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I'm trying to understand the relation between maximizing sum of utilities and finding Pareto efficient allocations. According to https://econ.ucsb.edu/~tedb/Courses/UCSBpf/pflectures/chap2.pdf (page 13), concavity plays an important role in whether maximizing the sum of utilities will help obtain the efficient allocations or not. Could someone explain this further?

A similar question is posted at Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?, but the counterexample in this answer seems contradictory to the result stated in the above document.

Specifically, my questions are:

  1. When can we maximize the sum of utilities to find all the efficient allocations?
  2. What criteria can be used to locate all the efficient allocations in corner/edge cases?
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  • $\begingroup$ What do you mean when you write "the efficient allocation"? There are usually several Pareto-efficient allocations. Do you just want to find one? $\endgroup$ – Giskard Jul 30 at 9:04
  • $\begingroup$ Your first link says on another page "it is true that maximizing the sum of utilities is a sufficient condition for Pareto optimality. But maximizing the sum of utilities is not a necessary condition for Pareto optimality." $\endgroup$ – Henry Jul 30 at 9:58
  • $\begingroup$ @Giskard Have edited my question. I am interested in finding the Pareto set, all efficient allocations. $\endgroup$ – PGupta Jul 30 at 11:42
  • $\begingroup$ @Henry Yes, so my understanding is that if the sum of utilities is maximized at an allocation, it is Pareto efficient. But the discussion at the same link seems to suggest that this can lead to wrong answers, hence the confusion. $\endgroup$ – PGupta Jul 30 at 11:46
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Finding Pareto efficient allocations via social planner's problems is a special case of scalarization in convex optimization.

Suppose agents' utility functions are $u_i:\mathbb{R}^n \rightarrow \mathbb{R}$, for $i = 1, 2, \cdots, m$, and the feasible allocations are given by $g(x) \geq 0$ for some $g:\mathbb{R}^n \rightarrow \mathbb{R}^p$. The general result is as follows.

Sufficiency under general conditions

If an allocation $x$ solves the social planner's problem with strictly positive social weights, then it is Pareto. This is true with no assumptions on utility functions and feasibility constraint. In other words, if $$ x \in \arg\max_{g(x') \geq 0} \sum_{i = 1}^m \lambda_i u_i(x'), \mbox{ for some } \lambda \in \mathbb{R}^m_{++}, \quad (*) $$ then it is Pareto optimal.

This is easy to see. If $y$ Pareto-improves $x$, then $$ \sum_{i = 1}^m \lambda_i u_i(y) > \sum_{i = 1}^m \lambda_i u_i(x) $$ for any $\lambda \in \mathbb{R}^m_{++}$. It is also easy to see that zero entry in $\lambda$ cannot be allowed---it would make the statement false.

The converse is not true in general---this sufficient condition is not necessary. There are Pareto allocations that do not solve $(*)$. This is because the set of achievable utility values $$ \mathcal{U} = \{ (u_1 (x), \cdots, u_m(x)); g(x) \geq 0 \} $$ is in general not a convex subset of $\mathbb{R}^m$. The social planner's weights $\lambda$ corresponds to certain supporting hyperplanes of $\mathcal{U}$. If $\mathcal{U}$ is not convex, its Pareto frontier cannot be recovered by varying $\lambda$.

However, there is a partial converse.

Necessity under concavity

Assume $u_i$'s and $g$ are concave. If an allocation $x$ is Pareto, then it solves a social planner's problem with nonnegative social weights. In other words, if $x$ is Pareto then $$ x \in \arg\max_{g(x') \geq 0} \sum_{i = 1}^m \lambda_i u_i(x') \mbox{ for some } \lambda \in \mathbb{R}^m_{+}. \quad (**) $$ This is true because of convex sets and separating hyperplanes.

This is a partial converse because it's necessary but not sufficient. There are allocations that solve $(**)$ but not Pareto.

Comment

Even under concavity, $(**)$ is a only a partial converse of $(*)$. Notice the gap between strictly positive and nonnegative social weights.

To recover the Pareto frontier, the general procedure is to first find solutions to $(*)$. Then check the solutions to $(**)$ case-by-case for Pareto optimality. Alternatively, one can also take the closure (the limit points) of solutions to $(*)$.

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