How can we ensure the allocation of resources is optimized?
General equilibrium theory is a mathematical framework that helps us understand how markets interact with each other to reach a state of equilibrium, where supply equals demand for all goods and services in the economy. This theory is fundamental in understanding how prices are determined in a competitive market and how the allocation of resources is optimized.
It is based on the Walrasian system, which assumes that there is perfect competition, no externalities, and all agents have perfect information. In this system, all markets are interconnected, and the prices of goods and services are determined by the interaction of supply and demand in all markets simultaneously.
To understand how the general equilibrium theory works, let's consider an example problem: imagine a small economy with two goods: apples and oranges. There are 100 consumers who want to buy these goods, and there are two firms that produce them. The production costs for each firm are as follows:
It costs Firm A $1 to produce an apple and $2 to produce an orange
It costs Firm B $2 to produce an apple and $1 to produce an orange
The consumers' utility functions are as follows:
Utility from an apple: U(A) = 10 - A
Utility from an orange: U(O) = 10 - O
The problem is to find the equilibrium prices and quantities of each good. To solve it using the general equilibrium theory, we need to find the price at which the quantity demanded equals the quantity supplied in both markets. We can start by assuming some initial prices and quantities and then adjust them until we reach the equilibrium.
Let's say we start with a price of $1 for apples and $2 for oranges. At these prices, Firm A will produce 50 apples and 25 oranges, while Firm B will produce 25 apples and 50 oranges. The total demand for apples and oranges is also 75 each, so there is an excess supply of oranges and excess demand for apples.
To eliminate the excess supply and demand, we need to adjust the prices. We can increase the price of apples and decrease the price of oranges until the excess supply and demand are eliminated. After several iterations, we can find that the equilibrium prices are $1.50 for apples and $1.50 for oranges. At these prices, each firm will produce 37.5 apples and 37.5 oranges, and the total demand for each good will also be 75, resulting in a state of equilibrium.
This example shows how the general equilibrium theory can help us achieve stability in a Walrasian system. By using mathematical models to understand the interactions of supply and demand in all markets, we can find the prices and quantities that will ensure that all goods and services are allocated efficiently, and the economy is in a state of equilibrium. By using math to study these phenomena, we can gain insights that are not always visible through observation alone and develop effective policies to promote economic stability and growth.
