Hicksian Demand
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In
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. M ...
, 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 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 ...
. The Hicksian demand function illustrates how a consumer would adjust their
demand In economics, demand is the quantity of a goods, good that consumers are willing and able to purchase at various prices during a given time. In economics "demand" for a commodity is not the same thing as "desire" for it. It refers to both the desi ...
for a good in response to a price change, assuming their income is adjusted (or compensated) to keep them on the same
indifference curve In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
—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 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 s ...
, which accounts for both the substitution effect and the reduction in real income caused by price changes. The function is named after
John Hicks Sir John Richard Hicks (8 April 1904 – 20 May 1989) was a British economist. He is considered one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economics ...
.


Relationship to other functions

Hicksian demand functions are often convenient for mathematical manipulation because they do not require representing income or wealth. Additionally, the function to be minimized is linear in the x_i, which gives a simpler optimization problem. However,
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 s ...
s of the form x(p, w) that describe demand given prices ''p'' and income w are easier to observe directly. The two are related by :h(p, u) = x(p, e(p, u)), \ where e(p, u) is the
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 prefere ...
(the function that gives the minimum wealth required to get to a given utility level), and by :h(p, v(p, w)) = x(p, w), \ where v(p, w) is 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 ...
(which gives the utility level of having a given wealth under a fixed price regime). Their derivatives are more fundamentally related by the
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 ...
. Whereas Marshallian demand comes from the Utility Maximization Problem, Hicksian Demand comes from the Expenditure Minimization Problem. The two problems are mathematical duals, and hence the Duality Theorem provides a method of proving the relationships described above. The Hicksian demand function is intimately related to the
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 prefere ...
. If the consumer's utility function u(x) is locally nonsatiated and strictly convex, then by
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 ...
it is true that h(p, u) = \nabla_p e(p, u).. Note that if there is more than one vector of quantities that minimizes expenditure for the given utility, we have a Hicksian demand correspondence rather than a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
.


Hicksian demand and compensated price changes

Marshallian demand curves show the effect of price changes on quantity demanded. As the price of a good rises, ordinarily, the quantity of that good demanded will fall, but not in every case. The price rise has both a
substitution effect In economics and particularly in consumer choice theory, the substitution effect is one component of the effect of a change in the price of a good upon the amount of that good demanded by a consumer, the other being the income effect. When a ...
and an
income effect 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 ...
. The substitution effect is the change in quantity demanded due to a price change that alters the slope of the budget constraint but leaves the consumer on the same indifference curve (i.e., at the same level of utility). The substitution effect always is to buy less of that good. The income effect is the change in quantity demanded due to the effect of the price change on the consumer's total buying power. Since for the Marshallian demand function the consumer's nominal income is held constant, when a price rises his real income falls and he is poorer. If the good in question is a ''
normal good In economics, a normal good is a type of a Good (economics), good which experiences an increase in demand due to an increase in income, unlike inferior goods, for which the opposite is observed. When there is an increase in a person's income, for ...
'' and its price rises, the income effect from the fall in purchasing power reinforces the substitution effect. If the good is an ''
inferior good In economics, inferior goods are those goods the demand for which falls with increase in income of the consumer. So, there is an inverse relationship between income of the consumer and the demand for inferior goods. There are many examples of infe ...
'', the income effect will offset in some degree to the substitution effect. If the good is a ''
Giffen good In microeconomics and consumer theory, a Giffen good is a product that people consume more of as the price rises and vice versa, violating the law of demand. For ordinary goods, as the price of the good rises, the substitution effect makes ...
'', the income effect is so strong that the Marshallian quantity demanded rises when the price rises. The Hicksian demand function isolates the substitution effect by supposing the consumer is compensated with exactly enough extra income after the price rise to purchase some bundle on the same indifference curve. If the Hicksian demand function is steeper than the Marshallian demand, the good is a normal good; otherwise, the good is inferior. Hicksian demand always slopes down.


Mathematical properties

If the consumer's utility function u(x) is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
and represents a locally nonsatiated preference relation, then the Hicksian demand correspondence h(p,u) satisfies the following properties: * Homogeneity of degree zero in ''p'': For all a>0 , h(ap,u)=h(p,u). This is because the same ''x'' that minimizes \sum_i p_i x_i also minimizes \sum_i ap_i x_i subject to the same constraint.Silberberg E. (2008) Hicksian and Marshallian Demands. In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95121-5_2702-1 * No excess demand: The constraint u(hx) \geq \bar holds with strict equality, u(x) = \bar . This follows from the continuity of the utility function. Informally, they could simply spend less until utility was exactly \bar .


See also

*
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 s ...
*
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 ...
*
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, pr ...
*
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 ...
*
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 th ...
* Hicks–Marshall laws of derived demand


References

* {{DEFAULTSORT:Hicksian Demand Function Demand Eponyms in economics