EconS Substitution E ects

Transcription

1 EconS Substitution E ects Eric Dunaway Washington State University eric.dunaway@wsu.edu September 25, 2015 Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

2 Introduction Last time, we introduced the income e ect, which tells us how much of an equilibrium quantity s change can be attributed to changes in income. Today, we are looking at the substitution e ect, which tells us how much of an equilibrium quantity s change can be attributed to changes in relative prices. Announcement: Exam 1 will be a week from today, on October 2nd. Everything we have covered through next Monday s lecture (Lecture 15) will be on the exam. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

3 When prices change, we will often be dealing with both an income and substitution e ect. The substitution e ect is the change in our bundle when we hold utility constant. With di erent relative prices, people will change their consumption. This should make sense. Buyers don t want to buy more expensive goods! In this case, the income e ect is typically much harder to calculate, so we will primarily focus on calculating the substitution e ect, then taking the di erence of the total and substitution e ects as the income e ect. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

4 z A B x A x B Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

5 Step 1: Calculate the optimal bundles with the old prices and the new prices. The di erence between those two bundles will be the total e ect. Note that the sign of the total e ect is important. Take the new bundle minus the old bundle. The old bundle can be represented as point A on the previous gure, while the new bundle is point B. Thus, the total e ect for good x would be Total E ect = x B x A Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

6 Step 2: We need to gure out what the consumer would choose under the new prices if they had enough money to reach their old indi erence curve. This could entail either giving them money or taking money away. In the case of our example, we need to decrease the income level of the consumer until the "pseudo-budget" line is tangent to the old indi erence curve. It will be parallel to the new budget line. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

7 z A B x A x B Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

8 z A C B x A x C x B Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

9 What we can see is that under the new prices and old utility, the consumer would actually choose bundle C rather than bundle A. In this case, bundle A isn t even obtainable under our psuedo-budget line. The point is that with bundle C, we can achieve the same utility level that we did with bundle A under the old prices. This captures the substitution e ect. When having enough money taken away that they can only reach their old utility, they will still choose a new bundle. To calculate the substitution e ect, we just take the di erence between the intermediate bundle (C), and the original bundle (A). Substitution E ect = x C x A Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

10 How do we calculate the pseudo budget line? It s a little bit complicated. I m going to show you how to do it at the end of the lecture. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

11 Step 3: Now that we have both the total and substitution e ects, we know that the income e ect is what s left over. Income E ect = Total E ect Substitution E ect Income E ect = (x B x A ) (x C x A ) = x B x C which should make sense since the distance between the nal point (B) and the intermediate point (C) would be the distance that was left over. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

12 z A B C x A x C x B x Total Effect Substitution Effect Income Effect Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

13 A few notes: The substitution e ect s sign will always be known. If we are dealing with a price decrease, it will be positive. If we are dealing with a price increase, it will be negative. Why? We always substitute away from whichever good gets more expensive. The income e ect s sign will vary. If we have a normal good, it will be the same as the substitution e ect. If we have an inferior good, it will be the opposite. Let s look at an inferior good. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

14 z A x A x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

15 z B A x A x B Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

16 z B A C x A x B x C Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

17 Notice that this time, x C is on the other side of x B than where it was when we had a normal good. This implies that the income e ect, x B x C will be negative. Intuitively, if we just looked at the gure without the price change, with just the pseudo budget line being the starting point, and the new budget line being the end point; the wealth expansion path would indicate that we have an inferior good. Since the price reduction of good x makes the consumer better o, rst they substitute away from good z to good x. Then, because they e ectively have more wealth, the income e ect means that they will buy less of good x since it is inferior. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

18 z B A C x A x B x C x Total Effect Substitution Effect Income Effect Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

19 Note that the total e ect still goes the same way as the substitution e ect. This is because the substitution e ect is larger than the income e ect. Again, it is theoretically possible that the income e ect could be larger than the substitution e ect. This would make the total e ect go the opposite direction from the substitution e ect. This would also be a Gi en good. For an example, see page 125 (Solved problem 5.5) in Perlo. Without the presence of a Gi en good, however, the total e ect will have the same sign as the substitution e ect. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

20 Let s look at a price increase with a normal good. All of the steps remain the same, just this time, we can expect the total, income and substitution e ects to all be negative. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

21 z A x A x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

22 z B A x B x A Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

23 z B A x B x A Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

24 z B C A x B x C x A Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

25 z B C A x B x C x A x Total Effect Substitution Effect Income Effect Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

26 Basically, the price increase behaved the same way as the price decrease, just in the opposite direction. An few important things to remember: The substitution e ect always comes rst. We move along the original utility curve to get from our initial bundle to our intermediate bundle. The income e ect comes second. We jump from our original utility curve to our new one. Both of these e ects come together to form the total e ect. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

27 Pseduo Budget Line Let s calculate the pseudo budget line. We re going to need two new tools to gure this out, The indirect utility function The expenditure function Don t worry! They re easy to get! Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

28 Pseduo Budget Line z A C B x A x C x B Total Effect x Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

29 Pseduo Budget Line The rst thing we need to do is gure out the indirect utility function. We do this by substituting the demand functions back into the utility function. Recall the example we worked on a few lectures ago. Ū = x 0.75 z 0.25 x = 3Y 4p x z = Y 4p z We can substitute x and z into the Ū function to obtain the indirect utility function 3Y 0.75 Y 0.25 Ū = = 4p x 4p z Y 4 (p x ) 0.75 (p z ) 0.25 Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

30 Pseduo Budget Line What the indirect utility function does is tell us what utility level we will achieve as a function of our parameters: the prices and income. It gets rid of all of the x s and z s that we calculate from within the model. It s a one stop method for calculating a utility level. Plugging in the values from the example, p x = 3, p z = 4 and Y = 64, we are left with Ū = (64) 4 (3) 0.75 (4) 0.25 = 8p Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

31 Pseduo Budget Line Another useful thing we can do is derive the expenditure function. This tells us the minimum amount of income needed to reach a given level of utility, given prices. The good news is that all we need to do to get the expenditure function is to solve the indirect utility function for Y Ū = Y 4 (p x ) 0.75 (p z ) 0.25 Y = 4 (p x ) 0.75 (p z ) 0.25 Ū Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

32 Pseduo Budget Line x = 3Y 4p x z = Y 4p z Remember our original bundle happened where p x = 3, p x = 4 and Y = 64. This leads to bundles Old Income z} { xa = 3( 64 ) 4( 3 ) = 16 z B = 64 4(4) = 4 {z} Old Price Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

33 Pseduo Budget Line x = 3Y 4p x z = Y 4p z Let s say that the price of good x increases to p x = 6 Our new equilibrium bundles are New Income z} { xb = 3( 64 ) 4( 6 ) = 8 z B = 64 4(4) = 4 {z} New Price Thus, our total e ect is xb xa = 8 16 = 8. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

34 Pseduo Budget Line To gure out our intermediate bundle, we need to gure out how much wealth the consumer needs to reach their original utility curve with the new (part B) prices. To gure out how much wealth we would need for the pseudo budget line, we can just plug these new prices (p x = 6, p z = 4), and the old utility level Ū = 8 p 2 to obtain Y C = 4 (p x ) 0.75 (p z ) 0.25 p Ū = 4(6)0.75 (4) 0.25 (8 2) = Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

35 Pseduo Budget Line Now, since the pseudo budget line is parallel to the new budget line, we can just use the new prices and our calculated level of wealth. Thus, the equation for our pseudo budget line is 6x + 4z = and, using our demand functions, we can solve for the intermediate points Pseudo Income xc = 3( z } { ) 4( 6 ) {z} New Price This leads to the substitution e ect of and an income e ect of = z C = (4) x C x A = 2.54 x B x C = 5.46 = 6.73 Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

36 Pseduo Budget Line What about good z? Recall the values we calculated. What is going on here? z A = 4 z B = 4 z C = 6.73 The substitution e ect, zc za = 2.73 is being countered by an opposite income e ect of zb zc = We should expect the substitution e ect to be positive, since x got more expensive (making people buy more z) We should expect the income e ect to be negative, since people can t a ord as much as before. They actually cancel each other out and our total e ect is zb za = 0. This is a neat feature of Cobb-Douglass utility. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

37 Pseduo Budget Line Pseudo budget lines are complicated. This is because we aren t using calculus. Calculus actually makes this process a lot easier. There s actually a really useful intuition to be gained from the value of Y C Y. This is the amount of money that you would have to give someone (or take away) for them to be indi erent after a price change. We call this the compensating variation, and it is both a measure of welfare, and an important facet of cost-of-living adjustments. Compensating variation has a partner, the equivalent variation, which is the amount of money that you would have to give someone (or take away) for them to be indi erent before a price change. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

38 Summary Income e ects measure how the equilibrium changes when prices are held constant, and people are made better or worse o. Substitution e ects measure how the equilibrium changes when prices change, but people are kept the at the same level of well being. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

39 Preview for Monday Consumer Choice Applications Cost of Living Adjustments We re looking at a really neat Human Resources problem. Real world stu! Fun with Labor Supply Ever wondered about supply side economics? Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40

40 Assignment 3-3 (1 of 1) See handout posted on the website. Eric Dunaway (WSU) EconS Lecture 14 September 25, / 40