Exercise #1. Andy consumes two goods: food (measured in dollars) and other things (also measured in dollars). Let x1 be the amount that Andy spends on food in a given month and let *2 be the amount that Andy spends on other things in a given month. Andy's preferences over consumption bundles (X1,X2) are summarized by the utility function: U (X1,X2) = X1X2- Andy's monthly income is $400. (a) [10 pts.] What is Andy's optimal consumption bundle? Show your work. Illustrate your answer with a neat and clear diagram showing Andy's budget line and indifference curves. Label the points at which the budget line intersects the axes and identify the optimal bundle. (b) [10 pts.] Suppose now that the government implements a subsidy program for food. Specif- ically, for each dollar that Andy spends on food, the government will give Andy $0.50 in cash, with the restriction that the total amount of cash that Andy receives from the government cannot exceed $100. In a neat and clear diagram, graph Andy's budget line. Label the points at which the budget line intersects the axes and determine the coordinates of the kink point. Exercise #2. Barb's preferences over consumption bundles (x,y) are summarized by the following utility function: u (X1,X2) = 16x - 2x2 + 4y, where x is the amount of good x that Barb consumes and y is the amount of good y that Barb consumes. Let Px and Py be the prices of goods x and y, respectively. Let m be Barb's income. Barb's goal is to maximize her utility subject to her budget constraint. (a) [10 pts.] Find an algebraic expression for Barb's marginal rate of substitution between goods x and y. Show your work. In addition, give a coincise explanation of the meaning of the marginal rate of substitution. (b) [10 pts.] Suppose that Px = Py = 2 and m = 24. Use your answer from part (a) to determine Barb's optimal consumption bundle. Show your work. (c) [10 pts.] Suppose now that Px increases to 6, while Py and m remain the same. What are Barb's optinal choices for x and y in this case? Is Barb better or worse off than she was in part (b)? Explain. d) [10 pts.] How much extra income must Barb be given in order to compensate her for the increase in the price of good x in part (c)? Explain. Exercise # 3. Consider the following statements and say whether they are true or false and why. To get credit you should provide a clear justification for your answers. (a) [4 pts.] If two goods are perfect complements and the price of one of them increases, the quantity demanded of both goods decreases. (b) [4 pts.] A non-transitive preference relation > can be represented by some utility function. (c) [4 pts.] Consider two goods x and y. If preferences are strictly convex, the absolute value of the marginal rate of substitution between x and y is decreasing along an indifference curve as x increases. (d) [4 pts.] The following Cobb-Douglas utility functions represent two different preference relations: U1 (x,y) = 0.3 log(x) + 0.6 log(y) U2 (x,y) = 0.6 log(x) + 1.2log(y). (e) [4 pts.] If a consumer is making an optimal choice between two goods x and y, then, independently of his preferences, the following condition must always hold: Px = MRS (x,y) . Py (f) [4 pts.] If the following condition holds Px = MRS (x,y) Py then a consumer must be making the optimal choice between x and y, independently of his preferences. (g) [4 pts.] A cigar is a luxury good for a consumer that has Cobb-Douglas preferences over cigars and food. (h) [4 pts.] Consider two goods x and y, with prices Px and Py, respectively. A 0.07 percent value tax on these two goods does not affect the relative price of x in terms of y. (i) [4 pts.] The marginal rate of substitution measures the rate at which the market is willing to substitute one good for the other. (j) [4 pts.] An indifference curve represents the collection of all the bundles that a consumer can buy.Exercise #1. [10 pts.] A monopolist has an inverse demand curve given by p(y) = 12-y and a cost curve given by c(y) = y". Describe how to determine its profit-maximizing level of output, and compute it. Show your work: to receive full credit you should show how you compute the optimal quantity of output. Exercise #2. [15 pts.] You are the manager of a new amusement park that has monopoly power over the service it offers: roller coaster rides. You have figured that the park will attract 1,000 people per day, and each person will take a = 50 - 50p rides, where p is the price of a ride. The marginal cost of a ride is essentially zero. (a) If admission to the park were free and you had to choose the profit maximizing price of a ride p", which value would you set? (b) Suppose now that you can use a two-part tariff, i.e., you can set a price for admission to the park, and another one for each roller coaster ride. Which admission fee and price of a ride would you choose? Show your work: to receive full credit you should show how you compute the optimal prices. Exercise #3. [20 pts.] Your company has monopoly over product H. It sells it in the UK and in the US. The cost function for your firm is c(q) = 10q. Your marketing department has identified the UK and the US demand curves for H to be qUS = 50,000 - 2,000pus 4UK = 10,000 - 500PUK; where qus and quy represent the quantities demanded by US and UK consumers respectively, and pus and pux the (US dollar) prices charged for the product in the US and in the UK, respectively. (a) If you were to charge the same price in both countries, how many units of H should you sell, and what price should you charge in order to maximize your company's profits? (b) If you could charge different prices in the US and the UK what prices would you choose and how many units would you sell in the US and how many in the UK? Exercise #4. [20 pts.] A company sells product A in a competitive market. Its long-run cost function is given by c(y) = y' + 10 for y > 0 c(y) = 0 for y = 0, where y represents the quantity of good A. (a) What is the lowest price at which this company will supply a positive amount of product A in the long-run? (b) Suppose that the market price for A is p = 2v10, and that 100 firms are operating in this market in the long-run. What is the equilibrium market demand for the product? Part II Exercise #5. [17 pts.] The rental price of machinery K (measured in machine-hours) is $10 per hour, while the hourly wage rate for labor, L (measured in labor-hours), is $6. Find the cost function associated with the following technology: y = 10K + L, where K represents machinery ( measured in machine-hours) and L labor (measured in labor-hours). Part I Exercise #6. [18 pts.] John likes books and restaurant meals. His utility function is u(It. I'm) = (15)02 (1m)0.4. where r, represents the quantity of books that John "consumes" in a month, and I'm the amount of restaurant meals that he consumes. Suppose that both books and restaurant meals are perfectly divisible goods. The relative price of books in terms of restaurant meals is 0.4, i.e., buying a book costs 40% of the price of a restaurant meal. The nominal price of a restaurant meal is $40. John's income is $1,000 per month. Compute the quantity of books and restaurant meals that John consumes in a month