A: Since we are assuming that each hamburger comes with 1 ounce of grease that has to be picked up, we can think of this as a single input production process (using only labor) that produces 2 outputs — hamburgers and grease — in equal quantities.
(a) On a graph with hours of labor on the horizontal axis and hamburgers on the vertical, illustrate your production frontier assuming decreasing returns to scale. Then illustrate the profit maximizing plan assuming for now that it does not cost anything to have grease picked up (i.e. assume q = 0.)
(b) Now suppose q > 0. Can you think of a way of incorporating this into your graph and demonstrating how an increase in q changes the profit maximizing production plan?
(c) Illustrate the marginal cost curves with and without q — and then illustrate again how the cost of having grease picked up (i.e. q > 0) alters the profit maximizing production choice.
(d) With increasing fuel prices, the demand for hybrid cars that run partially on gasoline and partially on used cooking grease has increased. As a result, fast food chains report that they no longer have to pay to have grease picked up — in fact, they are increasingly being paid for their grease. (In essence, one of the goods you produce used to have a negative price but now has a positive price.) How does this change how many hamburgers are being produced at your fast food restaurant?
(e) We have done all our analysis under the assumption that labor is the only input into ham- burger production. Now suppose that labor and capital were both needed in a homothetic, decreasing returns to scale production process. Would any of your conclusions change?
(f) We have also assumed throughout that producing one hamburger necessarily entails producing exactly one ounce of grease. Suppose instead that more or less grease per hamburger could be achieved through the purchase of fattier or less fatty hamburger meat. Would you predict that the increased demand for cooking grease in hybrid vehicles will cause hamburgers at fast food places to increase in cholesterol as higher gasoline prices increase the use of hybrid cars?
B: Suppose that the production function for producing hamburgers x is x = f (ℓ) = Aℓα where α < 1. Suppose further that, for each hamburger that is produced, 1 ounce of grease is also produced.
(a) Set up the profit maximization problem assuming that hamburgers sell for price p and grease costs q (per ounce) to be hauled away.
(b) Derive the number of hours of labor you will hire as well as the number of hamburgers you will produce.
(c) Determine the cost function (as a function of w, q and x).
(d) Derive from this the marginal cost function.
(e) Use the marginal cost function to determine the profit maximizing number of hamburgers and compare your answer to what you got in (b).
(f) How many hours of labor will you hire?
(g) How does your production of hamburgers change as grease becomes a commodity that people will pay for (rather than one you have to pay to have hauled away)?