Q) You produce diamond-studded dog collars. Your production function is: Y = AL^0.5K. Assume you have a large stockpile of diamonds and other collar material. Then the total number of collars you can produce in each hour (Y) is the product of your labor force (L in individuals) and capital (K, tables for working, space for tables, etc.). "A" is a variable that captures any innovation you can create or adopt to make more collars per hour with the same number of workers and tables.

a. Thanks to the recent surge in demand for diamond-studded collars, the equilibrium price of collars has increased to $100/collar. You are wondering if your company can supply more to sell more at this new, higher price. You can really only change your labor force in the short-run. A calculus whiz on your economics team has determined that your MPL = 0.5AL^-0.5K. Say your "A" = 1, your current labor force is 100 people, and you have 50 units of capital (space for production, tables, glue guns, etc.). New workers will cost you $50/hr. Will you scale up? Why or why not?

b. At what production point will it no longer be worth it for you to scale up? How many laborers will you be employing?

c. You and your co-CEO anticipate that this demand for your product will continue next period. Your co-CEO is really excited about your booming business and suggests that you should scale-up for future production by changing your capital stock. She states that the only options are to either buy more capital or invest in some new tech to make your current capital more productive. You suspect there is at least one other option for how to change your capital. What is it? What key idea is your co-CEO missing?