2. Aggregation and Social Planner's Problem

Chapter theme: aggregation and the social planner's problem. Linking the "centralized (social planner)" and "decentralized (competitive equilibrium)" problems, and discussing aggregation. §2.1 Equivalence of centralized & decentralized problems: Definition 2.1 utility possibility set \(\mathbf U\); Assumption CC (\(u^i\) concave); Theorem 2.1 under HH/CC/FF + \(u^i\) strictly increasing, \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is Pareto optimal \(\iff\) there exists a weight \(\boldsymbol\lambda\in\mathbb R^I_+\) under which it solves the weighted problem \(W:\max\sum_i\lambda_i u^i(\mathbf x^i)\); the representative agent \(u(\mathbf x)=\max\sum_i\lambda_i u^i(\mathbf x^i)\) s.t. \(\sum_i\mathbf x^i=\mathbf x\) (Remark 2.1: two angles on the same economy). §2.2 Pure exchange economy: aggregate endowment \(\bar{\mathbf e}=\sum_i\mathbf e^i\), feasibility \(\bar{\mathbf e}\ge\sum_i\mathbf x^i\). §2.2.1 planner's-problem FOC \(\lambda_i\frac{\partial u^i}{\partial x^i_\ell}=\gamma_\ell\) (2.3) (MRS equal across agents, free of \(\boldsymbol\lambda\); \(\gamma\) = social shadow value). §2.2.2 household-problem FOC \(\frac{\partial u^i}{\partial x^i_\ell}=\mu_i p_\ell\) (2.4). §2.2.3 competitive eq \(\Rightarrow\) planner's problem (set \(\lambda_i=1/\mu_i\), \(\gamma_\ell=p_\ell\)) \(\Rightarrow\) Pareto optimal. §2.2.4 Pareto optimal \(\Rightarrow\) planner's problem (set \(\mu_i=1/\lambda_i\), \(p_\ell=\gamma_\ell\)) \(\Rightarrow\) competitive eq. §2.2.5 Aggregation: Definition 2.2 aggregation (price \(\mathbf p\) free of \(\boldsymbol\lambda\) / individual endowments); Definition 2.3 homotheticity; Proposition 2.1 homothetic utility + proportional endowments \(\mathbf e^i=\delta^i\bar{\mathbf e}\) \(\Rightarrow\) aggregation, autarky \(\bar{\mathbf x}^i=\delta^i\bar{\mathbf e}\) (2.9), prices \(p_\ell=\kappa\frac{\partial h(\bar{\mathbf e})}{\partial x_\ell}\) (2.10).

Definition 2.1 (Utility possibility set) Define the utility possibility set \(\mathbf U\) as the set of all utilities that are attainable in some feasible allocation: $$\mathbf U\equiv\left\{\mathbf u=\begin{pmatrix}u^1\\u^2\\\vdots\\u^I\end{pmatrix}\in\mathbb R^I:u^i\le u^i(\mathbf x^i),\ \text{for }\forall i\in\mathbf I\text{ and for all feasible allocation }\{\mathbf x^i,\mathbf y^j\}\right\}$$

Assumption CC The utility function \(u^i(\mathbf x^i)\) is concave for \(\forall i\in\mathbf I\).

Theorem 2.1 Suppose that \(u^i\) is strictly increasing for \(\forall i\), and also impose assumptions HH, CC and FF. Then \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is a Pareto optimal allocation if and only if there exists a vector \(\boldsymbol\lambda\in\mathbb R^I_+\) such that \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves the problem \(W\): $$\max_{\{\mathbf x^i,\mathbf y^j\}}\sum_{i\in\mathbf I}\lambda_i u^i(\mathbf x^i)\quad\text{s.t.}\quad\{\mathbf x^i,\mathbf y^j\}\text{ is a feasible allocation}$$

Proof (\(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves \(W\) \(\Rightarrow\) Pareto optimal) Again, we will prove it by contradiction. Suppose that \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is not Pareto optimal, then there must be a feasible allocation \(\{\mathbf x^i,\mathbf y^j\}\) that Pareto dominates it, which means $$u^i(\mathbf x^i)\ge u^i(\bar{\mathbf x}^i),\ \text{for }\forall i\in\mathbf I;\qquad u^{i'}(\mathbf x^{i'})>u^{i'}(\bar{\mathbf x}^{i'}),\ \text{for some }i'\in\mathbf I$$ Suppose that the weights for agents \(i'\) are strictly positive, i.e. \(\lambda_{i'}>0\), then $$\sum_{i\in\mathbf I}\lambda_i u^i(\bar{\mathbf x}^i)<\sum_{i\in\mathbf I}\lambda_i u^i(\mathbf x^i)$$ which contradicts with the fact that \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves the problem \(W\).

So the only way to avoid this immediate contradiction is by setting the weights for agents \(i'\) who are strictly better off with \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) to zero and all agents with positive weights must be indifferent between the two allocations. But then, since the agents who are better off now have zero weight, we can simply take away all their endowments and redistribute to whoever has strictly positive weights. By the local non-satiation assumption, the agents with positive weights who receive the redistributed extra endowments will be strictly better off, which should strictly increase the value of \(\sum_{i\in\mathbf I}\lambda_i u^i(\mathbf x^i)\) since people who lose their endowments have zero weights and thus make no difference. So this redistribution plan based on \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\), which is feasible, yields higher value of \(\sum_{i\in\mathbf I}\lambda_i u^i(\mathbf x^i)\) and it again contradicts with the fact that \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves the problem \(W\). \(\blacksquare\)

Proof (Pareto optimal \(\Rightarrow\) \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves \(W\) for some \(\boldsymbol\lambda\in\mathbb R^I_+\)) Define $$\mathbf A\equiv\left\{\hat{\mathbf u}\in\mathbb R^I:\mathbf u=\begin{pmatrix}u^1=u^1(\bar{\mathbf x}^1)\\u^2=u^2(\bar{\mathbf x}^2)\\\vdots\\u^I=u^I(\bar{\mathbf x}^I)\end{pmatrix}\right\}\ \text{for }\forall i\in\mathbf I,\qquad\mathbf B\equiv\text{int}(\mathbf U)$$ Since \(u^i\) is strictly increasing, \(\mathbf A\) is a singleton (a set with only one element). Since \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) is a Pareto optimal allocation, it must be the case that \(\mathbf A\cap\mathbf B=\varnothing\), otherwise \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) would not be Pareto optimal by non-local satiation.

Assumptions HH, CC and FF imply that both \(\mathbf A\) and \(\mathbf B\) are convex. Then we can use the Separating Hyperplane Theorem to argue that there must be a vector \(\boldsymbol\lambda^\star\in\mathbb R^I\), \(\boldsymbol\lambda^\star\ne\mathbf 0\) such that $$\boldsymbol\lambda^\star\cdot\hat{\mathbf u}\ge\boldsymbol\lambda^\star\cdot\mathbf u,\quad\text{for }\forall\mathbf u\in\mathbf B\tag{2.1}$$ Since elements in \(\mathbf B\) can be arbitrarily close to its closure, and \(\mathbf U\) is the closure of \(\mathbf B\), by taking limits we can substitute \(\mathbf U\) for \(\mathbf B\) and the inequality still holds weakly, i.e. $$\boldsymbol\lambda^\star\cdot\hat{\mathbf u}\ge\boldsymbol\lambda^\star\cdot\mathbf u,\quad\text{for }\mathbf u\in\mathbf U\tag{2.2}$$ Therefore, \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves the problem \(W\). Then we only need to show that we can choose non-negative weights. For \(i'\) such that \(\lambda^\star_{i'}<0\), it must be true that \(\hat u^{i'}=0\), otherwise there would be a feasible allocation \(\{\bar{\mathbf x}'_0,\bar{\mathbf y}'_0\}\) where everyone has exactly the same allocation as \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) except that all the endowment of agent \(i'\) is transferred to some agent with strictly positive weight; denote such allocation's utility singleton as \(\hat{\mathbf u}_0\), and it satisfies \(\boldsymbol\lambda^\star\cdot\hat{\mathbf u}_0\ge\boldsymbol\lambda^\star\cdot\hat{\mathbf u}\), contradicting that \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves \(W\). So everyone with negative weight must have zero utility, and thus we can change that weight from negative to zero without changing the LHS of (2.2) while making the RHS weakly smaller, so (2.2) still holds. Thus, \(\{\bar{\mathbf x}^i,\bar{\mathbf y}^j\}\) solves \(W\) for non-negative weights \(\boldsymbol\lambda\). \(\blacksquare\)

Remark 2.1 The economy of many agents and the economy with only one representative agent are not really different economies in the sense that both economies have the same set of agents and their utility functions. Even the representative agent has his utility function \(u\) defined on the same set of agents as in the counterpart. So the "two economy" statement just means we can view the same economy from two angles, both of which give us the same solution of the equilibrium condition.

Remark 2.2 Each element \(\lambda_i\) in the weight vector \(\boldsymbol\lambda\) is economically the marginal value of an extra dollar for agent \(i\). This is intuitive because for agents with higher weight, they will naturally end up consuming more of each good, which makes their marginal utility lower than the agents with lower weight.

Remark 2.3 Each element \(\gamma_\ell\) in the Lagrangian vector \(\boldsymbol\gamma\) can be interpreted as the marginal utility to the society for one extra unit of good \(\ell\) in equilibrium. Note that this is true because every agent has the same \(\gamma_\ell\) for \(\forall\ell\in\{1,2,\dots,m\}\), which means that the economy as a whole will increase its value by the same amount whoever the extra unit of good is given to.

Remark 2.4 If the solution \(\{\bar{\mathbf x}^i\}\) to the social planner's problem \(W\) is a function of weight vector \(\boldsymbol\lambda\), then by the ratio above the vector \(\boldsymbol\gamma\) is thus also a function of weight vector \(\boldsymbol\lambda\) (which means that there is no aggregation as we will introduce later); on the opposite, if \(\{\bar{\mathbf x}^i\}\) does not involve weight vector \(\boldsymbol\lambda\), then the vector \(\boldsymbol\gamma\) does not involve weight vector \(\boldsymbol\lambda\) either.

Remark 2.5 For agents with higher Lagrangian multiplier \(\mu_i\), their marginal utility of one extra dollar is higher, which, by assumption CC, means that they must consume relatively less.

Remark 2.6 Each element \(p_\ell\) in the price vector \(\mathbf p\) can be interpreted as the marginal utility measured in dollars to the society for one extra unit of good \(\ell\). Note that this is true because every agent has the same \(p_\ell\) for \(\forall\ell\in\{1,2,\dots,m\}\).

Remark 2.7 If the solution \(\{\bar{\mathbf x}^i\}\) to the social planner's problem \(W\) is a function of vector \(\boldsymbol\mu\), then the vector \(\mathbf p\) is a function of \(\boldsymbol\mu\); on the opposite too.

Remark 2.8 Since \(\lambda_i=\frac{1}{\mu_i}\), now it becomes clear that agents assigned with high weight end up consuming more than other agents and thus have lower marginal utility for an extra unit of dollar.

Definition 2.2 (Aggregation) There is aggregation in the economy if the equilibrium price vector \(\mathbf p\) (or the shadow value of social endowment, vector \(\boldsymbol\gamma\)) do not involve the weight vector \(\boldsymbol\lambda\), i.e. independent of individual endowments \(\{\mathbf e^i\}\).

Remark 2.9 When there is aggregation in the economy, each agent making their own decisions will yield exactly the same allocation as a representative agent, so we can treat each agent in that economy as a representative agent.

Definition 2.3 (Homotheticity) A homothetic function is defined as a monotonic transformation of a function that is homogeneous of degree one (h.o.d. 1). I.e. \(u^i\) is homothetic if \(u^i(\mathbf x)=g^i(h(\mathbf x))\) where \(g^i\) is strictly increasing and concave, and \(h\) is h.o.d. 1, i.e. \(h(\alpha\mathbf x)=\alpha h(\mathbf x)\) for \(\forall\alpha>0\), \(\forall\mathbf x\).

Proposition 2.1 If utility functions are homothetic and endowments are proportional to aggregate endowment, i.e. \(\exists\delta^i>0\) s.t. \(\mathbf e^i=\delta^i\bar{\mathbf e}\) for \(\forall i\in\mathbf I\), then there is aggregation in the economy. And the competitive equilibrium allocation, which is Pareto optimal, is autarky, i.e. $$\bar{\mathbf x}^i=\delta^i\bar{\mathbf e},\quad\text{for }\forall i\in\mathbf I\tag{2.9}$$ and the equilibrium prices are $$p_\ell=\kappa\frac{\partial h(\bar{\mathbf e})}{\partial x_\ell},\quad\text{for }\forall\ell\in\{1,2,\dots,m\}\tag{2.10}$$ where \(\kappa>0\) is an arbitrarily chosen constant. Equation (2.9) and equation (2.10) imply that neither \(\mathbf p\) nor \(\{\bar{\mathbf x}^i\}\) depend on the weight vector \(\boldsymbol\lambda\).

Proof We want to show that the allocation \(\{\bar{\mathbf x}^i\}\) and the price vector \(\mathbf p\) depicted in (2.9) and (2.10) satisfy the f.o.c. of competitive equilibrium.

Recall the f.o.c. for the household's utility maximization problem given by equation (2.4), and plug in the specific (homothetic) utility function: $$\frac{\partial g^i(h(\mathbf x^i))}{\partial h}\frac{\partial h(\mathbf x^i)}{\partial x_\ell}=\mu_i p_\ell$$ for \(\forall\ell\in\{1,2,\dots,m\}\) and \(\forall i\in\mathbf I\). Then we want to verify that the allocation \(\{\bar{\mathbf x}^i\}\) and the price vector \(\mathbf p\) depicted in (2.9) and (2.10) satisfy this f.o.c., so plug them in and we hope the following is true: $$\begin{aligned}&\frac{\partial g^i(h(\delta^i\bar{\mathbf e}))}{\partial h}\frac{\partial h(\delta^i\bar{\mathbf e})}{\partial x_\ell}=\mu_i\kappa\frac{\partial h(\bar{\mathbf e})}{\partial x_\ell}\\[2pt]\Big[\because\ \frac{\partial h(\bar{\mathbf e})}{\partial x_\ell}\text{ is h.o.d. 0}\Big]\Rightarrow{}&\frac{\partial g^i(h(\delta^i\bar{\mathbf e}))}{\partial h}\frac{\partial h(\bar{\mathbf e})}{\partial x_\ell}=\mu_i\kappa\frac{\partial h(\bar{\mathbf e})}{\partial x_\ell}\\[2pt]\Rightarrow{}&\frac{\partial g^i(h(\delta^i\bar{\mathbf e}))}{\partial h}=\mu_i\kappa\end{aligned}$$ (\(h\) is h.o.d. 1, so its partials \(\frac{\partial h}{\partial x_\ell}\) are h.o.d. 0, hence \(\frac{\partial h(\delta^i\bar{\mathbf e})}{\partial x_\ell}=\frac{\partial h(\bar{\mathbf e})}{\partial x_\ell}\).) This is easily true if we set $$\mu_i=\frac{1}{\lambda_i}=\frac{\kappa}{\partial g^i(h(\delta^i\bar{\mathbf e}))/\partial h}\quad\text{for }\forall i\in\mathbf I$$

So, now we have found an allocation \(\{\bar{\mathbf x}^i\}\) and an equilibrium price vector \(\mathbf p\) that give us a competitive equilibrium (and a Pareto optimal allocation) that are independent of the weight vector \(\boldsymbol\lambda\) (i.e. independent of individual endowments \(\{\mathbf e^i\}\)). \(\blacksquare\)

Remark 2.10 When we want to check whether an economy has aggregation, simply start with the f.o.c. for the social planner's problem, and then calculate any two entries of vector \(\boldsymbol\gamma\) and take the ratio between them (which is the relative price) to see if there is aggregation. Or equivalently, check if each entry in vector \(\boldsymbol\gamma\) involves individual weight (involving total weight may still have aggregation).