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Utility Function

A mathematical representation U(x,y)U(x, y) that assigns a scalar utility value to every combination of goods (x,y)(x, y). It formalizes the consumer's preferences over consumption bundles.
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The statement of the theorem

Let x=(x1,x2,…,xn)∈R+n\mathbf{x} = (x_1, x_2, \dots, x_n) \in \mathbb{R}_+^n be a consumption bundle. A utility function U:R+n→RU: \mathbb{R}_+^n \to \mathbb{R} is defined such that U(x)U(\mathbf{x}) assigns a scalar utility value to x\mathbf{x}. For the function to represent well-behaved preferences, it must be strictly monotonic (i.e., ∂U∂xi>0\frac{\partial U}{\partial x_i} > 0 for all ii) and typically assumed to be concave to ensure diminishing marginal utility.