\) aggregates to \(\gamma_l\propto(\bar\theta_l-\bar e_l)^\sigma\), with a \(\lambda\) factor common to all goods. So \(\gamma_l/\gamma_k=\big(\frac{\bar\theta_l-\bar e_l}{\bar\theta_k-\bar e_k}\big)^\sigma\) is independent of \(\lambda\). On the demand side, \(x^i_l=\theta^i_l-\frac{p_l^{1/\sigma}}{\sum_kp_k^{1+1/\sigma}}(p\cdot\theta^i-w^i)\) is linear in \(w^i\) with a common slope. (This assumes an interior solution, \(w^i\lt p\cdot\theta^i\).)
False. Slide 33: identical \(\frac{x_1^{1-\rho}}{1-\rho}+\frac{x_2^{1-\beta}}{1-\beta}\) with \(\rho\ne\beta\) gives \(\gamma_1/\gamma_2\propto\big(\sum\lambda_i^{1/\rho}\big)^\rho/\big(\sum\lambda_i^{1/\beta}\big)^\beta\), which depends on the distribution of \(\lambda\). Engel curves are non-linear, so a transfer from rich to poor changes aggregate demand.
False. Non-identical Cobb–Douglas: \(p_l\bar e_l=\sum_i\beta^i_lw^i\), so moving wealth toward households with high \(\beta^i_l\) raises \(p_l\). Exact aggregation needs a common homothetic ordering \(h\).
False. It is not homothetic: the MRS at \(tx\) differs from the MRS at \(x\) because \(\theta^i\) does not scale. It is homothetic in \(x-\theta^i\) (quasi-homothetic). It aggregates because, after translating each origin, everyone shares the same homothetic preferences over \(\hat x^i\) and feasibility becomes \(\sum\hat x^i=\bar e-\bar\theta\).
False. Proportionality is sufficient. Demand is \(x^i=w^id(p)\), so aggregate demand is \(\big(\sum_iw^i\big)d(p)\) for any endowments, and clearing gives \(p\propto\nabla h(\bar e)\). Slide 30 notes that all that is needed is \(p\cdot e^i=\delta^ip\cdot\bar e\), which holds with \(\delta^i\equiv p\cdot e^i/p\cdot\bar e\).
False. Stone–Geary with common \(\beta\) and different \(\theta^i\) aggregates (§5). So do CARA expected utilities \(\sum_s\pi_s(-e^{-\rho_ix_s}/\rho_i)\) with different \(\rho_i\) and common beliefs, whose wealth slope \(1/\sum_sp_s\) is common (Exercise 25.7). What is needed is the Gorman form: linear Engel curves with a common slope.
False. \(\gamma_l=\sum_i\lambda_i\beta^i_l/(\bar e_l-\bar\theta_l)\), so \(\gamma_l/\gamma_k\) contains \(\sum_i\lambda_i\beta^i_l/\sum_i\lambda_i\beta^i_k\), which depends on \(\lambda\) when the \(\beta^i\) differ. Common \(\theta\) does not help. It is the \(\beta\)'s (the shape) that must be common, while the \(\theta\)'s (the origins) may differ.
True. Every household can be given its bliss level \(\theta^i_l\), and the remaining \(\bar e_l-\bar\theta_l\) is disposed of. Nobody's marginal utility of good \(l\) is positive there, so the planner's multiplier is \(\gamma_l=0\), and so is \(p_l=\gamma_l/\kappa\). This is why the exam's formulas need \(\bar e\ll\bar\theta\).
A pure endowment economy has \(m\) goods and \(I\) consumers with
$$u^i(x^i)=-\sum_{l=1}^m\pi_l\frac{(\theta^i_l-x^i_l)^{1+\sigma}}{1+\sigma}\quad\text{on }X^i=\{x\in\mathbb R^m:x_l\le\theta^i_l\},\qquad\pi_l,\sigma,\theta^i_l>0 .$$The planner maximizes \(\sum_i\lambda_iu^i(x^i)\) with \(\lambda_i>0\), subject to \(\sum_ix^i_l=\bar e_l\) for each \(l\). Let \(\bar\theta_l=\sum_i\theta^i_l\), and assume \(\bar e_l\lt\bar\theta_l\) for every \(l\) (the exam uses this without stating it).
Work with the shortfall \(z^i_l=\theta^i_l-x^i_l\ge0\). For (h), write \(\delta^i=a_i/(a_i+A_{-i})\) with \(a_i=\lambda_i^{-1/\sigma}\), and remember that \(\bar e_l-\bar\theta_l\lt0\). For (k), use \(p=\gamma/\kappa\) from Lesson 23.
(a) Each term is \(v(c)=-\pi_l(\theta-c)^{1+\sigma}/(1+\sigma)\), with \(v'(c)=\pi_l(\theta-c)^\sigma>0\) for \(c\lt\theta\) and \(v''(c)=-\sigma\pi_l(\theta-c)^{\sigma-1}\lt0\). The Hessian of \(u^i\) is diagonal with negative entries, so \(u^i\) is strictly concave, and it is strictly increasing in each argument below satiation. (The derivative vanishes only at the boundary \(x_l=\theta^i_l\).) Note: the official answer key writes \(v''=\sigma(\theta-c)^{\sigma-1}\lt0\), dropping the minus sign and \(\pi_l\). The expression above is the correct one.
(b) \(\sigma=1\).
(c) \(\theta^i_l\) is consumer \(i\)'s satiation (bliss) level of good \(l\).
(d) \(\lambda_i\pi_l(\theta^i_l-x^i_l)^\sigma=\gamma_l\).
(e) \(x^i_l=\theta^i_l-(\gamma_l/\pi_l)^{1/\sigma}(1/\lambda_i)^{1/\sigma}\).
(f) Summing over \(i\): \(\bar\theta_l-\bar e_l=(\gamma_l/\pi_l)^{1/\sigma}\sum_j\lambda_j^{-1/\sigma}\), so \(\gamma_l=\pi_l\dfrac{(\bar\theta_l-\bar e_l)^\sigma}{\big(\sum_j\lambda_j^{-1/\sigma}\big)^\sigma}\).
(g) \(x^i_l-\theta^i_l=(\bar e_l-\bar\theta_l)\dfrac{\lambda_i^{-1/\sigma}}{\sum_j\lambda_j^{-1/\sigma}}\).
(h) \(\delta^i=\lambda_i^{-1/\sigma}/\sum_j\lambda_j^{-1/\sigma}>0\), and the \(\delta^i\) sum to one by construction. With \(a_i=\lambda_i^{-1/\sigma}\), \(\delta^i=a_i/(a_i+A_{-i})\) is increasing in \(a_i\), and \(a_i\) is decreasing in \(\lambda_i\). So \(\delta^i\) falls as \(\lambda_i\) rises. Since \(\bar e_l-\bar\theta_l\lt0\), \(x^i_l=\theta^i_l+\delta^i(\bar e_l-\bar\theta_l)\) rises: a higher weight means a smaller share of the aggregate shortfall.
(i) \(\bar u(x)=-\sum_l\pi_l(\bar\theta_l-x_l)^{1+\sigma}/(1+\sigma)\), so \(\bar u_l(\bar e)=\pi_l(\bar\theta_l-\bar e_l)^\sigma\).
(j) From (f), the factor \(\big(\sum_j\lambda_j^{-1/\sigma}\big)^{-\sigma}\) is common to all goods and cancels: $$\frac{\gamma_l}{\gamma_k}=\frac{\pi_l(\bar\theta_l-\bar e_l)^\sigma}{\pi_k(\bar\theta_k-\bar e_k)^\sigma}=\frac{\bar u_l(\bar e)}{\bar u_k(\bar e)}.$$ They are equal for every \(\lambda\). This is strong (Gorman) aggregation: a representative agent whose satiation point is the sum \(\bar\theta\) reproduces the shadow prices, which depend only on aggregates.
(k) Every CE is a planner optimum for some \(\lambda\), with \(p=\gamma/\kappa\). So, whatever the endowment distribution, $$\frac{p_l}{p_k}=\frac{\pi_l}{\pi_k}\left(\frac{\bar\theta_l-\bar e_l}{\bar\theta_k-\bar e_k}\right)^{\sigma},$$ which is decreasing in \(\bar e_l\) (good \(l\) becomes less scarce relative to satiation) and increasing in \(\bar e_k\).
There are \(m\) goods and \(I\) agents with \(u^i(c)=H(c-\theta^i)\), where \(\theta^i\in\mathbb R^m\) and \(H\) is HD1, strictly increasing and strictly quasi-concave. Assume \(H\) is defined on \(\mathbb R^m_+\) (consumption at or above \(\theta^i\)) and \(\bar e-\bar\theta\gg0\), where \(\bar\theta=\sum_i\theta^i\).
Change variables to \(z^i=c^i-\theta^i\). An HD1 quasi-concave function that is positive on the cone is concave, hence superadditive: \(H(a+b)\ge H(a)+H(b)\). For (d), compute the indirect utility \(v^i(p,w)\) and differentiate in \(w\).
(a) \(U(\bar e;\lambda)=\max_{\{c^i\}}\sum_i\lambda_iH(c^i-\theta^i)\) subject to \(\sum_ic^i=\bar e\).
(b) Let \(z^i=c^i-\theta^i\ge0\), so the constraint becomes \(\sum_iz^i=Z\equiv\bar e-\bar\theta\). By superadditivity, \(\sum_i\lambda_iH(z^i)\le(\max_i\lambda_i)\sum_iH(z^i)\le(\max_i\lambda_i)H(Z)\), and giving all of \(Z\) to a top-weight agent attains this. So $$U(\bar e;\lambda)=\Big(\max_i\lambda_i\Big)\,H\big(\bar e-\bar\theta\big).$$ With equal weights \(\lambda_i=\lambda\), the case that matters for equilibria (see (d)), \(U=\lambda H(\bar e-\bar\theta)\), attained by any split along the ray \(z^i=s_iZ\). The "simple transformation" is the sum \(\bar\theta=\sum_i\theta^i\). Either way, \(\nabla_{\bar e}U\propto\nabla H(\bar e-\bar\theta)\), independent of \(\lambda\), which is aggregation.
(c) One agent: with \(I=1\), the constraint forces \(c^1=\bar e\), so \(U=\lambda_1H(\bar e-\theta^1)\). The proposed formula gives \(H(\bar e-\lambda_1\theta^1)\), which differs whenever \(\lambda_1\ne1\). Homogeneity: any value of the form \(\max\sum_i\lambda_iu^i\) is HD1 in \(\lambda\), i.e. \(U(\bar e;k\lambda)=kU(\bar e;\lambda)\), but \(H(\bar e-\sum_ik\lambda_i\theta^i)\) is not. The underlying mistake is to write \(\lambda_iH(z^i)=H(\lambda_iz^i)\) and then act as if the constraint were on \(\sum_i\lambda_iz^i\). It is on \(\sum_iz^i\).
(d) Let \(P(p)=\min\{p\cdot z:H(z)\ge1\}\). By homogeneity, agent \(i\)'s indirect utility is \(v^i(p,w^i)=(w^i-p\cdot\theta^i)/P(p)\), so its marginal utility of income is \(\mu_i=1/P(p)\), the same for every agent. The Negishi weights \(\lambda_i=1/\mu_i=P(p)\) are therefore equal. With linear \(H\)-utility, utility is transferable at the same rate for everyone. By §7, prices are \(p=\kappa\nabla H(\bar e-\bar\theta)\) for any endowment distribution with \(p\cdot e^i\ge p\cdot\theta^i\).
For each family, (i) say whether preferences are identical and whether they are homothetic, (ii) derive \(\gamma_l/\gamma_k\) from the planner's FOCs, and (iii) decide whether aggregation holds. Assume interior solutions, and \(\bar e\gg\bar\theta\) or \(\bar e\ll\bar\theta\) as appropriate.
(a) Not identical; homothetic. \(\gamma_l=\sum_i\lambda_i\beta^i_l/\bar e_l\), so \(\frac{\gamma_l}{\gamma_k}=\frac{\sum_i\lambda_i\beta^i_l}{\sum_i\lambda_i\beta^i_k}\frac{\bar e_k}{\bar e_l}\). No aggregation unless \(\beta^i\equiv\beta\).
(b) Not identical; not homothetic (quasi-homothetic). \(\gamma_l=\beta_l\sum_i\lambda_i/(\bar e_l-\bar\theta_l)\), so \(\frac{\gamma_l}{\gamma_k}=\frac{\beta_l(\bar e_k-\bar\theta_k)}{\beta_k(\bar e_l-\bar\theta_l)}\). Aggregation, provided \(\bar e\gg\bar\theta\).
(c) Not identical; not homothetic. FOC \(\lambda_i\beta_l(\theta^i_l-x^i_l)=\gamma_l\), so \(\bar\theta_l-\bar e_l=\frac{\gamma_l}{\beta_l}\sum_i\lambda_i^{-1}\) and \(\frac{\gamma_l}{\gamma_k}=\frac{\beta_l(\bar\theta_l-\bar e_l)}{\beta_k(\bar\theta_k-\bar e_k)}\). Aggregation, provided \(\bar e\ll\bar\theta\). This is Exercise 25.2 with \(\sigma=1\).
(d) Identical; not homothetic. \(\frac{\gamma_1}{\gamma_2}=\frac{(\sum_i\lambda_i^{1/\rho})^\rho}{(\sum_i\lambda_i^{1/\beta})^\beta}\frac{\bar e_2^{\,\beta}}{\bar e_1^{\,\rho}}\): HD0 in \(\lambda\) but dependent on its distribution. No aggregation.
(e) Not identical; not homothetic (quasi-homothetic). \(\gamma_l=\beta_l\big(\sum_i\lambda_i^{1/\sigma}\big)^\sigma(\bar e_l-\bar\theta_l)^{-\sigma}\), so \(\frac{\gamma_l}{\gamma_k}=\frac{\beta_l}{\beta_k}\big(\frac{\bar e_k-\bar\theta_k}{\bar e_l-\bar\theta_l}\big)^\sigma\). Aggregation. The sharing rule is \(x^i_l=\theta^i_l+\frac{\lambda_i^{1/\sigma}}{\sum_j\lambda_j^{1/\sigma}}(\bar e_l-\bar\theta_l)\).
Pattern: (b), (c) and (e), like Exercise 25.2, are all \(g(H(\pm(x-\theta^i)))\) with \(g\) and the HD1 \(H\) common. (a) breaks "identical" and (d) breaks "homothetic".
Periods \(t=1,2,\dots\). Each generation lives two periods and contains types \(i=1,\dots,I\), with preferences as in Exercise 25.2 over \((x^{t,i}_t,x^{t,i}_{t+1})\), weights \((\pi_y,\pi_o)\) and type-specific satiation points \((\theta^i_y,\theta^i_o)\). Type \(i\) is endowed with \(\alpha^i_y\) when young and \(\alpha^i_o\) when old. The initial old have \(\alpha^i_o\) and utility \(x^{0,i}_1\). Let \(\bar\alpha_y,\bar\alpha_o,\bar\theta_y,\bar\theta_o\) be the cohort sums, with \(\bar\alpha\ll\bar\theta\).
(a) No / Yes / Yes. Types trade within the cohort because their autarky MRSs differ. The initial old eat their endowment, and induction on cohort totals rules out trade across generations, so each cohort consumes \((\bar\alpha_y,\bar\alpha_o)\). Every cohort is the same two-good economy, so \(r\) is constant. (The exam text prints "\(I-\bar\alpha\)" in the second question. That is a leftover from the problem-set version, where young endowments were \(1-\alpha_n\). The intended quantity is \(\bar\alpha_y\).) Full solutions are in Exercise 16.2 and Exercise 10.5.
(b) By Exercise 25.2(k) with goods \((y,o)\) and aggregate \((\bar\alpha_y,\bar\alpha_o)\): \(\dfrac{1}{1+r}=\dfrac{\pi_o}{\pi_y}\Big(\dfrac{\bar\theta_o-\bar\alpha_o}{\bar\theta_y-\bar\alpha_y}\Big)^\sigma\).
(c) \(1/(1+r)=\pi_o/\pi_y\lt1\), so \(r>0\). The value of the aggregate endowment is finite, so the OLG version of the First Welfare Theorem applies and the equilibrium is Pareto optimal.
(d) By §6, \(\frac{p_t}{p_{t+1}}=\frac{\gamma_y}{\gamma_o}=\frac{\beta_y}{\beta_o}\Big(\frac{\bar\alpha_o-\bar\theta_o}{\bar\alpha_y-\bar\theta_y}\Big)^\sigma\), so $$1+r=\frac{\beta_y}{\beta_o}\left(\frac{\bar\alpha_o-\bar\theta_o}{\bar\alpha_y-\bar\theta_y}\right)^{\sigma}.$$ The equilibrium is not Pareto optimal when \(r\lt0\), i.e. when \(\beta_y(\bar\alpha_o-\bar\theta_o)^\sigma\lt\beta_o(\bar\alpha_y-\bar\theta_y)^\sigma\): the cohort is relatively rich in supernumerary terms when young, or values old age highly. Then the value of the aggregate endowment is infinite, and a small transfer from every young to every old at every date improves everyone (Lesson 11).
Let \(u^i(x)=g^i(h(x-\theta^i))\) with \(h\) HD1, increasing and concave, and \(g^i\) strictly increasing.
(a) The budget is \(p\cdot\hat x\le w-p\cdot\theta^i\equiv\hat w\) with \(\hat x=x-\theta^i\). Since \(g^i\) is increasing, \(i\) maximizes \(h(\hat x)\) subject to it. If \(d(p)\) solves this at \(\hat w=1\), then for \(\hat w>0\) the bundle \(\hat w\,d(p)\) is affordable, and any affordable \(\hat x\) satisfies \(h(\hat x)=\hat w\,h(\hat x/\hat w)\le\hat w\,h(d(p))=h(\hat w\,d(p))\). So \(\hat x^i=\hat w\,d(p)\).
(b) Summing, \(\sum_i\hat x^i=\big(\sum_i\hat w^i\big)d(p)=p\cdot(\bar e-\bar\theta)\,d(p)\). Clearing requires this to equal \(\bar e-\bar\theta\), so \(\bar e-\bar\theta\) must be the optimal translated bundle at prices \(p\). Its FOC is \(\nabla h(\bar e-\bar\theta)=\nu p\). Hence \(p\propto\nabla h(\bar e-\bar\theta)\), which involves only aggregates.
(c) \(\hat w^i=p\cdot(e^i-\theta^i)>0\): each household must be able to afford its own translation bundle at the equilibrium prices. Otherwise its problem has no interior solution (Stone–Geary utility is \(-\infty\) below subsistence), the linear demand formula fails, and aggregation can break. Gorman aggregation is always "for interior demands".
(d) \(x^i_l=\theta^i_l+(w-p\cdot\theta^i)d_l(p)\) contains a fixed component \(\theta^i_l\) that is bought regardless of wealth. The elasticity is \(\frac{\partial x^i_l}{\partial w}\frac{w}{x^i_l}=\frac{d_l w}{\theta^i_l+(w-p\cdot\theta^i)d_l}\), which is below 1 iff \(\theta^i_l>p\cdot\theta^i\,d_l\). With a single translated good this holds automatically, since then \(p\cdot\theta^i=p_l\theta^i_l\) and \(p_ld_l\lt1\). Generally, goods with relatively large subsistence requirements have wealth elasticities below one: they are necessities.
For (c), solve \(\pi_se^{-\rho_ix_s}=\mu_ip_s\) for \(x_s\), substitute into the budget to find \(\log\mu_i\), and substitute back. (This is inspired by a 2022 problem set on risk sharing.)
(a) Optimal translated consumption is \(\hat x=\hat w\,d(p)\) with \(h(d(p))=1/P(p)\): the cheapest unit of \(h\) costs \(P\), so a dollar buys \(1/P\) units. Hence \(v^i=h(\hat w\,d)=\hat w/P=(w-p\cdot\theta^i)/P(p)\). Roy's identity: \(\partial v/\partial w=1/P\) and \(\partial v/\partial p_l=-\theta^i_l/P-(w-p\cdot\theta^i)P_l/P^2\), so $$x^i_l=-\frac{\partial v/\partial p_l}{\partial v/\partial w}=\theta^i_l+(w-p\cdot\theta^i)\frac{P_l(p)}{P(p)},\qquad\frac{\partial x^i_l}{\partial w}=\frac{P_l(p)}{P(p)},$$ which is common to all \(i\) (the Gorman form with \(b=1/P\)).
(b) \(x^i_l=\beta^i_lw/p_l\), so \(\partial x^i_l/\partial w=\beta^i_l/p_l\), which differs across \(i\). A dollar moved from \(j\) to \(i\) changes aggregate demand for \(l\) by \((\beta^i_l-\beta^j_l)/p_l\ne0\), so prices must adjust.
(c) The FOC gives \(x^i_s=-\frac{1}{\rho_i}\log\frac{\mu_ip_s}{\pi_s}\). The budget \(\sum_sp_sx^i_s=w\) gives \(\log\mu_i=-\frac{\rho_iw+\sum_sp_s\log(p_s/\pi_s)}{\sum_sp_s}\). Substituting back, $$x^i_s=\frac{w}{\sum_{s'}p_{s'}}+\frac{1}{\rho_i}\Big[\frac{\sum_{s'}p_{s'}\log(p_{s'}/\pi_{s'})}{\sum_{s'}p_{s'}}-\log\frac{p_s}{\pi_s}\Big].$$ The wealth slope \(1/\sum_{s'}p_{s'}\) is the same for everyone: the Gorman form. So aggregate demand depends only on aggregate wealth, and so do equilibrium prices. On the planner side, \(\gamma_s/\gamma_{s'}=\frac{\pi_s}{\pi_{s'}}\exp\big(-(\bar e_s-\bar e_{s'})/\sum_j\rho_j^{-1}\big)\), independent of \(\lambda\). The representative agent is CARA with risk tolerance \(\sum_j1/\rho_j\). (The domain must allow negative consumption, which CARA does.)
Two goods, households with \(u^i=\beta^i\log x_1+(1-\beta^i)\log x_2\) and proportional endowments \(e^i=\delta^i\bar e\), with \(\sum_i\delta^i=1\).
(a) Let \(W=p\cdot\bar e\), so \(w^i=\delta^iW\). Clearing good 1: \(p_1\bar e_1=\sum_i\beta^iw^i=\bar\beta_\delta W\), where \(\bar\beta_\delta\equiv\sum_i\delta^i\beta^i\). Similarly \(p_2\bar e_2=(1-\bar\beta_\delta)W\). So $$\frac{p_1}{p_2}=\frac{\bar\beta_\delta}{1-\bar\beta_\delta}\cdot\frac{\bar e_2}{\bar e_1},$$ which depends on the distribution \(\delta\) through the wealth-weighted taste \(\bar\beta_\delta\), unless \(\beta^i\equiv\beta\).
(b) With log utility and taste weights summing to one, \(V^i=\log w^i+\text{terms in }p\), so \(\mu_i=\partial V^i/\partial w=1/w^i\) and \(\lambda_i=w^i=\delta^iW\). Richer households get larger weights, as in Lesson 23.
(c) From §2, \(\frac{\gamma_1}{\gamma_2}=\frac{\sum_i\lambda_i\beta^i}{\sum_i\lambda_i(1-\beta^i)}\frac{\bar e_2}{\bar e_1}=\frac{W\bar\beta_\delta}{W(1-\bar\beta_\delta)}\frac{\bar e_2}{\bar e_1}\), which matches (a).
(d) No. Household \(i\) consumes \(x^i_1=\beta^iw^i/p_1=\frac{\beta^i}{\bar\beta_\delta}\delta^i\bar e_1\), which is not \(\delta^i\bar e_1\) unless \(\beta^i=\bar\beta_\delta\). Households with a stronger taste for good 1 trade good 2 for it. With identical tastes, nobody wants to trade away from a proportional endowment, which is why the Lesson 24 CE is autarky.