Indifference curve

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 give 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.[10] 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.[10] A line connecting all points of tangency between the indifference curve and the budget constraint is called the expansion path.[11]

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  Figure 1: An example of an indifference map with three indifference curves represented
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  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.
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  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 the less 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 ${\displaystyle \scriptstyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha },0\leq \alpha \leq 1}$. The negative slope of the indifference curve incorporates the willingness of the consumer 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 ${\displaystyle \scriptstyle U\left(x,y\right)=\alpha x+\beta y}$.

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 include - 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 Leontief function: ${\displaystyle \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}}$.

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.[nb 2]

Let

${\displaystyle A\;}$ be a set of mutually exclusive alternatives among which a consumer can choose.

${\displaystyle a\;}$ and ${\displaystyle b\;}$ be generic elements of ${\displaystyle A\;}$.

In the language of the example above, the set ${\displaystyle A\;}$ is made of combinations of apples and bananas. The symbol ${\displaystyle a\;}$ is one such combination, such as 1 apple and 4 bananas and ${\displaystyle b\;}$ is another combination such as 2 apples and 2 bananas.

A preference relation, denoted ${\displaystyle \succeq }$, is a binary relation define on the set ${\displaystyle A\;}$.

The statement

${\displaystyle a\succeq b\;}$

is described as '${\displaystyle a\;}$ is weakly preferred to ${\displaystyle b\;}$.' That is, ${\displaystyle a\;}$ is at least as good as ${\displaystyle b\;}$ (in preference satisfaction).

The statement

${\displaystyle a\sim b\;}$

is described as '${\displaystyle a\;}$ is weakly preferred to ${\displaystyle b\;}$, and ${\displaystyle b\;}$ is weakly preferred to ${\displaystyle a\;}$.' That is, one is indifferent to the choice of ${\displaystyle a\;}$ or ${\displaystyle b\;}$, meaning not that they are unwanted but that they are equally good in satisfying preferences.

The statement

${\displaystyle a\succ b\;}$

is described as '${\displaystyle a\;}$ is weakly preferred to ${\displaystyle b\;}$, but ${\displaystyle b\;}$ is not weakly preferred to ${\displaystyle a\;}$.' One says that '${\displaystyle a\;}$ is strictly preferred to ${\displaystyle b\;}$.'

The preference relation ${\displaystyle \succeq }$ is complete if all pairs ${\displaystyle a,b\;}$ can be ranked. The relation is a transitive relation if whenever ${\displaystyle a\succeq b\;}$ and ${\displaystyle b\succeq c,\;}$ then ${\displaystyle a\succeq c\;}$.

For any element ${\displaystyle a\in A\;}$, the corresponding indifference curve, ${\displaystyle {\mathcal {C}}_{a}}$ is made up of all elements of ${\displaystyle A\;}$ which are indifferent to ${\displaystyle a}$. Formally,

${\displaystyle {\mathcal {C}}_{a}=\{b\in A:b\sim a\}}$.

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

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 ${\displaystyle \left(x,y\right)}$ there is a unique relation, ${\displaystyle U\left(x,y\right)}$, representing the utility (satisfaction) relation associated with ${\displaystyle \left(x,y\right)}$. The relation ${\displaystyle \left(x,y\right)\to U\left(x,y\right)}$ 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 ${\displaystyle U(x,y)\geq U(x',y')}$, then the bundle ${\displaystyle \left(x,y\right)}$ is described as at least as good as the bundle ${\displaystyle \left(x',y'\right)}$. If ${\displaystyle U\left(x,y\right)>U\left(x',y'\right)}$, the bundle ${\displaystyle \left(x,y\right)}$ is described as strictly preferred to the bundle ${\displaystyle \left(x',y'\right)}$.

Consider a particular bundle ${\displaystyle \left(x_{0},y_{0}\right)}$ and take the total derivative of ${\displaystyle U\left(x,y\right)}$ about this point:

${\displaystyle dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_{0}\right)dy}$

or, without loss of generality,

${\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0}){\frac {dy}{dx}}}$ (Eq. 1)

where ${\displaystyle U_{1}\left(x,y\right)}$ is the partial derivative of ${\displaystyle U\left(x,y\right)}$ with respect to its first argument, evaluated at ${\displaystyle \left(x,y\right)}$. (Likewise for ${\displaystyle U_{2}\left(x,y\right).}$)

The indifference curve through ${\displaystyle \left(x_{0},y_{0}\right)}$ must deliver at each bundle on the curve the same utility level as bundle ${\displaystyle \left(x_{0},y_{0}\right)}$. 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 ${\displaystyle x\,}$ by ${\displaystyle dx\,}$, without moving off the indifference curve, one must also change the quantity of ${\displaystyle y\,}$ by an amount ${\displaystyle dy\,}$ such that, in the end, there is no change in U:

${\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0}$, or, substituting 0 into (Eq. 1) above to solve for dy/dx:

${\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0\Leftrightarrow {\frac {dy}{dx}}=-{\frac {U_{1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}}}$.

Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve at point ${\displaystyle \left(x_{0},y_{0}\right)}$. This ratio is called the marginal rate of substitution between ${\displaystyle x\,}$ and ${\displaystyle y\,}$.

If the utility function is of the form ${\displaystyle U\left(x,y\right)=\alpha x+\beta y}$ then the marginal utility of ${\displaystyle x\,}$ is ${\displaystyle U_{1}\left(x,y\right)=\alpha }$ and the marginal utility of ${\displaystyle y\,}$ is ${\displaystyle U_{2}\left(x,y\right)=\beta }$. The slope of the indifference curve is, therefore,

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

Observe that the slope does not depend on ${\displaystyle x\,}$ or ${\displaystyle y\,}$: the indifference curves are straight lines.

A class of utility functions known as Cobb-Douglas utility functions are very commonly used in economics for two reasons:

1. They represent ‘well-behaved’ preferences, such as more is better and preference for variety.

2. They are very flexible and can be adjusted to fit real-world data very easily. If the utility function is of the form ${\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }}$ the marginal utility of ${\displaystyle x\,}$ is ${\displaystyle U_{1}\left(x,y\right)=\alpha \left(x/y\right)^{\alpha -1}}$ and the marginal utility of ${\displaystyle y\,}$ is ${\displaystyle U_{2}\left(x,y\right)=(1-\alpha )\left(x/y\right)^{\alpha }}$.Where ${\displaystyle \alpha <1}$. The slope of the indifference curve, and therefore the negative of the marginal rate of substitution, is then

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

A general CES (Constant Elasticity of Substitution) form is

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

where ${\displaystyle \alpha \in (0,1)}$ and ${\displaystyle \rho \leq 1}$. (The Cobb–Douglas is a special case of the CES utility, with ${\displaystyle \rho \rightarrow 0\,}$.) The marginal utilities are given by

${\displaystyle U_{1}(x,y)=\alpha \left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}x^{\rho -1}}$

and

${\displaystyle U_{2}(x,y)=(1-\alpha )\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}y^{\rho -1}.}$

Therefore, along an indifference curve,

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

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

As used in biology, the indifference curve is a model for how animals 'decide' whether to perform a particular behavior, based on changes in two variables which can increase in intensity, one along the x-axis and the other along the y-axis. For example, the x-axis may measure the quantity of food available while the y-axis measures the risk involved in obtaining it. The indifference curve is drawn to predict the animal's behavior at various levels of risk and food availability.