Review of Previous Lectures

Transcription

1 Review of Previous Lectures 1 Main idea Main question Indifference curves How do consumers make choices? Focus on preferences Understand preferences Key concept: MRS Utility function The slope of the indifference curve Represent preferences

2 Budget Constraint 2 Example 1 Johanna has 10 to spend, the price of an apple is 1 and the price of a banana is 2. What are her options? Should she buy only apples? Should she spend all her money? How many apples does she need to give up to get a banana?

3 Budget Set 3 Definition: The budget set is the set of baskets that are affordable given the limits of the available income. The budget line is the set of baskets that can be purchased if the consumer uses his/her entire income. Price of x is P x and price of y is P y. The total expenditure on basket is (X,Y) is P X X + P Y Y The basket is affordable if total expenditure does not exceed total income I: P X X + P Y Y I The budget line solves:

4 Budget Constraint 4 The negative slope of the budget line equals the price ratio P x /P y. Y I/P Y It tells us how many units of y does the consumer need to trade to acquire an additional unit of x. x Y -P X /P Y I/P X X

5 Budget Constraint 5 Example 2: Ivan has I = 800, prices are P x = 20 and P y = 40. How many units of Y must the consumer give up for another unit of x?

6 Change in income 6 Y Decrease in income to I I /P Y I/P Y Increase in income to I > I I /P Y I /P X -P X /P Y I/P X I /P X X

7 Change in price 7 Y I/P Y A decrease in the price P Y : P Y < P Y I/P Y -P X /P Y -P X /P Y I/P X X

8 Consumer s Problem 8 The consumer s problem is to choose a basket that is affordable. Y I/P Y C Question: Which basket will the consumer choose? A B D I/P X X

9 Consumer s Problem 9 The consumer s problem is to choose a basket that is affordable. Y I/P Y C A D I/P X X

10 Consumer s Problem 10 The consumer s problem is to choose a basket that is affordable. Y I/P Y C A D I/P X X

11 Consumer s Problem 11 The consumer s problem is to choose a basket that is affordable. Y I/P Y C A D I/P X X

12 X Optimal Choice 12 Intuitively, the optimal bundle lies on the most north-eastward lying indifference curve on the budget line Y A B

13 Optimal Choice Finding the consumer s optimal bundle by comparing MRS to the slope of the budget line. Y Would the consumer prefer to get bundle B instead of A? I/P Y How much Y is the A x B How much Y does the consumer need to give up for an additional x? consumer willing to give up for an additional x? X I/P X

14 Optimal Choice 14 When changes are small we can compare MRS to the slope of the budget line. Y Would the consumer prefer to get bundle B instead of A? How much Y is the consumer willing to give up for an additional x? I/P Y A x B I/P X How much Y does the consumer need to give up for an additional x? X

15 Optimal Choice 15 Y x A At any point on the budget line we compare the MRS with the slope of the budget line. Rule 1: If they are not equal, the consumer can improve by either giving up some x for y, or vice versa. D x x C Question: Where is the optimal bundle? X

16 Optimal Choice 16 I/P Y Y A At any point on the budget line we compare the MRS with the slope of the budget line. Rule 2: If they are equal, and if the indifference curves are convex (MRS diminishes), the consumer can do no better. x B X

17 Optimal Choice 17 When indifference curves are convex (and do not intersect the axes), the consumer s optimal bundle is such that: the rate at which X and Y are exchanged in the marketplace is the same as the rate at which the consumer would be willing to exchange X for Y. MRSx,y = P x /P y

18 Optimal Choice Finding the consumer s optimal bundle by solving the utilitymaximization problem.

19 Optimal Choice 19 Technical derivation: Solution: MRSx,y = MU x /MU y = P x /P y

20 Example 20 U (x,y) = xy and I = 1 000, P x = 50 and P y = 200 Questions: Suppose the consumer gets basket A, containing 4 units of each good A= (x=4, y=4). Is she better off trading some y for more x? Vice versa? Is the basket optimal?

21 Example 21 Suppose the consumer listened, and traded 2 units of good Y, got 400 and bought 8 more units of good x. Now she has basket B=(x=12, y=2) Is she better off trading some y for more x? Vice versa? Is the basket optimal? What is the optimal bundle?

22 Example 2 22 Suppose that Ben says that I only enjoy a cup of coffee with a glass of water and I only enjoy a glass of water with a cup of coffee. Kim says three glasses of water is as good as one cup of coffee. Each has income I = 12 and prices are P c = 2 and P w = 1. Questions: (a) What kind of goods are water and coffee for Ben and Kim? (b) Suppose each is given aa bundle with 3 cups of coffee and 6 glasses of water.. Is Ben better off giving up some water for more coffee (or vice versa)? What about Kim? (c) What is the optimal bundle of Ben and of Kim?

23 23 Optimal Choice

24 Example 3 24 U (X,Y) = XY and I = 12, P x = 1 and P y = 2. Solve for the optimal basket. Now, suppose the price of y triples to 6. Solve for the optimal basket.

25 Optimal Choice: Corner Points 25 Corner Points: One good is not being consumed at all. Optimal basket lies on the axis. Y I/P Y C A D I/P X

26 Examples 26 I = 1000, P x = 50 and P y = 100 Find the optimal bundle for each consumer below: a. Johanna has a Cobb-Douglas utility function: U = x 0.25 y 0.75 b. For Frank the goods are perfect substitutes: U = 3x+4y c. Also for Isabel, but her utility function is: U = 2x+7y d. For Daniel the goods are perfect complements: U = min{x,y} e. Mary has the following utility function: U = xy+12x

27 Example a 27 Johanna has a Cobb-Douglas utility function: U = x 0.25 y 0.75

28 Example b 28 For Frank the goods are perfect substitutes: U=3x+4y He should consume more x. He is willing to trade ¾ of y for an additional x, but he only has to trade ½ of y for an additional x.

29 Example c 29 For Isabel the goods are also perfect substitutes: U=2x+7y She should consume more y. She is willing to trade 2/7 of y for an additional x, but she has to give up ½ of y for an additional x.

30 Example d 30 For Daniel the goods are perfect complements: U = min{x,y} Daniel always chooses a bundle where x=y. More of x does not help him if it doesn t come with more of y. 50x+100y=1000 and x=y implies that x=y=1000/150

31 Example e 31 Mary s utility function is U = xy+12x MRS = P x / P Y gives x = 2y+24 and budget constraint 1000 = 50(2y+24) +100y. Solution is y=-1, x=22. Is this the optimal choice? In this case, the optimal bundle is a corner solution: spend all on x gives U(20,0)=240. Check: spend all on y gives U(0,10)=0.

32 Takeaways Optimal consumption is determined by 3 factors: i. Preferences (or indifference curves). ii. Price ratio. iii. Income. 2. The MRS tells us what trades the consumer is willing to make. The price ratio tells us what trades the consumer can make. Comparing the two gives us direction of improvement. 3. How to find the optimal bundle.

33 Composite Good 33 Useful idea: Compare specific good to the expenditure on all other goods. Composite Good: A good that represents the collective expenditure on every other good except the commodity being considered.

34 Composite Good 34 Useful idea: Compare specific good to the expenditure on all other goods