office hours: 8:00AM – 8:50AM tuesdays LUMS C85 ECON 102 Tutorial: Week 4 Ayesha Ali www.lancaster.ac.uk/postgrad/alia10/econ102.html a.ali11@lancaster.ac.uk office hours: 8:00AM – 8:50AM tuesdays LUMS C85

A few things we need to know: Marginal Rate of Substitution definition: How many units of good 2 you’re willing to give up (substitute) for 1 more unit of good 1, while keeping your utility constant. equation: 𝑀𝑅𝑆 1, 2 = 𝑀𝑈 1 𝑀𝑈 2 Rational Spending Rule Utility is maximized where the marginal rate of substitution of two goods is equal to the price ratio of those two goods. Mathematically, we can write it as: : benefit of 1 additional unit of good 1 (expressed in units of good 2) cost of 1 additional unit of good 1 (in units of good 2) If MRS1,2 > P1/P2 then buy more of good 1 and less of good 2. If MRS1,2 < P1/P2 then buy more of good 2 and less of good 1.

Q1: Chapter 4 Problem 2 You are having lunch at an all-you-can-eat buffet. If you are rational, what should be your marginal utility from the last morsel of food you swallow?

Q1: Chapter 4 Problem 2 You are having lunch at an all-you-can-eat buffet. If you are rational, what should be your marginal utility from the last morsel of food you swallow? Since the marginal cost of an additional morsel of food is zero, a rational person will continue eating until the marginal benefit of the last morsel (its marginal utility) falls to zero.

Q1: Chapter 4 Problem 3 Martha’s current marginal utility from consuming orange juice is 75 utils per centilitre and her marginal utility from consuming coffee is 50 utils per centilitre. If orange juice costs 25 cents per centilitre and coffee costs 20 cents per centilitre, is Martha maximising her total utility from the two beverages? If so, explain how you know. If not, how should she rearrange her spending?

Q1: Chapter 4 Problem 3 Martha’s current marginal utility from consuming orange juice is 75 utils per centilitre and her marginal utility from consuming coffee is 50 utils per centilitre. If orange juice costs 25 cents per centilitre and coffee costs 20 cents per centilitre, is Martha maximising her total utility from the two beverages? If so, explain how you know. If not, how should she rearrange her spending? We can find Martha’s Marginal Rate of Substitution between orange juice and coffee: Martha’s MRSO,C : 𝑀𝑅𝑆 𝑂,𝐶 = 𝑀𝑈 𝑂 𝑀𝑈 𝐶 = 75 50 =1.5 We know the Rational Spending Rule states that utility is maximized if MRSO,C = PO/PC. We can find the price ratio: 𝑃 𝑂 𝑃 𝐶 = .25 .20 =1.25 Because 𝑀𝑅𝑆 𝑂,𝐶 ≠ 𝑃 𝑂 𝑃 𝐶 , Martha is not maximizing her utility. The Rational Spending Rule states that: If 𝑀𝑅𝑆 𝑂,𝐶 > 𝑃 𝑂 𝑃 𝐶 , then Martha should spend more on orange juice and less on coffee.

Q1: Chapter 4 Problem 8 Tom has a weekly allowance of €24, all of which he spends on pizza and movie rentals, whose prices are €6 per slice and €3 per rental, respectively. If slices of pizza and movie rentals are available only in whole-number amounts, list all the possible combinations of the two goods that Tom can purchase each week with his allowance.

Q1: Chapter 4 Problem 8 Tom has a weekly allowance of €24, all of which he spends on pizza and movie rentals, whose prices are €6 per slice and €3 per rental, respectively. If slices of pizza and movie rentals are available only in whole-number amounts, list all the possible combinations of the two goods that Tom can purchase each week with his allowance. no. of movies price: £3/movie slices of pizza price: £6/slice total cost £0 4 £24 1 £3 3 £18 £21 2 £6 £9 £12 5 £15 6 7 8 no. of movies price: £3/movie slices of pizza price: £6/slice total cost

Q1: Chapter 4 Problem 8 Tom has a weekly allowance of €24, all of which he spends on pizza and movie rentals, whose prices are €6 per slice and €3 per rental, respectively. If slices of pizza and movie rentals are available only in whole-number amounts, list all the possible combinations of the two goods that Tom can purchase each week with his allowance. no. of movies price: £3/movie slices of pizza price: £6/slice total cost £0 4 £24 1 £3 3 £18 £21 2 £6 £9 £12 5 £15 6 7 8

utils/week from movies Q1: Chapter 4 Problem 9 Refer to Problem 8. Tom’s total utility is the sum of the utility he derives from pizza and movie rentals. If these utilities vary with the amounts consumed as shown in the table below, and pizzas and movie rentals are again consumable only in whole-number amounts, how many pizzas and how many movie rentals should Tom consume each week? pizzas per week utils/week from pizza movies per week utils/week from movies 1 20 40 2 38 46 3 54 50 4 68 5 80 56 6 90 57 7 98 8 104

Combinations of pizza and rentals that cost €24 per week Q1: Chapter 4 Problem 9 Refer to Problem 8. Tom’s total utility is the sum of the utility he derives from pizza and movie rentals. If these utilities vary with the amounts consumed as shown in the table below, and pizzas and movie rentals are again consumable only in whole-number amounts, how many pizzas and how many movie rentals should Tom consume each week? Assuming that Tom spends his entire weekly allowance, £24, the affordable combinations and their corresponding utilities are as listed in the table below: From this table, we can see that Tom gets the most utility from 3 pizzas per week and 2 movie rentals. This is his optimal combination. Combinations of pizza and rentals that cost €24 per week Total Utility 0 pizzas, 8 rentals 0 + 57 = 57 1 pizza, 6 rentals 20 + 57 = 77 2 pizzas, 4 rentals 38 + 54 = 92 3 pizzas, 2 rentals 54 + 46 = 100 4 pizzas, 0 rentals 68 + 0 = 68

Question 2(a) Ann buys only orange juice and yogurt. In 2014, Ann earns £20,000, orange juice is priced at £2 a carton, and yogurt is priced at £4 a tub. Draw Ann’s budget constraint. (For each part of this question, put the quantity of orange juice on the y-axis, and the quantity of yogurt on the x-axis)

Question 2(a) Ann buys only orange juice and yogurt. In 2014, Ann earns £20,000, orange juice is priced at £2 a carton, and yogurt is priced at £4 a tub. Draw Ann’s budget constraint. (For each part of this question, put the quantity of orange juice on the y-axis, and the quantity of yogurt on the x-axis) The diagram above show’s Ann’s budget constraint. If Ann buys only orange juice, she can afford 10,000 cartons. If she buys nothing buy yogurt she can afford 5,000 tubs. Note that we can calculate the slope of this line by: -10,000/5,000 = -2. XOJ XY 10,000 5,000

Question 2(b) Suppose that all prices increase by 10 percent in 2015, and Ann’s salary remains constant. Draw Ann’s new budget constraint. We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub.

Question 2(b) Ann’s new budget line is depicted in red. Suppose that all prices increase by 10 percent in 2015, and Ann’s salary remains constant. Draw Ann’s new budget constraint. We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub. So then, in 2015: No change to earnings; Poj = £2.20 a carton; Py = £4.40 a tub. XOJ XY 9,090 4,545 Ann’s new budget line is depicted in red. Since both prices have increased by 60%, the relative price of yogurt to orange juice did not change. This means that the slope of the budget line is the same as in 2014. However, since both prices have increased, Ann is not able to afford the same bundles as before.   After the price increase, the price of orange juice is £2.2, and the price of yogurt is £4.4. So, if Ann buys nothing but orange juice, she can afford 20,000/2.2 = 9,090.90 cartons. However, since we can’t buy .9 of a carton of juice, Anne can afford only 9,090 cartons. If Ann buys only yogurt, she can afford 20,000/4.4 = 4,545.45 tubs. Again, because if we can’t buy .45 of a tub of yogurt, Anne can afford 4,545 tubs of yogurt. Notice that a proportional increase in all prices (leaving income unchanged) has the same effect as a decrease in income (leaving all prices unchanged).

Question 2(c) Now suppose that all prices increase by 10 percent in 2015 and that Ann’s salary increases by 10 percent as well. Draw Ann’s new budget constraint. How will Ann’s optimal combination of orange juice and yogurt in 2015 compare to her optimal combination in 2014? We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub.

Question 2(c) Now suppose that all prices increase by 10 percent in 2015 and that Ann’s salary increases by 10 percent as well. Draw Ann’s new budget constraint. How will Ann’s optimal combination of orange juice and yogurt in 2015 compare to her optimal combination in 2014? We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub. So then, in 2015: Ann earns £22,000; Poj = £2.2 a carton; Py = £4.4 a tub. Ann’s new budget constraint is exactly the same as in part a. When all prices increase by 10 percent, and her income increases by 10 percent, she is able to afford exactly the same bundles as in 2014. Since her budget constraint is unchanged (and nothing about her preferences has changed), her optimal combination of goods is exactly the same as in 2014. XOJ XY 10,000 5,000

Question 3 Tim’s marginal rate of substitution between goods 1 and 2 is given by: 𝑀𝑅𝑆 1,2 = 2𝑥 1 3𝑥 2 The price of good 1 is $3/unit, and the price of good 2 is $1/unit. How many units of each good will Tim purchase if he has $20 to spend?

Rational Spending Rule If consumption bundle (x1*, x2*) maximises the utility of an individual, given her budget set, then it satisfies the condition benefit of 1 additional unit of good 1 (expressed in units of good 2) cost of 1 additional unit of good 1 (in units of good 2) If MRS1,2 > P1/P2 then buy more of good 1 and less of good 2. If MRS1,2 < P1/P2 then buy more of good 2 and less of good 1.

Question 3 Tim’s marginal rate of substitution between goods 1 and 2 is given by: 𝑀𝑅𝑆 1,2 = 2𝑥 1 3𝑥 2 The price of good 1 is $3/unit, and the price of good 2 is $1/unit. How many units of each good will Tim purchase if he has $20 to spend?

Question 3 So, we set MRS1,2 = p1/p2, which gives: 𝑀𝑅𝑆 1,2 = 𝑝 1 𝑝 2 Tim’s marginal rate of substitution between goods 1 and 2 is given by: 𝑀𝑅𝑆 1,2 = 3𝑥 2 2𝑥 1 The price of good 1 is $3/unit, and the price of good 2 is $1/unit. How many units of each good will Tim purchase if he has $20 to spend? We know that Tim is rational and wants to maximize utility. To maximize utility, the rational spending rule tells us that MRS1,2 should equal the price ratio, p1/p2. So, we set MRS1,2 = p1/p2, which gives:   𝑀𝑅𝑆 1,2 = 𝑝 1 𝑝 2 2𝑥 1 3𝑥 2 = 3 1 We can cross multiply to get: 9 𝑥 2 =2 𝑥 1 This can be simplified to: 𝑥 1 ∗ = 9 2 𝑥 2 ∗   The optimal bundle must exhaust Tim’s income. So the budget constraint is: 𝑝 1 𝑥 1 ∗ + 𝑝 2 𝑥 2 ∗ =𝑚, where m is the $20 he has available to spend. Plugging in the prices, we can re-write this as: 3 𝑥 1 ∗ + 𝑥 2 ∗ =20 Now, we have a system of two linear equations, with two unknowns. We can substituting 𝑥 1 ∗ = 9 2 𝑥 2 ∗ from t he rational spending rule into the second equation, Tim’s budget constraint:   3 ( 9 2 𝑥 2 ∗ )+ 2𝑥 2 ∗ =20 Solving this equation gives x*2 = 40 29 . We can then plug x*2 = 40 29 in to either of our first two equations to find x*1 = 180 29 .

Question 3 Tim’s marginal rate of substitution between goods 1 and 2 is given by: 𝑀𝑅𝑆 1,2 = 3𝑥 2 2𝑥 1 The price of good 1 is $3/unit, and the price of good 2 is $1/unit. How many units of each good will Tim purchase if he has $20 to spend? Note: My version of the solutions had the numerator and denominator in this problem switched around. If you want the extra practice, try solving this problem and check your answer in the next slide, once you’ve completed it.

Question 3 So, we set MRS1,2 = p1/p2, which gives: 𝑀𝑅𝑆 1,2 = 𝑝 1 𝑝 2 Tim’s marginal rate of substitution between goods 1 and 2 is given by: 𝑀𝑅𝑆 1,2 = 3𝑥 2 2𝑥 1 The price of good 1 is $3/unit, and the price of good 2 is $1/unit. How many units of each good will Tim purchase if he has $20 to spend? We know that Tim is rational and wants to maximize utility. To maximize utility, the rational spending rule tells us that MRS1,2 should equal the price ratio, p1/p2. So, we set MRS1,2 = p1/p2, which gives:   𝑀𝑅𝑆 1,2 = 𝑝 1 𝑝 2 3𝑥 2 2𝑥 1 = 3 1 We can cross multiply to get: 3 𝑥 2 =6 𝑥 1 This can be simplified to: 𝑥 2 ∗ =2 𝑥 1 ∗   The optimal bundle must exhaust Tim’s income. So our budget constraint is: 𝑝 1 𝑥 1 ∗ + 𝑝 2 𝑥 2 ∗ =𝑚, where m is the $20 he has available to spend. Plugging in the prices, we can re-write this as: 3 𝑥 1 ∗ + 𝑥 2 ∗ =20 Now, we have a system of two linear equations, with two unknowns. We can substituting 𝑥 2 ∗ =2 𝑥 1 ∗ from t he rational spending rule into the second equation, Tim’s budget constraint:   3 𝑥 1 ∗ + 2𝑥 1 ∗ =20 Solving this equation gives x*1 = 4. We can then plug x*1 = 4 in to either of our first two equations to find x*2 = 8.

Question 4 Draw two indifference curves that intersect at a point and label that point, “A”. Provide an argument for why such an intersection would violate our assumptions on individual preferences (specifically, monotonicity and transitivity). First, what do we know about indifference curves? An indifference curve shows: a bundle of goods between which a consumer is indifferent, that is they have no preference for one bundle of goods on the indifference curve than for another. There is an indifference curve for each level of utility. Indifference curves move farther from the origin as utility increases. Indifference curves never cross. The slope of an indifference curve is the MRS between two goods. Next, let’s define a couple of assumptions on individual preferences: Monotonicity This is often summarized as “more is better.” Transitivity If A>B, and B>C, then it is true that A>C. Note: there are 6 assumptions that we make, for a full re-cap, see Chapter 4, pg 125-126.

Question 4 Draw two indifference curves that intersect at a point and label that point, “A”. Provide an argument for why such an intersection would violate our assumptions on individual preferences (specifically, monotonicity and transitivity).

Question 4 Draw two indifference curves that intersect at a point and label that point, “A”. Provide an argument for why such an intersection would violate our assumptions on individual preferences (specifically, monotonicity and transitivity). Bundle A lies on both I1 and on I2. Since A is on indifference curve I1 and the bundle C is also on I1 this means that the consumer is indifferent between bundle A and C. We can write this as: A~C. But now since A is on I2 and B is on I2 this means that the consumer is also indifferent between bundles A and B: A~B.   So, we have A~B and A~C. Transitivity of preferences would then mean that the consumer is indifferent between bundles B and C: B~C. But this is clearly in violation of monotonicity of preferences since bundle C has more X1 and more X2 than the bundle B. So, what we’ve done here is we’ve proved that indifference curves cannot cross each other if preferences are both monotonic and transitive.

Next Week Please check Moodle for next week’s worksheet and for maths questions Links to videos for some additional examples on the basics: indifference curves and MRS, budget line, optimal point on a budget line. For extra practice, Rietzke assigned some Optional Questions -- Chapter 4, problems: 4, 5, and 10.