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Indirect Utility
__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 wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interactions of Agent (economics), economic agents and how economy, economies work. Microeconomics analyses what is viewed as basic elements within economy, economies, including individual agents and market (economics), markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyses economies as systems where production, distribution, consumption, savings, and Expenditure, investment expenditure interact; and the factors of production affecting them, such as: Labour (human activity), labour, Capital (economics), capital, Land (economics), land, and Entrepreneurship, enterprise, inflation, economic growth, and public policies that impact gloss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish to maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original utilitarian concept, developed by moral philosophers such as Jeremy Bentham and John Stuart Mill. * In a descriptive context, the term refers to an ''apparent'' objective function; such a function is revealed by a person's behavior, and specifically by their preferences over lotteries, which can be any quantified choice. The relationship between these two kinds of utility functions has been a source of controversy among both economists and ethicists, with most maintaining that the two are distinct but generally related. Utility function Consider a set of alternatives among which a person has a preference ordering. A utility fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Income
Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For example, a person's income in an economic sense may be different from their income as defined by law. An extremely important definition of income is Haig–Simons income, which defines income as ''Consumption + Change in net worth'' and is widely used in economics. For households and individuals in the United States, income is defined by tax law as a sum that includes any wage, salary, profit, interest payment, rent, or other form of earnings received in a calendar year.Case, K. & Fair, R. (2007). ''Principles of Economics''. Upper Saddle River, NJ: Pearson Education. p. 54. Discretionary income is often defined as gross income minus taxes and other deductions (such as mandatory pension contributions), and is widely used as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Maximization Problem
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my utility?" It is a type of Optimal decision, optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a Natural borrowing limit, constraint on total spending (income), the prices of the goods and their Preference (economics), preferences. Utility maximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are modelled as being Rational choice theory, rational, they seek to extract the most benefit for themselves. However, due to bounded rationality and other biases, consumers sometimes pick bundles that do not necessarily maximize their utility. The utility maximization bundle of the consumer is also not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Function
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar (mathematics), scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the ''degree''. That is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. This is also referred to a ''th-degree'' or ''th-order'' homogeneous function. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain of a function, domain and codomain are vector spaces over a Field (mathematics), field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconvex Function
In mathematics, a quasiconvex function is a real number, real-valued function (mathematics), function defined on an interval (mathematics), interval or on a convex set, convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. ''Univariate'' Unimodality, unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple argument of a function, arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with Star_domain, star-convex sublevel sets can be unimodal without being quasiconvex. Def ... [...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] |
Expenditure Function
In microeconomics, the expenditure function represents the minimum amount of expenditure needed to achieve a given level of utility, given a utility function and the prices of goods. Formally, if there is a utility function u that describes preferences over ''n ''goods, the expenditure function e(p, u^*) is defined as: :e(p, u^*) = \min_ p \cdot x where p is the price vector u^* is the desired utility level, \geq(u^*) = \ is the set of providing at least utility u^*. Expressed equivalently, the individual minimizes expenditure x_1p_1+\dots +x_n p_n subject to the minimal utility constraint that u(x_1, \dots , x_n) \ge u^*, giving optimal quantities to consume of the various goods as x_1^*, \dots x_n^* as function of u^* and the prices; then the expenditure function is :e(p_1, \dots , p_n ; u^*)=p_1 x_1^*+\dots + p_n x_n^*. Properties Suppose u is a continuous utility function representing a locally non-satiated preference relation on \textbf R^n_+. Then e(p, u^*) is # Homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gorman Polar Form
Gorman polar form is a functional form for indirect utility functions in economics. Motivation Standard consumer theory is developed for a single consumer. The consumer has a utility function, from which his demand curves can be calculated. Then, it is possible to predict the behavior of the consumer in certain conditions, price or income changes. But in reality, there are many different consumers, each with his own utility function and demand curve. How can we use consumer theory to predict the behavior of an entire society? One option is to represent an entire society as a single "mega consumer", which has an aggregate utility function and aggregate demand curve. But in what cases is it indeed possible to represent an entire society as a single consumer? Formally: consider an economy with n consumers, each of whom has a demand function that depends on his income m^i and the price system: :x^i(p,m^i) The aggregate demand of society is, in general, a function of the price system ... [...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] |
Value Function
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters of the problem. In a controlled dynamical system, the value function represents the optimal payoff of the system over the interval , t1/var> when started at the time-t state variable x(t)=x. If the objective function represents some cost that is to be minimized, the value function can be interpreted as the cost to finish the optimal program, and is thus referred to as "cost-to-go function." In an economic context, where the objective function usually represents utility, the value function is conceptually equivalent to the indirect utility function. In a problem of optimal control, the value function is defined as the supremum of the objective function taken over the set of admissible controls. Given (t_, x_) \in , t_\times \mathbb^, a typical optimal control problem is to : \text \quad J(t_, x_; u) = \int_^ I(t,x(t), u(t)) \, \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
