problems 1 and 2 are based on the given scenario:
Ruritania’s currency is the euro (€), and its only coal-mining company is a price-taker in the euro-area market for coal. The company sells at €90 per tonne to specialist wholesalers and faces no transportation costs on the coal it produces. It’s just variable costs relate to two distinct labor types, skilled (engineers and so on) and unskilled. Skilled workers command an hourly wage of €25, while that of unskilled workers is €8. Fixed costs are avoided. The firm’s production function is quadratic in both skilled and unskilled labor hours (correspondingly symbolized S and U):
q_{c }= 100S – 2S^{2} + 200U - 5U^{2} + 2SU
problem 1: By using the above information:
a) Derive an expression for the firm’s profit solely in terms of S and U.
b) By applying the implicit function theorem suitably (that is, by making use of a total differential for profit), get an algebraic expression for the slope of any given iso- profit contour in S, U space. Depict the firm’s iso-profit map in S, U space (that is, with S on the horizontal axis and U on the vertical) and describe how the slope expression gives insight into the shape of iso-profit contours.
c) Obtain first-order conditions for the firm’s optimal combination of S and U. Solve these equations for S and U by means of Cramer’s rule and state the amount of profit related with your solutions (which must be stated to 3 d.p.).
d) Get a second-order total differential (that is, quadratic form in dS and dU), express this in matrix form, and investigate whether the second-order conditions for the maximum or minimum are satisfied. Give a short prose talk of this procedure.
problem 2: Now suppose that Ruritania’s government passes legislation that restricts each firm to hire no more than 30 hours of skilled labor.
a) Set up the Lagrangian function and proceed to get and resolve by means of Cramer’s rule the first-order conditions for the optimal S, U input combination when the firm is subject to this constraint. Comment in brief on how your solutions for S and U, and the associated profit, compare with such obtained earlier for the unconstrained case.