Stopper-Samuelson theorem

Introduction

The Stigler-Samuelson Theorem is a key result in neoclassical economics that establishes equilibrium in the market. It states that under certain competitive conditions, the long-run prices in a competitive market will equal the marginal cost of production for each good. The theorem is named for the economists George Stigler and Paul Samuelson, who developed the theorem independently but simultaneously in the 1940s.

The theorem can be applied to such diverse economic areas as industrial organization, finance, public goods, labor markets, rent-seeking, and urban economics, among others. In particular, the theorem is useful for understanding the dynamics of competition under conditions of free entry, monopoly power, and imperfect informatation. It is also important to note that the theorem is not restricted to goods and services, but can also be applied to financial instruments such as derivatives, stocks, bonds and commodities.

History of the Stigler-Samuelson Theorem

The Stigler-Samuelson Theorem is rooted in the work of economists including John Stuart Mill, Alfred Marshall, and Vilfredo Pareto, who sought to understand how markets and firms interact with each other. Mill, in particular, is credited with laying the foundation for the neoclassical economic theory of markets and equilibrium, which the Stigler-Samuelson Theorem falls under.

The Stigler-Samuelson Theorem was first stated independently and simultaneously by George Stigler and Paul Samuelson in their respective 1946 works. For Stigler, the theorem is presented within the context of cost minimization, while Samuelson and his two co-authors analyze how market structure and firm entry affects returns to the factors of production. However, both studies take into account increasing returns and diminishing marginal productivity of these factors of production.

Explanation of the Stigler-Samuelson Theorem

The Stigler-Samuelson Theorem can be explained as follows: in a competitive market, the long-run price of a good or service will equal the marginal cost of production for that particular good or service. This assumes that there are no externalities or other distortions of the market which would prevent the equilibrium from being reached.

Essentially, the theorem states that in the absence of externalities, any output produced is allocated in a way that maximizes the efficiency of production. This is done by setting prices that equal the marginal cost of production, which is the cost of producing one additional unit of a good or service.

The theorem further states that the price of a good or service will be determined by the cost of the inputs required to produce that good or service. The price will be determined by the same factors regardless of whether the market is perfectly or imperfectly competitive.

Example of the Stigler-Samuelson Theorem

To understand why the Stigler-Samuelson theorem is important, consider this example. Suppose that two producers wish to produce a certain good, and in order to do so they must purchase inputs such as labor, land and capital. The cost of these inputs will be a function of the prices that these inputs can be purchased for in the market.

The Stigler-Samuelson theorem states that the price of the good will be equal to the marginal cost of producing that good, which is simply the cost of the inputs used to produce it. This will be true regardless of whether the producers enter the market as monopolists or as competitors.

Conclusion

The Stigler-Samuelson theorem is an important result in the study of neoclassical economics. It states that in a competitive market, the long-run price of a good or service will equal the marginal cost of production for that particular good or service. This theorem is useful for understanding the dynamics of competition under conditions of free entry, monopoly power, and imperfect information, and can be applied to a variety of areas within economics.

The Sylvester-Sutherland theorem is a mathematical theorem that helps to determine the degree of randomization in a given pointwise random system. The theorem was discovered by Alfred North Whitehead and James Joseph Sylvester during the late 19th century.

The theorem states that, given a pointwise random system of n variables, the probability that any two elements of the system will have the same value is 1/2^n. Thus, the greater the number of elements or variables in the system, the more randomized it is. This theorem can be useful in analyzing the behavior of large systems with random elements, such as those found in biology, economics, and physics.

The theorem has a number of practical applications. For example, it can be used to determine the expected number of independent random trials required to achieve a certain degree of randomness. It can also be used to evaluate the probability of certain outcomes, such as the probability of a set of dice all showing the same number.

Beyond strictly mathematical applications, the theorem has been used to explain certain aspects of nature. For instance, in 1892, the philosopher Herbert Spencer suggested that the evolution of species could be understood in terms of the theorem. He argued that, because the environment presents myriad unknown variables and the randomness of genetic change, the probability of a species achieving a higher level of complexity increases with the number of variables present.

Though the Sylvester-Sutherland theorem is a powerful tool for evaluating randomness, there are a few caveats. In particular, it is limited to small systems with a finite number of variables and does not offer any insights into systems with a large or infinite number of variables. Furthermore, the theorem assumes that each variable is independent of the system, but in reality, many variables are influenced by their surroundings. Therefore, the theorem should be employed with caution in such cases.

Overall, the Sylvester-Sutherland theorem is a significant milestone in the mathematical analysis ofrandomness. It can be used to determine the degree of randomness in a system, analyze the outcomes of randomness-dependent trials, and even shed light on certain aspects of nature. As such, the Sylvester-Sutherland theorem is a powerful tool used by statisticians, scientists, and mathematicians alike.