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Theory: 1) Nicholson 3.2 (Note that: a log... x/ax =1/(xIna), 3.4, 3.7 (Hint: observe how these three functions are related), 4.2, 4.5 (Note: use low-tech math and common sense here), 5.2 and 5.9. 2) Let U(x, y) = -1/x -1/y . Suppose that prices are P, and P., and income is I . (A) Calculate the utility-maximizing choices of x and y, that is, the Marshallian demand functions d, (P,, P,, I) and d, (P., P,, 1). (B) Calculate "indirect utility," i.e., the utility at the optimal choices, V(P, , P. , I). (C) For a given utility level U., solve the dual expenditure-minimization problem, and compute the optimal choices of X and Y, i.e., the "compensated demand functions" h, (P,, P,, U. ) and h, (P, P,, U.). D) Calculate the minimum expenditure function E(P,, P., U.) . Show that the expenditure and indirect utility functions you have calculated are inverses of one another, i.e., show that E(P. . P,, V(P:, P,; D)= I and V(P, , P,, E(P.,P,,U.))=U.. (E) Consider the Slutsky equation for consumer theory (see equations 5.22 and 5.26 in Nicholson). State a verbal interpretation of the Slutsky equation. (F) Verify directly that the Slutsky equation holds in this problem. Application: Dwyer and Lindsay (Irish Potato) article 3) Draw a diagram that illustrates the surprising properties of a Giffen good. Your diagram should illustrate a consumer's 2-good utility maximization problem for two different budget sets, where P, is higher under the second constraint (and x is the Giffen good). 4) Consider Figure 1 of Dwyer and Lindsay. (A) Explain why the top diagram is consistent with your diagram from the previous question. (B) Why is the lower diagram a better description of the Irish potato famine? (C) What is the evidence on what happened to potato prices during this period? Is this consistent with potatoes being a Giffen good? 5) Dwyer and Lindsay state that "Inferiority is necessary for a good to be Giffen." (A) Demonstrate mathematically why this statement must be true. (B) Why is it unlikely that potatoes were inferior during the Irish potato famine? 6) Dwyer and Lindsay also claim that "For a good to be Giffen, some normal good must be displaced by the inferior good as the price rise lowers real income." (A) Prove this statement. (B) Why is it also unlikely that this statement was applicable to potatoes during the Irish potato famine? 7) Someone who has read Dwyer and Lindsay says to you, "I don't see the big deal about Giffen goods - they're everywhere. Take gasoline, for example. Practically every summer, the price of gas increases and people buy more of it. Looks to me like a Giffen good." (A) Evaluate the logic of this assertion. (B) Can you think of a plausible 'natural experiment' (as in the Card & Krueger paper) that would allow you to test whether gasoline is a Giffen good? (C) What would you expect to find regarding the demand for gasoline?Exercises Line and Equation of a Line/Solving Two Equations 1. Consider a line given by the equation y = x + 2. a) Does this line intersect with a line going through (1, -3) with a slope of 3? If yes, solve for the intersection point. If no, why? b) Does this line intersect with a line going through (0, 4) and (2, 6)? If yes, solve for the intersection point. If no, why? c) Does this line intersect with a line given by the equation 4x - 4y = -8? 2. A policeman has $100 to spend by either buying x donuts ($5/donut) or y cups of coffee ($10/cup). Assuming he spends all $100, write the equation for the line that shows all combinations of donuts and coffee he can consume. (This is called "Budget Line.") Give your answer in both forms of Ax + By = C and y = mx + c. What are the x- and y-intercepts? 3. A line is given by the equation y = -2x + 5. If it is shifted up by 2 units, what is the new equation of this line? Instead, if it is shifted to the left by 2 units, what is the new equation of this line? 4. Repeat question 3 when the equation of a line is given by y = 2x + 5. 5. Suppose y is the distance travelled in miles, and x is the time travelled in hours. If the relationship between distance and time is given by equation y = 3x, what is the unit of the slope? Percentages and Weighted Averages 6. Alice has received her salary raise from $10 to $12.5, what is the percentage change in her salary? What about Bryan who has received his salary cut from $12.5 to $10? 7. In ECON 666, homework, midterm 1, midterm 2 and final exams worth 10, 25, 25 and 40 percent of final grade, respectively. Each item worth 100 points. You have received full points on homework, 90 points for midterm 1, and 80 points for midterm 2. To receive an A, you need to score 85 points. What is the minimum you need to score in final exam to receive A? Area of Triangle, Rectangle, and Trapezoid 8. What is the area of a triangle with the base of 3 units and height of 6 units? 9. What is the area of a rectangle with the width of 4 units and height of 8 units?Problems 1. Suppose a monopolistic local utility company faces a demand curve given by P = 120 - 40. Total cost for this firm is given by TC = 400 + 40, and MC is fixed at $4 per unit. a. Does the technology of a firm represent economies of scale? b. What is the fixed cost? Does this indicate high barriers to entry? c. What is the socially optimal level of production and price? d. Suppose this industry operates as a monopoly. Find the equilibrium price and quantity. e. The government, bowing to public pressure to regulate monopolies, decides to force firms to charge their marginal cost just like they would in perfect competition. How much will the monopolist produce? What is the profit for this monopolist? Is it sustainable? f. Suppose the government instead chooses to force the monopolist to charge a price equal to their average total cost, this monopolist will supply 25 units. What will be their profits? 2. Plastic molding has both industrial and dental uses. Consider a monopolist producer of this good with constant marginal cost MC = 4. The demand curves for the two market segments are given below Dental users P - 100 - 20 Industry users: P - 50 - 0.5Q a. If a monopolist can practice third-degree price discrimination, what price will they set in the two markets? What is the consumer surplus for each market? b. Now suppose the monopolist cannot price discriminate Instead, they must charge a single price in both markets. What price will they charge? c. Is consumer surplus higher or lower without price discrimination? d. (True story) Facing a market like this, one supplier of the plastic molding methyl methacrylate considered mixing arsenic with the product sold to industrial users. You might think about why this could be advantageous to the seller. 3. Anna and Boris went to the state fair together, but now can't find each other. They'd like to meet up, but can go to only one of two events to find the other. They can go to the horse show or to the truck rally. Neither person will enjoy the event if alone, but Amma would prefer the horse show while Boris would prefer the truck rally. This is modeled as a game in the table below Boris Horse Show Truck Kally 2, 1 Anna Horse Show 0.0 Truck Kully 0 , 0 1, 2 a. Does Anna have a dominant strategy? Does Boris? b. What are the equilibria in this game
