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 pr ...
, a consumer's preferences are called homothetic if they can be represented by a
utility function
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 economics, normative context, utility refers to a goal or ob ...
which is
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
of degree 1.
For example, in an economy with two goods
, homothetic preferences can be represented by a utility function
that has the following property: for every
:
::
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a homothetic function is a
monotonic transformation of a function which is
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
;
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 as ...
functions are only defined up to an increasing
monotonic transformation, 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
In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within their given income. Consumer theory uses the concepts of a budget constraint and a preference map ...
, the ratios of goods demanded by consumers will depend only on
relative prices, 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. F ...
or scale. This translates to a linear
expansion path in income: the slope of indifference curves is constant along rays beginning at the origin.
This is to say, the
Engel curve for each good is linear.
Furthermore, the
indirect utility function can be written as a linear function of wealth
:
::
which is a special case of the
Gorman polar form. 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) is a common specification of many production functions and utility function
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term ...
(CES) are homothetic. They can be represented by a utility function such as:
:
This function is homogeneous of degree 1:
:
Linear utilities,
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 are not always homothetic. E.g, the function
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 is a branch of economics that deals with the performance, structure, behavior, and decision-making of an economy as a whole. This includes regional, national, and global economies. Macroeconomists study topics such as output (econ ...
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 wit ...
). 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
In economics, elasticity of intertemporal substitution (or intertemporal elasticity of substitution, EIS, IES) is a measure of responsiveness of the Economic growth, growth rate of consumption (economics), consumption to the real interest rate. If ...
, 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 (mathematics), transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point ...
*
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 p ...
References
{{DEFAULTSORT:Homothetic Preferences
Utility function types