Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 1 of 10 ECON 20002 Intermediate Microeconomics Tutorial 2 solutions Consumer Theory: Preferences and Choices Problem 1. Octopuses understand Consumer Theory toooooooo. a) Draw the budget line for Margot and several of her indifference curves. Let’s start with indifference curves – combinations of bundles that give the consumer exactly the same utility. Assume that Margot has 5 right shoes and 3 left shoes. What happens if we add more right shoes Her utility doesn’t change because there is no matching shoes – so she is not wearing them, they are in a special “useless stuff” drawer. Therefore, the horizontal line to the right is on the same indifference curve. The same for lefts shoes – the vertical line is on the indifference curve. The indifference curves are L shaped – see the graph with four indifference curves depicted below. Note that the kink is where she consumes shoes in correct proportions – 5 right shoes to 3 left shoes. The utility increases up and to the right – if she gets more of right and left shoes at the same time, she is better off. For mathematically inclined: there is some problem with the indivisibility of shoes – you can’t walk around in 1/8 of the shoe. We are going to disregard it for the sake of simplicity (think that she is buying them in millions) and assume that shoes are perfectly divisible. How to find where the kink is from the formula: if utility is U(q ,q ) = min[3q ,5q ], then the kink is where 3q =5q . Common mistakes: if the consumer wants to buy multiples of (5 right shoes and 3 left shoes), that means that at the optimal point (kink) 3q =5q (not 5q =3q ) The budget line: Margot can afford 70/4=17.5 right shoes only or 70/5=14 left shoes only; the slope of the budget line is minus the price ratio: -p /p =-4/5=-0.8 b) Calculating the optimal bundle: As you can see from the picture, the optimal bundle is on the budget line, at the kink of the indifference curve, where 3q =5q . That gives us two conditions (equations) that we can solve together to find optimal consumption of right and left shoes: kink: 3q = 5q budget line: p q + p2q2 = I Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 2 of 10 Let’s substitute prices and income in the budget constraint: 4q + 5q2 = 70. From kink condition q2 = 0.6q ; plug it into the budget constraint: 4q + 5*0.6q = 70 => (4 + 3)q = 70 => q = 10; substitute back into kink condition q2 = 0.6q = 0.6*10 = 6 As we need to repeat this calculation 4 more times for different prices/income, I suggest we derive a general formula, but feel free to just repeat the above procedure. Calculating the optimal bundle – a general formula: kink: 3q = 5q budget line: p q + p2q2 = I From kink condition q2 = 0.6q ; substitute it into the budget constraint: Income = $70; p2 = $5 p1 4 11 0.5 q1, right shoes 10 5 20 q2, left shoes 6 3 12 What happens when price changes e) The price-consumption curve shows the consumption bundles chosen for different values of price. For each price of right shoe p we can find the optimal consumption bundle (keeping income and other price constant) – by finding the bundle on the highest indifference curve that she can afford (i.e. on the budget line) – the same way as in part a). By plotting corresponding prices and quantities of right shoes demanded we can draw the demand curve for right shoes. You can also draw the demand curve just by using the formula of demand for right shoes, but we are going to do it graphically instead. When the price of right shoes decreases, the budget line rotates around the point (q = 0, q =14) anticlockwise. Why At the point (q = 0, q = 14) Margot spends all her money on left shoes, and how many of them she can afford does not depend on the price of right shoes. Therefore, this bundle is a corner point on a budget line for any p , corresponding to the maximum number of left shoes she can buy. The maximum number of right shoes she can afford will depend on the price p . When p decreases, the maximum number of right shoes she can afford increases – the budget line rotates anticlockwise. The corresponding optimal point moves along the line q = 3q /5 up and to the right. The ( )1 1 2 1 1 2 1 1 2 1 1 2 0.6 => 0.6 => ; 0.6 0.6 Ip q p q I p p q I q q q p p + = + = = = + 1 1 2 1 1 2 1 1 ) 11 70 70 70 70= 5; 0.6 0.6*5 3 0.6 0.6*5 3 11 3 14 c p Iq q q p p p p = = = = = = = = = + + + + 1 1 2 1 1 ) 0.5 70 70 70= 20; 0.6 0.6*20 12 3 0.5 3 3.5 d p q q q p = = = = = = = + + Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 3 of 10 price-consumption curve connects the kink points – it is the straight line q = 3q /5. Extra: what happens if p = 0 All income is spent on left shoes: q = 14; to match that amount, she will get q = 70/3. Any more of right shoes is fine, but they will not bring her higher utility. You can draw the price consumptions curve as the horizontal line to the right from this point. f) Both goods are ordinary (not Giffen) – the higher the price the fewer shoes she buys – the demand curve for right shoes is downward sloping. Here we are talking about the change in quantity demanded of a good in response to its price change – i.e. the movement is along the demand curve; the demand curve itself does not shift! What happens when income changes p1 = $4; p2 = $5 1 2 1 2 1 1 2 ) 4, 5, 35 35= 5; 0.6 0.6*5 3 0.6 4 0.6*5 7 7 g p p I I I Iq q q p p = = = = = = = = = = + + 1 2 1 2 1 ) 4, 5, 105 105 15; 0.6 0.6*15 9 7 7 h p p I Iq q q = = = = = = = = = Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 4 of 10 income 70 35 105 q1, right shoes 10 5 15 q2, left shoes 6 3 9 i) Income-consumption curve shows the consumption bundles chosen for different values of income (keeping prices constant). When income increases from I to I to I to I , the budget line moves up and to the right: from B to B to B to B (bottom graph). The optimal bundle is always on the budget line where the kink of the corresponding (tangent) indifference curve is, i.e. on the line q = 3q /5. The straight line q = 3q /5 is the income-consumption curve. It shows the optimal bundles of right shoes and left shoes for different values of income. It is very easy to draw the corresponding Engel curve (how quantity demanded of right shoes depends on income) – you can also use formula if you want. The quantity demanded is proportional to income (if Margot’s income is doubled, she will buy twice as many shoes), so Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 5 of 10 Engel curve is a straight line. The first graph is for part (k). j) Both goods are normal – the higher the income the more shoes she buys, i.e. the Engel curve is upward sloping. k) If you keep prices constant and increase income, the quantity of good 1 demanded will increase, i.e. when income increases the demand curve shifts to the right. The graph is posted in part (i). For normal good, quantity demanded is increasing in income, Engel curve is upward sloping, and the demand curve shifts to the right when income increases. Problem 2. Melbourne fabulous dining (in-kind vs. cash). a) Assume we want Nicholas to dine out more often. What would be the best way to go about it Should we give him cash or (non-tradable) meal vouchers Here is Nicholas’ initial optimal bundle: He currently consumes 4 meals and 600 of OG, that must be the tangency point between his budget line and an indifference curve. After receiving 4 vouchers, his budget line will shift up by 4. This is a “truncated” budget line: he cannot consume less than 4 meals because he cannot cash in (sell) the vouchers. If we assume that both OG and meals are normal goods for Nicholas, then he will buy more of both goods. His new bundle is somewhere in the area shown on the graph. Therefore, given vouchers for 4 more meals, he will buy some more OG (potentially including alcohol and gambling) and consume some extra meals (but less than 4). If you think that giving him 4 meal vouchers would make him consume four more meals (8 Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 6 of 10 meals total), you are so wrong! He can substitute meals for OG. He can use 4 vouchers to buy meals, buy fewer meals with his own money, and spend money saved on more alcohol and gambling. Therefore, Sally is incorrect when she is saying that Nicholas will have 4 more meals and no extra spending on OG as a result of her policy. b) If he is given extra $160, his budget line will move outwards. When given extra cash, Nicholas can buy more meals, more OG, or more of both. If we assume that both OG and meals are normal goods for Nicholas, then he will buy more of both goods. His new bundle is somewhere in the area shown on the graph. Therefore, given extra $160 to buy 4 more meals, he will buy some more OG and some extra meals (but less than 4). c) Vouchers and cash have the same effect when both goods are normal because of substitution. Nicholas consumes 4 meals initially; so when he is given 4 meal vouchers, he can potentially “turn” them all into OG just by buying 4 less meals himself. Thus, there is no difference between Sally’s and David’s policies – in both cases Nicholas will consume more OG and more meals, but the number of extra meals will be less than four. (Both policies have the same effect when OG is an inferior good and meals are normal good – check for yourself. Both policies will lead to the same decrease in consumption of OG and the same increase in consumption of meals.) d) Things change if meals is an inferior good. Now as Nicholas’ income rises, he would like to consume less than 4 meals and more OG. He will be able to achieve this under David’s scheme but under Sally’s scheme he will end up at the ‘corner’ where he just uses his vouchers for meals and spends all his money on OG. Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 7 of 10 Thus, in this situation the policies have different effect and Sally’s policy does make sure that Nicholas spends less money on OG (potentially, on alcohol and gambling) compared to David’s policy. That said, both policies fail if the idea was to get Nicholas to dine out more often. Under Sally’s policy he still has just 4 meals but has extra OG. Under David’s scheme Nicholas has fewer than four meals and even more OG. Finally, note that Nicholas prefers David’s policy. So, if the aim is to make Nicholas better off in his own opinion then he has more utility under David’s policy because he is not as constrained. This should lead to a bit of debate about whether we think that Nicholas’ own preferences should be used to judge his welfare. Should government implement policies that help us to consume what we want (even if it is alcohol, tobacco or other drugs) or should they implement policies that make us consume what the government wants us to consume (even if we don’t want to) Particularly, should we tell poor people, whom we give money, what they should consume Is it a bit paternalistic Just because they are poor, doesn’t mean they can’t make their own choices. Conclusion: ‘vouchers’ and ‘cash’ have the same effect unless meals is an inferior good: When both meals and OG are normal goods, both policies lead to increase in consumption of meals and OG but the number of extra meals is smaller than the number of meal vouchers provided. When meals is a normal good and OG is an inferior good, both policies lead to increase in the number of meals by more than the number of vouchers and decrease in Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 8 of 10 consumption of OG (but it is highly unlikely bordering on improbable that OG is an inferior good). When meals is an inferior good, cash and vouchers lead to different outcomes – under voucher policy, Nicholas will consume 4 meals (=vouchers) and a lot of OG, but under cash policy he will consume even less than 4 meals and even more OG. Neither policy could lead to four or more extra meals (unless OG is an inferior good). Nicholas will prefer cash policy when meals is an inferior good for him, and indifferent between cash and vouchers otherwise. Amusing thing: if there are only two goods and your preferences satisfy “more is better”, why can’t both goods be inferior e) “In-kind” budget set is always a truncation of the “cash” budget set, as you are not allowed to sell “in-kind vouchers”. Therefore, any consumer will be either better off with cash than vouchers, or indifferent. The reason for giving vouchers might be paternalistic – we want you to consume what we want you to consume, not what you want to consume. Government might also be influenced by lobby, special interest group for a particular industry. For example, if government provides milk vouchers, that would increase the demand for milk, and the profits of milk industry. Christmas/b-day presents – might be a huge loss of welfare (as utility is lower compared to getting cash instead of unwanted presents), but maybe there is some extra pleasure/utility when your loved ones understand you/ get you what you want. Salary packaging – may have something to do with taxation, if you have to pay tax on cash, but not on fringe benefits (like using jet). If the tax is the same, then you have the same discussion as above. Problem 3. (Additional problem) Let’s agree to disagree. This is a fairly straightforward question on choice. The five people have the same choice set but differ in their preferences. So they make different choices. (a) The budget set is standard. The end points are 15 cups of tea and 12 cups of coffee. It is drawn below. Remember that the whole budget set includes the interior, while the budget line is the outer boundary of the set. The slope of the budget line is (minus) price ratio = -$2/$2.5 = -0.8. Qcoffee Qtea 12 15 Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 9 of 10 (b) Anushka views tea either as neutral (if she has free disposal) or as a bad. So she will just buy 12 cups of coffee. Her preferences (assuming free disposal of tea) and her optimal choice are drawn below. Anushka’s indifference curves are drawn a bit thicker (to identify them easier) and labelled. The labels have weird numbers as they are ordinal – and it is useful to remember that it is only the order of the indifference curves that matters. Note than for her MRS = 0. (c) and (d) Brian and Cathy both view tea and coffee as perfect substitutes, but in different ratios. Brian views them as equivalent, so his optimum is all tea. Cathy views 1 coffee as equal to 2 teas so she buys all coffee. They are drawn below. Note that the slope of Brian’s indifference curves is -1 (MRS = 1). The slope of Cathy’s is -0.5 (MRS = 0.5). Qcoffee Qtea 12 15 Optimal bundle Brian Qcoffee Qtea 12 15 Optimal bundle Anushka 10 8 2 1 -100 Econ 20002_2022_SM1. Tutorial 2. Preferences and Choices T2 sol. Page 10 of 10 (e) Dee has concave to the origin indifference curves. So the tangency will not be optimal. Rather she will choose either all tea or all coffee – it just depends on the shape of her indifference curves. Her indifference curves exhibit increasing MRS. (f) Euripides will consume on the 45 degree line to ‘balance’ tea and coffee. Assuming free disposal his preferences look like ‘perfect complements’ as he will only consume equal amounts of both goods. His consumption bundle is 6.666 teas and 6.666 coffees. MRS is 0 on the horizontal part of the indifference curve and infinite on vertical. Qcoffee Qtea 12 15 Optimal bundle Cathy
