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Indifference Curve (IC)

A locus of points representing combinations of goods that yield the same level of utility. Mathematically, U(x,y)=UˉU(x, y) = \bar{U}, where Uˉ\bar{U} is a constant utility level.
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The statement of the theorem

Define the indifference curve IUI_U as the set of all consumption bundles x=(x1,x2)\mathbf{x} = (x_1, x_2) such that U(x1,x2)=UˉU(x_1, x_2) = \bar{U}, where Uˉ\bar{U} is a constant utility level. Formally, IU={x∈R+2∣U(x1,x2)=Uˉ}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.