Financial Economics

Economic Questions
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July 20, 2023
Question 1.
First Welfare Theorem
At the heart of economic theory lies the First Welfare Theorem, a central proposition that forms the cornerstone of our understanding of market dynamics and resource allocation. Often equated to Adam Smith's conceptualization of the "invisible hand," this theorem makes a profound statement: any competitive equilibrium, also known as a Walrasian equilibrium, gives rise to a Pareto-efficient distribution of resources (Dupont & Durham, 2021).
A Pareto-efficient distribution is a state where resources have been allocated so no individual can be better off without making someone else worse off. This delicate balance is a hallmark of perfect competition, an idealized state where all consumers and producers operate as price takers and market forces flow unhindered.
However, let us delve deeper into the mathematical architecture of this theory to appreciate its complexity and elegance. Imagine an economy populated by a set of consumers, which we will denote as I, and a variety of goods represented by J. Each consumer, identified as I within the set I, has its unique utility function, u_i(x_i), where x_i stands for the number of goods. The utility function is an economic construct that expresses a consumer's satisfaction or happiness derived from consuming a certain quantity of goods. Additionally, each consumer arrives in the market with an initial endowment of goods, ω_i.
In this perfectly competitive market, each consumer I has one overarching goal: to choose a certain quantity of goods, x_i, that will maximize their utility, u_i(x_i). However, this choice is not unlimited. It is constrained by the consumer's budget, which forms a boundary of affordability. Mathematically, this is represented by the budget constraint px_i ≤ pω_i, where p stands for the price vector.
Now, a remarkable thing happens when such a market reaches equilibrium – a state where supply meets demand. The goods are distributed in a way that fulfills Pareto optimality conditions. In essence, it becomes impossible to improve any individual without inflicting a disadvantage on someone else. This is the crux of the First Welfare Theorem.
Let us solidify this understanding with a formal mathematical representation. We say there is no other allocation (x'_i, y') that can satisfy u_i(x'_i) ≥ u_i(x_i) for all i, with strict inequality for some i, and ∑ x'_i ≤ y' + ∑ ω_i. This inequality essentially encapsulates the idea of Pareto optimality, painting a picture of an economic tableau where efficiency reigns supreme.
However, it is essential to remember what the First Welfare Theorem does not say. While it underscores the efficiency of free markets, it remains silent on the question of equity. It does not make any assertions about the fairness of the distribution of goods.
Consider, for instance, a hypothetical economy consisting of two consumers, A and B. Let us assume that in a perfectly competitive market, the equilibrium distribution of goods is such that A gets the majority of goods while B gets a much smaller share. According to the First Welfare Theorem, this is a Pareto-...