Solve all of the following in details
Exercise #1. The supply ofprivate apartments for rent around Chill is given by the following function '35 = lm. where Q3 denotes the quantity of apartments supplied in a given year. The yearly demand for apartments is given by QD = 1., mm _ PI. where Q1: denotes the quantity of apartments demanded in a given year and 1] denotes the monthly rent for one apartment. {a} [5 pts.] Find the equilibrium price and quantity in this market. {b} [10 pts.] The university enacts a policy that subsidiaes rent for students that decide to rent a private apartment around CM'U. Specically, the university will subsidize rent by paying 31m to a student every time hefshe shows a receipt of monthly rent payment signed by a landlord. lIEompute the equilibrium prices and quantity in this case. [Hint there wiii ire two prices. One paid by the student out ofhisfher cum pocket, p5, and the other received by the seiner of the apartment, P3r with Pa = P3 l] {c} [1|] pts.] [in a diagram that has quantity of apartments on the x-axis and the monthly rent on the y-axis, draw the inve1se demand and supply curves and mark the equilibrium prices of points {a} and {b}. {d} [15 pts.] Compare the equilibrium in [all with the equilibrium in {h}: i} what is the change in consumer's surplus induced by the policy of the university? ii} \"that is the change in supplier's surplus? iii} How much does the university spend to subsidize rents? iv} \"hat is the deadweight loss induced by this policy? Exercise #2. Consider a rm which produces a single output using two inputs according to the following production function: y = min {2 KL} 1 where y is the rm's output1 K is machinery {measured in machine-hours} and L is labor supply [measured in person-hours}. Let r be the cost of one machine-hour and let to be the wage rate [i.e. the cost of one person-hour}. {a} [ll] pts.] Does the rm's technology exhibit increasing returns to scale? Explain carefully. {b} [5 pts.] In a neat and clear diagram1 draw the isoquant corresponding to output of II] units. Be sure to label important features in the diagram. {c} [1|] pts.] Suppose that r = 16 and w = 4. Determine the rm's cost function mfg}. {d} [10 pts.] Clo a diagram that has costs on the y-axis and output and the x-ain's, plot the rm's cost curve cy}, average cost curve Ay}, and marginal cost curve J'lfC{y}l. Exercise #3. Short questions. To get full credit you should justify your answers. {a} [5 pts.] Show that~ in the short run1 if the price of the xed factor is increased, prots will decrease. {b} [10 pts.] If pJ'rIPl } all, [where p is the output price~ MP1 the marginal product of factor l~ and an the rental rate of factor 1}\" should the rm increase or decrease the amount of factor 1 in order to increase prots? Explain carefully. {c} [III] pts.] If a rm had everywhere increasing returns to scale1 what would happen to its prots if prices remained xed and Hit doubled its scale of operations? Show your work. Problem Set # 1 As indicated on the course syllabus, this problem set is due at the beginning of your recitation section on Friday, February 2. Exercise # 1. In 1998, the Adams family was living in Concord, New Hampshire. John Adams, working as a high school teacher earned a salary of 30,000$ before taxes. As a police officer, Mary Adams earned a salary of 30,000$ a year gross of tax The items on which the Adams spend their income are: i) food, specifically hamburgers; ii) clothes. The price of hamburgers in Concord is 50 cents per hamburger, while clothes are 30$ each. There is no sales tax in New Hampshire. However, the Adams pay 50% of their incomes in federal taxes. Let Xn and Xc denote the amount of hamburgers and clothes that the Adams buy in a year. (a) What is the relative price of hamburgers in terms of clothes in Concord in 1998? (b) Write down the equation that represents the Adams' budget constraint in 1998. (c) Represent the Adams' budget line in 1998 on a clearly-labelled graph which has the quantity of clothes on the x-axis and the quantity of hamburgers on the y-axis. Shade in the region on the graph which contains the set of hamburgers-clothes bundles that the Adams can "afford" (d) A president is elected at the end of the year 1998 that reduces the federal income tax rate for the following years from 50% to 10%. Write down the equation that represents the Adams' new budget constraint in 1999 and represent it on the same graph of point (c) (Assume that both the Adams's income and the goods' prices are unchanged with respect to 1998.) Shade in the region on the graph which contains the set of hamburgers-clothes bundles that the Adams can "afford". On January 1, 2000 the Adams moved from Concord, NH to Pittsburgh, PA. John and Mary kept working in Pittsburgh as high school teacher and police officer, respectively. Their yearly incomes remain the same and so do their consumption habits. Also the prices of hamburgers and clothes remain the same. However, now their purchase of clothes is subject to a 7% sales tax. (e) What is the relative price of hamburgers in terms of clothes in Pittsburgh in 2000? Are hamburgers more expensive in terms of clothes in Concord or in Pittsburgh? (f) Write down the equation that represents the Adams' budget constraint in 2000. (g) Represent the Adams' budget line in 2000 on a clearly-labelled graph which has the quantity of clothes on the x-axis and the quantity of hamburgers on the y-axis. Shade in the region on the graph which contains the set of hamburgers-clothes bundles that the Adams can "afford" and compare it to the region that you have shaded at point (c). How many dollars would the Adams pay in sales taxes in the year 2000, if they spent 30,000$ in hamburgers and the rest of their income (net of federal taxes) in clothes? Exercise #2. Do Problem 3.9 in the Workouts book. Do not hand in the sheets from the Workouts book; write your answers instead on a separate sheet to be handed in. Be sure to show your work. Exercise #3. Do Problem 4.7 in the Workouts book. Do not hand in the sheets from the Workouts book; write your answers instead on a separate sheet to be handed in. Be sure to show your work. Exercise #4. Do Problem 4.11 in the Workouts book. Do not hand in the sheets from the Workouts book; write your answers instead on a separate sheet to be handed in. Be sure to show your work.Problem Set # 2 As indicated on the course syllabus, this problem set is due at the beginning of your recitation section on Friday, February 9. Exercise # 1. Anna's preferences over consumption bundles (X1,X2) are summarized by the utility function U (x1,X2) = X1 (x2 + 1)". (a) Derive an algebraic expression for the marginal utility MU1 (X1,*2) of good 1. (b) Derive an algebraic expression for the marginal utility MU2 (x1,X2) of good 2. (c) Use your answers from parts (a) and (b) to derive an algebraic expression for Anna's marginal rate of substitution (MRS) of good 2 for good 1. If Anna is currently consuming 3 units of good 1 and 5 units of good 2, what is the value of her MRS? d) Suppose again that Anna is currently consuming the bundle (3,5). Is Anna willing to exchange 1 unit of good 2 for 1 unit of good 1? [Hint: notice that an extra 1 unit of good 1 in this context is not a marginal change. Evaluate Anna's utility at the bundle (3,5) and the bundle (4,4) to realize she is not indifferent between the two.] (e) Suppose now that Anna is currently consuming 27 units of good 1. How many units of good 2 must she consume in order to leave her just as well off as she was in part (c)? What is the value of Anna's MRS at this consumption bundle? Compare your answer to the corresponding answer in part (c). Are your answers consistent with the convexity of Anna's preferences? Exercise # 2. The government of Bahnanas collects every year 10 million Bahnanas dollars (BS) worth of taxes. The constitution of the state of Bahnanas requires that in each year the government's budget be balanced. At the beginning of 1999 the government has to decide how to allocate tax revenues to two purposes. The government can pay down a certain amount of national debt. Let Xa denote how many millions of dollars are allocated to the payment of the national debt in 1999. The government can also contribute, to some extent, to its 1,000,000 citizens' medical expenditures. The government has computed that during 1999 around 50% of its population will make an appointment (and only one) with a doctor. Each visit to a doctor costs B$100. The government's health plan will cover a certain fraction Xh of the cost of a visit to a doctor. Notice that this variable is measured in percentage terms (e.g.: a value of 1 indicates 100% coverage). The government's budget line then reads (choose units so that all numbers in this equation represent millions of B$) 10 = Xd + 50Xh. The constitution of Bahnanas grants the prime minister the authority of choosing Xq and Xh. His preferences over policy bundles (Xa,X) are represented by the Cobb-Douglas utility function: H u (Xa,Xh) = 1000 xx (1) (a) What is the "relative price" of Xa in terms of Xh from the prime minister's point of view? "b) Find the policy bundle that maximizes the prime minister's utility subject to the govern- ment's budget constraint. (c) What is the share of the government's 1999 budget that the prime minister decides to spend on health care? (d) How would your answer to question (c) change if the prime minister's preferences were represented by the following utility function: V (Xd,Xn) = 0.5 log (Xd) + 0.5 log (xn) , instead of the one in equation (1)? Exercise #3. Do Problem 5.5 in the Workouts book. Do not hand in the sheets from the Workouts book; write your answers instead on a separate sheet to be handed in. Be sure to show your work
