Information about Indifference Curve
An indifference curve in microeconomic theory is a graph showing different bundles of goods, each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. In other words, they are all equally preferred. One can equivalently refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. Utility is then a device to represent preferences rather than something from which preferences come (Geanakoplis, 1987, p. 117). The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles (Böhm and Haller, 1987, p. 785).
A graph of indifference curves for an individual consumer associated with different utility levels is called an indifference map. Points yielding different utility levels are each associated with distinct indifference curves. An indifference curve describes a set of personal preferences and so can vary from person to person.
Indifference curves are typically represented to be:
Rationality (called an ordering relationship in a more general mathematical context): Completeness + transitivity. For given preference rankings, the consumer can choose the best bundle(s) consistently among a, b, and c from lowest on up.
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.
Of the remaining properties above, suppose, property (5) (convexity) is violated by a bulge of the indifference curves out from the origin for a particular consumer with a given budget constraint. Consumer theory then implies zero consumption for one of the two goods, say good Y, in equilibrium on the consumer's budget constraint. This would exemplify a corner solution. Further, decreases in the price of good Y over a certain range might leave quantity demanded unchanged at zero beyond which further price decreases switched all consumption and income away from X and to Y. The eccentricity of such an implication suggests why convexity is typically assumed.
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 he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), 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, describing the negative substitution effect. As price rises for a fixed money income, the consumer seeks less the expensive substitute at a lower indifference curve. The substitution effect is reinforced through the income effect of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb-Douglas function
.
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
.
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
.
The different shapes of the curves imply different responses to a change in price as shown from demand analysis in consumer theory. The results will only be stated here. A price-budget-line change that kept a consumer in equilbrium on the same indifference curve:
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.
is made of combinations of apples and bananas. The symbol 
is one such combination, such as 1 apple and 4 bananas and 
is another combination such as 2 apples and 2 bananas.
A preference relation, denoted
, is a binary relation define on the set .
The statement is weakly preferred to .' That is, is at least as good as (in preference satisfaction).
The statement is weakly preferred to , and is weakly preferred to .' That is, one is indifferent to the choice of or , meaning not that they are unwanted but that they are equally good in satisfying preferences.
The statement is weakly preferred to , but is not weakly preferred to .' One says that ' is strictly preferred to .'
The preference relation is complete if all pairs 
can be ranked. The relation is a transitive relation if whenever 
and 
then 
.
Consider a particular element of the set, such as 
. Suppose one builds the list of all other elements of which are indifferent, in the eyes of the consumer, to . Denote the first element in this list by 
, the second by 
and so on... The set 
forms an indifference curve since 
for all 
.
of the set is made of two numbers: The number of apples, call it 
and the number of bananas, call it 
In utility theory, the utility function of an agent is a function that ranks all pairs of consumption bundles by order of preference (completeness) such that any set of three or more bundles forms a transitive relation. This means that for each bundle
there is a unique relation, 
, representing the utility (satisfaction) relation associated with . The relation 
is called the utility function. The range of the function is a set of real numbers. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if 
, then the bundle is described as at least as good as the bundle 
. If 
, the bundle is described as strictly preferred to the bundle .
Consider a particular bundle
and take the total derivative of about this point:
where
is the partial derivative of with respect to its first argument, evaluated at . (Likewise for 
)
The indifference curve through must deliver at each bundle on the curve the same utility level as bundle . In other words, if one is to change the quantity of 
by 
, one must also change the quantity of 
by an amount 
such that, in the end, there is no change in U:
. This ratio is called the marginal rate of substitution between and .
then the marginal utility of is 
and the marginal utility of is 
. The slope of the indifference curve is, therefore,
or : Indifference curves are straight lines.
the marginal utility of is 
and the marginal utility of is 
. The marginal rate of substitution, and therefore the slope of the indifference curve is then
and 
. (The Cobb-Douglas is a special case of the CES utility, with 
.) The marginal utilities are given by
)
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 can be derived from ordinal utility theory, which posits that individuals can always rank any consumption bundles by order of preference.Map and properties of indifference curves
)
Figure 1: An example of an indifference map with three indifference curves represented
Indifference curves are typically represented to be:
- 1. defined only in the positive (+, +) quadrant of commodity-bundle quantities.
- 2. negatively sloped. That is, 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). Equivalently, satiation, such that more of either good (or both) is equally preferred to no increase, is excluded. (If utility U = f(x, y), U, in the third dimension, does not have a local maximum for any x and y values.)
- 3. complete, such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated).
- 4. transitive with respect to points on distinct indifference curves. That is, if each point on I2 is (strictly) preferred to each point on I1, and each point on I3 is preferred to each point on I2, each point on I3 is preferred to each point on I1. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.
- 5. (strictly) convex (sagging from below). With (2), convex preferences implies a bulge toward the origin of the indifference curve. As a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged, the substitution effect.
Assumptions
Let a, b, and c be bundles (vectors) of goods, such as (x, y) combinations above, with possibly different quantities of each respective good in the different bundles. The first assumption is necessary for a well-defined representation of stable preferences for the consumer as agent; the second assumption is convenient.Rationality (called an ordering relationship in a more general mathematical context): Completeness + transitivity. For given preference rankings, the consumer can choose the best bundle(s) consistently among a, b, and c from lowest on up.
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.
Of the remaining properties above, suppose, property (5) (convexity) is violated by a bulge of the indifference curves out from the origin for a particular consumer with a given budget constraint. Consumer theory then implies zero consumption for one of the two goods, say good Y, in equilibrium on the consumer's budget constraint. This would exemplify a corner solution. Further, decreases in the price of good Y over a certain range might leave quantity demanded unchanged at zero beyond which further price decreases switched all consumption and income away from X and to Y. The eccentricity of such an implication suggests why convexity is typically assumed.
Application
- Consumer theory uses indifference curves and budget constraints to generate consumer demand curves.
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 he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), 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, describing the negative substitution effect. As price rises for a fixed money income, the consumer seeks less the expensive substitute at a lower indifference curve. The substitution effect is reinforced through the income effect of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb-Douglas function
.
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
.
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
.
The different shapes of the curves imply different responses to a change in price as shown from demand analysis in consumer theory. The results will only be stated here. A price-budget-line change that kept a consumer in equilbrium on the same indifference curve:
- in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.
- in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the budget constraint) or would change quantity demanded from one end of the budget constraint to the other.
- in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.
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.
Preference relations
Let- = a set of mutually exclusive alternatives among which a consumer can choose
- and = generic elements of .
is made of combinations of apples and bananas. The symbol is one such combination, such as 1 apple and 4 bananas and is another combination such as 2 apples and 2 bananas.
A preference relation, denoted
, is a binary relation define on the set .
The statement
is weakly preferred to .' That is, is at least as good as (in preference satisfaction).
The statement
is weakly preferred to , and is weakly preferred to .' That is, one is indifferent to the choice of or , meaning not that they are unwanted but that they are equally good in satisfying preferences.
The statement
is weakly preferred to , but is not weakly preferred to .' One says that ' is strictly preferred to .'
The preference relation
is complete if all pairs can be ranked. The relation is a transitive relation if whenever and then .
Consider a particular element of the set
, such as . Suppose one builds the list of all other elements of which are indifferent, in the eyes of the consumer, to . Denote the first element in this list by , the second by and so on... The set forms an indifference curve since for all .
Formal link to utility theory
In the example above, an element of the set is made of two numbers: The number of apples, call it and the number of bananas, call it
In utility theory, the utility function of an agent is a function that ranks all pairs of consumption bundles by order of preference (completeness) such that any set of three or more bundles forms a transitive relation. This means that for each bundle
there is a unique relation, , representing the utility (satisfaction) relation associated with . The relation is called the utility function. The range of the function is a set of real numbers. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if , then the bundle is described as at least as good as the bundle . If , the bundle is described as strictly preferred to the bundle .
Consider a particular bundle
and take the total derivative of about this point:
or, without loss of generality,
(Eq. 1)
where
is the partial derivative of with respect to its first argument, evaluated at . (Likewise for )
The indifference curve through
must deliver at each bundle on the curve the same utility level as bundle . In other words, if one is to change the quantity of by , one must also change the quantity of by an amount such that, in the end, there is no change in U:
, or, substituting 0 into (Eq. 1) above to solve for dy/dx:
.
. This ratio is called the marginal rate of substitution between and .
Examples
Linear utility
If the utility function is of the form then the marginal utility of is and the marginal utility of is . The slope of the indifference curve is, therefore,
or : Indifference curves are straight lines.
Cobb-Douglas utility
If the utility function is of the form the marginal utility of is and the marginal utility of is . The marginal rate of substitution, and therefore the slope of the indifference curve is then
CES utility
A general CES (Constant Elasticity of Substitution) form is
and . (The Cobb-Douglas is a special case of the CES utility, with .) The marginal utilities are given by
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.
- Volker Böhm and Hans Haller (1987). "demand theory," The , v. 1, pp. 785-92.
- John Geanakoplis (1987). "Arrow-Debreu model of general equilibrium," The New Palgrave: A Dictionary of Economics, v. 1, pp. 116-24.
See also
Microeconomics (or price theory) is a branch of economics that studies how individuals, households, and firms make decisions to allocate limited resources,[1] typically in markets where goods or services are being bought and sold.
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graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc.
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A good or commodity in economics is any object or service that increases utility, directly or indirectly, not to be confused with good in a moral or ethical sense (see Utilitarianism and consequentialist ethical theory).
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Preference (or "taste") is a concept, used in the social sciences, particularly economics. It assumes a real or imagined "choice" between alternatives and the possibility of rank ordering of these alternatives, based on happiness, satisfaction, gratification, enjoyment, utility
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In economics, utility is a measure of the relative satisfaction or desiredness from consumption of goods. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility.
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Preference (or "taste") is a concept, used in the social sciences, particularly economics. It assumes a real or imagined "choice" between alternatives and the possibility of rank ordering of these alternatives, based on happiness, satisfaction, gratification, enjoyment, utility
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Mathematical problem may mean two slightly different things, both closely related to mathematical games:
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- general meaning
- a question that can be answered with the help of mathematics ; formal meaning : any tuple (S, C( ), r
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supply and demand describe market relations between prospective sellers and buyers of a good. The supply and demand model determines price and quantity sold in the market.
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Francis Ysidro Edgeworth (né Ysidro Francis 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.
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Vilfredo Federico Damaso Pareto [vil'fre:do pa're:to] (July 15, 1848, Paris – August 19, 1923, Geneva) was a French-Italian sociologist, economist and philosopher. He made several important contributions especially in the study of income distribution and in the analysis of
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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. Goods are often considered in ‘bundles’ or ‘baskets’.
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Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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A inverse or negative relationship is a mathematical relationship in which one variable decreases as another increases. For example, there is an inverse relationship between education and unemployment — that is, as education increases, the rate of unemployment
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In mathematics, a total order, linear order, simple order, or (non-strict) ordering on a set X is any binary relation on X that is antisymmetric, transitive, and total.
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convex, or concave up, if for any two points x and y in its domain C and any t in [0,1], we have
In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is
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In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is
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Convex preferences refer to a property of utility functions commonly represented in an indifference curve as a bulge toward the origin for normal goods (for unwanted goods, the curve bulges away from the origin).
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composite good is an abstraction used in economics that represents all consumption goods besides the one in question.
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Purpose
Consumer demand theory shows how the composite may be treated as if it were only a single good as to properties hypothesized about demand...... Click the link for more information.
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn.
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In mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc.) is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to
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In economics, an agent is an actor in a model that (generally) solves an optimization problem. In this sense, it is equivalent to the term player, which is also used in economics, but is more common in game theory.
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In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
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Consumer theory is a theory of economics. It relates preferences (through indifference curves and budget constraints) to consumer demand curves. The models that make up consumer theory are used to represent prospectively observable demand patterns for an individual
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A corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. The more usual solution will lie in the non-zero interior at the point of tangency between the objective function and the
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Consumer theory is a theory of economics. It relates preferences (through indifference curves and budget constraints) to consumer demand curves. The models that make up consumer theory are used to represent prospectively observable demand patterns for an individual
..... Click the link for more information.
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supply and demand describe market relations between prospective sellers and buyers of a good. The supply and demand model determines price and quantity sold in the market.
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In economics, the marginal rate of substitution (MRS) is the least-favorable rate at which an agent is willing to exchange units of one good or service for units of another.
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Consumer theory is a theory of economics. It relates preferences (through indifference curves and budget constraints) to consumer demand curves. The models that make up consumer theory are used to represent prospectively observable demand patterns for an individual
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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.
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A complementary good or complement good in economics is a good which is consumed with another good; its cross elasticity of demand is negative. This means that, if goods A and B were complements, more of good A being bought would result in more of good B also being bought.
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Consumer theory is a theory of economics. It relates preferences (through indifference curves and budget constraints) to consumer demand curves. The models that make up consumer theory are used to represent prospectively observable demand patterns for an individual
..... Click the link for more information.
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