Cobb-Douglas Maximization With Unequal Exponents

Linear Algebra

Optimization

Medium

Maximize

x^{0.4} y^{0.6}

subject to

x + y = 100

Hint 1

Take logs to convert the objective into a sum, which is easier to optimize with a Lagrange multiplier.

Hint 2

For a Cobb-Douglas objective under a linear budget constraint, the optimal shares equal the exponents.

Show worked solution

1. Set up the first-order conditions
Maximizing $0.4 ln x + 0.6 ln y$ subject to $x + y = 100$ gives $\frac{0.4}{x} = \frac{0.6}{y}$ , so $y = 1.5 x$ .
2. Apply the constraint
Combining $y = 1.5 x$ with $x + y = 100$ gives the optimal allocation.
$x = 40, y = 60$

Answer: x=40, y=60

1. Set up the first-order conditions