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Integrability Of Demand
In microeconomic theory, the problem of the integrability of demand functions deals with recovering a utility function (that is, consumer preferences) from a given walrasian demand function. The "integrability" in the name comes from the fact that demand functions can be shown to satisfy a system of partial differential equations in prices, and solving (integrating) this system is a crucial step in recovering the underlying utility function generating demand. The problem was considered by Paul Samuelson in his book Foundations of Economic Analysis, and conditions for its solution were given by him in a 1950 article. More general conditions for a solution were later given by Leonid Hurwicz and Hirofumi Uzawa. Mathematical formulation Given consumption space X and a known walrasian demand function x: \mathbb_^ \times \mathbb_ \rightarrow X , solving the problem of integrability of demand consists in finding a utility function u: X \rightarrow \mathbb such that : x(p, w) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microeconomic Theory
Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. Microeconomics focuses on the study of individual markets, sectors, or industries as opposed to the economy as a whole, which is studied in macroeconomics. One goal of microeconomics is to analyze the market mechanisms that establish relative prices among goods and services and allocate limited resources among alternative uses. Microeconomics shows conditions under which free markets lead to desirable allocations. It also analyzes market failure, where markets fail to produce Economic efficiency, efficient results. While microeconomics focuses on firms and individuals, macroeconomics focuses on the total of economic activity, dealing with the issues of Economic growth, growth, inflation, and unemployment—and with national policies relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PDEs
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shephard's Lemma
Shephard's lemma is a result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost-minimizing point of a given good (i) with price p_i is unique. The idea is that a consumer will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market. The lemma is named after Ronald Shephard, who proved it using the distance formula in his book ''Theory of Cost and Production Functions'' in 1953. The equivalent result in the context of consumer theory was first derived by Lionel W. McKenzie in 1957. It states that the partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods. Similar results had already been derived by John Hicks (1939) and Paul Samuelson (1947). Definition In consumer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hicksian Demand Function
In microeconomics, a consumer's Hicksian demand function (or compensated demand function) represents the quantity of a good demanded when the consumer minimizes expenditure while maintaining a fixed level of utility. The Hicksian demand function illustrates how a consumer would adjust their demand for a good in response to a price change, assuming their income is adjusted (or compensated) to keep them on the same indifference curve—ensuring their utility remains unchanged. Mathematically, :h(p, \bar) = \arg \min_x \sum_i p_i x_i : \ \ u(x) \geq \bar . where h(p,u) is the Hicksian demand function or commodity bundle demanded, at price vector p and utility level \bar. Here p is a vector of prices, and x is a vector of quantities demanded, so the sum of all p_ix_i is the total expenditure on all goods. The Hicksian demand function isolates the effect of relative prices on demand, assuming utility remains constant. It contrasts with the Marshallian demand function, which acco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indirect Utility Function
__NOTOC__ In economics, a consumer's indirect utility function v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of income w. It reflects both the consumer's preferences and market conditions. This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices. A consumer's indirect utility v(p, w) can be computed from their utility function u(x), defined over vectors x of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector x(p, w) by solving the utility maximization problem, and second, computing the utility u(x(p, w)) the consumer derives from that bundle. The resulting indirect utility function is :v(p,w)=u(x(p,w)). The indirect utility function is: *Continuous on R''n''+ × R+ where ''n'' is the number of goods; *Decreasing in prices; *Strictly increasing in income; * Homogenous wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expenditure Minimization Problem
In microeconomics, the expenditure minimization problem is the dual of the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts. Given a consumer's utility function, prices, and a utility target, * how much money would the consumer need? This is answered by the expenditure function. * what could the consumer buy to meet this utility target while minimizing expenditure? This is answered by the Hicksian demand function. Expenditure function Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function u defined on L commodities. Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices p that give utility of at least u^*, :e(p, u^*) = \min_ p \cdot x where :\geq = \ is the set of all packages that give utility at least as good as u^*. Hicksian demand correspondence Hicksian demand is define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (optimization)
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slutsky Equation
In microeconomics, the Slutsky equation (or Slutsky identity), named after Eugen Slutsky, relates changes in Marshallian (uncompensated) demand to changes in Hicksian (compensated) demand, which is known as such since it compensates to maintain a fixed level of utility. There are two parts of the Slutsky equation, namely the substitution effect and income effect. In general, the substitution effect is negative. Slutsky derived this formula to explore a consumer's response as the price of a commodity changes. When the price increases, the budget set moves inward, which also causes the quantity demanded to decrease. In contrast, if the price decreases, the budget set moves outward, which leads to an increase in the quantity demanded. The substitution effect is due to the effect of the relative price change, while the income effect is due to the effect of income being freed up. The equation demonstrates that the change in the demand for a good caused by a price change is the resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
