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1 office hours: 8:00AM โ€“ 8:50AM tuesdays LUMS C85
ECON 102 Tutorial: Week 4 Ayesha Ali office hours: 8:00AM โ€“ 8:50AM tuesdays LUMS C85

2 A few things we need to know:
Marginal Rate of Substitution definition: How many units of good 2 youโ€™re willing to give up (substitute) for 1 more unit of good 1, while keeping your utility constant. equation: ๐‘€๐‘…๐‘† 1, 2 = ๐‘€๐‘ˆ 1 ๐‘€๐‘ˆ 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.

3 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?

4 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.

5 Q1: Chapter 4 Problem 3 Marthaโ€™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?

6 Q1: Chapter 4 Problem 3 Marthaโ€™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 Marthaโ€™s Marginal Rate of Substitution between orange juice and coffee: Marthaโ€™s MRSO,C : ๐‘€๐‘…๐‘† ๐‘‚,๐ถ = ๐‘€๐‘ˆ ๐‘‚ ๐‘€๐‘ˆ ๐ถ = =1.5 We know the Rational Spending Rule states that utility is maximized if MRSO,C = PO/PC. We can find the price ratio: ๐‘ƒ ๐‘‚ ๐‘ƒ ๐ถ = =1.25 Because ๐‘€๐‘…๐‘† ๐‘‚,๐ถ โ‰  ๐‘ƒ ๐‘‚ ๐‘ƒ ๐ถ , Martha is not maximizing her utility. The Rational Spending Rule states that: If ๐‘€๐‘…๐‘† ๐‘‚,๐ถ > ๐‘ƒ ๐‘‚ ๐‘ƒ ๐ถ , then Martha should spend more on orange juice and less on coffee.

7 Q1: Chapter 4 Problem 8 Tom has a weekly allowance of โ‚ฌ24, all of which he spends on pizza and movie rentals, whose prices are โ‚ฌ6 per slice and โ‚ฌ3 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.

8 Q1: Chapter 4 Problem 8 Tom has a weekly allowance of โ‚ฌ24, all of which he spends on pizza and movie rentals, whose prices are โ‚ฌ6 per slice and โ‚ฌ3 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: ยฃ3/movie slices of pizza price: ยฃ6/slice total cost ยฃ0 4 ยฃ24 1 ยฃ3 3 ยฃ18 ยฃ21 2 ยฃ6 ยฃ9 ยฃ12 5 ยฃ15 6 7 8 no. of movies price: ยฃ3/movie slices of pizza price: ยฃ6/slice total cost

9 Q1: Chapter 4 Problem 8 Tom has a weekly allowance of โ‚ฌ24, all of which he spends on pizza and movie rentals, whose prices are โ‚ฌ6 per slice and โ‚ฌ3 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: ยฃ3/movie slices of pizza price: ยฃ6/slice total cost ยฃ0 4 ยฃ24 1 ยฃ3 3 ยฃ18 ยฃ21 2 ยฃ6 ยฃ9 ยฃ12 5 ยฃ15 6 7 8

10 utils/week from movies
Q1: Chapter 4 Problem 9 Refer to Problem 8. Tomโ€™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

11 Combinations of pizza and rentals that cost โ‚ฌ24 per week
Q1: Chapter 4 Problem 9 Refer to Problem 8. Tomโ€™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, ยฃ24, 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 โ‚ฌ24 per week Total Utility 0 pizzas, 8 rentals = 57 1 pizza, 6 rentals = 77 2 pizzas, 4 rentals = 92 3 pizzas, 2 rentals = 100 4 pizzas, 0 rentals = 68

12 Question 2(a) Ann buys only orange juice and yogurt.
In 2014, Ann earns ยฃ20,000, orange juice is priced at ยฃ2 a carton, and yogurt is priced at ยฃ4 a tub. Draw Annโ€™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)

13 Question 2(a) Ann buys only orange juice and yogurt.
In 2014, Ann earns ยฃ20,000, orange juice is priced at ยฃ2 a carton, and yogurt is priced at ยฃ4 a tub. Draw Annโ€™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 showโ€™s Annโ€™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

14 Question 2(b) Suppose that all prices increase by 10 percent in 2015, and Annโ€™s salary remains constant. Draw Annโ€™s new budget constraint. We know that in 2014: Ann earned ยฃ20,000; Poj = ยฃ2 a carton; Py = ยฃ4 a tub.

15 Question 2(b) Annโ€™s new budget line is depicted in red.
Suppose that all prices increase by 10 percent in 2015, and Annโ€™s salary remains constant. Draw Annโ€™s new budget constraint. We know that in 2014: Ann earned ยฃ20,000; Poj = ยฃ2 a carton; Py = ยฃ4 a tub. So then, in 2015: No change to earnings; Poj = ยฃ2.20 a carton; Py = ยฃ4.40 a tub. XOJ XY 9,090 4,545 Annโ€™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 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 ยฃ2.2, and the price of yogurt is ยฃ4.4. So, if Ann buys nothing but orange juice, she can afford 20,000/2.2 = 9, cartons. However, since we canโ€™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, tubs. Again, because if we canโ€™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).

16 Question 2(c) Now suppose that all prices increase by 10 percent in 2015 and that Annโ€™s salary increases by 10 percent as well. Draw Annโ€™s new budget constraint. How will Annโ€™s optimal combination of orange juice and yogurt in 2015 compare to her optimal combination in 2014? We know that in 2014: Ann earned ยฃ20,000; Poj = ยฃ2 a carton; Py = ยฃ4 a tub.

17 Question 2(c) Now suppose that all prices increase by 10 percent in 2015 and that Annโ€™s salary increases by 10 percent as well. Draw Annโ€™s new budget constraint. How will Annโ€™s optimal combination of orange juice and yogurt in 2015 compare to her optimal combination in 2014? We know that in 2014: Ann earned ยฃ20,000; Poj = ยฃ2 a carton; Py = ยฃ4 a tub. So then, in 2015: Ann earns ยฃ22,000; Poj = ยฃ2.2 a carton; Py = ยฃ4.4 a tub. Annโ€™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

18 Question 3 Timโ€™s marginal rate of substitution between goods 1 and 2 is given by: ๐‘€๐‘…๐‘† 1,2 = 2๐‘ฅ 1 3๐‘ฅ 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?

19 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.

20 Question 3 Timโ€™s marginal rate of substitution between goods 1 and 2 is given by: ๐‘€๐‘…๐‘† 1,2 = 2๐‘ฅ 1 3๐‘ฅ 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?

21 Question 3 So, we set MRS1,2 = p1/p2, which gives: ๐‘€๐‘…๐‘† 1,2 = ๐‘ 1 ๐‘ 2
Timโ€™s marginal rate of substitution between goods 1 and 2 is given by: ๐‘€๐‘…๐‘† 1,2 = 3๐‘ฅ 2 2๐‘ฅ 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: ย  ๐‘€๐‘…๐‘† 1,2 = ๐‘ 1 ๐‘ 2 2๐‘ฅ 1 3๐‘ฅ 2 = 3 1 We can cross multiply to get: ๐‘ฅ 2 =2 ๐‘ฅ 1 This can be simplified to: ๐‘ฅ 1 โˆ— = 9 2 ๐‘ฅ 2 โˆ— The optimal bundle must exhaust Timโ€™s income. So the budget constraint is: ๐‘ 1 ๐‘ฅ 1 โˆ— + ๐‘ 2 ๐‘ฅ 2 โˆ— =๐‘š, where m is the $20 he has available to spend. Plugging in the prices, we can re-write this as: ๐‘ฅ 1 โˆ— + ๐‘ฅ 2 โˆ— =20 Now, we have a system of two linear equations, with two unknowns. We can substituting ๐‘ฅ 1 โˆ— = 9 2 ๐‘ฅ 2 โˆ— from t he rational spending rule into the second equation, Timโ€™s budget constraint: ย  3 ( 9 2 ๐‘ฅ 2 โˆ— )+ 2๐‘ฅ 2 โˆ— =20 Solving this equation gives x*2 = We can then plug x*2 = in to either of our first two equations to find x*1 =

22 Question 3 Timโ€™s marginal rate of substitution between goods 1 and 2 is given by: ๐‘€๐‘…๐‘† 1,2 = 3๐‘ฅ 2 2๐‘ฅ 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 youโ€™ve completed it.

23 Question 3 So, we set MRS1,2 = p1/p2, which gives: ๐‘€๐‘…๐‘† 1,2 = ๐‘ 1 ๐‘ 2
Timโ€™s marginal rate of substitution between goods 1 and 2 is given by: ๐‘€๐‘…๐‘† 1,2 = 3๐‘ฅ 2 2๐‘ฅ 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: ย  ๐‘€๐‘…๐‘† 1,2 = ๐‘ 1 ๐‘ 2 3๐‘ฅ 2 2๐‘ฅ 1 = 3 1 We can cross multiply to get: ๐‘ฅ 2 =6 ๐‘ฅ 1 This can be simplified to: ๐‘ฅ 2 โˆ— =2 ๐‘ฅ 1 โˆ— The optimal bundle must exhaust Timโ€™s income. So our budget constraint is: ๐‘ 1 ๐‘ฅ 1 โˆ— + ๐‘ 2 ๐‘ฅ 2 โˆ— =๐‘š, where m is the $20 he has available to spend. Plugging in the prices, we can re-write this as: ๐‘ฅ 1 โˆ— + ๐‘ฅ 2 โˆ— =20 Now, we have a system of two linear equations, with two unknowns. We can substituting ๐‘ฅ 2 โˆ— =2 ๐‘ฅ 1 โˆ— from t he rational spending rule into the second equation, Timโ€™s budget constraint: ย  3 ๐‘ฅ 1 โˆ— + 2๐‘ฅ 1 โˆ— =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.

24 Question 4 Draw two indifference curves that intersect at a point and label that point, โ€œAโ€. 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, letโ€™s define a couple of assumptions on individual preferences: Monotonicity This is often summarized as โ€œmore is better.โ€ 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

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

26 Question 4 Draw two indifference curves that intersect at a point and label that point, โ€œAโ€. 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 weโ€™ve done here is weโ€™ve proved that indifference curves cannot cross each other if preferences are both monotonic and transitive.

27 Next Week Please check Moodle for next weekโ€™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.


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