Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11

Economics 2020a / HBS 4010 / HKS API-111 Fall 2011 Practice Problems for Lectures 1 to 11 LECTURE 1: BUDGETS AND REVEALED PREFERENCE 1.1. Quantity Discounts and the Budget Constraint Suppose that a consumer has wealth W = 32 and that there are two goods x and y with prices p = 4, q = 4 respectively. In addition, the per-unit price of good x is reduced to p = 1 for purchases beyond Q* units. Thus, if Q* = 4, the consumer can afford the bundle (8, 3), paying a total of 16 for the first four units of x, a total of 4 for the last four units of x and a total of 12 for the three units of y. (a) Find the equation(s) for the budget constraint for the following values of Q*: (a1) Q* = 5; (a2) Q* = 16/3; (a3) Q* = 6. (b) For what values of Q* does the quantity discount affect the budget constraint? (c) For what values of Q* is the budget set convex? 1.2. Discounts on Initial Units and the Budget Constraint Suppose that a consumer has wealth W = 24 and that there are two goods x and y with prices p = 4, q = 4 respectively. In addition, the per-unit price of good x is reduced to p = 2 for purchases of less than Q* units. Thus, if Q* = 4, the consumer can afford the bundle (6, 2), paying a total of 8 for the first four units of x, a total of 8 for the last two units of x and a total of 8 for the two units of y. (a) Find the equation(s) for the consumer s budget constraint with (a1) Q* = 2; (a2) Q* = 4; (a3) Q* = 6. (b) For what values of Q* is the budget set convex?

1.3. Revealed Preference and Transitivity Suppose that a consumer has wealth W = 60 to spend on goods x and y. Consider the following three possible budget lines where p k denotes the price of good x and q k denotes the price of good y for budget line k Budget Line 1: p 1 = 4, q 1 = 1; Budget Line 2: p 2 = 2, q 2 = 2; Budget Line 3: p 3 = 1, q 3 = 4. Denote the consumer s optimal bundle for budget line k as (x* k, y* k ), Assume throughout the problem that Walras Law holds i.e. that the consumer spends the entire budget of W = 60 given any of these three budget lines. (a) Find the equations for budget lines 1 through 3 and represent all three budget lines in a single graph. Use the principle of revealed preference in answering the questions in the remainder of the problem. (b) For what values of (x* 1, y* 1 ) can you conclude that the consumer prefers budget line 2 than to face budget line 1? Are there any values of (x* 1, y* 1 ) that demonstrate conclusive that the consumer prefers budget line 1 than to face budget line 2? For parts (c) and (d), assume that consumer chooses (6, 24) given budget line 2. (c) What bundles could be optimal for budget line 1; what bundles could not be optimal for budget line 1? What bundles could be optimal for budget line 3; what bundles could not be optimal for budget line 3? (d) Does the consumer prefer budget line 1 to budget line 2, or is it impossible to tell? (e) Suppose that the consumer chooses (16, 11) given budget line 3 but that you do not know what bundles the consumer chooses for either budget line 1 or budget line 2. Can you tell if the consumer prefers budget line 2 to budget line 3? Can you tell if the consumer prefers budget line 1 to budget line 3? (f) Suppose that the consumer chooses (6, 24) given budget line 2 and (16, 11) given budget line 3. Can you tell if the consumer prefers budget line 1 to budget line 3?

1.4 (Not) The Subsidy Principle A student wonders why it isn t possible to use the logic for the Lump Sum Principle to conclude that subsidies must be preferred to lump sum transfers. This problem is designed to explore that question. Suppose in a two-good world that p = 2, q = 2 and consider a consumer with initial wealth W = 60. After a lump sum transfer of T = 20, the consumer is observed to choose the bundle L = (20, 20). NOTE: These conditions match those from the example in lecture 1, except that in lecture we assumed that the consumer chose bundle (20, 20) in response to a subsidy, whereas this problem assumes that the consumer chose bundle (20, 20) in response to a lump sum transfer. (a) What subsidy s would have to be given to the consumer on good 2 (changing its price to q = q - s) so that the consumer could afford bundle L with original wealth W = 60? (b) Graph the budget lines through L = (20, 20) for the lump sum transfer and the subsidy from (a). (c) Can you conclude that the consumer (weakly) prefers the subsidy from (a) to the lump sum transfer of W = 20? (d) Explain why the logic indicated in parts (a) through (c) does not demonstrate that subsidies should be preferred to lump sum transfers. 1.5. Revealed Preference and Compensated Demand Suppose that there are two goods and that a consumer with wealth 100 selects the bundle (5, 5) at prices p 1 = q 1 = 10. You can assume throughout the problem that the consumer spends all wealth on goods. (a) According to the Law of Compensated Demand, what bundles are ruled out if prices change to p 2 = 8 and q 2 = 12 and wealth stays at 100? What bundles are ruled out if prices change to p 3 = 16 and q 3 = 24 and wealth increases to 200? (b) Now suppose that there are three goods with respective prices p, q and r, and that a consumer with wealth 150 selects the bundle (5, 5, 5) at prices p 1 = q 1 = r 1 = 10. Suppose that wealth stays at 150 and prices change to p 2 = 8, q 2 = 10, and r 2 = 12. For which of the three goods, if any, can you determine that demand goes up or demand goes down if the consumer chooses a bundle that is different from (5, 5, 5)? MWG: Revealed Preference with Unspecified Wealth 2.F.3 (a) through (d) LECTURE 2: INDIFFERENCE CURVES AND UTILITY FUNCTIONS

2.1 Properties of Preferences Consider the following preferences. Do they satisfy continuity, transitivity, monotonicity and local nonsatiation? (a) Lexicographic preferences. (x, y) (x, y ) if x > x or if x = x and y > y. (b) (x, y) (x, y ) if (x + y x y ) >1; (x, y) ~ (x, y ) if x + y x y < 1; (x, y ) (x, y) if (x + y x y) > 1. (c) (x, y) (x, y ) if (x-5) 2 (y-5) 2 > (x -5) 2 (y -5) 2 (x, y) ~ (x, y ) if (x-5) 2 (y-5) 2 = (x -5) 2 (y -5) 2 (x, y ) (x, y) if (x -5) 2 (y -5) 2 > (x-5) 2 (y-5) 2 (d) (x, y) (x, y ) if (x-5) (y-5) 2 > (x -5) (y -5) 2 (x, y) ~ (x, y ) if (x-5) (y-5) 2 = (x -5) (y -5) 2 (x, y ) (x, y) if (x -5) (y -5) 2 > (x-5) (y-5) 2 2.2 Convex Preferences and Quasiconcave Utility Functions Consider the following different rules for indifference curves: (1) All bundles (x, y) with the same value of xy are on the same indifference curve; (2) All bundles (x, y) with the same value of x + y are on the same indifference curve; (3) All bundles (x, y) with the same value of x + y 2 are on the same indifference curve. (a) Assume that preferences are strictly increasing in x and in y in each case. Explain why this implies that bundles on a higher indifference curve are strictly preferred to bundles on a lower indifference curve, where higher indifference curves lie up and to the right of lower indifference curves. (b) Draw a graph (or graphs) with indifference curves for each of these rules. Which of them appears to correspond to convex preferences? (c) Use the definition of convex preferences to determine which of these rules produces (strictly) convex preferences. 2.3 Skiing & Violins A friend tells you that according to her preferences for ski trips (x) and violin lessons (y), bundles (x, y) with the same value of 10 x + y 2 are on the same indifference curve and that her preferences are strictly increasing in each of x and y.

(a) Draw the indifference curves through the points (10, 10), (0, 15), and (30, 0). (b) Argue that the preferences associated with these indifference curves are not convex. For parts (c) and (d), assume that W = 300, p = 10 and q = 20. (c) Use a graph to identify the optimal bundle for the consumer given these prices. (d) Explain how the nonconvexity of preferences affects the nature of the optimal bundle in (c). (e) Fix q = 20 and let p range from 0.1 to 100. Find the optimal bundle as a function of p for all values of p in this range. 2.4 Piecewise Linear Indifference Curves and Optimal Bundles Suppose that there are two goods x and y and that a consumer has a preference for equal consumption as well as for more of each good. This particular consumer is indifferent between a bundle (x, y) with x > y and a bundle (z, z) with equal quantities of each good, where z = x - 2 (x- y) / 3. So, for example, this consumer is indifferent between (8, 5) and (6, 6). Suppose that the relationship is the same for (x, y) with y > x so that the consumer is also indifferent between (5, 8) and (6, 6). In words, the consumer would give up two units of the more abundant good in any bundle in order to increase the quantity of the less abundant good by one unit. (a) Find an equation to identify the consumer s preference between two bundles (x 1, y 1 ) and (x 2, y 2 ). Assume that x 1 > y 1 and x 2 > y 2. (b) On a graph, show the consumer s indifference curves through (4, 4) and (8, 8). (c) Use a graph to identify the optimal bundle for the consumer if prices p = q = 4 and wealth W = 32. (d) Use a graph to identify the optimal bundle for the consumer if p = 4, q = 1, W = 12. (e) Describe the intuition for the different results in (c) and (d). 2.4. Continuity and Indifference Curves (a) Consider a two-good world and let A = (2, 2), B = (2, 6), C = (0, 2). Draw a line between B and C and define bundle D = (1, 4), which is on this line. Suppose that (1) if x = (x 1, x 2 ) is on this line and x 1 > 1, then x is strictly preferred to A and (2) if x = (x 1, x 2 ) is on this line and x 1 < 1, then A is strictly preferred to x. Show that the continuity axiom in combination with (1) and (2) implies A ~ D.

(b) Part (a) demonstrates the existence of a single bundle D on the line between B and C that is on the same indifference curve as bundle A. In this part of the problem, we want to show that the indifference curve including bundle A must be a continuous curve at bundle D. Maintain the assumptions from (a) and now assume that there is a sequence of bundles y 1, y 2, y 3, that converges to a point x on the line between B and C. Argue that the preferences would not be consistent with the continuity axiom if y 1 ~ y 2 ~ y 3 ~ A and x (1, 4). Optional Problem MWG 2.F.4. Revealed Preference over Time

LECTURE 3: TRADEOFFS AND THE CONSUMER PROBLEM 3.1 Transformations of Utility Functions (a) Consider the utility function u(x, y) = xy. Find a transformation function f(u) to create the new utility function v(x, y) = f(u(x,y)) with the property that v(k, k) = k. Show that the marginal ratio of substitution is the same for both u and v. (b) Consider the utility function u(x, y) = 2x + y 2. Find a transformation function f(u) to create the new utility function v(x, y) = f(u(x,y)) with the property that v(k, k) = k. Show that the marginal ratio of substitution is the same for both u and v. 3.2. Cobb-Douglas Utility: Consider the utility function u(x, y) = x α y 1-α (a) Derive the consumer s optimal bundle at prices p and q. (b) Derive the value of the Lagrange multiplier. (c) How does the optimal bundle change in response to changes in parameters α, p, q? (d) Consider the transformation of this utility function v(x, y) = ln(u(x,y)) = α ln x + (1-α) ln y. Show that the consumer s optimal bundle is the same for u and for v. Is the Lagrange multiplier the same for u and for v? 3.3. Skiing & Violins revisited A friend tells you that her utility function for ski trips (x) and violin lessons (y) is u(x, y) = 10 x + y 2. Assume that W = 300, p = 10 and q = 20. NOTE: These conditions match the conditions of problem 2.3. (a) Find the first order conditions associated with this problem. (b) Explain why these conditions do not identify the optimal bundle. What do they identify instead? (c) What is the optimal bundle under these conditions? 3.4 Comparative Statics with One Good A consumer consumes only one good, bananas. Let b denote the number of bananas eaten. The consumer s willingness to pay (measured in dollars) for b bananas is log(b+1), where log

denotes the natural logarithm (as it always does in economics). Let p > 0 denote the price of a banana. Thus, the total surplus (i.e., utility) of the consumer who buys and eats b bananas is given by: u b log b 1 pb. (a) Given the consumer s objective function, u(b), derive the first-order conditions for the consumer s optimal choice of b, which you may denote b*. Note: be sure to address the fact that the optimal choice of b must be non-negative. (b) Explain whether you can be sure that the b* that satisfies the condition you found in part (a) is really a global maximum. (c) How will the consumer s optimal choice of b respond to an increase in the price of bananas? Does your answer depend on p? (d) Write down an expression for the optimized value of the consumer s objective function, and explain how the optimized level of the consumer s utility reacts to a change in p. Does your answer depend on p? Explain and interpret your answer. 3.5 Comparative Statics and Marginal Rate of Substitution Suppose that there are two goods x and y and consider the utility function u(x, y) = x 1/2 + y 1/3 Suppose that p = 1 and that wealth W = 9 + 8q is a function of q, the price of good y. Note that for any value of q, the bundle (9, 8) is on the budget line. (a) Find the value q* such that (9, 8) is the consumer s optimal bundle? (b) Argue that for p = 1, q < q*, the consumer s optimal bundle sets x < 9, y > 8. (c) Argue that for p = 1, q > q*, the consumer s optimal bundle sets x > 9, y < 8. For parts (d) through (f), suppose instead that p = q = 2 and that wealth is a fixed value W rather than a function of q. (d) Find the value W = W* for which the consumer s optimal bundle sets x = 1.5. (e) Argue that if W < W*, the consumer s optimal bundle sets x < 1.5. (f) Argue that if W > W*, the consumer s optimal bundle sets x > 1.5. NOTE: It is much simpler to apply the first-order conditions in the form of the marginal rate of substitution condition than to use a formal Lagrangian approach.

3.6 Utility Maximization with Quantity Discount Suppose that a consumer has wealth W = 32 and that there are two goods x and y with prices p = 4, q = 4 respectively. The consumer s utility function is given by u(x, y) = xy. In addition, the per-unit price of good x is reduced to p = 1 for purchases beyond Q* units. Thus, if Q* = 4, the consumer could then afford to consume (8, 3), paying a total of 16 for the first four units of x, a total of 4 for the last four units of x and a total of 12 for the three units of y. NOTE: These conditions match the conditions of problem 1.1. (a) Suppose that Q* = 6. Find the consumer s utility maximizing bundle. (b) Suppose that Q* = 5. Find the consumer s utility maximizing bundle. (c) Suppose that Q* = 16/3. Find the consumer s utility maximizing bundle(s). (d) Explain the intuition for the relationship of the solutions for (a) through (c). (e) When Q* < 8, the budget set is not convex. Explain how this property of the budget set affects the relationship of the optimal bundles in (a) through (c). 3.7 Utility Maximization with Discounts on Initial Units Suppose that a consumer has wealth W = 24 and that there are two goods x and y with prices p = 4, q = 4 respectively. The consumer s utility function is given by u(x, y) = xy. In addition, the per-unit price of good x is reduced to p = 2 for purchases of less than Q* units. Thus, if Q* = 4, the consumer could then afford to csonsume (6, 2), paying a total of 8 for the first four units of x, a total of 8 for the last two units of x and a total of 8 for the two units of y. NOTE: These conditions match the conditions of problem 1.2 (a) Suppose that Q* = 2. Find the consumer s utility maximizing bundle. (b) Suppose that Q* = 4. Find the consumer s utility maximizing bundle. (c) Suppose that Q* = 6. Find the consumer s utility maximizing bundle. (d) Explain the intuition for the relationship of the solutions for (a) through (c). (e) The budget set is convex for each value of Q* in (a) through (c). Explain how this property of the budget set affects the relationship of the optimal bundles in (a) through (c).

LECTURE 4: WEALTH AND CONSUMPTION 4.1 Wealth Elasticities (a) Suppose that u(x, y) = x α y 1-α. Verify that the wealth elasticity of each good is equal to 1 for all combinations of W, p, q, with W, p, q > 0. (b) Suppose that u(x, y) = x + ln(y). Find the wealth elasticity of demand for goods x and y for each combination of W, p, q > 0. Then verify the identity B x ε x, w + B y ε y, w = 1 where B x is the budget share of good x : B x = px / W. 4.2 Identifying Homothetic Preferences Which of the following utility functions represent homothetic preferences? (a) u(x, y) = x 1/2 + y 1/2 (b) u(x, y) = x 1/3 + y 2/3 (c) u(x, y) = ln (xy) + x 1/3 y 2/3 4.3 Homothetic Utility and Price Indices Suppose that a consumer has wealth 8 in each of two periods and faces prices (p=1, q=1) in period 1 and prices (p=1, q=4) in period 2 for goods x and y. The consumer has identical Cobb- Douglas preferences u(x, y) = xy for consumption in each period and chooses bundles for each period independently to maximize utility in that period. (a) What are the consumer s optimal bundles in periods 1 and 2.? (b) Compute the CPI for these bundles and prices based on the Laspeyres index, the Paasche index, and the Fisher ideal index. (c) Now find the exact change in wealth necessary to achieve the same utility in each period. What value for the price index corresponds to this cost of living adjustment? Suppose instead that the consumer has a quasilinear utility function v(x,y) = x + 8y 1/2 (d) What are the consumer s optimal bundles in periods 1 and 2.? (e) Compute the CPI for these bundles and prices based on the Laspeyres index, the Paasche index, and the Fisher ideal index. (f) Now find the exact change in wealth necessary to achieve the same utility in each period. What value for the price index corresponds to this cost of living adjustment? 4.4 Price Indices and Revealed Preference (somewhat related to MWG 2.F.4) Suppose that a consumer chooses bundle (x 1, y 1 ) in period 1 at prices (p 1, q 1 ) and then chooses bundle (x 2, y 2 ) in period 2 at prices (p 2, q 2 ), that at least one price changes from period 1 to period 2 and that the consumer spends the same level of wealth W in each period (i.e. p 1 x 1 + q 1 y 1 = p 2 x 2 + q 2 y 2 = W).

(a) Use a graph to show the relationship of (x 1, y 1 ) and (x 2, y 2 ) when the Laspeyres index is less than 1. Use a separate graph to show the relationship of (x 1, y 1 ) and (x 2, y 2 ) when the Paasche index is less than 1. (b) Use revealed preference to show that if the Laspeyres index based on these purchases is less than 1, then the consumer must strictly prefer (x 2, y 2 ) to (x 1, y 1 ). (c) Show that if the Laspeyres index based on these purchases is exactly equal to 1 than the Paasche index must be strictly less than 1 unless x 1 = x 2. How does this finding relate to a standard theoretical result in consumer theory? (d) Suppose that the Laspeyres index is less than 1. Is it possible for the Paasche index to be greater than 1? 4.5 Quasilinear Utility and Boundary Solutions (a) Suppose that u(x, y) = x + 10 y 1/2 and suppose that the price of good x is p = 1. For what prices q for good 2 will the consumer purchase a positive amount of good y? Explain why your answer does not depend on the consumer s wealth W (assuming only that W > 0). (b) Suppose that u(x, y) = x + 10 (y+1) 1/2 and suppose that the price of good x is p = 1. For what prices q for good 2 will the consumer purchase a positive amount of good y? Explain why your answer does not depend on the consumer s wealth W (assuming only that W > 0). (c) Explain intuitively why there is a qualitative difference in the results for (a) and (b). 4.6 Quasilinear Utility and First-Order Conditions Suppose that u (x 1, x 2, x 3 ) = A x 1 + x 2 1/2 + ln (x 3 ), where A is a positive constant. (a) What must be true of prices (p 1, p 2, p 3 ), wealth W, and the parameter A for the consumer to purchase positive quantities of all three goods? (b) Suppose that prices = (3, 4, 5). What is the maximum level of wealth such that the consumer does not purchase any of good 1? (c) Suppose that prices = (3, 4, 5) and A = 1. Use the quasilinearity property of the utility function to identify the optimal bundle for each level of wealth without setting up a Lagrangian. Then solve for the optimal bundle separately by using the Lagrangian method to verify that the answer is the same in both cases.

4.7 The Lancaster Characteristics Model Usually we think about the consumer s problem in terms of how many units of commodities are consumed. Alternatively we can focus on the services these commodities provide. In the case of nutrition, we might think of the calories, vitamins, minerals or taste as the characteristics of the food. What the consumer really cares about are these characteristics, not the commodities themselves. A calorie is a calorie, whether it comes from meat or vegetables. This approach was first used by Lancaster (1966) and also Becker (1965). To see how such a model would work, assume that food has two characteristics: calories, c, and taste, t. Taste is meant to capture the non-nutritive aspects of food, and could also be thought of as quality. Assume the consumer has a choice between only two foods: rice, r, and meat, m. Let (cr,tr) and(cm,tm) be the calories and taste provided by a unit of rice and meat, respectively. Let p > 0 be the price of a unit of rice and normalize the price of meat to 1. (a) As is often the case, a dollar spent on meat provides more taste but fewer calories than a dollar spent on rice. Write down the inequalities that capture this fact. (b) Suppose that consumers have wealth w and minimum calorie requirement c*. Let r and m denote the number of units of rice and meat the consumer purchases. Write down the consumer s budget constraint and minimum calorie constraint. (c) Draw the consumer s consumption set in the calorie-taste space, i.e., the set of bundles (r,m) that satisfy the two constraints. [Hint: begin by drawing in the points corresponding to spending all wealth on r and all wealth on m.] How do changing p, w and c* affect the set of bundles that satisfy both constraints? (d) Suppose the consumer maximizes a well-behaved utility function u(c,t) subject to the budget and calorie constraints. How is the point the consumer chooses likely to depend on the levels of w and c*? Explain your answer. REFERENCES: Lancaster, Kelvin J. (1966) A new approach to consumer theory, Journal of Political Economy 74(2): 132-57. Becker, Gary S. (1965) A Theory of the Allocation of Time, The Economic Journal 75(299): 493-517. OPTIONAL PROBLEMS FOR LECTURE 4: MWG problems: 2.F.3 parts (e) and (f), 3.D. 7.

LECTURE 5: DUALITY 5.1 Hicksian Demand and MRS Let u(x, y) = x 1/2 + y 1/2, p = 1, q = 1. (a) Find the minimum wealth required to achieve utility u = 6. (b) Suppose that initial wealth W = 18. Find the Hicksian compensation associated with a price increase from p = 1 to p = 2. (c) Explain why MRS must hold for the solution to the Consumer Problem for all values p, q, W > 0. (d) Use MRS to solve more generally for demand functions x(p, q, W), y(p, q, W). (e) Use the MRS to find the HIcksian compensation associated with a change from (p, q, W) to (p, q, W). 5.2 Cobb-Douglas Utility: Consider the utility function: a1 a2 u x, x x x 1 2 1 2 Let p 1 and p 2 be the (strictly positive) prices of x 1 and x 2, respectively, and assume the consumer s wealth is w. (a) Derive the consumer s Hicksian demand functions. (b) Derive the consumer s expenditure function. (c) Derive the consumer s indirect utility function, v(p,w). (d) Verify that h x (p,v(p,w)) = x(p,w) and e(p,v(p,w)) = W. 5.3 The Linear Expenditure System (adapted from Mas-Colell, Whinston, and Green, 3.D.6) Consider a three-good world and a consumer who has utility function u (x 1, x 2, x 3 ) = (x 1 b 1 ) a1 (x 2 b 2 ) a2 (x 3 b 3 ) a3 Assume that w p1b 1 p2b2 p3b 3.. (a) Show that you can use an appropriate transformation of this utility function to assure that the exponents sum to 1. (b) Derive the Walrasian demand functions and indirect utility functions.

(c) Compare the Walrasian demand functions from (b) to the Walrasian demand functions for a Cobb-Douglas function. Is the current utility function homothetic? If not, explain its relationship to a homothetic utility function. (d) How would you explain the consumer s income expansion path (the change in consumption as wealth increases) in words. What does this explanation indicate about the nature of the consumer s underlying preferences? 5.4 The Envelope Theorem (Problem 3 on Problem Set 3) Consider a consumer with utility function u(x) = x ln( y). (a) Consider the expenditure minimization problem at prices (p, q) and minimum utility level u*. Use the first-order conditions to identify an equation that characterizes the Hicksian demand h y as a function of p, q, and u*. (NOTE: This equation should not include h x.) (b) Solve for the Hicksian demands h x and h y at p = 1, q = 2 and u* = 1. What expenditure is required to achieve u* = 1 at these prices? (c) Use the equation from (a) to approximate the Hicksian demands h x and h y at (1) p = 1, u* = 1 and q = 2.02. (2) p = 1, u* = 1 and q = 2.2. Then approximate the expenditure function e(p, u) for each of these conditions. NOTE: It may be helpful to use Trial and Error methods in Excel (or some other equivalent computer aid) to approximate the Hicksian demands. It is best to produce answers that are correct up to three decimal places. e (d) Now compute the average value of based on your approximations for the expenditure q function for each of the price changes in (c). (e) Compare the results from (d) to the result from the envelope theorem identity do you learn from these comparisons? e q hy. What e (f) Our theoretical results from class use the identity hy to approximate e(p, u). What do q your findings in (e) indicate about this application of the envelope theorem? OPTIONAL PROBLEMS FOR LECTURE 5: MWG 3.E.6, 3.G.15 (a) through (c)

LECTURE 6: WELFARE EFFECTS OF PRICE CHANGES 6.1 Hicksian Compensation, CV, and EV Return to the conditions of Problem 5.1: u(x, y) = x 1/2 + y 1/2, p = 1, q = 1 and W = 18. (a) Consider a price change from p = 1 to p = 2. Find CV and EV that would be associated with this price change in the new problem in Lecture 5, first by identifying the optimal bundles under different conditions and then using those optimal bundles to compute CV and EV. (b) Now verify your computations from (a) by integrating Hicksian demand function to find CV and EV. 6.2 CV, EV and Wage Negotiations (Based on Varian, Microeconomic Analysis, ex. 10.2) Consider a consumer with utility function 1/ 2 1/ 2 u x, x x x and income of $500 per week. Suppose that prices are p 1 = 1 and p 2 = 1 and 1 2 1 2 that the consumer spends his entire paycheck each week. The consumer s boss asks him to move to a new city that is identical to his current city except that p 1 = 1 and p 2 = 2. The boss offers no raise in pay. The consumer tells his boss: Asking me to move is just like if I stayed here and you cut my pay by $A. I would be willing to move to the new city, but you would have to pay me $B more for me to be willing to do it. Find values for A and B, and relate them to concepts we discussed in class. If A = B, explain why. If not, explain why not. 6.3 Quasilinear Preferences and Welfare Analysis A consumer has quasilinear preferences with u(x, y) = x + 36 y 1/2. Assume that price p = 1 throughout the problem. (a) Find the consumer s Walrasian demand functions x(q, w) and y(q, w) for all combinations of price q for good y and wealth w. (b) Consider a price change from q = 6 to q = 9 with wealth fixed at w = 200. Will EV or CV be larger in magnitude for this price change? Explain your answer without performing any calculations. (c) Compute EV for this price change; if you find it difficult or impossible to complete this calculation, then estimate EV for the price change instead. For part (d) assume that u(x, y) is slightly altered to v(x, y) where v(x, y) = x + 36 y 1/2 if p < 100, but v(x, y) = x + 36 y 1/2 ε p (p 100) if p > 100. Assume that ε is a sufficiently small positive constant that v (x) > 0 for all feasible levels of x.

(d) Repeat part (c) for a consumer with utility function v. That is, consider a price change from q = 6 to q = 9 with wealth fixed at w = 200. Will EV or CV be larger in magnitude for this change for a consumer with utility function v? Explain your answer without performing any calculations. 6.4 Perfect Complements and Welfare Analysis Consider a consumer with utility function U(x 1, x 2 ) = min[2x 1, x 2 ]. Suppose that the consumer s wealth is W and that the prices for goods 1 and 2 are p 1 = 4 and p 2 = 2 respectively. Suppose that W = 80 and that the government imposes a tax of 1 per unit on good 2 so that the (net) price per-unit of good 2 increases from 2 to 3. (a) What is the effect of this tax on the consumption bundle for this consumer and on the conssumer s utility? (b) How much tax revenue does the government raise from purchases by this consumer? (c) Calculate CV and EV of this price change. (d) Explain the relationship of CV and EV for this tax in words. (e) Suppose that the government imposes a tax of t per unit on good 1 instead of on good 2. What tax rate per unit of good 1 would raise the same amount of revenue for the governaent as the tax in part (d) for good 2? OPTIONAL PROBLEMS FOR LECTURE 6: MWG 3.G.3 parts (a) through (c)

LECTURE 7: TAXES AND WELFARE ESTIMATION 7.1 Taxation and Deadweight Loss Suppose that a consumer has Cobb-Douglas utility u(x, y) = xy 2, that the consumer has wealth W = 18 and faces prices p = 2, q = 3. (a) Suppose that the government sets tax T 1 = 1 on good 1 so that the consumer faces prices p = 2 + T 1 = 3 and q = 3. What bundle will the consumer choose? How much tax revenue will the government raise? (b) Suppose instead that the government sets tax T 2 on good 2 so that the consumer faces prices p = 2 and q = 3 + T 2. What choice of T 2 will raise the same revenue as the tax T 1 = 1 in part (a)? (c) Now compute the utilities achieved by the consumer given the optimal bundles in (a) and (b). Does one produce more deadweight loss than the other? (d) Suppose instead that the government charges a lump sum tax equal to the tax revenue R from part (a). Under the lump sum tax, the consumer has wealth 18 R and faces the original prices p = 2, q = 3. What bundle will the consumer choose? Verify that the consumer does better with the lump sum tax than with either of the individual taxes (T 1, T 2 ) given that the taxes are calibrated to produce the same revenue for the government. (e) Suppose that the government wishes to induce the same outcome as the lump sum tax with a combination of taxes (T 1, T 2 ) on goods 1 and 2. Under these conditions, the consumer has wealth 18 and faces prices p T = 2 + T 1, q T = 3 + T 2. Find the combination of values T 1 and T 2 such that the consumer chooses the same bundle as in (d). 7.2 Comparing Two Taxes The government must raise taxes to generate additional revenue. It has decided to impose a 10% tax either on apples or bananas. Current data for a typical consumer is given in the table below. Price Quantity dxi dp i dx i dw Apples 1 50-100 0 Bananas 2 30-50 0.5 The government has asked for your help in determining which tax it should implement. Throughout the question, you may use linear approximations to the consumer s demand curves.

a. Briefly explain why it is more appropriate to use EV to determine which tax is better rather than CV. b. Estimate the EV of the apple tax. c. Estimate the EV of the banana tax. d. Based on parts b and c, which tax should the government implement? Explain your answer. 7.3 Comparative Statics and Welfare Measures Based on empirical data, you believe the following about a typical consumer s consumption of widgets (widgets are an imaginary good economists use you do not need to know what a widget is to correctly answer this question). Currently, the price of widgets is p 0 = $8, and a typical consumer consumes x 0 = 11 widgets. You have estimated the following parameters of the consumer s utility function. dx 0.5 and dx dw 0.01. dp Suppose the government is considering imposing a tax that will raise the price of widgets $2 to p 1 = $10. The remainder of this question asks you to use this information to estimate the welfare impact of this price change. In estimating the various quantities, you may assume that Walrasian and Hicksian demand curves are linear. (a) Estimate how many widgets this consumer consumes after the price increase. (b) Estimate the equivalent variation of this price change. (c) Estimate the compensating variation of this price change. (d) Estimate the change in consumer surplus associated with this price change. (e) Why is CV the most appropriate measure of the welfare change? (f) Is the change in consumer surplus a good estimate of CV in this case? (g ) Compute the tax revenue. Also compute the deadweight losses associated with each of the three welfare measures. Is the deadweight loss computed using the change in consumer surplus a good approximation of the deadweight loss computed using CV? 7.4 Welfare Analysis of a Price Change: Does the Path Matter? (Adapted from MWG 3.I.7) There are three commodities (L = 3), of which the third is a numeraire commodity with price p 3 = 1. A consumer s Walrasian demand functions are given by: x 1 (p, w) = 10 2p 1 + p 2 x 2 (p, w) = 10 ap 2 + p 1 a) Estimate the Equivalent Variation (EV) for a change in prices from (1, 1) to (2, 2) according to the following two methods (read part c before doing the computations, since you will need them later). Hint many of the calculations for this problem can be done by computing the area of an appropriate triangle.

i) Prices follow the path (1, 1) to (1, 2) to (2, 2) ii) Prices follow the path (1, 1) to (2, 1) to (2, 2) b) Under what conditions are the solutions to parts a.i and a.ii the same? How does this relate to the implications of utility maximization ( or expenditure minimization)? c) Assume that a = 2. Let EV 1 be the EV for the price change (1, 1) to (2, 1). Let EV 2 be the EV for the price change (1, 1) to (1, 2). Let EV 2 be the EV for the price change from (1, 1) to (2, 2). Compare EV T with EV 1 + EV 2. Interpret the difference. d) Suppose that the price increases in part c are due to taxes. Denote the deadweight loss for each price change by DW 1, DW 2, and DW T. Compare DW T with DW 1 + DW 2. Interpret the difference. LECTURE 8: UTILITY UNDER UNCERTAINTY 8.1 Mixtures Suppose that there are three possible outcomes, x 1, x 2, x 3 and suppose that a consumer is indifferent between a certainty of outcome x 2 and a 50-50 lottery between x 1 and x 3. (a) Explain how the independence axiom rules out the possibility that the consumer both strictly prefers x 1 to x 2 and strictly prefers x 3 to x 2. (b) Now suppose in addition, that x 1 is strictly preferred to x 2. Make a graph of all combinations of the probabilities in all lotteries L for which the consumer is indifferent between L and a certainty of x 2. (c) Explain the meaning of the slope of the indifference curve in (b). (d) Make a graph of all combinations of the probabilities in all lotteries L for which the consumer is indifferent between L and the lottery (0.5, 0.5, 0). How is this graph related to the graph from (b)? 8.2 Mixtures and the Independence Axiom Suppose that there are three possible outcomes, x 1, x 2, x 3, and consider two lotteries over these final states. L 1 = (1/3, 0, 2/3); i.e. P(x 1 ) = 1/3, P(x 2 ) = 0, P(x 3 ) = 2/3; L 2 = (0, 3/4, 1/4). P(x 1 ) = 0, P(x 2 ) = 3/4, P(x 3 ) = 1/4. (a) Rewrite L 1 and L 2 as mixtures in the following form:

L 1 = α L 3 + (1- α) L 1 ; L 2 = α L 3 + (1- α) L 2. (b) Calculate EU(L 1 ) and EU(L 2 ) and show that EU(L 1 ) > EU(L 2 ) if and only if EU(L 1 ) > EU(L 2 ). How is this finding related to the independence axiom? (c) Now suppose that L 1 ~ L 2. What is the set of all lotteries L such that L ~ L 1? What is the set of all lotteries L such that L is strictly preferred to L 1? You can assume that x 1 is strictly preferred to x 2 and x 2 is strictly preferred to x 3. 8.3. Contradictory Preferences Consider three monetary payoffs, $0, $100, and $200. Consider three lotteries over these payoffs, as defined by probabilities of each outcome. L 1 = (1/3, 1/3, 1/3); i.e. P($0) = 1/3, P($100) = 1/3, P($200) = 1/3; L 2 = (1/2, 0, 1/2); P($0) = 1/2, P($100) = 0, P($200) = 1/2; L 3 = (0, 3/4, 1/4). P($0) = 0, P($100) = 3/4, P($200) = 1/4; Suppose a consumer tells you that L 1 is strictly preferred to both L 2 and L 3. (a) Show that this consumer s preferences contradict expected utility maximization. (b) How does the result of (a) relate to the independence axiom?

8.4 Linear and Non-Linear Transformations of VNM Utility Functions Suppose that there are three possible outcomes x 1, x 2, x 3. A consumer tells you that she has identified her utility function: U(x 1 ) = 9, U(x 2 ) = 4, U(x 3 ) = 1 and that this utility function satisfies the conditions of the Expected Utility Theorem. (a) Define lotteries L p as having probability p of outcome 1 and probability 1-p of outcome 3. For what values of p will this consumer prefer L p to a certainty of outcome 2? Consider the following functions f 1 (y) = 2y, f 2 (y) = y 2, f 3 (y) = SQRT(y). Define the following utility functions: V 1 (L) = f 1 [EU(L)], V 2 (L) = f 2 [EU(L)], V 3 (L) = f 3 [EU(L)]. (b) Compute V 1 (L), V 2 (L), and V 3 (L) for the lotteries L p with p = ¼ and separately with p = ¾. Show that these transformations maintain the same preference ranking of L p and x 2 (with certainty) as in (a). (c) Consider the separate utility functions U k (x j ) = f k [U(x j )]. So for example, U 1 (x 1 ) = 2*9 = 18, U 1 (x 2 ) = 2*4 = 8, U 1 (x 3 ) = 2*1 = 2. Show that expected utility calculations for U 1 will yield an identical ranking of lotteries L p in comparison to x 2 as you found in (a). Use counterexamples to show that U 2 and U 3 produce different rankings of lotteries L p and x 2 than those from (a). (d) Suppose that for some p, the expected utility calculations for U and U 2 produce different preference rankings between L p and x 2. Show that the expected utility calculations for U 3 for this particular L p must yield the same preference ranking for L p and x 2 that is given by the expected utility calculations for U. (e) Explain the intuition for the result in (d) that the expected utility rankings of L p and x 2 based on U always agree with the expected utility rankings of L p and x 2 for at least one of U 2 and U 3. OPTIONAL PROBLEM FOR LECTURE 8: MWG: 6.B.4

LECTURE 9: RISK AVERSION 9.1 Certainty Equivalents Consider two lotteries L 1, which has equal probabilities of outcomes 0 and 16 and L 2 which has equal probabilities of outcomes 0 and 256. Consider three separate VNM utility functions: u 1 (w) = w 1/2, u 2 (w) = w 1/4, u 3 (w) = w 3/2. (a) Find the certainty equivalents and risk premia for lotteries L 1 and L 2 for a consumer with utility function u 1 (w) = w 1/2. What proportion of the expected value of each lottery is represented by RP? (b) Find the certainty equivalents and risk premia for lotteries L 1 and L 2 for a consumer with utility function u 1 (w) = w 1/4. What proportion of the expected value of each lottery is represented by RP? (c) Find the certainty equivalents and risk premia for lotteries L 1 and L 2 for a consumer with utility function u 1 (w) = w 3/2. What proportion of the expected value of each lottery is represented by RP? (d) Compare the values that you calculated in (a) through (c) and explain the ranking of certainty equivalents and risk premia in terms of the risk attitudes represented by these utility functions. Why do you find that the risk premium is the same proportion of expected value for each specific utility function for lotteries L 1 and L 2? 9.2 Playing with House Money Part 1: A person who is about to go on vacation to Las Vegas has budgeted $200 for gambling on the trip. Let w denote the amount of gambling money he has at the end of his trip (i.e. w < 200 if he loses money, w > $200 if he wins money.) He says that he is risk averse for values of w less than $200 and risk loving for values of w greater than $200 and that he will quit gambling if he reaches $1,000. (a) Consider a class of functions denoted by u(a, w) = ln(w) + Aw. What values of A are consistent with this description of his preferences? (b) Consider a class of functions denoted by v(b, w) = ln(w) + Bw 2. What values of B are consistent with this description of his preferences?

9.3 Full Insurance vs. No Insurance Consider a risk-averse consumer with continuous and strictly increasing utility function u(x). The consumer faces a risk of losing D with probability p. The consumer can buy full insurance for this risk at price π but cannot purchase partial insurance. Assume that u is continuous and differentiable. (a) Holding p and D fixed, show that the consumer will buy insurance if π < π*, where π* is a particular positive constant. (b) Holding π and D fixed, show that the consumer will buy insurance for p > p*, where p* is a particular positive constant. You can assume that 0 < π < D. (c) Interpret these results in words. 9.4. Measures of Risk Aversion Consider a consumer with each of the following utility functions: i) ii) u x e u x log x x iii) u x x 1 2 (a) For each of the utility functions above, compute the coefficient of absolute risk aversion and the coefficient of relative risk aversion. The consumer faces one of two possible risks: La: with probability ½, no loss occurs, and with probability ½, a loss of $10 occurs. Lb: with probability ½, no loss occurs, and with probability ½, a loss of 10% of wealth occurs. (b) Compute the maximum a consumer with utility function (i) above will pay for full insurance against risk La when initial wealth is w = 100 and when initial wealth is w = 200. (c) Compute the maximum a consumer with utility function (ii) above will pay for full insurance against risk Lb when initial wealth is w = 100 and when initial wealth is w = 200. (d) Compute the maximum a consumer with utility function (iii) above will pay for full insurance against risk La when initial wealth is w = 100 and when initial wealth is w = 200.

(e) Explain briefly how (b) illustrates constant absolute risk aversion, (c) illustrates constant relative risk aversion and (d) illustrates decreasing absolute risk aversion. OPTIONAL PROBLEMS for LECTURE 9: MWG 6.B.7, 6.C.2, 6.C.18

LECTURE 10: COMPARATIVE STATICS AND INVESTMENTS 10.1 Choice under uncertainty A risk averse consumer is considering an investment of money in a risky asset with price x per unit. Each unit of the asset is worth 100 + x with probability p/2, x with probability 1 p, and x 50 with probability p/2. So after accounting for the price of the asset, it is +100 with probability p/2, -50 with probability p/2, and 0 with probability 1 p. The consumer has initial wealth w and von-neumann-morgenstern utility function u(final wealth) = ln(final wealth) (a) Set up and solve the maximization problem for the consumer s optimal number of units of the asset, y. Assume that it is possible to buy fractional units of the asset. (b) How do you know that the first-order conditions for this maximization problem identify the optimal choice of y? (c) How does the level of investment vary with w? Explain the intuition for this answer. (d) How does the level of investment vary with p? Explain the intuition for this answer. 10.2 An Investment Problem Consider an investment problem with two possible outcomes. A consumer with Von-Neumann Morganstern utility function u(w) individual has initial wealth V to invest in a risky asset and a safe asset. If the consumer invests ß dollars in the risky asset (and thus V ß dollars in the safe asset), then the final wealth is (V-ß) + ßr, where ß is the return on the risky asset. The final wealth can also be written as V + ß (r-1). The return on the risky asset is known to be r 1 with probability p 1, and r 2 with probability 1 p 1, where r 1 >1, and 0 < r 2 < 1. a) What are the first-order conditions for the optimal choice of ß? b) Denote the optimal choice of ß as ß*. Under conditions can you conclude that ß* > 0? What is the economic meaning of these conditions? c) Assume that u(w) = ln(w). Show that dß* / dp 1 > 0, dß* / dr 1 > 0, dß* / dr 2 > 0. d) What results for the insurance problem are analogous to these comparative static results dß* / dp 1 > 0, dß* / dr 1 > 0, dß* / dr 2 > 0?

10.3 Equivalence of Insurance and Investment Problems This problem returns to the investment problem in 10.1 to compare it to the insurance problem in MWG. MWG Example 6.C.1 describes an insurance problem with initial wealth W, probability π of a loss of D dollars (and probability 1- π of no loss) and price q per dollar of insurance. To summarize notation, W = initial wealth in insurance problem V = initial wealth in investment problem D = possible loss in insurance problem r 1, r 2 = high and low return in investment problem p 1 = probability of high outcome in investment problem π = probability of loss in investment problem ß = number of dollars invested in risky asset in insurance problem α = number of units of insurance purchased (α = repayment if there is a loss) q = price per unit of insurance purchased Assume throughout the problem that W > V > W-D and that π = 1- p 1 so that the probability of the bad outcome is the same in both problems. (a) Find conditions on p 1 and π so that the probabilities of the good state are the same in both problems. (b) Find conditions on r 1 and r 2 so that if the consumer invests all of her money in the risky asset, the possible outcomes are W and W-D. (c) Find conditions on the parameters from the insurance problem so that full insurance produces certain wealth equal to V. (d) Assume that the conditions from (a) through (c) hold and that the price of insurance is actuarially fair. Show that the expected return for the risky asset must be equal to 1 in the investment problem. (e) Assume that the conditions from (a) through (c) hold. Show that the problems are equivalent in the sense that the distribution of insurance outcomes for each possible level of insurance α < D can be exactly replicated in the investment problem for some possible level of investment ß. 10.4 Insurance through Trading Suppose that there are two consumers and two possible states of the world. In state 1, consumer 1 has wealth $10 and consumer 2 has wealth $0. In state 2, consumer 1 has wealth $0 and consumer 2 has wealth $20. These states are equally likely and each consumer has utility function u(w) = w 1/2. In advance of learning the state, the consumers decide to make a contingent trade: consumer 1 will pay consumer 2 if state 1 occurs, while consumer 2 will pay consumer 1 if state 2 occurs.

In effect, each consumer buys insurance from the other. Suppose that there is a price p per unit of trade, which works as follows. If consumer 1 buys x units of insurance, he ends up with 10 x in state 1 and px in state 2. If consumer 2 buys y units of insurance, she ends up with y in state 1 and 20 py in state 2. (a) Suppose that consumer 1 chooses x and consumer 2 chooses y to maximize (individual) expected utility. Solve for consumer 1 s optimal choice of x and consumer 2 s optimal choice of y as a function of p. (b) What value p* produces x = y when the consumers choose these values optimally? (c) Does p* correspond to an actuarially fair or actuarially unfair price? Explain why p* takes this form, given what we know about how risk averse consumers purchase insurance at actuarially fair and actuarially unfair prices. 10.5. Correlated Stock Returns and Investment Decisions Suppose that a risk averse consumer is considering an investment of money in two companies. The price of each company s stock is 1 per unit of stock. There are two possible outcomes for the value of each stock. Each stock increases by A (per unit of stock) with probability p and falls by A with probability 1-p. The consumer has decided to invest the same amount of money, x, in each company and wants to determine the optimal choice of x. The joint distribution of returns across the companies is shown below. Probability Company 1 Company 2 q + A + A p q - A + A p q + A - A 1-2p + q - A - A Suppose that p is fixed and greater than 1/2, but let q vary. (a) Note that q must be between (2p - 1) and p for all the probabilities in the table to fall between 0 and 1. Explain the relationship between the results for the two companies for the separate cases q = 2p 1 and q = p. (Note that q = p 2 if the results are statistically independent.) (b) Suppose that the consumer has chosen a fixed level of investment x in each company. Show that the consumer s expected utility is declining in q. Explain the intuition for this result. (c) Assume that the consumer has decided to invest the same amount x in each company, but has not determined the value of x. Show that the optimal choice of x is declining in q. Explain the intuition for this result. (d) How would the results of parts (a) through (c) differ for a risk loving consumer?

10.6 Playing with House Money Part 2. A small business owner currently has $W 0 in cash and total debt D = $100. He is offered an opportunity to purchase shares in a very risky venture. Each share costs $1 and will yield return 0 with probability ¾ and return 2 with probability ¼. He can purchase up to W 0 shares given his cash on hand (and ignoring his debt). If he purchases x shares (x < W 0 ), then with probability ¾, he will end up with W 0 x in cash and with probability ¼, he will end up with W 0 + x in cash. Let W 1 denote his final level of cash (not adjusted for repayment of debt) and suppose that his utility function is given by u(w 1 ) = 0 if W 1 < D; u(w 1 ) = W 1 D if W 1 > D. (a) Graph the utility function. What attitudes towards risk are embodied by this utility function? (b) Explain how this utility function might arise given the business owner s debts. (c) Suppose that W 0 > D/2 and W 0 < D. Show that the business owner maximizes his expected utility by choosing x = W 0. (d) Assume that W 0 > D. Show that there exists a value W* > D such that if W 0 < W*, it is optimal for the business owner to choose x = W 0, but if W 0 > W*, it is optimal for the business owner to choose x = 0. (e) Explain the intuition for the results in (c) and (d). OPTIONAL PROBLEMS FOR LECTURE 10: MWG 6.C.15, 6.C.16

LECTURE 11: STOCHASTIC DOMINANCE 11.1. Stochastic Dominance and the Independence Axiom Sample problem 8.3 asked you to consider the following lotteries with monetary payoffs $0, $100, $200. L 1 = (1/3, 1/3, 1/3); L 2 = (1/2, 0, 1/2); L 3 = (0, 3/4, 1/4). In this problem, the values are ordered from lowest to highest, so L 3 yields a payoff of $100 with probability 3/4 and an outcome of $200 with probability 1/4. In Problem 8.3, you showed that it is not possible for an expected utility maximizer to both strictly prefer L 1 to L 2 and to strictly prefer L 1 to L 3. This problem draws a connection between that result and first-order stochastic dominance. (a) Show that L 1 is not first-order stochastically dominated by either L 2 or L 3. (b) Show that L 1 is first-order stochastically dominated a mixture of L 2 and L 3. (c) Explain why (b) means that an expected utility maximizer cannot both strictly prefer L 1 to L 2 and strictly prefer L 1 to L 3. 11.2. Stochastic Dominance and Lotteries Classify each of the following pairs of lotteries into one of the following three categories. Category 1: Every consumer with (weakly) increasing utility function for money would prefer lottery 1 to lottery 2 Category 2: Every risk averse consumer with (weakly) increasing utility function for money would prefer lottery 1 to lottery 2, but some consumers who are not risk averse would prefer lottery 2. Category 3: Some risk averse consumers would prefer lottery 1 to lottery 2, while other risk averse consumers would prefer lottery 2 to lottery 1 a) Lottery 1 has equal probabilities of the outcomes $100, $40, and $10. Lottery 2 has a 0.3 probability of the outcome $80 and 0.7 probability of the outcome $0. b) Lottery 1 has equal probabilities of the outcomes $ 75 and $ 25. Lottery 2 has an 0.25 probability of the outcome $100, an 0.5 probability of the outcome $50, and an 0.25 probability of the outcome $ 0. c) Lottery 1 gives $50 with certainty. Lottery 2 has equal probabilities of the outcomes $60 and $20. d) Lottery 1 has equal probabilities of $70 and $30. Lottery 2 gives $80 with probability 0.25 and $40 with probability 0.75.

11.3 Correlated Stock Returns and Investment Decisions Revisited Suppose that a risk averse consumer is considering an investment of money in two companies. The price of each company s stock is 1 per unit of stock. There are two possible outcomes for the value of each stock. Each stock increases by A (per unit of stock) with probability p and falls by A with probability 1-p. The consumer has decided to invest the same amount of money, x, in each company and wants to determine the optimal choice of x. The joint distribution of returns across the companies is shown below. (a) Fix q and consider two possible values of p with p 1 > p 2. Show that if the consumer invests the same amount x in company 1 and in company 2, then the distribution of returns given p 1 first-order stochastically dominates the distribution of returns given p 2. (b) Consider two possible values of q with q 1 > q 2. Show that if the consumer invests the same amount x in company 1 and in company 2, then the distribution of returns given q 2 second-order stochastically dominates the distribution of returns given q 1. (c) Fix p and q. Suppose that the consumer has decided to invest a total of 2x in the two companies. Show that the distribution of returns based on investment of x in each company second-order stochastically dominates the distribution of returns based on any other combination of investments x A, x B where x A + x A = 2x, x A x B. (d) Use the results from parts (a) through (c) to argue that the consumer s expected utility is increasing in p and decreasing in q.

11.4. Stochastic Dominance and Retirement Savings Suppose that a risk-averse consumer has wealth w to invest in two assets, a safe asset and a risky asset. Each unit of the safe asset costs one unit of wealth and has a known value of 1. Each unit of the risky asset costs p units of wealth and has value r. There are n possible values of r -- r 1, r 2,, r n, with P(r = r j ) = π j >0. Note that an investment in the risky asset yields a loss if the resulting value r is less than p and yields a gain if r > p. Assume r 1 < r 2 < < r n, r 1 < p and E(r) = π j r j > p. That is, the risky asset yields an expected profit, but at the risk of a net loss. The consumer s budget constraint is xp + y = w, where x is the number of shares of the risky asset and y is the number of shares of the safe asset chosen by the consumer. The two assets can be held in positive or negative quantities (no restrictions on x or y). If the risky asset has realized value of r j, then the consumer has final wealth w + x(r j p). (a) Show that the consumer holds a strictly positive amount of the risky asset, i.e. x* > 0, where x* represents the consumer s optimal choice of x. Now suppose that the consumer realizes that exogenous wealth is in fact stochastic and that the realization of w is connected to the realization of r. Specifically, suppose that there are two states of the world, where (1) r = r 1, w = w 1 in state 1; (2) r = r 2 and w = w 2 in state 2. Assume that r 1 < p < r 2, that w 1 < w 2 and let w* = E(w) = π 1 w 1 + π 2 w 2. (COMMENT: This problem is motivated by the example of Enron or Lehman Brothers employees who held disproportionate amounts of their retirement savings in shares of the company where they worked.) (b) Show that, for any fixed positive level of share purchases (i.e. for any x > 0), the distribution of resulting wealth under these conditions is second-order stochastically dominated by the distribution of resulting final wealth when exogenous wealth is known in advance and equal to w*. Explain why this result depends on the relationships w 1 < w 2 and r 1 < r 2. (c) Show, under the conditions given, that the optimal purchase of shares in the risky asset is lower when exogenous wealth is random than it is when exogenous wealth is known in advance. (d) How would the results from (b) and (c) change if there were n possible values for the exogenous wealth and n possible values for the asset s value with w 1 < w 2 <... < w n and r 1 < r 2 <... < r n? HINT: For parts (b) through (d), you may find it useful to use the condition for SOSD given by equation 6.D.2 in MWG. 11.5. Professor Kohlberg s Rule for Stochastic Dominance

Suppose that there are two lotteries L 1 and L 2 with outcomes as shown below. Assume that y 1 < y 2 < y 3. Outcome Probability L 1 L 2 Worst 1/3 x 1 = 20 y 1 Middle 1/3 x 2 = 50 y 2 Best 1/3 x 3 = 80 y 3 Professor Kohlberg suggested an approach to stochastic dominance based on weighted sums of outcomes. In this case, assume weights w 1, w 2, w 3, for worst, middle, and best outcomes respectively where w 1, w 2, w 3 > 0 and w 1 + w 2 + w 3 = 1. (a) Find conditions on y 1, y 2, y 3 so that the following inequality holds for all possible combinations of weights (w 1, w 2, w 3 ): w 1 (x 1 - y 1 ) + w 2 (x 2 y 2 ) + w 3 (x 3 y 3 ) > 0 (b) Demonstrate that you can write L 1 = L 2 + L 3 where L 3 is a lottery with only weakly positive outcomes (i.e. L 1 FOSD L 2 ) if and only if the conditions from (a) hold. For parts (c) and (d), suppose that (y 1 + y 2 + y 3 ) / 3 = 50 (i.e. Expected value of L 2 = 50). (c) Find conditions on y 1, y 2, y 3 so that the following inequality holds for all possible combinations of weights (w 1, w 2, w 3 ) with w 1 > w 2 > w 3. w 1 (x 1 - y 1 ) + w 2 (x 2 y 2 ) + w 3 (x 3 y 3 ) > 0 (d) Demonstrate that you can write L 2 can be written as a mean preserving spread of lottery L 1 (i.e. L 1 SOSD L 2 ) if and only if the conditions from (c) hold. (e) Relate the test for SOSD from (c) and (d) to equation 6.D.2 in MWG.