Mathematical Economics

The fundamental theorem of welfare economics states that an allocation is Pareto efficient if and only if it is socially optimal, meaning that it maximizes the total welfare of society.

The Nash equilibrium is a set of strategies for the players in a game such that no player can improve their outcome by unilaterally changing their strategy. This means that each player's strategy is a best response to the strategies of the other players.

The expected utility hypothesis states that individuals make decisions based on the expected value of the outcomes of their actions. This means that they weigh the probability of each possible outcome by the value they place on that outcome and then choose the action that has the highest expected value.

The Cobb-Douglas production function is a production function that exhibits constant returns to scale. This means that if all inputs are increased by a certain percentage, output will increase by the same percentage.

The Solow growth model is a model of economic growth that focuses on the role of capital accumulation. The model assumes that the economy is characterized by constant returns to scale and that there is a fixed supply of labor. The model shows that the steady-state growth rate of the economy is determined by the rate of technological progress.

The Black-Scholes model is a model for pricing options. The model assumes that the underlying asset price follows a geometric Brownian motion and that there are no transaction costs or taxes. The model can be used to price a variety of options, including call options, put options, and straddles.

The efficient frontier is the set of all portfolios that have the highest expected return for a given level of risk. The efficient frontier is a graphical representation of the relationship between expected return and risk. Investors can use the efficient frontier to choose a portfolio that meets their risk and return objectives.

The capital asset pricing model (CAPM) is a model that explains the relationship between the expected return and risk of an asset. The CAPM assumes that investors are rational and that they diversify their portfolios. The CAPM shows that the expected return of an asset is equal to the risk-free rate plus a risk premium. The risk premium is determined by the asset's beta, which is a measure of the asset's systematic risk.

The arbitrage pricing theory (APT) is a model that explains the relationship between the expected return and risk of an asset. The APT assumes that investors are rational and that they diversify their portfolios. The APT shows that the expected return of an asset is equal to the risk-free rate plus a risk premium. The risk premium is determined by the asset's exposure to a number of risk factors. These risk factors are typically macroeconomic factors, such as inflation, interest rates, and economic growth.

The rational expectations hypothesis is the hypothesis that individuals form expectations about the future based on all available information. This means that individuals use all of the information that is available to them to make predictions about the future. The rational expectations hypothesis is important because it implies that individuals' expectations are not biased. This means that the economy is more likely to be stable and predictable.

The Lucas critique is the critique that economic policies that are successful in one context may not be successful in another context. This is because economic policies can change the structure of the economy, which can make the policies less effective. The Lucas critique is important because it implies that economic policies should be tailored to the specific context in which they are being implemented.

The time inconsistency problem is the problem that arises when a government cannot commit to a future policy. This is because the government may have an incentive to change its policy in the future, even if it knows that this will make the economy worse off. The time inconsistency problem is important because it implies that governments should be careful about making promises about future policies.

The principal-agent problem is the problem that arises when one party (the agent) has more information than the other party (the principal). This can lead to a conflict of interest between the two parties, as the agent may have an incentive to act in their own self-interest, even if this is not in the best interests of the principal. The principal-agent problem is important because it can lead to inefficiencies and market failures.

The adverse selection problem is the problem that arises when one party (the seller) has more information about the quality of a good or service than the other party (the buyer). This can lead to a situation where the seller is able to sell a good or service for a higher price than it is actually worth. The adverse selection problem is important because it can lead to inefficiencies and market failures.

The moral hazard problem is the problem that arises when one party (the insured) has more information about the likelihood of a loss than the other party (the insurer). This can lead to a situation where the insured is able to take more risks than they would if they were fully insured. The moral hazard problem is important because it can lead to inefficiencies and market failures.