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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] |
Microeconomics
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] |
Paul Samuelson
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "has done more than any other contemporary economist to raise the level of scientific analysis in economic theory". "In a career that spanned seven decades, he transformed his field, influenced millions of students and turned MIT into an economics powerhouse" Samuelson was one of the most influential economists of the latter half of the 20th century."Paul Samuelson: The last of the great general economists died on December 13th, aged 94" ''The Economist'', December 17, 2009 In 1996, he was awarded the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consumer Theory
The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption (as measured by their preferences subject to limitations on their expenditures), by maximizing utility subject to a consumer budget constraint. Factors influencing consumers' evaluation of the utility of goods include: income level, cultural factors, product information and physio-psychological factors. Consumption is separated from production, logically, because two different economic agents are involved. In the first case, consumption is determined by the individual. Their specific tastes or preferences determine the amount of utility they derive from goods and services they consume. In the second case, a producer has different motives to the consumer in that they are focussed on the profit they make. This is explained further by producer theory. The models ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Preferences
In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". This implies that the consumer prefers a variety of goods to having more of a single good. The concept roughly corresponds to the concept of marginal utility#Diminishing marginal utility, diminishing marginal utility without requiring utility functions. Notation Comparable to the greater-than-or-equal-to Order theory#Partially ordered sets, ordering relation \geq for real numbers, the notation \succeq below can be translated as: 'is at least as good as' (in Preference (economics), preference satisfaction). Similarly, \succ can be translated as 'is strictly better than' (in preference satisfaction), and Similarly, \sim can be translated as 'is equivalent to' (in preference satisfaction). Definition Use ''x'', ''y'', and ''z'' to denote three consumpti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hotelling's Lemma
Hotelling's lemma is a result in microeconomics that relates the supply of a good to the maximum profit of the producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. Specifically, it states: ''The rate of an increase in maximized profits with respect to a price increase is equal to the net supply of the good.'' In other words, if the firm makes its choices to maximize profits, then the choices can be recovered from the knowledge of the maximum profit function. Formal statement Let p denote a variable price, and w be a constant cost of each input. Let x:\rightarrow X be a mapping from the price to a set of feasible input choices X\subset . Let f:\rightarrow be the production function, and y(p)\triangleq f(x(p)) be the net supply. The maximum profit can be written by :\pi (p) = \max_ p\cdot y(p) - w \cdot x(p). Then the lemma states that if the profit \pi is differentiable at p, the maximizing net supply is given by :y^*(p) = \frac . Proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marshallian Demand Function
In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the standard demand function. It is a solution to the utility maximization problem of how the consumer can maximize their utility for given income and prices. A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in their real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect. Although Marshallian demand is in the context of partial equilibrium theory, it is sometimes called Walrasian demand as used in general equilibrium theory (named after Léon Walras). According to the utility maximization problem, there are L commodities with price vector p ... [...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] |
Roy's Identity
Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function. Specifically, denoting the indirect utility function as v(p,w), the Marshallian demand function for good i can be calculated as :x_^(p,w)=-\frac where p is the price vector of goods and w is income, and where the superscript ^m indicates Marshallian demand. The result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments. Roy's identity is akin to the result that the price derivatives of the expenditure function give the Hicksian demand functions. The additional step of dividing by the wealth derivative of the indirec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envelope Theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models. The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions \left\ _ that are optimized. Statement Let f(x,\alpha) and g_(x,\alpha), j = 1,2, \ldots, m be real-valued continuously differentiable functions on \mathbb^, where x \in \mathbb^ are choice variables and \alpha \in \mathbb^ are parameters, and consider the problem of choosing x, for a given \alpha, so as to: : \max_ f(x, \alpha) subject to g_(x,\alpha) \geq 0, j = 1,2, \ldots, m and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Factor Demands
In economics, a conditional factor demand is the cost-minimizing level of an input (factor of production) such as labor or capital, required to produce a given level of output, for given unit input costs ( wage rate and cost of capital) of the input factors. A conditional factor demand function expresses the conditional factor demand as a function of the output level and the input costs.Varian, Hal., 1992, ''Microeconomic Analysis'' 3rd Ed., W.W. Norton & Company, Inc. New York. The conditional portion of this phrase refers to the fact that this function is conditional on a given level of output, so output is one argument of the function. Typically this concept arises in a long run context in which both labor and capital usage are choosable by the firm, so a single optimization gives rise to conditional factor demands for each of labor and capital. Since the optimal mix of input levels depends on the wage and rental rates, these rates are also arguments of the conditional demand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Row And Column Vectors
In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some , consisting of a single row of entries, \boldsymbol a = \begin a_1 & a_2 & \dots & a_n \end. (Throughout this article, boldface is used for both row and column vectors.) The transpose (indicated by ) of any row vector is a column vector, and the transpose of any column vector is a row vector: \begin x_1 \; x_2 \; \dots \; x_m \end^ = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end and \begin x_1 \\ x_2 \\ \vdots \\ x_m \end^ = \begin x_1 \; x_2 \; \dots \; x_m \end. The set of all row vectors with entries in a given field (such as the real numbers) forms an -dimensional vector space; similarly, the set of all column vectors with entries forms an -dimensional vector space. The space of row vectors with entries can be regarded as the dual spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
