Maximise a Cobb-Douglas utility under a budget
Optimising x^0.5*y^0.5 subject to 2*x+3*y = 60 gives 12.247449 at (15, 10), with multiplier λ = 0.204124. Full worked solution and an editable calculator.
Optimising f = x^0.5*y^0.5 subject to 2*x+3*y = 60 gives f = 12.247449 at (15, 10), with λ = 0.204124. The economics case: λ is the marginal utility of income, and spending splits evenly between the goods.
The quantity to maximise or minimise
Left-hand side of g = c
Right-hand side
The system to solve
∂/∂x: ∂f/∂x = λ · ∂g/∂x ∂/∂y: ∂f/∂y = λ · ∂g/∂y constraint: 2*x+3*y = 60
The economics case: λ is the marginal utility of income, and spending splits evenly between the goods.
Critical points
| x | y | λ | f | Type |
|---|---|---|---|---|
| 15 | 10 | 0.204124 | 12.247449 | stationary point |
More Lagrange multiplier problems
Frequently asked questions
What is the answer to "Maximise a Cobb-Douglas utility under a budget"? ▾
The optimum is f = 12.247449 at (15, 10), with Lagrange multiplier λ = 0.204124. The economics case: λ is the marginal utility of income, and spending splits evenly between the goods.
What system of equations does this produce? ▾
Setting ∇f = λ∇g gives one equation per variable (x, y), plus the constraint 2*x+3*y = 60. That is 3 equations in 3 unknowns, counting λ.
What does λ mean here? ▾
λ = 0.204124 means that raising the constraint value from 60 by one unit would change the optimal value of f by roughly 0.204124.
About this Calculator
Optimising x^0.5*y^0.5 subject to 2*x+3*y = 60 gives 12.247449 at (15, 10), with multiplier λ = 0.204124. Full worked solution and an editable calculator.