A planner in a small, open economy has a utility function

A planner in a small, open economy has a utility function

u = x₁x₂

and production takes place according to the production functions

y₁ = 10L₁ − 0.5L₁²

y₂ = 10L₂ − L₂²

The world prices of the goods are p₁ = $10 and p₂ = $5. The total amount

of labor available in the economy is five units. Find the utility-maximizing

outputs, consumptions, and labor allocations, and the solution value of the

shadow wage rate. Compare your solution to that of the previous problem.

Now assume that the economy has 20 units of labor. Solve the problem again

and compare this solution to the previous two.