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Indifference Curve Theory

Sequence of Expressions

Definition

Let

\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{R}_+^n

be a consumption bundle. A utility function

U: \mathbb{R}_+^n \to \mathbb{R}

is defined such that

U(\mathbf{x})

assigns a scalar utility value to

\mathbf{x}

. For the function to represent well-behaved preferences, it must be strictly monotonic (i.e.,

\frac{\partial U}{\partial x_i} > 0

for all

i

) and typically assumed to be concave to ensure diminishing marginal utility.

Definition

Define the indifference curve

I_U

as the set of all consumption bundles

\mathbf{x} = (x_1, x_2)

such that

U(x_1, x_2) = \bar{U}

, where

\bar{U}

is a constant utility level. Formally,

I_U = \{\mathbf{x} \in \mathbb{R}_+^2 \mid U(x_1, x_2) = \bar{U}\}

. The assumption of convexity of the IC implies that the consumer exhibits diminishing marginal rate of substitution.

Theorem

The Marginal Rate of Substitution (MRS) between goods

x

and

y

is defined as the absolute value of the slope of the indifference curve at any point

(x, y)

MRS_{xy} = \left| \frac{dy}{dx} \right| = - \frac{dy}{dx} \bigg|_{U=\bar{U}} = \frac{MU_x}{MU_y} = \frac{\partial U / \partial x}{\partial U / \partial y}

Model

The utility maximization problem is formulated as a constrained optimization:

\text{Maximize } U(x, y) \text{ subject to } P_x x + P_y y = M

where

P_x

and

P_y

are the prices of goods

x

and

y

, and

M

is the consumer's fixed income (budget constraint). The solution requires the Lagrangian method or the tangency condition.

Theorem

The optimal consumption bundle

(x^*, y^*)

is the point of tangency between the highest attainable indifference curve

I_{U^*}

and the budget constraint

B

. This condition requires that the Marginal Rate of Substitution equals the ratio of the prices:

MRS_{xy} = \frac{P_x}{P_y}

This condition, combined with the budget constraint, yields the optimal solution

(x^*, y^*)