**Advanced Economics Theorems**

**Arrow’s Impossibility Theorem:** states that it is impossible to design a fair and efficient voting system that satisfies certain basic criteria.

**Aumann’s Agreement Theorem:** states that under certain conditions, two Bayesian rational agents with common prior beliefs cannot agree to disagree.

**Bishop–Cannings theorem:** states that any uncorrelated random process can be modeled as a moving average of a white noise process.

**Bondareva–Shapley theorem:** characterizes the core of a cooperative game, which is the set of feasible payoffs that cannot be improved upon by a subset of the players.

**Coase Theorem:** states that in a situation where private property rights are well-defined and transaction costs are low, the allocation of resources will be efficient regardless of the initial distribution of property rights.

**Debreu Theorems:** several theorems in mathematical economics, named after Gerard Debreu, that provide a rigorous foundation for the theory of general equilibrium.

**Dorfman–Steiner theorem:** states that in a cooperative game, the optimal solution is to divide the total surplus among the players in proportion to their marginal contributions to the game.

**Duggan–Schwartz theorem:** states that under certain conditions, market-clearing prices and a welfare-maximizing allocation will coincide.

**Edgeworth’s Limit Theorem:** states that in an exchange economy, the limit of the sequence of Walrasian equilibria approaches Pareto efficiency.

**Efficient Envy-Free Division:** refers to a way of dividing a resource among multiple individuals in a fair and efficient manner, such that no one envies any other person’s share.

**Envelope Theorem:** states that the change in the value of an optimization problem’s objective function is equal to the change in the value of the corresponding Lagrangian multiplier.

**Factor Price Equalization:** states that in a perfectly competitive global market, the price of any factor of production (such as labor or capital) will equalize across countries.

**Fisher Separation Theorem:** states that a firm can separate ownership from management by offering shares of stock to investors.

**Frisch–Waugh–Lovell theorem:** states that under certain conditions, the coefficients in a multiple regression model can be obtained by regressing the dependent variable on the residuals from a previous regression.

**Fundamental Theorems of Welfare Economics:** provide conditions under which a market economy will lead to a Pareto efficient allocation of resources, and conditions under which government intervention can improve upon the market allocation.

**Gibbard–Satterthwaite theorem:** states that in a voting system, it is impossible to design a system that is both strategy-proof and Pareto efficient.

**Gibbard’s Theorem:** states that in a voting system, it is impossible to design a system that is both strategy-proof and allows for more than three alternatives.

**Heckscher–Ohlin theorem:** states that a country will export the goods that it can produce most efficiently, and import the goods that it produces less efficiently.

**Henry George Theorem:** states that in a model of economic growth, the tax revenue generated by land will eventually equal the rental value of the land.

**Holmström’s Theorem:** states that in a principal-agent problem, the optimal contract will make the agent’s effort dependent on their own performance, but not on their ability or the performance of other agents.

**Intensity of Preference: r**efers to the degree of preference or attachment an individual has for a certain good or outcome.

**Kuhn’s Theorem:** states that in a non-cooperative game, any solution that is obtained through a sequence of best responses will be a Nash equilibrium.

**Lerner Symmetry Theorem:** states that in a market with perfectly competitive firms, the market demand curve is identical to the average cost curve.

**Liberal Paradox:** states that in a liberal democracy, individual freedom and equality can be in conflict with one another.

**Modigliani–Miller theorem:** states that under certain conditions, the total market value of a firm is independent of its financing structure (debt versus equity).

**Moving Equilibrium Theorem:** states that in a market with prices that are determined endogenously, a change in demand will cause the market to move to a new equilibrium.

**Mutual Fund Separation Theorem:** states that in a market with well-diversified investors, the market portfolio will be efficient and the prices of individual assets will reflect their risk.

**Nakamura Number:** refers to the maximum number of players in a cooperative game that can form a coalition.

**No-trade Theorem:** states that under certain conditions, there will be no trade between two countries, even if they have different endowments and production technologies.

**Okishio’s Theorem:** states that in a competitive market, an increase in productivity will lead to an increase in real wages.

**Roy’s Identity:** states that the wage rate and the reservation wage of a worker are equal.

**Rybczynski theorem:** states that an increase in the endowment of one factor of production will lead to an increase in the output of the good that uses that factor intensively.

**Shephard’s Lemma:** states that the optimal production plan for a firm can be obtained by equating the marginal rate of substitution between inputs with the relative prices of inputs.

**Sonnenschein–Mantel–Debreu theorem:** states that in a market with a large number of agents, the aggregate demand function will be smooth, regardless of the distribution of individual demands.

**Stolper–Samuelson theorem:** states that an increase in the price of a good will raise the real income of the factor of production that is intensively used in the production of that good.

**Topkis’s Theorem:** states that in a partially ordered set, the highest (or lowest) element of the set that is dominated (or dominating) by a given element can be found by comparing the element to the minimal (or maximal) elements of the set.

**Uzawa’s Theorem:** states that under certain conditions, a change in the endowment of one factor of production will have a positive effect on the output of all goods.

**Weller’s Theorem:** states that in a cooperative game, the value of the game is equal to the sum of the values of its subgames.