Are Exchange Economies Inherently Quasi-Linear?

External question — Economics Stack Exchange Author: Joseph Basford Original post: https://economics.stackexchange.com/questions/59825 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. When solving general equilibrium models, we can often appeal to the welfare theorems to reduce the dimension of the problem by removing prices and focusing solely on the allocation. Moreover, if the conditions are met such that the second welfare theorem holds and price quasi-equilibria are price equilibria (primarily requiring convexity of preferences, among other less stringent requirements), then an allocation being Paretian is necessary and sufficient for it to be supportable by some price vector as an equilibrium allocation. Usually when solving this sort of problem, I introduce arbitrary welfare weights for all individuals (as primarily warranted by strict convexity) and solve to get the Pareto allocations. However, I often hear it said that we can typically further restrict attention to allocations which maximise $\textbf{total}$ welfare, i.e., we need not consider arbitrary welfare weights but instead can consider a uniform distribution over individuals' utilities. Whilst I understand this has something to do with an inherent degree of quasi-linearity in these problems (see this question and answer for the connection to maximising total welfare) I cannot seem to get a proper grasp regarding from where it is appearing. I include my current (very shaky) understanding below. MWG Ch. 10, parts C and D seem closely related to what I am asking. From what I can tell, they solve the problem in a two-stage dual-primal approach. First, they fix some good indexed by $\ell$ within the economy. Second, they solve the expenditure minimisation problem of minimising expenditure on all other goods $j\neq \ell$ subject to achieving some utility goal $\bar{u}$ and given the agent is already consuming a fixed amount $x_{\ell}$ . More explicitly, this step solves $$e(p,\bar{u},x_{\ell})=\min_{\{x_j\}_{j\neq \ell}} \sum_{j\neq \ell}p_jx_k $$ $$st. \text{ } \begin{cases} u(x_1,\dots, x_L)\geq \bar{u} \\ x_{\ell},p \text{ are given} \end{cases}$$ Finally, they use this expenditure function and something like additive separability of utility to reduce the agent's problem to solving $$\max_{\bar{u},x_{\ell}} \bar{u}+u_{\ell}(x_{\ell})$$ $$st. \qquad e(p,\bar{u},x_{\ell})+p_{\ell}x_{\ell}\leq p\cdot \tilde{e}$$ Where $\tilde{e}$ is the agent's endowment. Now, assuming additive separability of the agent's utility function, I think the expenditure function $e(p,\bar{u},x_{\ell})$ can be adjusted to be independent of $x_{\ell}$ by replacing $\bar{u}$ with $\bar{u}-u_{\ell}(x_{\ell})$ . However, MWG eventually get that the optimisation problem is something of the form $$\max_{m,x_{\ell}} m+u_{\ell}(x)$$ $$m+p_{\ell}x_{\ell}\leq p\cdot \tilde{e}$$ where $m$ is the quasi-linearity numeraire. How do I get to this? And what assumptions do I really need to make to get this to work? It seems like additive separability is required, is there anything else? Edit I may have produced an XY problem in this question. I give an explanation of where my question comes from here. In a past paper I did a while ago there was the following question. Consider two consumers $j \in \{A, B\}$ , thought of as countries that can employ labour to produce output on their land. Denote land by $x_A,x_B\geq 0$ and normalize its quantity so that $x_A + x_B = 1$ . Labour can originate from either country $m \in {A, B}$ and be employed in either country $n \in \{A, B\}$ . We denote the labour originating from country $m$ and employed in country $n$ as $\ell_m^n\geq 0$ ; country $j$ 's net cross-country labour income is given by $\iota^j$ . For both $j \in \{A, B\}$ symmetric utility is $$u^j(x^j,\ell_A^A,\ell_B^A,\ell_A^B,\ell_B^B,\iota^j)=(x^j)^{\alpha}(\ell_A^j+\ell_B^j)^{1-\alpha}+\iota^j-\frac{(\ell_j^A+\ell_j^B)^2}{2}$$ We assume throughout that $\alpha \in (0, 1)$ . Additionally labour income $\iota^j$ is given by $$\iota^j=\ell_j^{-j}w^{-j}-\ell^{j}_{-j}w^j$$ One of the sub-questions was whether any Pareto outcome maximises the sum of utilities. In the exam I managed to convince myself it did, but the solutions said this followed by quasi-linearity of preferences. It is not obvious to me how utility is quasi-linear here, besides the fact labour income enters linearly. Why exactly is it quasi-linear?

External answer — Economics Stack Exchange Author: Michael Greinecker Original post: https://economics.stackexchange.com/a/59830 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. The relevant parts in MWG are concerned with a quasi-linear partial equilibrium model in which there is essentially a single good under focus and a composite good (the one that utilities are linear in). Here is a more abstract approach to the relationship between quasi-linearity and "surplus maximization." There is a set $X$ of outcomes and a set $N$ of $n$ agents with agent $i$ having a utility function over outcomes and money in quasi-linear form $(x,m_i)\mapsto v_i(x)+m_i$ . An allocation is an element of $X\times\mathbb{R}^n$ . We say the allocation $(x,m_1,\ldots,m_n)$ is efficient if there is no Pareto-better allocation $(x',m_1',\ldots,m')$ such that $\sum_i m_i=\sum_i m_i'$ . Theorem: The allocation $(x,m_1,\ldots,m_n)$ is efficient if and only if $\sum_i v_i(x)\geq\sum_i v_i(x')$ for all $x'\in X$ . Proof: First, we show that an allocation that does not solve the maximization problem cannot be efficient. Suppose there is $x'\in X$ such that $$\sum_{i=1}^n v_i(x)< \sum_{i=1}^n v_i(x').$$ Let $e=\sum_i v_i(x')-\sum_i v_i(x)$ . By assumption, $e$ is positive. Let $$m_i'=m_i+v_i(x)-v_i(x')+e/n$$ for each $i\in N$ . Then $$v_i(x')+m_i'=v_i(x')+m_i+v_i(x)-v_i(x')+e/n=v_i(x)+m_i+e/n >v_i(x)+m_i$$ and $$\sum_{i=1}^n m_i=\sum_{i=1}^n m_i +\sum_{i=1}^n v_i(x)-\sum_{i=1}^n v_i(x')+e=$$ $$=\sum_{i=1}^n (m_i+v_i(x)-v_i(x')+e/n)=\sum_{i=1}^n m_i'.$$ Next, we show that an allocation that is not efficient does not solve the maximization problem. Suppose $(x,m)=(x,m_1,\ldots,m_n)$ is not efficient. Then there exists some outcome $(x',m')=(x',m_1',\ldots,m_n')$ with $$v_i(x')+m_i'\geq v_i(x)+m_i$$ for all $i\in N$ , $$v_i(x')+m_i'> v_i(x)+m_i$$ for some $i\in N$ , and $$\sum_i m_i=\sum_i m_i'.$$ Summing up inequalities: $$\sum_{i=1}^n \Big(v_i(x')+m_i'\Big)>\sum_{i=1}^n \Big(v_i(x)+m_i\Big),$$ $$\sum_{i=1}^n v_i(x')+\sum_{i=1}^n m_i'>\sum_{i=1}^n v_i(x)+\sum_{i=1}^n m_i.$$ Since $\sum_i m_i=\sum_i m_i'$ , this is equivalent to $$\sum_{i=1}^n v_i(x')>\sum_{i=1}^n v_i(x).$$