University of Toronto Department of Economics ECO 204 Summer 2012 Ajaz Hussain TEST 2 SOLUTIONS TIME: 1 HOUR AND 50 MINUTES YOU CANNOT LEAVE THE EXAM ROOM DURING THE LAST 10 MINUTES OF THE TEST. PLEASE REMAIN SEATED UNTIL THE PROCTOR ANNOUNCES THAT YOU CAN LEAVE THE ROOM IF YOU DETACH ANY PAGES FROM THIS TEST THEN IT S YOUR RESPONSIBILITY TO RE-STAPLE THE PAGES DO NOT HAVE A CELL PHONE ON YOUR DESK OR ON YOUR PERSON. ONLY AID ALLOWED: A CALCULATOR THERE IS A WORKSHEET AT THE END OF THE TEST GOOD LUCK! LAST NAME (AS IT APPEARS ON ROSI) FIRST NAME (AS IT APPEARS ON ROSI): MIDDLE NAME (AS IT APPEARS ON ROSI) STUDENT ID # (AS IT APPEARS ON ROSI) SIGNATURE: Do Not Write Below Question Maximum Possible Points Score 1 15 2 35 3 25 4 25 Total Points = 100 Page 1 of 36
Question 1 [15 Points] Whenever you dine out at restaurants, you like have food and alcohol as perfect complements in a 1:1 ratio. As for alcohol, you perceive 2 glasses of wine to be a perfect substitute for a glass of gin. Assume food and alcohol can be consumed in any non-negative amount, that you are a price taker, and that all pecuniary terms are strictly positive. (a) [10 Points] Let food =, wine = and gin =. Solve your UMP for and. Show all calculations below. Answer [A similar question was done in the lectures and HW] You like to have food (good 1) and alcohol as perfect complements in a 1:1 ratio: Alcohol * (Food) As for alcohol you perceive 2 glasses of wine (good 2) as a perfect substitute for a glass of gin (good 3). It s helpful to think of wine and gin as inputs producing alcohol as the output. ( ) ( ) Since wine and gin are perfect substitutes the alcohol production function is linear what are the parameters of this production function? Use the hint that a unit of alcohol consists of wine and gin as perfect substitutes in a 2:1 ratio: Page 2 of 36
(Gin) 1 A unit of alcohol 2 (Wine) The slope of an iso-quant of the alcohol production function is ½ so that the alcohol production function is (why?): With this the utility function over food and alcohol becomes: ( ) ( ) Your UMP is: Since the utility function is not differentiable everywhere we cannot use calculus to solve the UMP. Instead, we utilize the fact that at the optimum the following two equations must be hold (why?): These are 2 equations in 3 unknowns. As shown in lectures and discussed in HW the trick to solving this problem is to express the budget constraint in terms of the objective and one of the inputs to alcohol -- suppose we choose wine. Substituting and into the budget constraint yields: Page 3 of 36
We have expressed the objective in terms of a single variable (wine). Ordinarily we would look for a stationary solution by the first order condition, i.e. but this won t work here because the objective function is linear (not strictly concave) and does not peak anywhere: Whether you satisfy your cravings for alcohol depends on whether utility is increasing or decreasing in wine. Notice that if the slope is strictly positive (which happens whenever ) then: Utility Page 4 of 36
Since your optimal utility is increasing in zero gin: (wine) you ll want to satisfy your alcohol cravings by drinking only wine and How much wine will you drink? From : We see # of glasses of wine = # of plates of food. How much food will you have? We know that from the budget constraint: so that In sum: Returning to the utility function: From this equation we see if the slope is strictly negative (which happens whenever ) then: Page 5 of 36
Utility Since your optimal utility is decreasing in and zero amounts of wine : you ll want to satisfy your cravings for alcohol cravings by drinking only gin How much gin will you drink? From: That is, # of glasses of gin = ½(# of plates of food). How much food will you have? We know that that from the budget constraint: so In sum: Page 6 of 36
Of course there s the possibility that. In this case, you can satisfy your alcohol cravings by drinking either gin only, or wine only, or any combination of wine and gin where 2 glasses of wine are combined with 1 glass of gin. Page 7 of 36
(b) [5 Points] Please answer this question according to the last digit of your ID #: If your ID # ends in 0, 2, 4, 6, or 8 When will you only drink wine at dinner? How many glasses of wine will you order per plate of food? Show all calculations. If your ID # ends in 1, 3, 5, 7, or 9 When will you only drink gin at dinner? How many glasses of gin will you order per plate of food? Show all calculations. Answer There is nothing to solve for here: we re being asked to summarize the answers from part (a) we know that: If your ID # ends in 0, 2, 4, 6, or 8 When will you only drink wine at dinner? How many glasses of wine will you order per plate of food? Show all calculations. You will drink wine only whenever # of glasses of wine = # of plates of food If your ID # ends in 1, 3, 5, 7, or 9 When will you only drink gin at dinner? How many glasses of gin will you order per plate of food? Show all calculations. You will drink gin only whenever # of glasses of gin = ½ (# of plates of food) Page 8 of 36
Question 2 [35 Points] Whenever you dine out at restaurants, you like have food and alcohol as complements in a 1:2 ratio (i.e. 1 unit of food combined with 2 units of alcohol). As for alcohol, you prefer to have wine and gin as perfect complements i.e. 1:1 ratio. Assume food and alcohol can be consumed in any non-negative amount, that you are a price taker, and that all pecuniary terms are strictly positive. (a) [10 Points] Let food =, wine = and gin =. Solve your UMP for and. Show all calculations below. Answer [This question looks scary but if you understood how to solve question 1 (done in lectures and HW) then it shouldn t be too hard to extend the model where alcohol consists of wine and gin as perfect substitutes to one where the wine and gin are combined as complements]. You like to have food (good 1) and alcohol as complements in a 1:2 ratio: Alcohol * (Food) As for alcohol, you perceive wine (good 2) and gin (good 3) as perfect complements (in a 1:1 ratio): ( ) ( ) Page 9 of 36
It s helpful to think of wine and gin as inputs producing alcohol as the output so that the alcohol production function is: (Ginl) (Wine) As such, the utility function over food and alcohol becomes: ( ) ( ) Your UMP is: ( ) Since the utility function is not differentiable everywhere we cannot use calculus to solve the UMP. Instead, we utilize the fact that at the optimum the following two equations must be hold (why?): ( ) These 2 equations are easier solve that in question 1. Notice that we can solve for optimal amount of food straightaway: use the fact that in the budget constraint: Page 10 of 36
And since this implies that: In turn: ( ) ( ) The complete solution is: Notice # glasses of wine = # glasses of gin (as it should be since wine and gin are perfect complements) and # glasses of wine = # glasses of gin = 2 # plates of food (as it should be since food and alcohol are complements with 2 units of alcohol with each plate of food). For example, if you have a plate of food, you ll order 2 glasses of wine and 2 glasses of gin. Clearly, you have a drinking problem. Seek help. Page 11 of 36
(b) [5 Points] Suppose. How much food, wine and gin will you order? What is the impact of a $1 income tax on your optimal utility and optimal consumption of food and alcohol (please give answers for the total changes in consumption and utility). Show all calculations. Answer From above: And since this implies: You ll get 2 plates of food and 4 glasses of wine and 4 glasses of gin each and your optimal utility will be: The impact of a $1 income tax can be calculated in several ways: one method is to calculate the post income tax optimal quantities and utility: And since this implies: Thus: A second method is to use derivatives to calculate the approximate change in optimal quantities and utility (remember calculus methods always give approximate answers): Page 12 of 36
This is in fact the same answer as the exact change in. Now: This is in fact also the same answer exact change in and. Finally: Again this is the same answer as the exact change in. Page 13 of 36
(c) [10 Points] Suppose. Calculate the excise tax rate on food which will generate the same amount of tax revenue as the income tax in part (b) and calculate its impact on your utility and the optimal consumption of food and alcohol (please give answers for the total changes in consumption and utility). Show all calculations. Answer An excise tax rate on food raises the price of food from to. We want the excise tax rate on food to raise the same amount of tax revenue as the income tax (= $1). Now the revenue from an excise tax rate on food is: - - We know that the pre-excise tax optimal quantity of food is: Thus the post-excise tax optimal quantity of food is: Thus: We want this to be the same as income tax revenue of $1: That is, levying a 2.5% excise tax rate on food will raise $1 in tax revenues. Let s confirm this: recalling that the optimal quantity of food following the excise tax on food is: Page 14 of 36
The revenue from an excise tax on food is: This confirms that it raises the same amount of revenue as an income tax of $1. The impact of this excise tax rate on food on be calculated in several ways: one method is to calculate the optimal quantities and utility again: And since this implies: Thus: Notice the change in optimal quantities and optimal utility due to an excise tax on food designed to raise the same amount of tax revenue as a $1 income tax = change in optimal quantities and optimal utility due to a $1 income tax. This is not a coincidence as you will see below. A second method is to use derivatives to calculate the approximate change in optimal quantities and utility (remember calculus methods always give approximate answers): Now: Page 15 of 36
This is in fact the same answer as the exact change in. Now: This is in fact also the same answer exact change in and. Finally: Again this is the same answer as the exact change in. Page 16 of 36
(d) [10 Points] Decompose the price effect of the excise tax on food in part (c) into the substitution and income effects. Show all calculations. Answer An excise tax on food reduces the consumption of food. With food as good 1 and alcohol as good 2 we can graph the price effect on food: Alcohol A C 1.98 2 Price Effect Price Effect of Food = Optimal Quantity of Food After Excise Tax - Optimal Quantity of Food Before Excise Tax This is a little different from the lectures and HWs: there we examined the price effect due to a decrease in the price of good 1 (so that price effect ) but here we have a price increase because of which we have a negative price effect. To decompose this into the substitution and income effects we just do the opposite of what we did in the lectures and HW: instead of taking away income until the original bundle becomes affordable we now give the consumer more income until the original bundle become affordable: Alcohol B A C 1.98 2 Price Effect Page 17 of 36
It s obvious from this graph that the substitution effect is zero (in fact, review the lecture slides, HWs and chapters and you ll notice that for the complements utility function, the substitution effect is always zero). This means that the price effect equals the income effect since: Let s confirm this. First, notice that: The three budget constraints are: Budget constraint before excise tax on food Budget constraint after excise tax on food Budget constraint after excise tax on food and income boost To find notice that according to the original budget constraint if you consumed bundle then since expenditure = income: Following the excise tax on food the budget constraint became and you consumed bundle. To decompose the price effect we imagine giving you an income boost so that the budget constraint becomes where the income level makes the original bundle affordable (i.e. the budget line passes through ). In that case, if you were to consume bundle on the budget constraint then since expenditure = income: Subtract the second equation from the first: The three budget constraints are: Budget constraint before excise tax on food Budget constraint after excise tax on food Budget constraint after excise tax on food and income boost Page 18 of 36
Now, what is bundle (the optimal choice on the budget constraint after excise tax on food and income boost, i.e. )? We know that: Using we have: That is, at bundle you consume the exact same amount of food as in bundle or that the substitution effect is zero so that the price effect equals the income effect. Page 19 of 36
Question 3 [25 Points] Using data from Apple s 2011 and 2012 10-K reports, an analyst has estimated the following linear demand curve for Apple ipads: Here $ price/ipad and # of ipads. (a) [5 Points] In 2012-Q2 the average price of an ipad was $558.57. According to the demand curve, how many ipads will be sold in 2012-Q2? What will be the price elasticity of an ipad in 2012-Q2? Show all calculations. Answer [A similar question was done in the lecture and HW] The demand curve is: To find the quantity, let s derive the demand function: Thus: (Incidentally, the actual # of ipads sold in 2012 Q2 was 11,798,000 so you can see that the demand model over predicts for 2012 Q2). For the price elasticity we use the point elasticity formula: This means if Apple raises the price of ipads by 1% quantity sold of ipads decreases by 3.54%. Also notice that we will discuss this later in the course. Page 20 of 36
(b) [5 Points] Calculate the representative Apple ipad consumer s utility level from consuming Apple ipads (not the utility level from consuming ipads and all other goods). You don t need to prove any equations used for this calculation. Show all calculations and state all assumptions. Answer: [A similar question was done in the lecture and HW] Suppose ipad = good 1 and everything else = good 2. Let s make good 2 the base good so that Recall from the lecture slides that if we assume the representative Apple ipad consumer has the following type of quasi-linear utility function: Then assuming an interior solution (i.e. ) the area under the demand curve of good 1 (between 0 and ) is the utility from good 1: Here is the proof (not required for the test): At interior solution:. Set good 2 as the base good : Thus, in 2012 Q2: Page 21 of 36
$716.4 $558.57 Apple ipad Demand curve This area is ( I am calculating this as the area of the triangle plus the area of the rectangle): As discussed in lectures, this is a pecuniary measure of utility. Page 22 of 36
(c) [5 Points] What is the impact of a 1% excise tax on the representative Apple ipad consumer s utility level from consuming Apple ipads and all other goods? You don t need to prove any equations used for this calculation. Show all calculations and state all assumptions. Answer: [A similar question was done in the lecture and HW] Suppose ipad = good 1 and everything else = good 2. Let s make good 2 the base good so that Recall from the lecture slides that if we assume the representative Apple ipad consumer has the following type of quasi-linear utility function: Then assuming an interior solution (i.e. ) we know that the consumer surplus is the representative Apple ipad consumer s total utility from ipads and everything else minus her income (we don t know nor her income): Here is the proof (not required for the test): But: ( ) = We are being asked for the impact on utility due to a 1% excise tax on Apple ipads: this is easily calculated by: Page 23 of 36
Now: After the 1% excise tax: Thus: ( ) $716.4 $716.4 $558.57 $(1.01)558.57 Apple ipad Demand curve Apple ipad Demand curve Thus, the impact of the 1% excise tax on the utility of the representative Apple ipad consumer is: We get this result without: knowing the actual equation of observing observing income. Page 24 of 36
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(d) [10 Points] Confirm your answer in part (c) by deriving and using the representative Apple ipad consumer s utility function of consuming Apple ipads and all other goods. Show all calculations and state all assumptions. Answer: [A similar question was done in the HW] Suppose ipad = good 1 and everything else = good 2. Let s make good 2 the base good so that Recall from the lecture slides that if we assume the representative Apple ipad consumer has the following type of quasi-linear utility function: From lectures, HWs and chapters we know that it s possible to derive the equation for from the demand curve for good 1. This technique is frequently used in economics because it s easy to derive get the demand curve for a good (we saw examples in the lectures where we derived demand functions and models through regression analysis) and from this we can derive the utility function for good 1. Here is the proof (not required for the test): At interior solution:. Set good 2 as the base good : Thus: [ ] To get the constant, set and assume that (i.e. 0 units of good 1 gives you zero utility from good 1). This implies that the constant is 0 so that: Page 26 of 36
Thus: Now, the representative Apple ipad consumer chooses optimal amounts of good 1 and good 2 by solving her UMP: For the sake of completeness, I ve thrown in non-negativity constraints and Now, the consumer solves the UMP by solving: because good 2 is the base good. [ ] [ ] By the envelope theorem, the impact on optimal utility due to an excise tax on ipads (good 1) is: You really need to know how we get this. Now: We need -- from lectures and HWs recall that the UMP FOC with respect to is: Thus the approximate change in utility due to a 1% excise tax on good 1 is: This differs from the exact change in utility of of error is: but in fact this is a close approximation since the margin Page 27 of 36
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Question 4 [25 Points] Consider a 3 period economy where at the beginning of each consumer receives (individualspecific) real incomes (in units of corn) respectively. In the first two periods, consumers can borrow or save corn at the prevailing real interest rate. Suppose a consumer has the following utility function over consumption at and : Assume and that the nominal interest rate and inflation rate are. (a) [15 Points] Assume a uniform rate of inflation and nominal interest rate between periods and. Solve the consumer s inter-temporal UMP for. Show all calculations. Hints: For we must have Answer Take the hint and transform the utility function: Notice, as we have repeatedly discussed in lectures and HWs, that we do not transform the utility but rather the utility function. Carrying on: Use the hint The agent solves the UMP: Now use the hint that or we must have : as such, the UMP becomes Now, the budget constraint is: Page 29 of 36
With a uniform real interest rate this becomes: For simplicity let so that the budget constraint becomes: The UMP becomes: Setup the Lagrangian function: [ ] The FOCs and KT conditions are: [ ] The KT condition gives rise to two possibilities: Possibility #1 which will happen when Now, to sign we need to know for which we need either or. Now, if then from the budget constraint we see that: Page 30 of 36
Let s solve for we need to express in terms of. From: And from: Equating the s we have: Substitute in budget constraint: [ ] From this we have: Now, we are ready to see when From: Thus whenever: Page 31 of 36
Thus, if then: Possibility #2 which will happen when To sign we need to know and in turn for this we need to know. From: From: Now, from the budget constraint we have: Page 32 of 36
Now whenever: This is the exact opposite condition for possibility #2. Thus, if then: Page 33 of 36
(b) [10 Points] Please answer this question according to the last digit of your ID #: If your ID # ends in 0, 2, 4, 6, or 8 Under what conditions will the consumer save corn at savings at if the real interest increases? Show all calculations. If your ID # ends in 1, 3, 5, 7, or 9 Under what conditions will the consumer borrow corn at to borrowings at if the real interest increases? Show all calculations.? What will happen to? What will happen Answer If your ID # ends in 0, 2, 4, 6, or 8 Under what conditions will the consumer save corn at savings at if the real interest increases? Show all calculations.? What will happen to The consumer will save corn at when: We need to consider the two possibilities. Now if then so that the consumer will save whenever: What happens to savings as? Now: [ ] [ ] (we need to assume ). Thus, savings increase when the real interest rate rises. Now if then so that the consumer will save whenever: Page 34 of 36
What happens to savings as? Now: (we need to assume ). Thus, savings increase when the real interest rate rises. If your ID # ends in 1, 3, 5, 7, or 9 Under what conditions will the consumer borrow corn at to borrowings at if the real interest increases? Show all calculations.? What will happen The consumer will borrow corn at when: We need to consider the two possibilities. Now if then so that the consumer will borrow whenever: What happens to borrowing as? Now: [ ] [ ] [ ] This will be positive whenever which means that the consumer will borrow less as real interest rates rise (remember that if consumer is borrowing then she has negative savings so that a positive derivative means that the negative savings becomes a larger number or that she borrows less). The expression will be negative whenever which means that the consumer will borrow more as real interest rates rise (remember that if consumer is borrowing then she has negative savings so that a negative derivative means that the negative savings becomes a smaller number or that she borrows more). Now if then so that the consumer will borrow whenever: Page 35 of 36
What happens to borrowing as? Now: This means that the consumer will borrow less as real interest rates rise (remember that if consumer is borrowing then she has negative savings so that a positive derivative means that the negative savings becomes a larger number or that she borrows less). Page 36 of 36