Suppose that there are two food stores in town. La Boulangerie sells bread and La Fromagerie sells cheese. It costs $1 to make a loaf of bread and $2 to make a pound of cheese. If La Boulangerie's price is PB dollars per loaf of bread and La Fromagerie's price is PF dollars per pound of cheese, their respective weekly sales, QB thousand loaves of bread and QF thousand pounds of cheese, are given by the following equations:

QB = 14 - PB - 0.5PF

QF = 19 - 0.5PB - PF

a.) What are profit functions for La Boulangerie, B, and La Fromagerie, F?

b.) What are the best response functions in terms of setting the prices for La Boulangerie and La Fromagerie?

c.) Graph the respective best-response functions. Plot La Boulangerie's price on the vertical axis and La Fromagerie's price on the horizontal axis.

d.) Calculate the Nash equilibrium

Rather than setting their prices independently, suppose now that the two stores collude to set prices jointly so as to maximize the sum of their profits. Over the next few questions, we are going to find the joint profit maximizing prices for the stores.

e.) Start by identifying the profit function that captures the sum of the profits for the two stores i.e. J = B + F.

f.) What price should the two stores set for La Boulangerie? In other words, what PB maximizes the joint profit function? This looks a little trickier because the joint profit function, if you've calculated it correctly, has two squared terms in it instead of one. Think about it this way, though. Recall that the generic quadratic equation looks like y = ax2 + bx + c. In this part of the question, the x we're interested in is PB. Any term in our joint profit function that doesn't have some kind of PB in it is just part of the constant, 'c'. The 'x' that maximizes 'y' is still -b/2a. With this in mind, what is the best response function for the price set by La Boulangerie, PB?

the quantity of customers served by Matt - the customer demand function for Matt's restaurant - was QM = 44 - 2PM + PS and the quantity of customers served by Sean was QS = 44 - 2PS + PM. The profits for each restaurant also depend on the cost of serving each customer. Suppose that Matt is able to reduce his costs to just $2 per customer by eliminating the wait staff. Suppose that Sean continues to incur a cost of $8 per customer.

a.) What is Sean's profit function, S?

b.) What is Matt's profit function (revenue per customer multiplied by number of customers), M?

c) Sean needs to choose his price to maximize his profit function. What is Sean's best response function?

d) Matt needs to choose his price to maximize his profit function. What is Matt's best response function? Explain what this best response function means.

e) Calculate the Nash equilibrium

f) Graph the two best response functions and indicate the location of the Nash equilibrium on the graph. To match what we did in class, plot Matt's price on the vertical axis and Sona's price on the horizontal axis.

g) Look at the graph of the best response functions that we came up with in class when the cost per consumer was $8 for both Matt and Sean. Indicate which best response function in your new graph has moved and by how much. Explain why these changes occurred in your graph.

In the late 90s it was observed that the relative price of equipment (capital) has declined at an average annual rate of more than 3 percent. There has also been a negative correlation (-0.46) between the relative price of new equipment and new equipment investment. This can be interpreted as evidence that there has been significant technological change in the production of new equipment. Technological advances have made equipment less expensive, triggering increases in the accumulation of equipment both in the short and long run. Concrete examples in support of this interpretation abound: new and more powerful computers, faster and more efficient means of telecommunication and transportation, robotization of assembly lines, and so on. In this problem we are going to extend the Solow Growth Model to allow for such investment specific technological progress. Start with the standard Solow model with population growth and assume for simplicity that the production function is Cobb-Douglas: Yt = Kt L1t , where the population growth rate is deltaLt/Lt= n. Similarly, just as in the basic model, assume that investment and consumption are constant fractions of output It = sYt and Ct = (1 s)Yt. However, assume that the relationship between investment and capital accumulation is modified to:

Kt+1 Kt = qtIt Kt

where the variable qt represents the level of technology in the production of capital equipment and grows at an exogenously given rate , i.e. deltaqt / qt= . Intuitively, when qt is high, the same investment expenditure translates into a greater increase in the capital stock. (Note: another way to interpret qt is as the inverse of the relative price between machinery and output: when qt is high, machinery is relatively cheaper). (a) Transform the model (the production function, the equations for consumption and investment, and the capital accumulation equation) in per-worker form.

c) Suppose that capital per worker kt grows at a constant rate (we do not know that yet, but we will make a guess). Divide the capital accumulation equation by kt and use this assumption to prove that qtk1t has to be constant over time.

Suppose we are planning power system operations for 4 hours. Demand in the four hours is 350, 480, 460, and 380 MW, respectively. We have a collection of zero-cost variable resources with output of 0, 80, 40, and 0 MW in the four hours. We also have a collection of thermal assets with a cost function f(x) = .05x 2 (e.g., it costs $500 to produce 100 MW for an hour and $2000 to produce 200 MW for an hour). As in Problem 2, we can also curtail load at a cost of $10,000/MWh. Construct a model in Excel describing this system and solve to determine an optimal production schedule for the three resources (variable, thermal, and demand). Since the problem is nonlinear, you will need to choose the GRG Nonlinear solving method rather than Simplex LP.

(Production functions, inputs are perfect complements) Fine epoxy is used to produce LEDs and other electrical components. To get stable, good qualities (durability, resistance, adhesion) epoxy, the epoxy resins (R) are cured ("linked") to hardeners (H) like amines and acids, at the following fixed proportion: to produce 1 unit of final epoxy, we need to cure 2 units of R with 1 unit of H. Let be the quantity of final epoxy produced, (, ).

Then, the production function of epoxy is given by: (, ) = 1 2 min(, ), for some positive numbers and .

a. What is and ?

b. If we want to produce 10 units of the final product epoxy, what would be the least amount needed of R and H?

c. Does the production function of epoxy exhibit increasing, constant, or decreasing returns to scale? (IRS, CRS, or DRS?)

d. Draw a map of some isoquants of this production function