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Utility Theory

Field: Consumer Theory

Sequence of Expressions

Let XX be the set of all consumption bundles. The preference relation ≻\succ defined on XX is a binary relation such that for any two bundles A,B∈X\mathbf{A}, \mathbf{B} \in X, the consumer can state A≻B\mathbf{A} \succ \mathbf{B} (A is strictly preferred to B), B≻A\mathbf{B} \succ \mathbf{A} (B is strictly preferred to A), or A∼B\mathbf{A} \sim \mathbf{B} (A is indifferent to B).
For any two bundles A∈X\mathbf{A} \in X and B∈X\mathbf{B} \in X, the preference relation ≻\succ must satisfy the completeness property: A≻B\mathbf{A} \succ \mathbf{B} or B≻A\mathbf{B} \succ \mathbf{A} or A∼B\mathbf{A} \sim \mathbf{B}.
The preference relation ≻\succ must be transitive. If A≻B\mathbf{A} \succ \mathbf{B} and B≻C\mathbf{B} \succ \mathbf{C}, then it must follow that A≻C\mathbf{A} \succ \mathbf{C}.
Define the utility function U:X→RU: X \to \mathbb{R} as a scalar mapping that assigns a utility value to every consumption bundle x=(x1,x2,…,xN)\mathbf{x} = (x_1, x_2, \dots, x_N). The consumer's objective is to find the bundle x∗\mathbf{x}^* that maximizes this function: max⁡x∈XU(x)\max_{\mathbf{x} \in X} U(\mathbf{x})
Let x=(x1,x2,…,xN)\mathbf{x} = (x_1, x_2, \dots, x_N) be the consumption bundle, P=(p1,p2,…,pN)P = (p_1, p_2, \dots, p_N) be the price vector, and MM be the consumer's income. The set of affordable bundles XX is defined by the budget constraint: ∑i=1Npixi≤M\sum_{i=1}^{N} p_i x_i \le M
The Marginal Rate of Substitution (MRS) between good x1x_1 and good x2x_2 is defined as the ratio of the marginal utilities, which represents the slope of the indifference curve: MRS1,2=MU1MU2=∂U/∂x1∂U/∂x2MRS_{1,2} = \frac{MU_1}{MU_2} = \frac{\partial U / \partial x_1}{\partial U / \partial x_2}
At the optimal consumption point x∗\mathbf{x}^*, the ratio of marginal utilities must equal the ratio of prices, ensuring the marginal utility per dollar spent is equal across all goods: MU1p1=MU2p2=⋯=MUNpN\frac{MU_1}{p_1} = \frac{MU_2}{p_2} = \dots = \frac{MU_N}{p_N}
The formal utility maximization problem is to find the optimal bundle x∗\mathbf{x}^* that maximizes the utility function U(x)U(\mathbf{x}) subject to the budget constraint: max⁡U(x)s.t.∑i=1Npixi≤M\max U(\mathbf{x}) \quad \text{s.t.} \quad \sum_{i=1}^{N} p_i x_i \le M
The total effect of a change in a parameter (e.g., price p1p_1) on the optimal demand x1∗x_1^* is decomposed into two parts: the Substitution Effect (SE) and the Income Effect (IE). Mathematically, this is often represented as: ∂x1∗∂p1=∂x1∗∂p1∣SE+∂x1∗∂M∣SE\frac{\partial x_1^*}{\partial p_1} = \frac{\partial x_1^*}{\partial p_1} \bigg|_{\text{SE}} + \frac{\partial x_1^*}{\partial M} \bigg|_{\text{SE}} (where the SE component is calculated by holding real income constant).