Homework 2 ECN205 Spring 2011 Wake Forest University Instructor: McFall

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1 Homework 2 ECN205 Spring 2011 Wake Forest University Instructor: McFall Instructions: Answer the following problems and questions carefully. Just like with the first homework, I ll call names randomly to put an answer on the board. Good answers will earn extra credit on the second test. Not so good answers will earn less. 1. Explain why two indifference curves cannot cross. If indifference curves cross, then the completeness rule of preference mapping has been broken. Completeness states that bundles of goods can be ranked in preference order, and if one bundle is preferred to another bundle, then all goods that lie on the preferred bundle s indifference curve MUST lie above the indifference curve of the less preferred bundles. 2. Justin consumes together two goods tortilla chips (t) and salsa (s). His utility function is given by U(t, s) = min(t, 2s). What is his utility if he has 3 units of tortillas and one unit of salsa? What would Justin have to be given in order for him to move up to another utility curve? Explain mathematically and graphically. If Justin has 3t and 1s, then his utility is given by min(3, 2*1) = 2. (One unit of tortilla is left uneaten.) If he is to move to another utility level, he d need at least ½ unit of salsa. Given that ½ unit of salsa, he d be able to move to a utility level of 3, min(3, 2*1.5), and utilize all of his tortilla resources. 3. Rob likes to consume coffee with cream. He refuses to drink the delicious brew unless he has six parts coffee (c) to one part cream (m). What is his utility function? Graph his utility function. Because Rob consumes coffee and cream together, we can consider these goods to be perfect complements. Given his preferred ratio of consumption, Rob s utility function is U(c, m) = min((1/6)c, m). (It could also be min(c, 6m).) 4. Amanda likes Snickers candy bars and chocolate chip cookies. When you take a Snickers from her, her level of utility remains constant so long as she is given two chocolate chip cookies. What is her utility function? Graph her utility function. What is her utility if she consumes two Snickers and two cookies? What is her utility if she consumes three Snickers and no cookies? Explain briefly. Amanda views Snickers and chocolate chip cookies as perfect substitutes. Given her substitution preference, her utility is U(s, c) = c + 2s, where c = # units of cookies and s = # units of Snickers. U(2, 2) = 2 + 2*2 = 6. U(3, 0) = 0 + 2*3 = 6.

2 5. Suppose Dude s utility function is given by U(w, b) = w 0.5 b 0.5, where w = # of white Russians consumed and b = games of bowling. Graph Dude s utility function. What is Dude s utility when he has four white Russians and bowls 3 games? Identify two different bundles of white Russians and games of bowling that would keep him at this same utility level. What is a bundle of white Russians and bowling that would double his utility level? Dude s utility given 4w and 3b = (4^.5)*(3^.5) = 2*3^.5 His utility is maintained when he consumes 3w and 4b or 1w and 12b. His utility is doubled when he consumes 3w and 16b. 6. What is Amanda s marginal rate of substitution when she has 50 Snickers and 10 chocolate chip cookies? What is her marginal rate of substitution when she has 10 Snickers and 10 chocolate chip cookies? Explain. Given 50s and 10c, Amanda s utility is 2*50 + 1*10 = 110. If we take a Snickers from her, we d have to give her 2 chocolate chip cookies in order for her utility to remain at 110. The same can be said when her bundle is 10s and 10c. Therefore, her MRS = What is Dude s marginal rate of substitution at U = U(3, 2)? What about at U = U(2, 3)? Dude s utility function is U(w, b) = w.5 b.5. To calculate his marginal rate of substitution at any bundle, we must evaluate U/ b/ U/ w = ½w.5 b.5 /½w.5 b.5 = w/b = w/ b. So, we can substitute values of w and b into the expression we evaluated in order to calculate MRS at any given bundle. This implies that at w=3 and b=2, MRS = 3/2. Similarly, at w=2 and b=3, MRS = 2/3. U(3,2) = = Imagine we took a half unit of bowling from him. To compensate him fully for his loss (keep him at U=2.45), we d have to solve w/ 0.1 = 3/2 = 0.15 unit of white Russian. For U(2, 3) = (2*3).5 = 2.45, if we take a tenth of a unit of bowling from Dude, we d have to give him w/ 0.1 = 2/3 = unit of white Russian. The difference between the compensation he d have to receive when U(2,3) and U(3,2) cuts to the heart of diminishing utility. Because he doesn t have that much w under U(2, 3) compared to U(3, 2), we don t have to give him as much w in order for him to remain at the same utility level. His marginal utility change is larger from a smaller amount of w because he doesn t have that much w to begin with. 8. The Dude can consume white Russians or bowling. The price of a white Russian is $5/unit. The price of a unit of bowling is $8/unit. His income is $80/week. A) Write an equation for his budget constraint. B) Graph his budget constraint. C) Describe how you would find Dude s marginal rate of transformation (MRT). D) What happens to his MRT when the price of bowling rises to $10/unit? Show this change on the graph. a) 100 = 5w + 8b b) See below

3 c) Dude s MRT is the ratio of the prices that he faces. This ratio describes the opportunity cost of consuming w or b. The marginal rate of transformation of bowling = 8/5 units of white Russians. The MRT of a white Russian is 5/8 unit of bowling. d) The MRT of white Russians falls (Dude has to give up less bowling to get a white Russian after the price change) to 1/2 unit of bowling. 9. Amanda wants to maximize the utility she earns from consuming Snickers bars and chocolate chip cookies. The price of a Snickers is $1/unit. The price of a unit of cookies is $3/unit. Her income is $12/week. How many candy bars and cookies should she consume? What should she do if the price of Snickers rose to $1.50/unit? As always, explain. Amanda s optimal bundle is a corner solution. She won t consume any cookies, because the price of cookies is too large (3/unit) compared to the marginal utility she earns from eating a cookie (1/unit). So, she ll consume 12 Snicker bars with her income and earn a utility level of 12 by doing so. If the price of Snickers changes to $1.50/unit, then Amanda will still find it in her best interest to consume as many Snickers as possible. She ll find this corner solution to be optimal so long as the price of Snickers is less than $3/unit. At this point, her utility curve will be identical to her budget constraint, and she will be indifferent to any bundle on that line. 10. Dude has a utility function of U(w, b) = w.5 b.5. His budget constraint is given in question 8. Set up his constrained optimization problem, solve for the optimal levels of bowling and white Russians he should consume. Explain how you arrived at your answer. Dude wants to maximize U = (XY).5 subject to his budget constraint, 100 = 5w + 8b. So, he ll find an amount of w and b that satisfies max (wb).5 + λ(100 5w 8b) To start, she ll want to find the marginal utility of a unit of w and b: U/ w = 0 =.5w.5 b.5 λ5 U/ w = 0 =.5w.5 b.5 λ8 Doing some algebra and simplifying gives us the marginal rate of substitution and the marginal rate of transformation. b/w = 5/8 = 8b = 5w. We can interpret the above equation as an expenditure rule for Dude. He ll want to spend as much on white Russians as he does on bowling. We can substitute b = (5/8)w into our budget constraint to get 100 = 5w + 8(5/8)w = 10w. Solving 100 = 10w gives us w* = 10. Substituting w = 5 into our budget constraint and solving for b gives us 100 = 5(10) + 8b 8b = 50 b* = What happens to Dude s optimal bundle if the prices of white Russians fall to $4/unit?

4 If Dude sees a change in the price of white Russians from 5 to 4, then the budget constraint becomes 100 4w 8b = 0. We ll take the same steps as before. We ll find the marginal rate of substitution and equate it to the marginal rate of transformation. This gives us b = (1/2)w. Performing the same substitution as before, we get 100 = 4w + 8(1/2)w w* = b* = (We d expect Dude to consume more white Russians if the price of white Russians fell Our answer makes intuitive sense. Whew!) What happens if his utility function changes to U(w, b) = w.4 b.6? Explain. Dude is going to wish to maximize U = w.4 b.6 subject to his budget constraint. This gives us max w.4 b.6 + λ(100 5w 8b = 0) Taking the ratio of the partial derivatives gives us MRS = MRT: (2/3)(w/b) = 5/8 = (16/15)b = w Substituting this into the budget constraint gives us 100 = 5(16/15)b + 8b 100 = (16/3)b + 8b 100 = 13.33b b* = 7.5 Solving for w gives w* 0 = 100 5w 7.5(8) w* = Show graphically Dude s substitution and income effects when he faces a 50% reduction in the price of white Russians.

5 U=1 2. s U=2 U=3 A = (1, ½ ) B = (2, 1) C = (3, 3/2) A B C t 3. m c 4. s c