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Encyclopedia > Indifference curve

An indifference curve is a graph showing different bundles of goods, each measured as to quantity, to which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another, as they render the same level of satisfaction (utility) for the consumer. Indifference curves are a device to represent levels of preference and are used in choice theory. Preference (or taste) is a concept, used in the social sciences, particularly economics. ... In economics, utility is a measure of the relative happiness or satisfaction (gratification) gained by consuming different bundles of goods and services. ... Preference (or taste) is a concept, used in the social sciences, particularly economics. ...

An example of how indifference curves are obtained as the level curves of a utility function

1 History
2 Preference Relations and Utility
- 2.1 Preference Relations
- 2.2 Formal link to Utility theory
  - 2.2.1 Examples
3 Map and properties of indifference curves
- 3.1 Assumptions
- 3.2 Examples of Indifference Curves
4 Application
5 References
6 See also

Image File history File links Download high-resolution version (966x2584, 288 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Indifference curve ... Image File history File links Download high-resolution version (966x2584, 288 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Indifference curve ...

History

The theory of indifference curves was developed by Francis Ysidro Edgeworth, Vilfredo Pareto and others in the first part of the 20th century. The theory is derived from ordinal utility theory, which posits that individuals can always rank two consumption bundles by order of preference. Edgeworth // Brief biography Francis Ysidro Edgeworth (February 8, 1845 - February 13, 1926) was an Irish polymath who studied at Trinity College, Dublin before obtaining a scholarship to Balliol College, Oxford where he subsequently became a professor. ... Please wikify (format) this article as suggested in the Guide to layout and the Manual of Style. ... Ordinal utility theory states that while the utility of a particular good and service cannot be measured using an objective scale; a consumer is capable of ranking different alternatives available. ...

Preference Relations and Utility

Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves.

The idea of an indifference curve is a straightforward one: If a consumer was equally satisfied with 1 apple and 4 bananas, 2 apples and 2 bananas, or 5 apples and 1 banana, these combinations would all lie on the same indifference curve. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

Preference Relations

Suppose that the set of alternatives among which a consumer can choose is called $A;$ . Denote a generic element of $A;$ by $a;$ or $b;$ . In the language of the example above, the set $A;$ is made of combinations of apples and bananas. The symbol $a;$ is one such combination, such as 1 apple and 4 bananas and $b;$ is another combination such as 2 apples and 2 bananas.

A preference relation, denoted $geq$ , is a binary relation define on the set $A;$ . In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...

The statement

$ageq b;$

is understood as ' $a;$ is weakly preferred to $b;$ .'

The statement

$asim b;$

is understood as ' $a;$ is weakly preferred to $b;$ , and $b;$ is weakly preferred to $a;$ .' One says that ' $a;$ is indifferent to $b;$ .'

The statement

$a>b;$

is understood as ' $a;$ is weakly preferred to $b;$ , but $b;$ is not weakly preferred to $a;$ .' One says that ' $a;$ is strictly preferred to $b;$ .

The preference relation $geq$ is complete if all pairs $a,b;$ can be ranked. The relation is a transitive relation if whenever $ageq b;$ and $bgeq c,;$ then $ageq c;$ . In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...

Consider a particular element of the set $A;$ , such as $a_0;$ . Suppose one builds the list of all other elements of $A;$ which are indifferent, in the eyes of the consumer, to $a_0;$ . Denote the first element in this list by $a_1;$ , the second by $a_2;$ and so on... The set ${a_i:igeq 0}$ forms an indifference curve since $a_isim a_j;$ for all $i,jgeq 0;$ .

Formal link to Utility theory

In the example above, an element $a;$ of the set $A;$ is made of two numbers: The number of apples, call it $x,;$ and the number of bananas, call it $y.;$

Utility theory posits that when agents can rank all pairs of consumption bundles by order of preference (and this ranking is a transitive relation) then there is a utility function. This means that for each bundle $left(x,yright)$ there is a unique number, $Uleft(x,yright)$ , representing the utility (satisfaction) level associated with $left(x,yright)$ . In economics, utility is a measure of the relative happiness or satisfaction (gratification) gained by consuming different bundles of goods and services. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... This article is about utility in economics and in game theory. ... In economics, utility is a measure of the relative happiness or satisfaction (gratification) gained by consuming different bundles of goods and services. ...

The relation $left(x,yright)to Uleft(x,yright)$ is called the utility function. The range of the function is the real line. The actual value of the function has no meaning. Only the ranking of those values has content for the theory. More precisely, when $U(x,y)geq U(x',y')$ then the bundle $left(x,yright)$ is considered to be at least as good as the bundle $left(x',y'right)$ . When $Uleft(x,yright)>Uleft(x',y'right)$ then the bundle $left(x,yright)$ is strictly preferred to the bundle $left(x',y'right)$ . This article is about utility in economics and in game theory. ...

Consider a particular bundle $left(x_0,y_0right)$ and take the total derivative of $Uleft(x,yright)$ about this point: In mathematics, a total derivative may be either. ...

$dUleft(x_0,y_0right)=U_1left(x_0,y_0right)dx+U_2left(x_0,y_0right)dy$

where $U_1left(x,yright)$ is the partial derivative of $Uleft(x,yright)$ with respect to its first argument, evaluated at $left(x,yright)$ . (Likewise for $U_2left(x,yright).$ )

The indifference curve through $left(x_0,y_0right)$ must deliver, all along, the utility of this particular bundle, that is $Uleft(x_0,y_0right)$ . In other words, if one is to change the quantity of $x,$ by $dx,$ , one must also change the quantity of $y,$ by an amount $dy,$ such that, in the end, there is no change in $U,$ , that is $dU=0,$ . The equation of the total derivative dictates then

$dUleft(x_0,y_0right)=0Leftrightarrowfrac{dx}{dy}=-frac{U_2(x_0,y_0)}{U_1(x_0,y_0)}$

Thus, the ratio of marginal utilities gives the slope of the indifference curve at point $left(x_0,y_0right)$ . This ratio is called the marginal rate of substitution between $x,$ and $y,$ . Look up Slope in Wiktionary, the free dictionary. ... In economics, the marginal rate of substitution (MRS for short) is the rate at which consumers are willing to give up units of one good in exchange for more units of another good. ...

Examples

Linear Utility

If the utility function is of the form $Uleft(x,yright)=alpha x+beta y$ then the marginal utility of $x,$ is $U_1left(x,yright)=alpha$ and the marginal utility of $y,$ is $U_2left(x,yright)=beta$ . The slope of the indifference curve is, therefore,

$frac{dx}{dy}=-frac{beta}{alpha}.$

Observe that the slope does not depend on $x,$ or $y,$ : Indifference curves are straight lines.

Cobb-Douglas Utility

If the utility function is of the form $Uleft(x,yright)=x^alpha y^{1-alpha}$ the marginal utility of $x,$ is $U_1left(x,yright)=alpha left(x/yright)^{alpha-1}$ and the marginal utility of $y,$ is $U_2left(x,yright)=(1-alpha) left(x/yright)^{alpha}$ . The marginal rate of substitution, and therefore the slope of the indifference curve is then In economics, the Cobb-Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. ... In economics, the marginal rate of substitution (MRS for short) is the rate at which consumers are willing to give up units of one good in exchange for more units of another good. ... Look up Slope in Wiktionary, the free dictionary. ...

$frac{dx}{dy}=-frac{1-alpha}{alpha}left(frac{x}{y}right).$

CES Utility

A general CES (Constant Elasticity of Substitution) form is In economics, more specifically econometrics or mathematical economics, there are production functions that describe the output given a certain combination of inputs (e. ...

$U(x,y)=left(alpha x^rho +(1-alpha)y^rhoright)^{1/rho}$

where $alphain(0,1)$ and $rholeq 1$ . (The Cobb-Douglas is a special case of the CES utility, with $rho=1,$ .) The marginal utilities are given by In economics, the Cobb-Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. ...

$U_1(x,y)=alpha left(alpha x^rho +(1-alpha)y^rhoright)^{left(1/rhoright)-1} x^{rho-1}$

and

$U_2(x,y)=(1-alpha)left(alpha x^rho +(1-alpha)y^rhoright)^{left(1/rhoright)-1} y^{rho-1}.$

Therefore, along an indifference curve,

$frac{dx}{dy}=-frac{1-alpha}{alpha}left(frac{x}{y}right)^{1-rho}.$

Map and properties of indifference curves

Figure 1: An example of an indifference map with three indifference curves represented

A graph of indifference curves for an individual consummer associated with different utility levels is called an indifference map. There are many indifference curves on an indifference map. A point on the indifference-curve is associated only one indifference curve (avoiding intersectionn with other indifference curves). But different points yielding different utility levels are each associated with distinct indifference curves. The rational consumer prefers the higher, or right most, indifference curve, since they represent bundles of goods providing higher quantities of goods. An indifference curve describes a set of personal preferences and so varies from person to person. Image File history File links Simple-indifference-curves. ... Image File history File links Simple-indifference-curves. ...

Indifference curves are typically assumed to be:

1. defined only in the positive (+, +) quadrant of commodity quantities.
2. negatively sloped. This represents the assumption that as quantity consumed of one good (X) increases, total satisfaction would increase if not offset by a decrease in the quantity consumed of the other good (Y).
3. stacked on positively-sloped rays (straight lines) from the origin. The positive slope of each ray is associated with higher levels of satisfaction at higher indifferent curves on the ray. (More of both goods increases satisfaction.) This is a representation of the non-satiation assumption. It is a way of tracing the income effect of increased quantities demanded of each good as a budget constraint shifts out parallel to itself.
4. convex and bulge toward the origin. This represents the assumption that as a consumer decreases consumption of one good in successive units, sucessively larger doses of the other good are required to keep satisfaction unchanged, the substitution effect. It is also consistent with a transitive ranking of commodity bundles from lowest level of satisfaction on up.
5. non-intersecting, as intersection would mean that a bundle of two goods renders two different levels of satisfaction, which is impossible. Also, non-intersection avoids intransitivity.
6. ubiquitous. In other words, there exists an indifference curve through any given point on an indifference map

Fig. ... A negative, or inverse relationship is a mathematical relationship in which one variable decreases as another rises. ... Three lines â€” the red and blue lines have same slope, while the red and green ones have same y-intercept. ... In mathematics and statistics, a direct relationship is a positive relationship between two variables in which they both increase or decrease in conjunction. ... Consumer theory relates preferences, indifference curves and budget constraints to consumer demand curves. ... Consumer theory is a theory of economics. ... In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ... Consumer theory relates preferences, indifference curves and budget constraints to consumer demand curves. ...

Assumptions

The first three assumptions are necessary, the next two are convenient.

Rationality: Consumers know their individual preferences and can choose between consumption bundle X and consumption bundle Y. They know either that X is preferred to Y, Y is preferred to X, or that they are indifferent between X and Y.

Consistency: If a consumer chooses bundle X to bundle Y in the first instance and his preferences and income are unchanged, he cannot choose bundle Y to bundle X in the second instance.

Transitivity: If a consumer prefers bundle X to bundle Y, and prefers bundle Y to bundle Z, then he must prefer bundle X to bundle Z.

Continuity: This means that you can choose to consume any amount of the good. For example, I could drink 11 mL of soda, or 12 mL, or 132 mL. I am not confined to drinking 2 liters or nothing. See also continuous function in mathematics. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

Non-satiation: This is the idea that more of any good is always preferred to less.

Convexity: The marginal value a person gets from each commodity falls relative to the other good. In a two-good world, if a consumer has relatively lots of one good he would be happier with a little less of that good and a little more of the other.

Examples of Indifference Curves

Figure 1 encore: An example of an indifference map with three indifference curves represented

Figure 2: Three indifference curves where Goods X and Y are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel.

Figure 3: Indifference curves for perfect complements X and Y. The "elbows" of the curves are collinear.

In Figure 1, the consumer would rather be on I3 than I2, and would rather be on I2 than I1, but does not care where they are on each indifference curve. The slope of an indifference curve, known by economists as the marginal rate of substitution, shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For most goods the marginal rate of substitution is not constant so their indifference curves are curved. The curves are convex to the origin as a result of the negative substitution effect (as price rises consumers seek less expensive substitutes) which is reinforced through the income effect (Beattie-LaFrance) and is required to satisfy the axiom of non-satiation. An example of a utility function that generates indifference curves of this kind is the Cobb-Douglas function $Uleft(x,yright)=x^alpha y^{1-alpha}, 0 leq alpha leq 1$ . Image File history File links Simple-indifference-curves. ... Image File history File links Simple-indifference-curves. ... Image File history File links Indifference-curves-perfect-substitutes. ... Image File history File links Indifference-curves-perfect-substitutes. ... Image File history File links Indifference-curves-perfect-complements. ... Image File history File links Indifference-curves-perfect-complements. ... In economics, the marginal rate of substitution (MRS for short) is the rate at which consumers are willing to give up units of one good in exchange for more units of another good. ... Consumer theory relates preferences, indifference curves and budget constraints to consumer demand curves. ... Consumer theory relates preferences, indifference curves and budget constraints to consumer demand curves. ...

If the goods are perfect substitutes then the indifference curves will be parallel lines since the consumer would be willing to trade at a fixed ratio. The marginal rate of substitution is constant. An example of a utility function that is associated with indifference curves like these would be $Uleft(x,yright)=alpha x + beta y$ . In economics, one kind of good (or service) is said to be a substitute good for another kind insofar as the two kinds of goods can be consumed or used in place of one another in at least some of their possible uses. ...

If the goods are perfect complements then the indifference curves will be L-shaped. An example would be something like if you had a cookie recipe that called for 3 cups flour to 1 cup sugar. No matter how much extra flour you had, you still could not make more cookie dough without more sugar. Another example of perfect complements is a left shoe and a right shoe. The consumer is no better off having several right shoes if she has only one left shoe. Additional right shoes have zero marginal utility without more left shoes. The marginal rate of substitution is either zero or infinite. An example of the type of utility function that has an indifference map like that above is $Uleft(x,yright)= min { alpha x, beta y }$ . A complement good (or complementary good) is a good that should be consumed with another good. ...

Application

Consumer theory uses indifference curves and budget constraints to produce consumer demand curves.

Consumer theory is a theory of economics. ... The supply and demand model describes how prices vary as a result of a balance between product availability at each price (supply) and the desires of those with purchasing power at each price (demand). ...

References

Bruce R. Beattie and Jeffrey T. LaFrance, “The Law of Demand versus Diminishing Marginal Utility” (2006). Review of Agricultural Economics. 28 (2), pp. 263-271.

Contents

History GA_googleFillSlot("encyclopedia_square");

Preference Relations and Utility

Preference Relations

Formal link to Utility theory

Examples

Linear Utility

Cobb-Douglas Utility

CES Utility

Map and properties of indifference curves

Assumptions

Examples of Indifference Curves

Application

References

See also

History