Kuhn–Tucker conditions for a Cobb–Douglas consumer

Maximize u(x₁, x₂) = x₁^α x₂^(1-α) subject to the budget constraint, then add an upper limit on how much of one good is available.

Availability limit applies to

Case 1: budget constraint only

max u s.t. p₁x₁ + p₂x₂ ≤ w

Case 2: budget and availability

Multipliers and optimal bundles

λ = ∂v/∂w is the marginal utility of income, and μ = ∂v/∂x̄ is the utility gained from one more unit of availability.