Suppose you are a production manager. There are two outputs for you to produce X and Y . Let the quantity produced of each be denoted x and y respectively. There are two units of inputs: labor and capital. Each unit of X takes 2 units of labor and 1 unit of capital to produce. Each unit of Y takes 1 unit of labor and 2 units of capital to produce. There are L units of labor and K units of capital to allocate between the production of the two outputs. The manager is evaluated by the senior leadership of the firm in how much is produced. Specifically, the manager's evaluation is increasing in the amount produced of each, but with diminishing returns. Thus, let B(x, y) = θln(x) + (1 - θ) ln(y) be the benefit received. The parameter θ ∈ (0, 1) be the relative importance of good X.
Identify the optimal production levels of X and Y.