Homothetic preferences
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In
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 pref ...
, a consumer's preferences are called homothetic if they can be represented by a
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
which is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of degree 1. For example, in an economy with two goods x,y, homothetic preferences can be represented by a utility function u that has the following property: for every a>0: ::u(a\cdot x,a\cdot y) = a\cdot u(x,y) In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a homothetic function is a
monotonic transformation In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...
of a function which is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
; however, since
ordinal utility In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to a ...
functions are only defined up to an increasing
monotonic transformation In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...
, there is a small distinction between the two concepts in consumer theory. In a model where competitive consumers optimize homothetic utility functions subject to a budget constraint, the ratios of goods demanded by consumers will depend only on
relative price A relative price is the price of a commodity such as a good or service in terms of another; i.e., the ratio of two prices. A relative price may be expressed in terms of a ratio between the prices of any two goods or the ratio between the price o ...
s, not on
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 ...
or scale. This translates to a linear
expansion path In economics, an expansion path (also called a scale lineJain, TR; Khanna OP (2008). ''Economics.'' VK Publications, ) is a path connecting optimal input combinations as the scale of production expands.Hirschey, Mark (2008). ''Managerial econom ...
in income: the slope of indifference curves is constant along rays beginning at the origin. This is to say, the
Engel curve In microeconomics, an Engel curve describes how household expenditure on a particular good or service varies with household income. There are two varieties of Engel curves. Budget share Engel curves describe how the proportion of household income s ...
for each good is linear. Furthermore, the
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 con ...
can be written as a linear function of wealth w: ::v(p_x,p_y,w) = f(p_x,p_y)\cdot w which is a special case of the
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. The ...
. Hence, if all consumers have homothetic preferences (with the same coefficient on the wealth term), aggregate demand can be calculated by considering a single "representative consumer" who has the same preferences and the same aggregate income.


Examples

Utility functions having
constant elasticity of substitution Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McK ...
(CES) are homothetic. They can be represented by a utility function such as: :u(x,y) = \left(\left(\right)^ + \left(\right)^\ \right)^ This function is homogeneous of degree 1: :u(a x,a y) = \left(a^r\left(\right)^ + a^r\left(\right)^\ \right)^ = (a^r)^ \left(\left(\right)^ + \left(\right)^\ \right)^ = a u(x,y)
Linear utilities In economics and consumer theory, a linear utility function is a function of the form: ::u(x_1,x_2,\dots,x_m) = w_1 x_1 + w_2 x_2 + \dots w_m x_m or, in vector form: ::u(\overrightarrow) = \overrightarrow \cdot \overrightarrow where: * m is the n ...
,
Leontief utilities In economics, especially in consumer theory, a Leontief utility function is a function of the form: u(x_1,\ldots,x_m)=\min\left\ . where: * m is the number of different goods in the economy. * x_i (for i\in 1,\dots,m) is the amount of good i in the ...
and Cobb–Douglas utilities are special cases of CES functions and thus are also homothetic. On the other hand,
quasilinear utilities In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n) whe ...
are not always homothetic. E.g, the function u(x,y) = x+\sqrt cannot be represented as a homogeneous function.


Intratemporally vs. intertemporally homothetic preferences

Preferences are intratemporally homothetic if, in the same time period, consumers with different incomes but facing the same prices and having identical preferences will demand goods in the same proportions. Preferences are intertemporally homothetic if, across time periods, rich and poor decision makers are equally averse to proportional fluctuations in consumption. Models of modern
macroeconomics Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics dealing with performance, structure, behavior, and decision-making of an economy as a whole. For example, using interest rates, taxes, and ...
and public finance often assume the constant-relative-risk-aversion form for within period utility (also called the power utility or
isoelastic utility In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with ...
). The reason is that, in combination with additivity over time, this gives homothetic intertemporal preferences and this homotheticity is of considerable analytic convenience (for example, it allows for the analysis of steady states in growth models). These assumptions imply that the
elasticity of intertemporal substitution Elasticity of intertemporal substitution (or intertemporal elasticity of substitution, EIS, IES) is a measure of responsiveness of the growth rate of consumption to the real interest rate. If the real interest rate rises, current consumption may de ...
, and its inverse, the coefficient of (risk) aversion, are constant.


Evidence

However, it is well known that in reality, consumption patterns change with economic affluence. This means that preferences are not actually homothetic. It has long been established that relative price changes affect people differently even if all face the same set of prices.


See also

*
Homothetic transformation In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by the ...
*
Homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...


References

{{DEFAULTSORT:Homothetic Preferences Utility function types