EconS Utility. Eric Dunaway. Washington State University September 15, 2015

Transcription

1 EconS Utility Eric Dunaway Washington State University eric.dunaway@wsu.edu September 15, 2015 Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

2 Introduction Last time, we saw how people formed their own preferences, and the basic rules that govern them. Today, we are going to take those rules and add some math to them, transforming preferences into utility. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

3 Economic History Before we get into our discussion on utility, I want to spend a few minutes talking about how the idea of utility was developed. A little bit of economic history and philosophy! Before the late half of the 18th century, economics didn t exist formally. Merchants had rules of thumb for pricing, but nothing to tell them why they should price the way they do. Several economic practices that are commonplace today were actually considered immoral. For example, it was illegal for Christians to make interest bearing loans (usury). Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

4 Adam Smith Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

5 Adam Smith In 1776, Scottish philosopher and the rst modern economist, Adam Smith, published a book called An Inquiry into the Nature and Causes of the Wealth of Nations. In his book, he proposed that people are motivated by incentives, and act in their own self-interest. "It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages." Essentially, he proposed that merchants don t provide goods to the market because they like you. They do it because they want something in return (your money, to buy the things they need). Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

6 Adam Smith We get the concept of supply and demand from Adam Smith, as well as the idea of the invisible hand. Remember that the invisible hand is what nudges market into an equilibrium. When we have excess supply or excess demand, prices adjust until that excess is eliminated. Adam Smith s ideas formalized the discipline of economics, and almost everything we have today is thanks to his work. He wasn t alone, though. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

7 Jeremy Bentham Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

8 Jeremy Bentham The concept of utility comes from English philosopher Jeremy Bentham. Bentham s life work was to nd a universal moral code, and his solution to morality was utility. He proposed that all events could be assigned a numerical level of happiness, and that people should do whatever maximizes that number. Furthermore, Bentham proposed that society should do whatever maximizes the sum of everyone s happiness. Anyone see a problem with this? Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

9 Jeremy Bentham First of all, recall that when we talked about welfare, we mentioned that utility is an ordinal property. This means that the number is really meaningless, but we just like whichever number is higher. The reason that utility is ordinal is because there is no consistent scale across all individuals to put a number on happiness. If I told you that I received 5 units of happiness from watching a cat video on the internet, would you actually know what I meant? Of course not! Bentham s utilitarianism had the problem of being di cult to measure. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

10 Jeremy Bentham There s another problem with Bentham s utilitarianism. Utilitarianism doesn t account for morally reprehensible acts. Bentham actually proposed to pick all of the beggars o the streets, and put them into forced labor so that the rich would not have to look at them. He argued that the gains in utility that the rich would receive would more than o set the hardship of the beggars, since their lives were bad anyways. As a thought experiment, what if I were to o er everyone but one person in this room to live in a utopia. That one person, however, will be forced to live in the worst conditions imaginable (I ll leave the imagination up to you). Would that be worth it? Would it be moral? Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

11 John Stuart Mill Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

12 John Stuart Mill John Stuart Mill was a student of Jeremy Bentham s, and he was able to solve the moral reprehensibility problem. Mill proposed that not all pleasures in life should be weighted the same. Speci cally, intellectual and moral pleasures should be weighted higher than physical pleasures. Mill lobbied the English parliament to grant extra voting power to university graduates, as he believed that they understood society better and would make better decisions. It didn t work, but was he right? Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

13 Utility So why use utility? Utility allows us to match preference levels to a number, which is incredible useful. I can take the indi erence curves that we talked about last time, and actually call them utility levels, associated with their respective values. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

14 Utility Apples I 1 I 2 I 3 I 4 E D 4 A C F G B H 0 4 Oranges Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

15 Utility Apples U 1 U 2 U 3 U 4 E D 4 A C F G B H 0 4 Oranges Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

16 Utility With utility, rather than giving all of the rankings for all of our alternatives, I can now just say that whatever gives me the highest number of utility is my most preferred bundle. But what about ordinality? It will take a few lectures to get there, but we re going to see that the answers we obtain (demand functions) are not actually functions of the utility level. We just use the concept of utility to create a framework to nd our solution, which is great! As a small note: There are some kinds of demand functions that are functions of utility levels. They are called Hicksian demand functions, and we will not be studying them. We will be studying Marshallian demand in this class. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

17 Utility Now, all we need to do is gure out some equations to make utility curves. It s math time! There are more kinds of utility functions than I can count. We are going to stick to one kind of utility function in this class, called Cobb-Douglass, named after the two economists that developed it. The math is the most straightforward for this kind, and it s always well-behaved. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

18 Utility Function We can express utility as a function of the quantity of goods that are consumed. For example, consider two goods, which we ll call good x and good y. The utility level will be some function of x and y, i.e., Ū = U(x, y) Let s make this a bit more simple. Let U(x, y) = x 0.5 y 0.5. If I had 16 units of x and 9 units of y, my utility level would be Ū = (16) 0.5 (9) 0.5 = 4 3 = 12 Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

19 Utility Function From the utility function, we can also choose di erent values for Ū and plot the indi erence curves. For example, let Ū = 12 (you can pick any number). The utility function becomes 12 = x 0.5 y 0.5 Dividing both sides by x 0.5 gives and squaring both sides gives y 0.5 = 12 x 0.5 y = 144 x which is fairly straightforward to plot. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

20 Utility Function y U 3 U 2 U 1 0 x Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

21 Utility Function A quick note on utility functions. For the most part, people don t actually know what their utility functions look like in the real world. We can get pretty close to the true shape by asking enough questions about preferences, though. We call this revealed preferences. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

22 Marginal Utility While the absolute levels of the utility level are useful, we also want to know how fast utility is changing. This is essentially the slope of the utility function, but don t worry too much about that in this class. This will tell us how much more utility we gain by increasing our consumption of a good by 1 unit. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

23 Marginal Utility We can get the marginal utilities by applying the power rule on the utility function for both x and y. Remember, rst you multiply it by the exponent, then decrease the exponent by 1. Recall the utility function Ū = x 0.5 y 0.5 For the marginal utility with respect to x, we multiply by x s exponent, 0.5, and then decrease x s exponent by 1 to obtain MU x = 0.5x 0.5 y 0.5 For the marginal utility with respect to y, we multiply by y s exponent, (also 0.5), and then decrease the exponent by 1 to obtain MU y = 0.5x 0.5 y 0.5 Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

24 Marginal Utility MU x = 0.5x 0.5 y 0.5 Let s analyze this for a moment. Let x = 1 and y = 1. This makes MU x = 0.5. What this means is that when I have one unit of x and one unit of y, an increase of 1 unit of x will increase my utility by 0.5. Now, let x = 1 and y = 16. This makes MU x = 2. Now, when I have one unit of x and 16 units of y, increaing the amount of good x I have by just one will increase my utility by 2. This makes sense because I have a lot of y and not very much x, so getting more of x would make me a lot happier. The same thing works for MU y, but it gives the changes in utility when we increase y by one unit. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

25 Marginal Rate of Substitution We can use marginal utilities to nd the marginal rate of substitution. Since we know how much they are changing, we can just substitute them in MRS = Change in utility with respect to x Change in utility with respect to y = MU x MU y Note that the negative sign is in there, because in order to get more of good x and remain at the same utility level, we would have to give up some of good y. We are going to use this formula a lot, so it s very important that you practice it! Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

26 Marginal Rate of Substitution Let s nd the marginal rate of substitution for our example. MRS = MU x = 0.5x 0.5 y 0.5 MU y 0.5x 0.5 y 0.5 We can do some tricks with exponents to simplify this a bit (I am going to show how to do this in the next example) MRS = y x Thus, the more of x that we have, the smaller the MRS will be, and the less we would get for giving up an additional unit of y. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

27 Example As a note, from here on out, I am going to replace the good y with good z. This is just to make notation easier going forward (We will be using the letter Y to denote income. It gets confusing). Everything we ve done stays the same. Let s go through that whole process again, but with some di erent numbers. Remember the steps to calculate the marginal rate of substitution: 1. Calculate the marginal utility with respect to good x by applying the power rule to the utility function with respect to x. 2. Calculate the marginal utility with respect to good z by applying the power rule to the utility function with respect to z. 3. Apply the marginal rate of substitution formula. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

28 Example Consider the following utility function Ū = x 0.6 z 0.4 What is the marginal rate of substitution? Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

29 Example Ū = x 0.6 z Calculate the marginal utility with respect to good x by applying the power rule to the utility function with respect to x. Applying the power rule, I multiply the equation by x s exponent, 0.6, then subtract 1 from x s exponent to obtain MU x = 0.6x 0.4 z 0.4 Note that I completely leave z alone. It s not a ected by the power rule in this case. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

30 Example Ū = x 0.6 z Calculate the marginal utility with respect to good z by applying the power rule to the utility function with respect to z. Applying the power rule, I multiply the equation by z s exponent, 0.4, then subtract 1 from z s exponent to obtain MU z = 0.4x 0.6 z 0.6 Note that I completely leave x alone. It s not a ected by the power rule in this case. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

31 Example MU x = 0.6x 0.4 z 0.4 MU z = 0.4x 0.6 z 0.6 Apply the marginal rate of substitution formula. MRS = MU x MU z Substituting in my results from the previous two steps (don t forget the negative sign!), MRS = 0.6x 0.4 z x 0.6 z 0.6 Let s look at simplifying this a little bit. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

32 Example MRS = 0.6x 0.4 z x 0.6 z 0.6 First, we have That s the same thing as = 3 2 = 1.5 I can just replace that in the marginal rate of substitution formula MRS = 1.5 x 0.4 z 0.4 x 0.6 z 0.6 Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

33 Example MRS = 1.5 x 0.4 z 0.4 x 0.6 z 0.6 Now we have the x terms x 0.4 x 0.6 There is a mathematics rule that when you have two terms that have the same base (where the base is x in this case) and you divide them, you can just take the numerator s exponent and subtract the denominator s exponent. This turns our fraction into Numerator z } { x 0.4 x 0.6 {z} Denominator = x = x 1 = 1 x Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

34 Example MRS = 1.5 x 0.4 z 0.4 x 0.6 z 0.6 I can perform the same steps with regards to z, Numerator z} { z 0.4 z 0.6 {z } Denominator = z 0.4 ( 0.6) = z 1 = z Combining what I did to x and z, we get our simpli ed marginal rate of substitution, MRS = 1.5 z x Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

35 Example It s very important to practice simplifying the marginal rate of substitution. With our Cobb-Douglass utility, it s always going to simplify into something nice. You are going to need it simpli ed to work with it more when we get more tools in the next few lectures. You could bet money that I m going to ask you to do this on the exam. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

36 Summary Utility is derived from preferences, and is a critical tool for economists to gure out how people make their choices. While we can t gure out utility functions directly in the real world, we can get pretty close by observing the choices that people make. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

37 Preview for Friday We re talking about Budget Constraints, i.e., why we can t have nice things. Homework is due Friday at the end of the lecture. Quiz will also be returned at the end of the lecture. Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38

38 Assignment 5 (1 of 1) 1. Consider the following utility function where x and z are both goods. Ū = x 0.3 z 0.7 a. What are the marginal utilities with respect to x and z? b. What is the marginal rate of substitution? (Remember to combine the exponents!) Eric Dunaway (WSU) EconS Lecture 10 September 15, / 38