Suppose that Target's production function isQ(L,K)=L^1/3K^2/3

. Use this information to answer the following questions.

1. Derive Target's marginal product of labor, marginal product of capital, and marginal rate of technical substitution.
2. Let the wage rate (price of labor) be $1 and the rental rate (price of capital) be $2. If Target must produce 10 units of output, how much labor and capital will they use? What will be their total cost of production?
3. Now suppose that, in the short-run, Target can only use 5 units of capital. How many units of labor must the company use to still produce 10 units of output? What is the total cost of production in this short-run scenario? (Assume the same prices as in the previous question).
4. How much money does Target save by being able to choose both inputs in the long-run, relative to being constrained to the fixed 5 units of capital in the short-run?

All else equal, is it always cheaper to produce in the long-run than in the short-run?