= Indifference Curve =

An '''indifference curve''' is a line or plane connecting the consumer bundles of equal utility.

A set of indifference curves from different utility levels all plotted to one graph forms a sort of topographic map called the '''indifference map'''.

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== Properties ==

An indifference curve in terms of two goods, ''x'' and ''y'', is plotted on a graph as a negative and convex curve.

The negativity derives from an assumption that a more is always better. This is called ''strong monotonicity''.

The convexity derives from an assumption that a mixed bundle is preferred to a homogenous one. This is called ''diminishing marginal rates of substitution''.

Other technical assumptions include completeness, reflexivity, transitivity, and continuity.

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== Bivariate Derivation ==

The utility function ''U(x, y)'' can be [[Calculus/PartialDerivatives|partially derived]] for each good to establish that marginal utilities are ''MU,,x,, = ∂U/∂x'' and ''MU,,y,, = ∂U/∂y''. It follows then that it can be totally derived as ''dU = MU,,x,,dx + MU,,y,,dy''.

Taken with respect to ''x'', the derivative of the utility function is:

{{attachment:derivative.svg}}

An indifference curve connects the consumer bundles of equal utility. In other words, on an indifference curve, utility is ''constant'' and the derivative of utility is ''0''.

{{attachment:mrs1.svg}}

{{attachment:mrs2.svg}}

For simplicity, the absolute value of this ratio is preferred. (Since an indifference curve is always negative, it is implicitly known that the actual slope is the negation.) This ratio is called the '''marginal rate of substitution''' ('''MRS''').

{{attachment:mrs3.svg}}

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== Usage ==

In neoclassical consumer choice theory, the optimal strategy is to pick the bundle where the [[Economics/BudgetConstraint|budget constraint]] is tangent to the indifference curve:

''m,,indif,, = m,,budget,,''

''-MRS,,xy,, = -MU,,x,,/MU,,y,, = -P,,x,,/P,,y,,''

''MU,,x,,/MU,,y,, = P,,x,,/P,,y,,''

''MU,,x,,/P,,x,, = MU,,y,,/P,,y,,''

The intuitive explanation is that the optimal choice exists where, for all goods, the marginal utility given price is equal, so there is no incentive to substitute.

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== Multivariate Derivation ==

More generally, the total derivative of the bivariate utility function can be expressed as the [[Calculus/GradientVector|gradient]] ''∇U = [∂U/∂x ∂U/∂y]''. This can be expanded for ''n'' goods like ''∇U = [∂U/∂x,,1,, ∂U/∂x,,2,, ... ∂U/∂x,,n,,]''.

The indifference curve at a given point '''''x''' = [x,,1,, x,,2,, ... x,,n,,]'' is [[LinearAlgebra/Orthogonality|orthogonal]] to the gradient vector ''∇U'' evaluated at '''''x'''''. 

Skipping ahead to usage: This gradient vector is orthogonal to the indifference curve; the price vector ('''''p''''') is orthogonal to the [[Economics/BudgetConstraint|budget constraint]]; the point where the budget constraint is tangent to the indifference curve is also the point where these two vectors are similar. The middleman of tangency can be cut out. Solve with the [[Calculus/LagrangianMethod|Lagrangian method]].

{{attachment:lagrangian1.svg}}

This necessarily means that, for each good ''i'':

{{attachment:lagrangian2.svg}}

... giving ''n'' equations for the system.

The last necessary constraint is the budget constraint: the [[LinearAlgebra/VectorMultiplication#Dot_Product|dot product]] '''''p''' · '''x''''' is equal to ''m''. (Note that the quantities are given as '''''x''''' instead of '''''q''''' here.)

This can be rewritten as a more conventional Lagrangian system by understanding that the budget constraint is ''g('''x''') = m - '''p''' · '''x'''''. Therefore the Lagrangian function is ''L('''x''', λ) = U('''x''') + λ(m - '''p''' · '''x''')''. An equivalent solution emerges from this.



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