Indifference curve

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An indifference curve is a graph showing combinations of goods for which a consumer is indifferent, that is, it has no preference for one combination versus another, as they render the same level of satisfaction for the consumer. Indifference curves are a device to represent preferences and are used in choice theory.

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History

The theory of indifference curves was developed by Francis Ysidro Edgeworth, Vilfredo Pareto and others in the first part of the 20th century. The theory is derived from ordinal utility theory, which posits that individuals can always rank two consumption bundles by order of preference.

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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.

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Preference Relations

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

A preference relation, denoted <math>\geq</math>, is a binary relation define on the set <math>A\;</math>.

The statement

<math>a\geq b\;</math>

is understood as '<math>a\;</math> is weakly preferred to <math>b\;</math>.'

The statement

<math>a\sim b\;</math>

is understood as '<math>a\;</math> is weakly preferred to <math>b\;</math>, and <math>b\;</math> is weakly preferred to <math>a\;</math>.' One says that '<math>a\;</math> is indifferent to <math>b\;</math>.'

The statement

<math>a>b\;</math>

is understood as '<math>a\;</math> is weakly preferred to <math>b\;</math>, but <math>b\;</math> is not weakly preferred to <math>a\;</math>.' One says that '<math>a\;</math> is strictly preferred to <math>b\;</math>.

The preference relation <math>\geq</math> is complete if all pairs <math>a,b\;</math> can be ranked. The relation is a transitive relation if whenever <math>a\geq b\;</math> and <math>b\geq c,\;</math> then <math>a\geq c\;</math>.

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

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Formal link to Utility theory

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

Utility theory posits that when agents can rank all pairs of consumption bundles by order of preference (and this ranking is a transitive relation) then there is a utility function. This means that for each bundle <math>\left(x,y\right)</math> there is a unique number, <math>U\left(x,y\right)</math>, representing the utility (satisfaction) level associated with <math>\left(x,y\right)</math>.

The relation <math>\left(x,y\right)\to U\left(x,y\right)</math> is called the utility function. The range of the function is the real line. The actual value of the function has no meaning. Only the ranking of those values has content for the theory. More precisely, when <math>U(x,y)\geq U(x',y')</math> then the bundle <math>\left(x,y\right)</math> is considered to be at least as good as the bundle <math>\left(x',y'\right)</math>. When <math>U\left(x,y\right)>U\left(x',y'\right)</math> then the bundle <math>\left(x,y\right)</math> is strictly preferred to the bundle <math>\left(x',y'\right)</math>.

Consider a particular bundle <math>\left(x_0,y_0\right)</math> and take the total derivative of <math>U\left(x,y\right)</math> about this point:

<math>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</math>

where <math>U_1\left(x,y\right)</math> is the partial derivative of <math>U\left(x,y\right)</math> with respect to its first argument, evaluated at <math>\left(x,y\right)</math>. (Likewise for <math>U_2\left(x,y\right).</math>)

The indifference curve through <math>\left(x_0,y_0\right)</math> must deliver, all along, the utility of this particular bundle, that is <math>U\left(x_0,y_0\right)</math>. In other words, if one is to change the quantity of <math>x\,</math> by <math>dx\,</math>, one must also change the quantity of <math>y\,</math> by an amount <math>dy\,</math> such that, in the end, there is no change in <math>U\,</math>, that is <math>dU=0\,</math>. The equation of the total derivative dictates then

<math>dU\left(x_0,y_0\right)=0\Leftrightarrow\frac{dx}{dy}=-\frac{U_2(x_0,y_0)}{U_1(x_0,y_0)}</math>

Thus, the ratio of marginal utilities gives the slope of the indifference curve at point <math>\left(x_0,y_0\right)</math>. This ratio is called the marginal rate of substitution between <math>x\,</math> and <math>y\,</math>.

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Examples

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Linear Utility

If the utility function is of the form <math>U\left(x,y\right)=\alpha x+\beta y</math> then the marginal utility of <math>x\,</math> is <math>U_1\left(x,y\right)=\alpha</math> and the marginal utility of <math>y\,</math> is <math>U_2\left(x,y\right)=\beta</math>. The slope of the indifference curve is, therefore,

<math>\frac{dx}{dy}=-\frac{\beta}{\alpha}.</math>

Observe that the slope does not depend on <math>x\,</math> or <math>y\,</math>: Indifference curves are straight lines.

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Cobb-Douglas Utility

If the utility function is of the form <math>U\left(x,y\right)=x^\alpha y^{1-\alpha}</math> the marginal utility of <math>x\,</math> is <math>U_1\left(x,y\right)=\alpha \left(x/y\right)^{\alpha-1}</math> and the marginal utility of <math>y\,</math> is <math>U_2\left(x,y\right)=(1-\alpha) \left(x/y\right)^{\alpha}</math>. The marginal rate of substitution, and therefore the slope of the indifference curve is then

<math>\frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right).</math>
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CES Utility

A general CES (Constant Elasticity of Substitution) form is

<math>U(x,y)=\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho}</math>

where <math>\alpha\in(0,1)</math> and <math>\rho\leq 1</math>. (The Cobb-Douglas is a special case of the CES utility, with <math>\rho=0\,</math>.) The marginal utilities are given by

<math>U_1(x,y)=\alpha \left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho-1} x^{\rho-1}</math>

and

<math>U_2(x,y)=(1-\alpha)\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho-1} y^{\rho-1}.</math>

Therefore, along an indifference curve,

<math>\frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right)^{1-\rho}.</math>
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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, as this would mean that a single combination of two goods will render two different levels of satisfaction, which is impossible. Also, this is a consequence of the assumption that the preference relation is transitive.
  • 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 words, there exists an indifference curve through any given point on an indifference map.
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Indifference Map

There are many indifference curves. To a given pair of goods is associated only one indifference curve (indifference curves do not intersect.) But different pairs of goods, yielding different utility level, will be associated with distinct indifference curves. The list of indifference curves associated with different utility level is called an Indifference Map. The rational consumer prefers the higher, or right most, Indifference curve, since they represent combinations of goods providing higher utility levels. 

  Indifference curve varies from one person to another, which is due to their personal preference.
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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 happier with a little less of that good and a little more of the other.

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Examples of Indifference Curves

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

Image:Indifference curve.png


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.

Image:Indifference curve perfect sub.png

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.

Image:Indifference curve perfect comp.png

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.

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Application

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See also

de:Indifferenzkurve fr:Courbe d'indifférence ko:무차별곡선 it:Curve di indifferenza hu:Közömbösségi görbe ja:無差別曲線 zh:無異曲線

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