In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. 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.[1] The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.[2]

There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an indifference map.

An example of an indifference map with three indifference curves represented

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 and is like a contour line on a topographical map. Each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction (assuming positive marginal utility for the goods) you are essentially climbing a mound of utility. The higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top", or a "bliss point", a consumption bundle that is preferred to all others

Indifference curves are typically represented to be:

1. Defined only in the non-negative quadrant of commodity quantities (i.e. the possibility of having negative quantities of any good is ignored).

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.) The negative slope of the indifference curve reflects the law of

diminishing marginal utility. That is as more of a good is consumed total utility increases at a decreasing rate - additions to utility per unit consumption are successively smaller. Thus as you move down the indifference curve you are trading consumption of units of Y for additional units of X.

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. With (2), convex preferences imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged.

[edit] Assumptions of consumer preference theory

Preferences are complete o Assume that there are two consumption bundles A and B each containing two

commodities x and y. A consumer can unambiguously determine that one and only one of the following is the case:

A is preferred to B ⇒ A p B[3]

B is preferred to A ⇒ B p A[3]

A is indifferent to B ⇒ A I B[3] Note that this axiom precludes the possibility that the consumer cannot

decide,[4] and that a consumer is able to make this comparison with respect to every conceivable bundle of goods.[3]

Preferences are reflexive o Means that if A and B are in all respect identical the consumer will recognize this fact

and be indifferent in comparing A and B A = B ⇒ A I B[3]

Preference are transitive[nb 1] o If A p B and B p C then A p C.[3]

o Also A I B and B I C then A I C.[3] This is a consistency assumption.

Preferences are continuous o If A is preferred to B and C is infinitesimally close to B then A is preferred to C.o A p B & C → B ⇒ A p B.

"Continuous" means infinitely divisible - just like there are an infinity of numbers between 1 and 2 all bundles are infinitely divisible. This assumption makes indifference curves continuous.

Preferences exhibit strong monotonicity. o if A has more of both x and y than B then A is preferred to B

this is assumption is commonly called the "more is better" assumption

an alternative version of this assumption is strong monotonicity which requires that if A and B have the same quantity of one good, but A has more of the other then A is preferred to B

Indifference curves exhibit diminishing marginal rates of substitution o This assumption assures that indifference curves are smooth and convex to the origin.o This assumption also set the stage for using techniques of constrained optimization.

Because the shape of the curve assures that the first derivative is negative and the second is positive.

o The marginal rate of substitution tells how much y a person is willing to sacrifice to get one more unit of x.

o This is also called the substitution assumption. This is the most critical assumption of consumer theory. Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up of one commodity to get one unit of other good is that amount which will leave the consumer indifferent between the new and old situations"[6] The negative slope of the indifference curves represents the willingness of the consumer to make a trade off. [6]

There are also many sub-assumptions: o Irreflexivity - for no x is xpxo Negative transivity if xnot-py then for any third commodity z, either xnot-pz or znot-py or both.

[edit] Application

To maximise utility, a household should consume at (Qx, Qy). Assuming it does, a full demand schedule can be deduced as the price of one good fluctuates.

Consumer theory uses indifference curves and budget constraints to generate consumer demand curves. For a single consumer, this is a relatively simple process. First, let one good be an example market e.g. carrots, and let the other be a composite of all other goods. Budget constraints gives a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line (illustrated). This follows from common sense: if the market values a good more than the household, the household will sell it; if the market values a good

less than the household, the household will buy it. The process then continues until the market's and household's marginal rates of substitution are equal.[7] Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve.[8]

[edit] Examples of indifference curves

Figure 1: 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 . The negative slope of the indifference curve incorporates the willingness of the consumer to means to make trade offs.[9]

If two goods are perfect substitutes then the indifference curves will have a constant slope since the consumer would be willing to switch between at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these would be .

If two goods are perfect complements then the indifference curves will be L-shaped. Examples of perfect complements include left shoes compared to right shoes: 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, so bundles of goods differing only in the number of right shoes they includes - however many - are equally preferred. 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 equilibrium 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.[nb 2]

[edit] Preference relations and utility

Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves showing combinations of equal preference to the consumer.

[edit] Preference relations

Let

= a set of mutually exclusive alternatives among which a consumer can choose

and = generic elements of .

In the language of the example above, the set 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 described as ' is weakly preferred to .' That is, is at least as good as (in preference satisfaction).

The statement

is described as ' 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 described as ' 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 .

[edit] 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 . That is, when preferences are represented by a utility function, the indifference curves are the level curves of the utility function. Therefore, if one is to change the quantity of by , without moving off the indifference curve, 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:

.

Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve

at point . This ratio is called the marginal rate of substitution between and .

[edit] Examples

[edit] 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,

Observe that the slope does not depend on or : the indifference curves are straight lines.

[edit] Cobb-Douglas utility

If the utility function is of the form the marginal utility of is

and the marginal utility of is .Where α < 1. The slope of the indifference curve, and therefore the negative of the marginal rate of substitution, is then

[edit] CES utility

A general CES (Constant Elasticity of Substitution) form is

where and . (The Cobb-Douglas is a special case of the CES utility, with .) The marginal utilities are given by

and

Therefore, along an indifference curve,

These examples might be useful for modelling individual or aggregate demand.

Assumptions of consumer preference theory

Preferences are complete -

o Assume that there are two consumption bundles A and B each containing two

commodities x and y. A consumer can unambiguously make the following comparisons:

o A is preferred to B ⇒ A p B B is preferred to A ⇒ B p A A is indifferent to B ⇒ A I B

Note that this axiom precludes the possibility that the consumer cannot decide. Indifference allowed indecision not. I can't make up my mind.

Note this assumption means that a consumer is able to make this comparison with respect to every conceivable bundle of goods.

Preferences are reflexive o Means that if A and B are in all respect identical the consumer will recognize this fact

and be indifferent in comparing A and B A = B ⇒ A I B

Preference are transitive

o If A p B and B p C then A p C.o Also A I B and B I C then A I C.

This is a consistency assumption.

Preferences are continuous

o If A is preferred to B and C is infinitesimally close to B then A is preferred to C.o A p B & C → B ⇒ A p B.

"continuous" means infinitely divisible - just like there are an infinity of numbers between 1 and 2 all bundles are infinitely divisible. This assumption allows the use of curves and areas all the basic requirements of differential and integral calculus.

Preferences exhibit non-satiation. o if A and B have identical amounts of x and A has a little more y than B then A is

preferred to B, this is the more is better assumption

Indifference Curves exhibit diminishing marginal rates of substitution o This assumption assures that indifference curves are smooth and convex to the origin.o This assumption also set the stage for using techniques of constrained optimization.

Because the shape of the curve assures that the first derivative is negative and the second is positive.

o The MRS tells how much y a person is willing to sacrifice to get one more unit of x.o This is also called the substitution assumption. This is the most critical assumption of

consumer theory. Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up of one commodity to get one unit of other good is that amount which will leave the consumer indifferent between the new and old situations" The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.

There are also many sub-assumptions: Irreflexivity - for no x is xpx negative transivity if xpy then for any third commodity z, either xpz or zpy or both.

Application

Consumer theory

Consumer theory

Consumer choice is a theory of microeconomics that relates preferences to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most complex relations in economics...

uses indifference curves and budget constraints to generate consumer demand curves

Supply and demand

Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...

.

Examples of indifference curves

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

Marginal rate of substitution

In economics, the marginal rate of substitution is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of satisfaction.-Marginal rate of substitution as the slope of indifference curve:...

, 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

Income effect

In economics, the income effect is the change in consumption resulting from a change in real income.With a higher income, there will be a new budget constraint line that intersects a higher indifference curve...

of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb-Douglas function . The negative slope of the indifference curve incorporates the willingness of the consumer to means to make trade offs.

If two goods are perfect substitutes

Substitute good

.A substitute good, in contrast to a complementary good, is a good with a positive cross elasticity of demand.This means a good's demand is increased when the price of another good is increased. Conversely, the demand for a good is decreased when the price of another good is decreased...

then the indifference curves will have a constant slope since the consumer would be willing to trade at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these would be .

If two goods are perfect complements

Complement good

.A complementary good, in contrast to a substitute good, is a good with a negative cross elasticity of demand. This means a good's demand is increased when the price of another good is decreased. Conversely, the demand for a good is decreased when the price of another good is increased...

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

Consumer theory

Consumer choice is a theory of microeconomics that relates preferences to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most complex relations in economics...

. The results will only be stated here. A price-budget-line change that kept a consumer in equilibrium 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

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight. Often curves in two-dimensional or three-dimensional Euclidean space are of interest....

.

Preference relations

Let = a set of mutually exclusive alternatives among which a consumer can choose and = generic elements of .In the language of the example above, the set 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

Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

define on the set .

The statementis described as ' is weakly preferred to .' That is, is at least as good as (in preference satisfaction).

The statementis described as ' 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 statementis described as ' 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

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

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

Utility

In economics, utility is a measure of relative satisfaction. 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...

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

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

. This means that for each bundle there is a unique relation, , representing the utility

Utility

In economics, utility is a measure of relative satisfaction. 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...

(satisfaction) relation associated with . The relation is called the utility function. The range

Range (mathematics)

In mathematics, the range of a function refers to the output of a function, but there is not universal agreement on the subject of whether the output is the range or is included in the range. This disagreement among mathematicans is illustrated by the function f with f =...

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 . That is, when preferences are represented by a utility function, the indifference curves are the level curves of the utility function. Therefore, if one is to change the quantity of by , without moving off the indifference curve, 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:.Thus, the ratio of marginal utilities gives the absolute value of the slope

Slope

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

of the indifference curve at point . This ratio is called the marginal rate of substitution

Marginal rate of substitution

In economics, the marginal rate of substitution is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of satisfaction.-Marginal rate of substitution as the slope of indifference curve:...

between and .

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

Cobb-Douglas

Cobb-Douglas

In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. It was proposed by Knut Wicksell , and tested against statistical evidence by Charles Cobb and Paul Douglas in 1900–1928.For production, the function iswhere:*...

 utility

If the utility function is of the form the marginal utility of is and the marginal utility of is .Where . The slope

Slope

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

of the indifference curve, and therefore the negative of the marginal rate of substitution

, is then

CES utility

A general CES (Constant Elasticity of Substitution

) form iswhere and . (The Cobb-Douglas

Cobb-Douglas

In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. It was proposed by Knut Wicksell , and tested against statistical evidence by Charles Cobb and Paul Douglas in 1900–1928.For production, the function iswhere:*...

is a special case of the CES utility, with .) The marginal utilities are given byandTherefore, along an indifference curve,These examples might be useful for modelling

individual or aggregate demand.

The aim of indifference curve analysis is to analyse how a rational consumer chooses between two goods. In other words, how the change in the wage rate will affect the choice between leisure time and work time.

Indifference analysis combines two concepts; indifference curves and budget lines (constraints)

The indifference curve

An indifference curve is a line that shows all the possible combinations of two goods between which a person is indifferent. In other words, it is a line that shows the consumption of different combinations of two goods that will give the same utility (satisfaction) to the person.

For instance, in Figure 1 the indifference curve is I1. A person would receive the same utility (satisfaction) from consuming 4 hours of work and 6 hours of leisure, as they would if they consumed 7 hours of work and 3 hours of leisure.

Figure 1: An indifference curve for work and leisure

An important point is to remember that the use of an indifference curve does not try to put a physical measure onto how much utility a person receives.

The shape of the indifference curve

Figure 1 highlights that the shape of the indifference curve is not a straight line. It is conventional to draw the curve as bowed. This is due to the concept of the diminishing marginal rate of substitution between the two goods.

The marginal rate of substitution is the amount of one good (i.e. work) that has to be given up if the consumer is to obtain one extra unit of the other good (leisure).

The equation is below

The marginal rate of substitution (MRS) = change in good X / change in good Y

Using Figure 1, the marginal rate of substitution between point A and Point B is;

MRS = -3 / 3 = -1 = 1

Note, the convention is to ignore the sign.

The reason why the marginal rate of substitution diminishes is due to the principle of diminishing marginal utility. Where this principle states that the more units of a good are consumed, then additional units will provide less additional satisfaction than the previous units. Therefore, as a person consumes more of one good (i.e. work) then they will receive diminishing utility for that extra unit (satisfaction), hence, they will be willing to give up less of their leisure to obtain one more unit of work.

The relationship between marginal utility and the marginal rate of substitution is often summarised with the following equation;

MRS = Mux / Muy

It is possible to draw more than one indifference curve on the same diagram. If this occurs then it is termed an indifference curve map (Figure 2).

Figure 2: An indifference map

The general rule is that indifference curves further too the right (I4 and I5) show combinations of the two goods that yield a higher utility, while curves to the left (I2 and I1) show combinations that yield lower levels of utility.

A Budget Line (budget constraints)

The budget line is an important component when analysing consumer behaviour. The budget line illustrates all the possible combinations of two goods that can be purchased at given prices and for a given consumer budget. Remember, that the amount of a good that a person can buy will depend upon their income and the price of the good.

This discussion outlines the construction of a budget line and how the change in the determinants will affect the budget line.

Figure 3 constructs a budget line for a given budget of £60, £2 per unit of x and £1 per unit of y.

With a limited budget the consumer can only consume a limited combination of x and y (the maximum combinations are on the actual budget line).

A change in consumer income and the budget line

If consumer income increases then the consumer will be able to purchase higher combinations of goods. Hence an increase in consumer income will result in a shift in the budget line. This is illustrated in Figure 4. Note that the prices of the two goods have remained the same, therefore, the increase in income will result in a parallel shift in the budget line.

Assume consumer income increased to £90.

Figure 4: An increase in consumer income

If consumer income fell then there would be a corresponding parallel shift to the left to represent a fall in the potential combinations of the two goods that can be purchased.

A change in the price of a good and the budget line

If income is held constant, and the price of one of the goods changes then the slope of the curve will change. In other words, the curve will pivot. This is illustrated in Figure 5.

Figure 5: A change in price

The reduction of the price of good x from £2 to £1 means that on a fixed budget of £60, the consumer could purchase a maximum of 60 units, as opposed to 30. Note that the price of good y has remained fixed, hence the maximum point for good y will remain fixed.

Indifference analysis combines two concepts; indifference curves and budget lines (constraints)

The first stage is to impose the indifference curve and the budget line to identify the consumption point between two goods that a rational consumer with a given budget would purchase.

The optimum consumption point is illustrated on Figure 6.

Figure 6: The optimum consumption point

A rational, maximising consumer would prefer to be on the highest possible indifference curve given their budget constraint. This point occurs where the indifference curve touches (is tangential to) the budget line. In the case of Figure 6, the optimum consumption point occurs at point A on indifference curve I3.

Indifference analysis can be used to analyse how a consumer would change the combination of two goods for a given change in their income or the price of the good.

The next section looks at the income and substitution effects of a change in price.

If we assume that the good is normal, then the increase in price will result in a fall in the quantity demanded. This is for two reasons; the income effect (have a limited budget, therefore can purchase lower quantities of the good) and the substitution effect (swap with alternative goods that are cheaper).

These two processes can be visualised using indifference analysis (see Figure 7).

Figure 7: An increase in the price of good x (a normal good)

Due to the price of good x increasing, the budget line has pivoted from B1 to B2 and the consumption point has moved.

The decrease in the quantity demanded can be divided into two effects;

The substitution effect

The substitution effect is when the consumer switches consumption patterns due to the price change alone but remains on the same indifference curve. To identify the substitution effect a new budget line needs to be constructed. The budget line B1* is added, this budget line needs to be parallel with the budget line B2 and tangential to I1.

Therefore, the movement from Q1 to Q2 is purely due to the substitution effect.

The income effect

The income effect highlights how consumption will change due to the consumer having a change in purchasing power as a result of the price change. The higher price means the budget line is B2, hence the optimum consumption point is Q3. This point is on a lower indifference curve (I2).

Therefore, in the case of a normal good, the income and substitution effects work to reinforce each other.

In microeconomics, an indifference curve is a graph showing combinations of two goods to which an economic agent (such as a consumer or firm) is indifferent, that is, it has no preference for one over the other. They are used to analyse the choices of economic agents.

For example, 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.

For a given pair of goods, many indifference curves can be drawn. The consumer is generally assumed to prefer combinations of goods representing higher levels of consumption. The rational consumer will make choices between the two goods to reach the highest indifference curve feasible given the choices available to her.

The theory of indifference curves was developed by Vilfredo Pareto and others in the first part of the 20th century. The theory was developed so that analysis of economic choices could be based upon preferences, which can be observed, rather than the older concept of utility which suffers from the disadvantage that it cannot be objectively measured.

Indifference Curve Properties

Indifference curves are typically assumed to have the following features:

Indifference curves do not cross. This is a consequence of the assumption that consumers will always prefer to have more of either good than to have less.

The curves are convex, which is a consequence of the assumption that as consumers have less and less of one good, they require more of the other good to compensate (corresponding to the law of diminishing marginal utility.

Assumptions

These properties follow mathematically from the first three of the following list of assumptions. These assumptions, which are troublesome, are made in conceiving of indifference curves and demand functions:

Completeness: This assumption rests on the assertions that choice-makers have (1) infinite knowledge not just about the details of any apparent options, but about all (other) existing possibilities, and how much they would cost (including all implicit costs not reflected in the price), and (2) infinite knowledge about the set of factors which affect the personal satisfaction inherent in the option. This is thus subject to another selection problem: the total utility of a given choice depends on how many tangible and intangible factors one takes into account. Does one want to know how a commodity was made, who or what was destroyed by its production, or what the alternatives might have been, given that such knowledge will likely affect the object's desirability (utility)? Since not buying something but rather waiting for a future alternative (which might radically change the attractiveness of existing options) is always one option, the current utility is not even theoretically assessable. All these (thoroughly impossible) conditions would together mean that every pair of options has a unique ordering of utility.

Transitivity: Essentially, this says the pair ordering above extends to more than two options and is unique.

Non-satiation: This is the idea that people always want more --- not of something, but of everything and anything! This has elsewhere been called the philosophy of the cancer cell. Non-satiation is frequently relaxed when the specifics of the market show that this is the case.

Satiation: The marginal value a person gets from each commodity falls with the number of units. This is also called convexity. If one has much of something, one is not very happy with even more (but, according to the previous assumption, still a little).

Robotic behaviour: This seems separate from the other "rationality" criteria above. This is the assumption that a consumer not only can order options according to preference, but acts on such a utility-based rationale rather than, say, primarily out of habit or subject to an influence like "training" (e.g. due to marketing). In other words, the relationship between "utility" and effective "preference" is quietly assumed, in contradiction to what any psychologist or advertising agent knows. In reality, people making choices are well aware that they are not acting on the sum of their knowledge, but rather on habit, bias, and impulse as well. Consumers do not make their purchase decisions all at once, and thus do not have a "budget line" for most decisions. This "irrationality" applies to mass behaviour (subject to "fashion") even more than it does to individuals. Note however that a preference based on habit, bias or fashion does not necessarily contradict the theory.

Independence of budget and desires: It is assumed that consumers do not have control over the amount of their income. In reality, many people are in a position to earn extra income when they need to purchase something big, and to emphasize work which does not earn monetary compensation when they have enough.

Independence of purchase choices from non-monetary choices: More generally, since the given formalism is used to represent money-transactions only --- i.e. in the context of a market --- there is an assumption that a consumer's set of choices which concern monetary costs is entirely independent from the set of choices which do not have any cost involved. Instead, humans are social beings and thus use heuristics such as morality and social influence to make these decisions. If this is true, the rational choice theory cannot be repaired with any perturbation; it is simply inapplicable.

Example Indifference Curves

Below is an example of three indifference curves:

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 indicating a diminishing marginal rate of substitution.

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.

If the goods are perfect complements then the indifference curves will be angled. 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 reach a higher cookie level if there was not enough 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.

2 Indifference Curve Properties

Indifference curves are typically assumed to have the following features:

An Indifference curve slopes downward from left to right (negative slope). The negative slope is a consequence of the fact that the demand for one commodity (X) increases while the demand for another commodity (Y) decreases (because of diminishing marginal utility of Y), which is necessary to maintain the total satisfaction.

Indifference curves do not intersect. This is a consequence of the assumption that consumers will always prefer to have more of either good than to have less.

The curves are convex, which is a consequence of the assumption that as consumers have less and less of one good, they require more of the other good to compensate (corresponding to the law of diminishing marginal utility).

The Indifference curves are ubiquitous throughout an indifference map. In other word, there exists an indifference curve through any given point on an indifference map.

2.1 Indifference Map

For a given pair of goods, many indifference curves can be drawn and placed next to each other. This representation is called an Indifference Map. The rational consumer is expected to prefer the higher or right most Indifference curve, since they represent combinations of goods providing higher levels of consumption.

3 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, then 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.

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 a happier with a little less of that good and a little more of the other.

4 Example Indifference Curves

Below is an example of an indifference map having three indifference curves:

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 indicating a diminishing marginal rate of substitution.

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.

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.