A company produces computers using machines (K) and workers (L). Their production is the following F(K, L)

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A company produces computers using machines (K) and workers (L). Their production is the following F(K, L) = K^1/3L^1/6 such that L workers and K machines allow the company to produce F(K, L) computer units. The usage of these inputs is compensated by paying an annual wage w=20 for each worker and an annual rent of r=80 for each machine. The firm just signed a lease to rent 100 machines and is therefore stuck with these 100 machines for the year. 1. The firm received a total computer order of 10 units for the year. What is the number of workers the firm needs to fill up this order? Why is this number independent from the wage of the workers? 2. By only comparing the current MRTS and the price ratio, do you think they should have acquired more or less machines? A big client just announced it would be willing to buy a number q of computers each year in the next five years. Therefore, the company decides to only produce for this client in the next years. 3. What will be the numbers of machines and workers that will minimize the total cost of the firm? Setup the Lagrangian of this problem and find the optimal number of machines and workers to produce a quantity q. 4. Show that the resulting cost of production is C(q) = 2 2 3 60q 2 ? 95q 2 . Plot this cost function for quantities ranging from 10 to 40. 5. Show that the marginal cost is equal to the Lagrange multiplier from question (3). 6. Would the firm make profit if the unit price of computers was 2000 and the order was 20 units? What if the unit price was 2000 but the order was 40 units? How do you explain this answer?