MIDTERM EXAM ANSWER KEY

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1 MIDTERM EXAM ANSWER KEY ECON 210 PROFESSOR GUSE Instructions. You have 2 hours to complete the exam. There are a total of 75 points on the exam. The exam is designed to take about 1 minute per point. You are allowed to have a single page of handwritten notes, 2-sided. You may not use any other notes, books or aids of any kind, be they human, electronic or mechanical. Calculations may be left in expression form for full credit. There is space provided for each question. If you need additional space, extra sheets are provided at the end. Please justify and explain your answers where needed and show your work. Please write your name on the exam itself and record the time you started and the time you finished. Finally, please turn in your cheat sheet with your exam. Name: Date and Time Started: Date and Time Finished: Full Pledge and signature: (1) (15 points) Frederica gets a weekly allowance of m and cares only about her current weekly consumption of bananas and licorice. The price of bananas is $1 each. Last week, the price of licorice was $1 per ounce and Frederica consumed 12 bananas and 12 ounces of licorice. (a) (2) What is m equal to? ANSWER. Assuming that Frederica s preferences exhibit the more is better trait, her choice of (12,12) would be on her budget line and therefore satisfy the budget line equation. m = p L x L +p B x B = = 24 where p L is the price of Licorice, x L is the quantity of licorice, x B is the quantity of bananas and the p s are their prices. (b) This week the price of licorice increased to $1.50 per ounce and Frederica s consumption of bananas remained the same. (i) (3) What is the (Slutsky) compensated income level? ANSWER To afford the original choice of 12 units of licorice and 12 bananas at the new Date: October 19,

2 Bananas 24 Comp d BL CC x c B IE SE OC 12 NC Original BL New BL 8 x c L IE SE Licorice Figure 1. Frederica s response to the price increase in licorice with a plausible compensated choice. OC, CC and N C stand for original, compensated and new choices, repsectively. IE and SE arrows indicate the directions of substitution and income effects for each good. prices, Frederica would need m c = = 30 (ii) (10) Are bananas a normal good, an inferior good or is it impossible to know with the information given? Explain using a diagram.answer We are told that Frederica s demand for bananas does not change. Hence the total effect on bananas was zero. Recall the Slutsky identity: TE = SE +IE since TE is zero in this case and SE must be positive (by the Law Of Comp d Demand), then IE must be negative. Such effects are illustrated in Figure 1. Raising the price of licorice makes Frederica poorer, a negative income effect is the normal response to a decrease in wealth. 2

3 (2) (35 points) Cookie Monster(CM) has preferences over bundles of milk(l) and cookies (C) which are represented by u(x L,x C ) = x 2 L x8 C where x L and x C are, respectively, the quantities of milk and cookies in a bundle. (a) (5 pts) What are CM s demands for milk and cookies as a functions of income (m), the price of milk (p L ), and the price of cookies (p C )? (Hint: either derive the demands, or just explain in a few words why your answer is right.) AN- SWER. The utility function is Cobb-Douglas. We know that CD preferences yield constant budget share demands. In this case, we can infer from the exponents on the factors in the utility function that, under standard budgets, Cookie Monster always spends 20% of his income on milk and 80% of his income on cookies. Demands are therefore x L (p L,p C,m) =.2m p L x C (p L,p C,m) =.8m p C (b) (10 pts) The price of milk is p L = $1.00 per gallon and the price of cookies is p C = $4.00 per pound. Draw a picture which shows Cookie Monster s income expansion path in x L x C space out to an income level of $100 per week.answer See Figure 2. (c) (2 pts) Suppose that Cookie Monster makes $100 per week. In the same picture, draw CM s budget line and label his optimal consumption bundle.answer See Figure 2. (d) (8 pts) Grover approaches Cookie Monster (who is just about to eat his optimally choosen bundle) and offers to give Cookie Monster 4 gallons of milk in exchange for a pound of cookies. How does CM respond? Explain. ANSWER. Cookie Monster declines Grover s offer. Grover is offering something that Cookie Monster could have already had, namely the bundle (x C = 19,x L = 24). We know that Grover s offer is on the budget line because the slope of the budget line is 4 gallons of milk per cookie, reflecting the price ratio. We also know that, among all the bundles on his budget line, CM can t do better than his optimal choice of bundle (20, 20). (e) Cookie Monster continued (5pts) Suppose that the CTW (CM s employer) starts a wellness program which augments CM s salary of $100 cash with an additional $60 in milk vouchers (like cash but can ONLY be used to buy milk). Draw CM s new budget and optimal bundle. ANSWER. See Figure 3. As we can see in Figure 3, Cookie Monster would choose to consume 25 cookies (the most he can buy with his cash), along with 60 gallons of milk. Note, this is more milk and fewer cookies than CM would have choosen if CTW has simply given him $60 in cash instead of milk stamps. (f) (5 pts) At the optimal choice, at what rate would CM be willing to accept cookies in exchange for giving up milk? How does this relate to the MRT in the market? Explain. ANSWER. The rate at which CM is willing to give up milk 3

4 Milk 100 Budget Line with m = 100 Grover s Offer Income Expansion Path Optimal Choice Cookies Figure 2. Cookie Monster s budget and optimal choice when m = 100 along with his Income Expansion Path (in magneta). for cookies is determined by his MRS function: MRS(x C,x L ) = du/dx C du/dx L = 8x2 L x7 C 2x L x 8 C = 4x L x C Evaluating his MRS at the optimal bundle, we get MRS(x C = 25,x L = 60) = which is roughly 10 gallons per cookie and obviously much greater than 4 gallons per cookie - the rate at which CM must give up milk for additional cookies as suggested by the price ratio. We can see this in Figure 3 which shows indifference curves that are steeper than the budget line with measured at Cookie s optimal choice. So why can t CM take the trade in the market? Because of the kink in his budget - which follows from the fact that he is not 4

5 160 Budget Line with m = 100 and $60 in milk vouchers Milk 100 Optimal Choice Budget Line with m = 100 Income Expansion Path Orig Choice Hypothetical Optimal with $160 cash budget Cookies Figure 3. Cookie Monster s Optimal Choice when $100 cash income is supplemented with $60 of milk stamps. allowedtospendhismilkvouchersoncookies. IfCTWhadgivenhimanaddition $60 in cash instead of milk vouchers, CM would have moved to a point where MRS = 4 and eaten more cookies and less milk. This is drawn in Figure 3 and labeled Hypothetical Optimal 5

6 (3) (15 Points) Arthur has preferences over present and future consumption according to ( ) 1 u(x 0,x 1 ) = logx 0 + logx 1 1+δ He expects income in the current and future period of (ω 0,ω 1 ) = (500,1600) and faces an interest rate of r =.2 whether he decides to borrow or save. (a) (3 points) Write down Arthur s budget line equation and draw a picture of his budget set. ANSWER Arthur s budget is x 0 (1.2)+x 1 = 500(1.2)+1600 (b) (3 points) Write down the first order condition for the solution of the consumer problem in this context. In other words, write down the equation that represents the tangency condition.answer MRS 0,1 = 1+r du/dx 0 du/dx 1 = 1+r 1 x 0 1 = 1+r (1+δ)x 1 (1+δ)x 1 = 1.2 x 0 (c) (4 points) Show that if δ =.2 then Arthur would perfectly smooth his consumption. (i.e. He would consume exactly the same amount in each period.) ANSWERPlugginginδ =.2inthefirstorderconditionderivedintheprevious part, we get (1.2)x 1 x 0 = 1.2 x 1 = x 0 This proves what was asked for. To get demands, just combine this equation with the budget line equation to get 2.2x 0 = 500(1.2)+1600 x 0 = 500(1.2)+1600 = x 1 = 1000 These demands make up the optimal choice at r =.2 in Figure 4. (d) (5 points) Suppose that δ =.2 and r increases from.2 to.3. Show (with as much precision as time allows) in your diagram the income and substitution effects on 6

7 future consumption. ANSWER We know that original choice is (1000,1000). Therefore his compensated budget line equation must be 1.3x C 0 +x C 1 = Combine this with the first order condition: (1.2)x 1 x 0 = x 1 = 1.3x 0 and we get compensated demands: 1.2x C 1 +x C 1 = 2300 x C 1 = > 1000 x C 0 = < 1000 These demands are represented in Figure 4 as the compensated choice labeled CC. The new budget line equation is 1.3x N 0 +x N 1 = Combine this with the first order condition to get new demands: 1.2x 1 +x 1 = 2250 x N 0 = < 1000 x N 1 = > 1000 Note that the direction of effect on current demand is unambiguous, since substition and income both are both negative along the current consumption axis. However, the total effect on future consumption is potential ambiguous, since subsitution is positive while income effect must be negative (since Arthur is poorer after the rate increase). However, without performing the entire calculation we can see that total effect on future consumption is somewhat positive since new demand is a bit over $1000 (original demand for it). Hence in Figure 4 we show SE along the future axis with a greater magnitude than the IE. 7

8 Future Consumption ω CC 45 degree (perfect smoothing) 1000 SE IE NC Compensated BL New Budget Line (r =.3) Optimal Choice at r =.2 Budget Line when r =.2 IE SE Current Consumption Figure 4. Arhtur s optimal choice at r =.2 (OC), r =.3 (NC) and compensated choice (CC). Substitution effects are represented by green arrows, income effect by blue arrows. 8