Aggregation and Market Demand: An Exterior Differential Calculus Viewpoint
用外微分工具研究向量场能否分解为梯度线性组合，证明满足瓦尔拉斯定律的解析映射可局部分解为n个个体效用最大化需求函数之和，且分解随映射连续变化。
We analyze under which conditions a given vector field can be disaggregated as a linear combination of gradients. This problem is typical of aggregation theory, as illustrated by the literature on the characterization of aggregate market demand and excess demand. We argue that exterior differential calculus provides very useful tools to address these problems. In particular, we show, using these techniques, that any analytic mapping in Rn satisfying Walras Law can be locally decomposed as the sum of n individual, utility-maximizing market demand functions. In addition, we show that the result holds for arbitrary (price-dependent) income distributions, and that the decomposition can be chosen such that it varies continuously with the mapping. Finally, when income distribution can be freely chosen, then decomposition requires only n/2 agents.