Goods 1 and 2 are available at dollar prices of p1 per unit of Good 1 and p2 per unit of Good 2. A utility function U(x₁, x₂) is a function representing the utility or benefit of consuming xj units of good j. The marginal utility of the jth good is ∂U/∂xj, the rate of increase in utility per unit increase in the jth good. Prove the following law of economics: Given a budget of L dollars, utility is maximized at the consumption level (a, b) where the ratio of marginal utility is equal to the ratio of prices:

  

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marginal utility of Good 1 marginal utility of Good 2 = Ux, (a, b) Ux (a, b) = P1 P2