Assignment
1. 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. fte coffee beans come in half-kilogram sacks, c, and she rents time on a machine by the hour, m. fte 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 a while, Laura knows that there is not much flexibility in the production process. fte 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) = macb,
(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, full time). 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 ma- chine 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).
2. In this exercise, you will explore the equilibrium of a market in which we know the cost function of the firms, as well as the demand function of consumers. You will first analyze the short-run equilibrium, and then the long-run equilibrium. Each firm in the industry has the same technology with cost function
c(y) = k2 + 2y + y2
where y is the quantity produced and k is some positive fixed cost. You should think of this cost function as coming out of a standard cost minimization problem with respect to the inputs.
(a) Derive the average cost c(y)/y, the average variable cost and the marginal cost c'(y) for y > 0. Graph all these cost curves in an appropriately labeled diagram.
(b) Draw the supply function in the graph, and write down the equation that represents it, y(p).
(c) Derive the aggregate supply function YS(p) by adding up the supply y(p) over the J firms that are in the market. Write down the expression for YS(p).
(d) Consider now the demand side of the market. For simplicity, assume a linear demand func- tion: Y D (p) = A - Bp, where A and B are positive numbers, and YD represents the total quantity demanded in the industry. Assume that this comes from aggregation of the individ- ual demand functions derived from maximization. Find the short-run equilibrium price p* by equating YD and YS. You can assume that A and B are such that we are on the increas- ing part of the supply function, i.e. quantity produced is strictly positive. Find the short-run equilibrium industry production Y* = YD(p*) = YS(p*).
(e) Under what conditions for A; B; J and k the firms will indeed produce a positive quantity of output? We maintain this assumption for the next items.
(f) In the short-run, what happens to Y* and p* as the fixed cost k increases? What is the intuition?
(g) Consider now the long-run equilibrium, in which firms are allowed to enter the market. Solve for the number of firms J* that will enter into the market.
(h) How does the number of firms J* depend on k, A and B?
(i) What is the equation of the long-run supply curve? Is it horizontal, increasing or decreasing?
3. Suppose that there are 100 identical firms, each with the following technology: f = K1/4L1/2. Suppose also that in the short run K = 1, with r = 1,and P = 4. For the short run:
(a) If we assume that each firm is a profit maximizer, what is the problem that each firm solves?
(b) Calculate the MPL and each firm's demand for labor.
(c) Calculate each firm's profits.
(d) Calculate the market demand of labor.
Suppose that after considering basic activities required for life, there are 200 hours in a month which an individual can distribute between leisure and hours worked, that is 200 = h+l where h is leisure and l is labor. Suppose there are 1000 identical individuals, each with the same preferences: U(h, x) = x1/2 + h1/2 over leisure and consumption x. Normalize the prize of consumption x to 1.
(e) State the individual budget restriction.
(f) State the individual's problem and the first order conditions.
(g) If ω = 1, find the optimum of leisure h and consumption x.
(h) Find the individual supply of l (do not assume ω = 1).
(i) Find the market supply of labor.
Assume that the amount of labor (hours worked) is traded in a competitive market.
(j) What will be the equilibrium wage rate ω*?
(k) What will be the equilibrium level of labor?