Laura just opened a coffee shop in NYC named Squirrel Coffee. Hoping to cater to the hipsters, she sells a special kind of cold brew infused with nitrogen, which she dubs ‘nitro coffee’. Assume for the purposes of this question, that nitro coffee requires coffee beans and a machine to add nitrogen and transform it in liquid form. The coffee beans come in half-kilogram sacks, c, and she rents time on a machine by the hour, m. The price of coffee beans is pc, the price of machine hours is pm. She sells her nitro coffee, q by the cup (only one size).
(a) After producing for awhile, Laura knows that there is not much flexibility in the production process. The machine takes half an hour to transform 500 grams of coffee beans into a cup of nitro coffee. Running the machine longer doesn’t make more without extra coffee beans, and extra coffee beans(if not allowed enough time in the machine) doesn’t get enough nitrogen to produce nitro coffee. Which of the following production functions explain this relationship?
(i) f1(m,c) = m^a c^b,
(ii) f2(m,c) = am+bc,
(iii) f3(m,c) =min{am,bc}.
What are the values of the parameters a and b for the production function that you chose?
(b) Laura had previously signed an agreement to rent the machine for exactly 8 hours per day (one machine, fulltime). Assuming that she can’t get out of the contract she’s already committed to, and that her shop only has room for one machine, find Laura’s fixed, variable, average variable, and marginal costs (as a function of the prices of inputs and the number of cups of nitro coffee she produces).
(c) Find the price at which Laura would shut-down assuming that she has already paid the machine rental.
(d) Find the price at which Laura would shut-down assuming her contract is up and she’s deciding whether or not to rent the machine again (using the same contract of 8 hours per day).