Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 5 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. 1 Note: In some questions I've given guidance on how to derive the demand curve from indifference curve map. In other questions, I have given the demand curve in these questions, it would behoove you to figure out how to get the result. If you get stuck, please do see me or Asad. Question 1 (Based on 20072008 Final Exam Question) In a marketing survey, customers were asked about their preferences over cigarettes and coffee. The results indicate that customers perceive cigarettes and coffee to be pleasurable (`good' goods) and have increasing marginal rate of substitution. (a) Will consumers engage in "addictive" behavior? If so, when will they consume coffee only? Answer: Yes, it is possible. Suppose cigarette and coffee prices are uniform (i.e. do not vary by volume), then it is possible to consume just cigarettes or just coffee ("addictive behavior") as depicted below. In these figures, I've deliberately drawn the budget line so that it gives only one corner solution. You should draw a case where the budget line can give two possible corner solutions and then attempt the questions below with this model). The addict will consume cigarettes only if the ratio of coffee to cigarette prices is high. Similarly, the addict will consume coffee only if the ratio of coffee to cigarette prices is low. You should convince yourself of this by checking when the budget line below is steep and flat. 1 Note for book: These figures created for HW 2 (20082009). 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Addicted to Cigarettes Addicted to Coffee (b) In part (a), what happens to coffee consumption as the price of coffee decreases? Answer: As the price of coffee decreases, the budget line become flatter since its slope: P1/P2 = Pcoffee /Pcigarettes falls. At the same time, the budget line swings out, since the xaxis intercept Y/P1 = Y/Pcoffee becomes larger. What happens to coffee consumption depends on the initial situation. Suppose the consumer is already addicted cigarettes (see figure above) so that the consumer is drinking zero cups of coffee. As the price of coffee falls, initially there is no change in 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain quantities of coffee consumed. But as the price drops low enough, there will come a point where the consumer switches from cigarettes to coffee: From an Addiction to Cigarettes to an Addiction to Coffee Thus, the demand curve for coffee will look like (the downward sloping part is for illustrative purposes: for example, the downward sloping part could be nonlinear): One of the implications here is that for some ranges of coffee prices the price elasticity of coffee (E = % change in coffee consumption/ % change in coffee prices) is zero. An analyst, working in this range of prices may conclude that coffee sales do not respond to coffee prices. This, as you can see, is not a global statement. On the other hand, suppose the consumer was already a coffee addict: in that case, the demand curve will be (assuming that consumer was a coffee addict from price of $0 on): 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (c) In part (a), what happens to cigarettes consumption as the price of coffee decreases? Answer: Again, the answer depends on whether the consumer was already a cigarettes or coffee addict. Take the first case: if she is already a cigarette addict, then as the price of coffee falls, initially she will not switch out to coffee. Eventually however (even addicts respond to prices!) the addict responds to the low coffee prices. Thus, the cross demand curve for cigarettes with respect to coffee prices is: For the case where the consumer is already a coffee addict, the cross demand curve for cigarettes is simply nil and is: 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (d) In part (a), what happens to coffee and cigarettes consumption as income increases? Answer: It again depends on whether the consumer was already a cigarettes or coffee addict. In the following figure for the case when the consumer is already a cigarettes addict more income results in more addictive smoking. In this figure for the case when the consumer is already a coffee addict more income results in more caffeine intake: 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (e) Will consumers engage in addictive behavior if coffee prices decrease with volume? Answer: It is possible to break addictive behavior so long as coffee prices increase (not decrease) with volume: for then, with one price "break" point, we'd have an opportunity to break the consumer's vile addiction to coffee and cigarettes: To understand this budget line, note that cigarette prices are uniform (and therefore constant) while coffee prices are increasing with volume. For the first few units of coffee, the price of coffee is low; the price then increases over some units of coffee. Thus, the budget line slope: (P1/P2 = Pcoffee /Pcigarettes ) is initially flat and then steep. 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 2 (Based on Summer 2008 Final Exam Question) If Ajax if doesn't have at least 5 pounds of food a day, he will die. In fact, with less than 5 pounds of food a day, he doesn't care about anything else. Suppose that once he has the threshold level of food: (a) He prefers food and everything else as perfect complements. Characterize Ajax's optimal choice. In particular is it an interior or corner solution? What happens to his consumption of food and everything else as the price of food decreases? Answer: Here are Ajax's indifference curves. From these, we can examine the effect of lower food prices on quantities of food and everything else: Observe how below 5 lbs of food, the optimal choice is a corner solution after which it is an interior solution. From these optimal choices, the demand curve for food and crossdemand for everything else are: 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) He prefers food and everything else as imperfect substitutes with decreasing MRS. Characterize Ajax's optimal choice in particular, is it an interior or corner solution? What happens to his consumption of food and everything else as the price of food decreases? Answer: In this figure, I have drawn a specific case of imperfect substitute curves you will get different results for other depictions of imperfect substitute preferences: From these, we can examine the effect of lower food prices on quantities of food and everything else: 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Here, note, how as the price of food decreases, consumption of everything else is initially nil, but once Ajax has enough food to sustain himself, consumption of everything else increases, but there comes a point where it decreases. (c) He prefers food and everything else as perfect substitutes with MRS of 1. Characterize Ajax's optimal choice. In particular, is it an interior or corner solution? What happens to his consumption of food and everything else as the price of food decreases? Answer: Ajax's indifference curves are shown below this figure shows the effect of lower food prices. Beginning with high food prices, the optimal choices are initially corner solutions, followed by one particular price at which the budget line is tangent to the indifference curve (so that there will be infinite number of solutions), followed once again by corner solutions: 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (d) He prefers food and everything else as perfect substitutes with MRS of 1/2. Characterize Ajax's optimal choice. In particular, is it an interior or corner solution? What happens to his consumption of food and everything else as the price of food decreases? Answer: Qualitatively, the solution looks the same as part (c). (e) He only cares about everything else. Characterize Ajax's optimal choice. In particular, is it an interior or corner solution? What happens to his consumption of food and everything else as the price of food decreases? Answer: The indifference curves are: 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Observe how the solutions are corner solutions. Beginning from high food prices, as the price of food decreases, Ajax first only consumes food and once he reaches the threshold level of food, consumes only everything else. Thus, the demand and cross demand curves are: (f) He reaches a "bliss point" at 10 lbs of food a day and 5 units of everything else. Characterize Ajax's optimal choice. In particular, is it an interior or corner solution? What happens to his consumption of food and everything else as the price of food decreases? Answer: I am going to assume that once Ajax has 5 lbs of food a day, his preferences are "circular" with an alternative model of "bliss" preferences you'll get different results. There ought to be more "circles" in the figure above I have drawn a few circles due to space (and time) limitations). 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 3 (Summer 2008 Final Exam Question) Meredith has preferences over wine and cheese represented by concentric circles centered around 10 units of cheese and wine. (a) Suppose her felicity is greatest at the center. What happens to Meredith's consumption of cheese as the price of cheese decreases? Answer: Answer: Here are Meredith's preferences: From this, we have the demand curve for cheese: 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) Suppose her felicity is lowest at the center. What happens to Meredith's consumption of cheese as the price of cheese decreases? Answer: Meredith's preferences are: From this, we have the demand curve for cheese: 13 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 4 In this question, you will go over the discussion about subsidizing education. This figure depicts a program in which the government provides a specific level of education for free: it depicts the private budget line and public budget point: In this question, you will analyze two variations of that question: (a) "coupons" that can only be used for education (exactly like gift certificates that can be spent only a certain store) and (b) a cash subsidy. Assume consumer has "imperfect substitutes" preferences over education and everything else, which are each good goods. (a) Suppose the government gives everyone a coupon (voucher) which can be used to obtain the up to 12 years of education for free. Analyze the impact of this policy on optimal choice, 14 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain "happiness" and education levels. Note: in this question, the consumer will have the option of 12 years of education for free and if she wants more than 12 years of education to pay for additional years of schooling. In the lecture question you had to either go to free public schools for 12 years or go to private schooling for any number of years. Answer: The coupon is an education "gift" allowing the recipient to get 12 years of education absolutely free. Let education be on the x axis and everything else on the y axis. The slope of the budget line is: P1/P2 = Peducation/Pelse. The coupon makes the first 12 years of education free; thus, the for the first 12 years, the slope of the budget line is 0. For any education beyond 12 years, the slope will Peducation/Pelse . In essence, the budget line shifts out, but leaves the maximum amount of other goods that can be purchased unchanged. The shift is almost as if the consumer has a higher income, except that she cannot spend the extra income on everything else. This figure depicts the initial and new budget constraints: The impact of the program depends on preferences and initial choices. Three cases are depicted in the following three figures: 15 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Here, the consumer avails herself of the voucher program by studying more and buying more of everything else here, everything else and education are normal goods. At the initial and new optimal choice, her MRS = slope of the budget line and she is happier than before. Here, the consumer avails herself of the voucher program by buying more of everything else but studying the same number of years as before. Here, everything is a normal good but education is neither a normal nor an inferior good. At the initial and new optimal choice, her MRS = slope of the budget line and she is happier than before. You should be able to easily draw a case where the consumer will study less when there is a voucher program (the case where everything else is a normal good and education is an inferior good). 16 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Here, the consumer avails herself of the voucher program by studying more and buying more of everything else. At the initial optimal choice her MRS = slope of the budget line but at the new optimal choice MRS slope of the budget line. She is happier than before. It appears from these three cases, that the voucher program results in varying effects on education (some study more, some less, some same) but all seem to result in more happiness. This is in contrast to the case of an (unrestricted) income subsidy where it was possible that switching to public education could potentially make some people less happy. Why is this not happening here? The answer is that the voucher makes available (education, else) choices that were previously unavailable to the consumer. Since the voucher makes more of education and else available by the more is better assumption this makes the consumer happier. Although she may consume less of one good, she won't consume less of both goods. (b) Suppose the government gives a cash payment guaranteeing 12 years of education which the consumer can spend however she wishes. Analyze the impact of this policy on optimal choice, "happiness" and education levels. In particular, compare the optimal choices in the coupon versus subsidy programs. Answer: The cash subsidy, denoted S, is tantamount to boosting consumer income from Y to Y + S, shifting the budget line out by S: 17 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Notice, the cash subsidy makes available more (education, else) combinations than the voucher program. The impact on optimal choice and happiness depends on preferences. Here are some interesting cases: In this figure, with the cash subsidy, the consumer is made better off; she consumes more education and everything else. But her choice with the cash subsidy is identical to the choice under the voucher program, despite the fact that the subsidy program makes more bundles available than the voucher program. In the figure below, with the cash subsidy, the consumer is made better off; she consumes more education and everything else and her choice with the cash subsidy is different from the choice under the voucher program. 18 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain In this case, the person is happier with the cash subsidy than the voucher program. Compared to the initial level of education, the subsidy program also results in more education but less than the voucher program. The figure below depicts the case of someone who is happier with the subsidy program but consumes less education for her, education is an inferior good. Question 5 In this question, you will practice the intertemporal consumption model. Let's use the 2 periods, noinflation, and capital markets model. A consumer has income Y1 and Y2 in T = 1, 2 respectively. The consumer can borrow or lend at nominal interest rate i. For simplicity, assume P1 = P2 = 1 (i.e. no inflation). (a) Suppose the consumer has the utility function U = min (C1, C2). Solve for the optimal consumption in T = 1, 2. 19 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: Since the consumer has complements utility, we can guess what the answer will be: C1 = C2. The question now is, what percentage of Y1 and Y2 will she save/borrow. Observe that we cannot solve the UMP by the Lagrangian because the utility function is not differentiable at the corner. Instead, we appeal to the fact that C1 = C2 and substitute this into the budget constraint. Formally, the UMP is: choose Q1 , Q2 to max U = min (C1, C2) subject to: (Consumption at T = 2) = (Income at T = 2) + (Savings from T = 1)(1 + i) The budget constraint can be rewritten as: C2 = Y2 + (Y1 C1)(1 + i) Now, substitute C1 = C2: C1 = Y2 + (Y1 C1)(1 + i) C1 = Y2 + Y1 (1 + i) C1(1 + i) C1 + C1(1 + i) = Y2 + Y1 (1 + i) C1 [1 + (1 + i)] = Y2 + Y1 (1 + i) C1 (2 + i) = Y2 + Y1 (1 + i) C1 = [Y2 + Y1 (1 + i)]/(2 + i) Since C1 = C2 this implies: C1 = C2 = [Y2 + Y1 (1 + i)]/(2 + i) By definition, this consumer is "smoothing consumption", i.e., consumption levels are constant across time. This result is clearly nonKeynesian: recall that in the Keynesian model, consumption is a function of current income. 2 (b) Interpret your answer for the case when i = 0 (i.e. nominal interest rates are nil). Answer: What is the consumer's consumption when nominal interest rates are 0? For real life examples, 2 For book, investigate if there are some preferences that will generate C = f(Y) even with capital markets. The Keynesian model is trivially true if there are no capital markets since consumers cannot borrow/lend. 20 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain look at the Japanese economy where nominal interest rates are almost 0 (see figure below) and with deflation, real interest rates are negative. Source: Economagic.com With i = 0, consumption becomes: C1 = C2 = [Y2 + Y1 ]/2 This is a nice result: it says that even if the consumer could borrow/lend, if nominal interest rates are nil, the consumer will in each period, consume the average lifetime income 3 . (c) Interpret your answer for the case when i = 1 (i.e. nominal interest rates are 100%). Answer: What is the consumer's consumption when nominal interest rates are 100%? With i = 1, consumption becomes: C1 = C2 = [Y2 + Y1 (1 + 1)]/(2 + 1) C1 = C2 = [Y2 + 2Y1]/3 C1 = C2 = (2/3)Y1 + (1/3) Y2 This is a nice result: it says that even if the consumer could borrow/lend at 100% nominal interest rates (gulp), the consumer will in each period, consume a weighted average of lifetime income: 2/3 of first year income and 1/3 of second year income. 3 For book, reconcile this with permanent income hypothesis. Check what Friedman said I seem to recall that his was a statically driven model across blacks and whites in the US. 21 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Obviously, the weighted average of income depends on the utility function: for example, you'd get different results if the utility function was U = min (C1, C2) for . As an exercise, see what would happen if U = min (C1, 2C2), so that each unit of consumption in the first period is to be had in conjunction with two units of consumption in the second period 4 . (d) Use the results for C1 = C2 from part (a). When will this consumer be a borrower (lender) in T = 1? Answer: If the consumer is a borrower in the first period, then she is consuming more than her income at T = 1: C1 > Y1. Now: C1 = [Y2 + Y1 (1 + i)]/(2 + i) which implies that: C1 > Y1 [Y2 + Y1 (1 + i)]/(2 + i) > Y1 Y2 + Y1 (1 + i) > Y1 (2 + i) Y2 > Y1 (2 + i) Y1 (1 + i) Y2 > Y1 [(2 + i) (1 + i)] Y2 > Y1 That's a nice result: if you have U = min (C1, C2) and know that future income will be greater than present income, you will borrow today and lend tomorrow. Similarly, if Y2 < Y1 then you will lend today and borrow in the future. Question 6 In this question consider once again the intertemporal consumption, 2 periods, noinflation, and capital markets model, where the consumer has income Y1 and Y2 in T = 1, 2 respectively. The consumer can borrow or lend at nominal interest rate i. You will investigate the effect of higher nominal interest rates on whether someone who is a lender (borrower) continues to be a lender (borrower), and, the impact on utility. You will show: If the consumer is a lender at T = 1, then as nominal interest rates rise, she will continue to be a lender. This should be intuitive: at the current interest rates, if I 4 For the book: do this model with inflation. Check Merton's continuous time finance book for results on consumption patterns. 22 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain am lending money, then as interest rates rise, lending becomes even more attractive and so I will continue to be a lender. The impact of rising interest rates on the lenders' utility is obvious: they will be "happier" If the consumer is a borrower at T = 1, then as nominal interest rates rise, she will may switch to being a lender 5 . This should be intuitive: at the current interest rates, if I am borrowing money, then as interest rates rise slightly, I may continue borrowing. However, there may come a threshold interest rate, where borrowing becomes too expensive or put it another way, lending becomes more attractive so that I will switch from being a borrower to a lender. The impact of rising interest rates on the borrower's utility, if they continue to be a borrower, is obvious: they will be "sadder" For this question assume the consumer has CobbDouglas utility function between C1 and C2: U = C1 C2 Intuitively, the consumer is willing to tradeoff consumption today against consumption tomorrow. She's not like the consumer with U = min (C1, C2) who will do consumption smoothing. In contrast the consumer U = C1 C2 won't necessarily smooth consumption she may or may not. For simplicity, assume P1 = P2 = 1 (i.e. no inflation). (a) Solve the consumer's intertemporal UMP. Hint: it's easier to work with the log transformation of the utility function. Answer: The Lagrangian is: Choose C1 , C2 to max L = C1 C2 [C2 + C1(1 + i) Y1 (1 + i) Y2] Where the budget constraint is the same as the precious question. We could get the first order conditions which will be easier if we work with the log transformation of U = C1 C2. The Lagrangian becomes: Choose C1 , C2 to max L = log C1 + log C2 [C2 + C1(1 + i) Y1 (1 + i) Y2] The first order conditions are: For the book: investigate general equilibrium effect: is it possible that everyone becomes a lender (which is impossible)? Or is there some market clearing condition that ensures that there will always be some lenders and borrowers may have implications for credit crunch and glut. Also investigate the threshold interest rate and the idea that borrower's utility goes down and when they switch to being a lender, their utility increases. Is there for example a "jump"? Actually, we have this result from the Lagrangian already. 5 23 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L/C1 = 0 /C1 (1 + i) = 0 /C1 = (1 + i) L/C2 = 0 /C2 = 0 /C2 = L/ = 0 C2 + C1(1 + i) = Y1 (1 + i) + Y2 Dividing the 1st and 2nd first order conditions yields: [/C1 ]/[/C2 ] = [ (1 + i)]/ [/C1]/[/C2 ] = (1 + i) (/)(C2 /C1) = (1 + i) (/)(C2 /C1) = (1 + i) (/) C2 /(1 + i) = C1 Thus: C2 = (/) C1 (1 + i) We can substitute this in the budget constraint: C2 + C1(1 + i) = Y1 (1 + i) + Y2 (/) C1 (1 + i) + C1(1 + i) = Y1 (1 + i) + Y2 C1 (1 + i) [(/) + 1] = Y1 (1 + i) + Y2 C1 (1 + i) ( + )/ = Y1 (1 + i) + Y2 C1 = {/( + )}[Y1 (1 + i) + Y2 ]/(1 + i) 24 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain C1 = {/( + )}[Y1 + Y2 /(1 + i)] That is, in period 1, she will consume a constant fraction of the PV income. Note this takes the form: {/( + )}(Income), a familiar result from UMP. Now: C2 = (/) C1 (1 + i) C2 = {/( + )}[Y1 (1 + i)+ Y2] That is, in period 2, she will consume a constant fraction of the FV income. Note this takes the form: {/( + )}(Income), a familiar result from UMP. (b) When is the consumer a borrower in period 1? What about a lender in period 1? Answer: To be a borrower at T = 1, this consumer would consume more than her income, so she'd be forced to borrow: C1 > Y1 {/( + )}[Y1 + Y2 /(1 + i)] > Y1 {/( + )}Y1 + {/( + )}Y2 /(1 + i) > Y1 {/( + )}Y2 /(1 + i) > Y1 {/( + )}Y1 {/( + )}Y2 /(1 + i) > Y1 [1 {/( + )}] {/( + )}Y2 /(1 + i) > Y1 [ /( + )] Y2 /(1 + i) > Y1 (/) (Y2 /Y1) > (1 + i) If this condition holds true, the consumer will always be a borrower. In contrast, she will be a lender if the opposite condition holds: (/) (Y2 /Y1) < (1 + i). (c) Suppose this consumer is a lender in period 1. Show that if the nominal interest rate rises, she will continue to be a lender. If it helps you can assume that = = , Y1 = 10, Y2 = 5, i = 10%. 25 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: If she is a lender, then it must be true that: (/) (Y2 /Y1) < (1 + i) Now if the nominal interest rate i increases, the right hand side of this inequality will increase, maintaining the inequality. Put simply, this condition always holds if i increases. Here is an example: assume that = = , Y1 = 5, Y2 = 10, i = 10%. The lender condition becomes: 0.5 < 1.1 So the consumer is a lender at T = 1. If interest rates rise, say to 20%, the condition becomes: 0.5 < 1.2 And so the consumer stays a lender. (d) Suppose this consumer is a borrower in period 1. Show that if the nominal interest rate rises, she may become a lender. If it helps you can assume that = = , Y1 = 5, Y2 = 10, i = 10%. Answer: If she is a borrower, then it must be true that: (/) (Y2 /Y1) > (1 + i) Now if the nominal interest rate i increases, the right hand side of this inequality will increase. At some high enough interest rate, this inequality will be violated (reversed) in which case the consumer becomes a lender. Here is an example: assume that = = , Y1 = 5, Y2 = 10, i = 10%. The borrower condition becomes: 2 > 1.1 So the consumer is a borrower at T = 1. If interest rates rise, say to 20%, the condition becomes: 2 > 1.2 The consumer is still a borrower. But now suppose interest rates rise to 110%. Then the condition becomes: 26 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 2 < 2.1 In which case the consumer becomes a lender. 27 ...
View
Full Document
 Fall '08
 HUSSEIN
 Economics, Microeconomics, Department of Economics, S. Ajaz Hussain, Ajaz Hussain

Click to edit the document details