Hicksian demand function
Encyclopedia
In microeconomics
, a consumer's Hicksian demand correspondence is the demand of a consumer over a bundle of goods that minimizes their expenditure while delivering a fixed level of utility
. If the correspondence
is actually a function
, it is referred to as the Hicksian demand function, or compensated demand function. The function is named after John Hicks
.
Mathematically,
where h(p,u) is the Hicksian demand function, or commodity bundle demanded, at price level p and utility level
. Here p is a vector of prices, and X is a vector of quantities demanded so that the sum of all pixi, is the total expense on goods X.
, which gives a simpler optimization problem. However, Marshallian demand function
s of the form
that describe demand given prices p and income 
are easier to observe directly. The two are trivially related by
where
is the expenditure function
(the function that gives the minimum wealth required to get to a given utility level), and by
where
is the indirect utility function
(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
.
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
. If the consumer's utility function
is locally nonsatiated and strictly convex
, then
, then the income effect from the rise in purchasing power from a price fall reinforces the substitution effect. If the good is an inferior good
, then the income effect will offset in some degree the substitution effect.
The Hicksian demand function is also downward sloping, but isolated the substitution effect by supposing the consumer is compensated exactly enough to purchase some bundle on the same indifference curve. Hicksian demand illustrates the consumer's new consumption basket after the price change while being compensated as to allow the consumer to be as happy as previously (to stay at the same level of utility). If the Hicksian demand function is "steeper" than Walrasian demand, the good is a normal good; otherwise, the good is inferior.
is continuous and represents a locally nonsatiated preference relation, then the Hicksian demand correspondence 
satisfies the following properties:
i. Homogeneity of degree zero in p: For all
, 
. This is because the same x that minimizes 
also minimizes 
subject to the same constraint.
ii. No excess demand: The constraint
holds with strict equality, 
. This follows from continuity of the utility function. Informally, they could simply spend less until utility was exactly 
.
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
, a consumer's Hicksian demand correspondence is the demand of a consumer over a bundle of goods that minimizes their expenditure while delivering a fixed level of utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
. If the correspondence
Correspondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...
is actually a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, it is referred to as the Hicksian demand function, or compensated demand function. The function is named after John Hicks
John Hicks
Sir John Richard Hicks was a British economist and one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economics were his statement of consumer demand theory in microeconomics, and the IS/LM model , which...
.
Mathematically,
where h(p,u) is the Hicksian demand function, or commodity bundle demanded, at price level p and utility level
. Here p is a vector of prices, and X is a vector of quantities demanded so that the sum of all pixi, is the total expense on goods X.Relationship to other functions
Hicksian demand functions are often convenient for mathematical manipulation because they do not require income or wealth to be represented. Additionally, the function to be minimized is linear in the, which gives a simpler optimization problem. However, Marshallian demand functionMarshallian demand function
In microeconomics, a consumer's Marshallian demand function specifies what the consumer would buy in each price and wealth situation, assuming it perfectly solves the utility maximization problem...
s of the form
that describe demand given prices p and income are easier to observe directly. The two are trivially related bywhere
is the expenditure functionExpenditure function
In microeconomics, the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices....
(the function that gives the minimum wealth required to get to a given utility level), and by
where
is the indirect utility functionIndirect utility function
In economics, a consumer's indirect utility functionv gives the consumer's maximal utility when faced with a price level p and an amount of income w. It represents the consumer's preferences over market conditions....
(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
Slutsky equation
The Slutsky equation in economics, named after Eugen Slutsky , relates changes in Marshallian demand to changes in Hicksian demand...
.
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
Expenditure function
In microeconomics, the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices....
. If the consumer's utility function
is locally nonsatiated and strictly convexConvex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
, then
Hicksian Demand and Compensated Price Changes
Downward sloping Marshallian demand curves show the effect of price changes on quantity demanded. As the price of a good rises, presumably the quantity of that good demanded will fall, holding wealth and other prices constant. However, this price changes due to both the income effect and the substitution effect. The substitution effect is 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.) By this effect, the consumer is posited to substitute toward the good that becomes comparatively less expensive. If the good in question is a normal goodNormal good
In economics, normal goods are any goods for which demand increases when income increases and falls when income decreases but price remains constant, i.e. with a positive income elasticity of demand...
, then the income effect from the rise in purchasing power from a price fall reinforces the substitution effect. If the good is an inferior good
Inferior good
In consumer theory, an inferior good is a good that decreases in demand when consumer income rises, unlike normal goods, for which the opposite is observed. Normal goods are those for which consumers' demand increases when their income increases....
, then the income effect will offset in some degree the substitution effect.
The Hicksian demand function is also downward sloping, but isolated the substitution effect by supposing the consumer is compensated exactly enough to purchase some bundle on the same indifference curve. Hicksian demand illustrates the consumer's new consumption basket after the price change while being compensated as to allow the consumer to be as happy as previously (to stay at the same level of utility). If the Hicksian demand function is "steeper" than Walrasian demand, the good is a normal good; otherwise, the good is inferior.
Mathematical Properties
If the consumer's utility function is continuous and represents a locally nonsatiated preference relation, then the Hicksian demand correspondence satisfies the following properties:i. Homogeneity of degree zero in p: For all
, . This is because the same x that minimizes also minimizes subject to the same constraint.ii. No excess demand: The constraint
holds with strict equality, . This follows from continuity of the utility function. Informally, they could simply spend less until utility was exactly .See also
- Marshallian demand functionMarshallian demand functionIn microeconomics, a consumer's Marshallian demand function specifies what the consumer would buy in each price and wealth situation, assuming it perfectly solves the utility maximization problem...
- Convex preferencesConvex preferencesIn economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...
- Expenditure minimization problemExpenditure minimization problemIn microeconomics, the expenditure minimization problem is another perspective on the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts...