Optimization with Budget Constraints

Business Calculus · Axiom Academy

ACTIVITY Budget Constraints and Optimization Interactive exploration of constrained choice A budget constraint shows all combinations of two goods that a consumer can afford given their income and prices: where p₁ = price of good 1, p₂ = price of good 2, I = income Blue line: budget constraint. Green curves: indifference curves (U = xy). Yellow dot: optimal choice. For utility function U(x, y) = xy, find the utility-maximizing bundle: For U = xy with budget p₁x + p₂y = I, the optimal solution always spends exactly half the budget on each good: x* = I / (2p₁) → Spending on x = p₁ · x* = I/2 y* = I / (2p₂) → Spending on y = p₂ · y* = I/2 Budget line shifts outward (parallel). Consumer buys more of both goods. Utility increases. Budget line pivots inward on the x-axis. Consumer buys less of good 1, may buy more or less of good 2. Budget line pivots outward on the y-axis. Consumer can afford more of good 2, total utility increases. If good 1 is subsidized, p₁ decreases, rotating the budget line and changing the optimal bundle. Consumer marketing: Understanding how price changes affect purchasing decisions Resource allocation: Distributing a fixed budget across multiple investments Production planning: Choosing input combinations subject to cost constraints Portfolio optimization: Allocating investment dollars across asset classes The Shadow Price Interpretation The Lagrange multiplier λ tells us the value of relaxing the budget constraint. In our example with λ = 10:

This is the written version of the interactive lesson above. See the full Business Calculus course.