Indifference curve

In microeconomic theory, an indifference curve 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).

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

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:

• 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

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 Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)=x^\alpha y^{1-\alpha}, 0 \leq \alpha \leq 1 .

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 Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)=\alpha x + \beta y .

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 Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)= \min \{ \alpha x, \beta y \} .

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.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): A\;

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

Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

and Failed to parse (Missing texvc executable; please see math/README to configure.): b\;
= generic elements of Failed to parse (Missing texvc executable; please see math/README to configure.): A\;

. In the language of the example above, the set Failed to parse (Missing texvc executable; please see math/README to configure.): A\;

is made of combinations of apples and bananas. The symbol Failed to parse (Missing texvc executable; please see math/README to configure.): a\;
is one such combination, such as 1 apple and 4 bananas and Failed to parse (Missing texvc executable; please see math/README to configure.): b\;
is another combination such as 2 apples and 2 bananas.
 

A preference relation, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \succeq , is a binary relation define on the set Failed to parse (Missing texvc executable; please see math/README to configure.): A\; .

The statement

Failed to parse (Missing texvc executable; please see math/README to configure.): a\succeq b\;

is described as 'Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

is weakly preferred to Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

.' That is, Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

is at least as good as Failed to parse (Missing texvc executable; please see math/README to configure.): b\;
(in preference satisfaction). 

The statement

Failed to parse (Missing texvc executable; please see math/README to configure.): a\sim b\;

is described as 'Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

is weakly preferred to Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

, and Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

is weakly preferred to Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

.' That is, one is indifferent to the choice of Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

or Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

, meaning not that they are unwanted but that they are equally good in satisfying preferences.

The statement

Failed to parse (Missing texvc executable; please see math/README to configure.): a\succ b\;

is described as 'Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

is weakly preferred to Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

, but Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

is not weakly preferred to Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

.' One says that 'Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

is strictly preferred to Failed to parse (Missing texvc executable; please see math/README to configure.): b\;

.'

The preference relation Failed to parse (Missing texvc executable; please see math/README to configure.): \succeq

is complete if all pairs Failed to parse (Missing texvc executable; please see math/README to configure.): a,b\;
can be ranked. The relation is a transitive relation if whenever Failed to parse (Missing texvc executable; please see math/README to configure.): a\succeq b\;
and Failed to parse (Missing texvc executable; please see math/README to configure.): b\succeq c,\;
then Failed to parse (Missing texvc executable; please see math/README to configure.): a\succeq c\;

.

Consider a particular element of the set Failed to parse (Missing texvc executable; please see math/README to configure.): A\; , such as Failed to parse (Missing texvc executable; please see math/README to configure.): a_0\; . Suppose one builds the list of all other elements of Failed to parse (Missing texvc executable; please see math/README to configure.): A\;

which are indifferent, in the eyes of the consumer, to Failed to parse (Missing texvc executable; please see math/README to configure.): a_0\;

. Denote the first element in this list by Failed to parse (Missing texvc executable; please see math/README to configure.): a_1\; , the second by Failed to parse (Missing texvc executable; please see math/README to configure.): a_2\;

and so on...  The set Failed to parse (Missing texvc executable; please see math/README to configure.): \{a_i:i\geq 0\}
forms an indifference curve since Failed to parse (Missing texvc executable; please see math/README to configure.): a_i\sim a_j\;
for all Failed to parse (Missing texvc executable; please see math/README to configure.): i,j\geq 0\;

.

Formal link to utility theory

In the example above, an element Failed to parse (Missing texvc executable; please see math/README to configure.): a\;

of the set Failed to parse (Missing texvc executable; please see math/README to configure.): A\;
is made of two numbers: The number of apples, call it Failed to parse (Missing texvc executable; please see math/README to configure.): x,\;
and the number of bananas, call it Failed to parse (Missing texvc executable; please see math/README to configure.): y.\;

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x,y\right)

there is a unique relation, Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)

, representing the utility (satisfaction) relation associated with Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x,y\right) . The relation Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x,y\right)\to U\left(x,y\right)

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 Failed to parse (Missing texvc executable; please see math/README to configure.): U(x,y)\geq U(x',y')

, then the bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x,y\right)

is described as at least as good as the bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x',y'\right)

. If Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)>U\left(x',y'\right) , the bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x,y\right)

is described as strictly preferred to the bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x',y'\right)

.

Consider a particular bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x_0,y_0\right)

and take the total derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)
about this point:

Failed to parse (Missing texvc executable; please see math/README to configure.): dU\left(x_0,y_0\right)=U_1\left(x_0,y_0\right)dx+U_2\left(x_0,y_0\right)dy

or, without loss of generality,  

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dU\left(x_0,y_0\right)}{dx}= U_1(x_0,y_0).1+ U_2(x_0,y_0)\frac{dy}{dx}

(Eq. 1)

where Failed to parse (Missing texvc executable; please see math/README to configure.): U_1\left(x,y\right)

is the partial derivative of Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)
with respect to its first argument, evaluated at Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x,y\right)

. (Likewise for Failed to parse (Missing texvc executable; please see math/README to configure.): U_2\left(x,y\right). )

The indifference curve through Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x_0,y_0\right)

must deliver at each bundle on the curve the same utility level as bundle Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x_0,y_0\right)

. In other words, if one is to change the quantity of Failed to parse (Missing texvc executable; please see math/README to configure.): x\,

by Failed to parse (Missing texvc executable; please see math/README to configure.): dx\,

, one must also change the quantity of Failed to parse (Missing texvc executable; please see math/README to configure.): y\,

by an amount Failed to parse (Missing texvc executable; please see math/README to configure.): dy\,
such that, in the end, there is no change in U:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dU\left(x_0,y_0\right)}{dx}= 0

, or, substituting 0 into (Eq. 1) above to solve for dy/dx:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dU\left(x_0,y_0\right)}{dx} = 0\Leftrightarrow\frac{dy}{dx}=-\frac{U_1(x_0,y_0)}{U_2(x_0,y_0)}

. Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve at point Failed to parse (Missing texvc executable; please see math/README to configure.): \left(x_0,y_0\right) . This ratio is called the marginal rate of substitution between Failed to parse (Missing texvc executable; please see math/README to configure.): x\,

and Failed to parse (Missing texvc executable; please see math/README to configure.): y\,

.

Examples

Linear utility

If the utility function is of the form Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)=\alpha x+\beta y

then the marginal utility of Failed to parse (Missing texvc executable; please see math/README to configure.): x\,
is Failed to parse (Missing texvc executable; please see math/README to configure.): U_1\left(x,y\right)=\alpha
and the marginal utility of Failed to parse (Missing texvc executable; please see math/README to configure.): y\,
is Failed to parse (Missing texvc executable; please see math/README to configure.): U_2\left(x,y\right)=\beta

. The slope of the indifference curve is, therefore,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dx}{dy}=-\frac{\beta}{\alpha}.

Observe that the slope does not depend on Failed to parse (Missing texvc executable; please see math/README to configure.): x\,

or Failed to parse (Missing texvc executable; please see math/README to configure.): y\,

Indifference curves are straight lines.

Cobb-Douglas utility

If the utility function is of the form Failed to parse (Missing texvc executable; please see math/README to configure.): U\left(x,y\right)=x^\alpha y^{1-\alpha}

the marginal utility of Failed to parse (Missing texvc executable; please see math/README to configure.): x\,
is Failed to parse (Missing texvc executable; please see math/README to configure.): U_1\left(x,y\right)=\alpha \left(x/y\right)^{\alpha-1}
and the marginal utility of Failed to parse (Missing texvc executable; please see math/README to configure.): y\,
is Failed to parse (Missing texvc executable; please see math/README to configure.): U_2\left(x,y\right)=(1-\alpha) \left(x/y\right)^{\alpha}

. The marginal rate of substitution, and therefore the slope of the indifference curve is then

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right).

CES utility

A general CES (Constant Elasticity of Substitution) form is

Failed to parse (Missing texvc executable; please see math/README to configure.): U(x,y)=\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha\in(0,1)

and Failed to parse (Missing texvc executable; please see math/README to configure.): \rho\leq 1

. (The Cobb-Douglas is a special case of the CES utility, with Failed to parse (Missing texvc executable; please see math/README to configure.): \rho=0\, .) The marginal utilities are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): U_1(x,y)=\alpha \left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{\left(1/\rho\right)-1} x^{\rho-1}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): U_2(x,y)=(1-\alpha)\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{\left(1/\rho\right)-1} y^{\rho-1}.

Therefore, along an indifference curve,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right)^{1-\rho}.

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

References and external links

• Anatomy of Cobb-Douglas Type Utility Functions in 3D
• Anatomy of CES Type Utility Functions in 3D
• 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 New Palgrave: A Dictionary of Economics, 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

• Budget constraint
• Consumer theory
• Convex preferences
• Homo economicus
• Microeconomics
• Rationalitycs:Indiferenční křivka

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