No Sudden Reversal of Prefrences Continuous Utility Function

In this article, we will understand the first half in how to derive the demand curve: how to model and understand Preferences intuitively, graphically, and mathematically.

Topic: Economics

Utility Maximization

To understand the demand curve, we need to know how the demand curve is formed. Consumer decision making will be based on utility maximization, which is how to spend your money in a way that maximizes your happiness/satisfaction/utility. Utility is used to describe the satisfaction that the consumer gains from a good or service.

Utility Maximization is based on 2 things: Consumer Preferences and Budget Constraints (what they can afford).

Preferences

For preferences, it's about modelling people's tastes. For this article, we will talk about unconstrained choices like you won the lottery to focus on understanding the first factor which is preferences.

There are 3 assumptions for preferences:

1)Completeness: You have preferences over a set of goods you choose from. You can like two goods as much as each other, but you can't say I don't know or not have an answer.

2)Transitivity: Basically, when you prefer A to B and prefer B to C, you prefer A to C. It's simple logic. If you like apples more than oranges and like oranges more than bananas, you like apples more than bananas.

3) Non-Satiation: More goods result in more satisfaction/utility. Consumers will always be happier with more goods than less goods.

Indifference Curves

We graph preference through indifference curves, which are graphical maps of preferences. We call it indifference curve because if there are 2 combinations of goods as points on the curve, you are indifferent to both combinations of goods. Both combinations give the same utility/satisfaction.

Let's take an example to understand. As I said we will speak as if you have won the lottery. You have an unconstrained choice without the pressures of a budget to not go over. Let's say you can only spend it on cookies and pizza slices. I know you can spend on a lot of things, but we want to focus on two variables that later we can apply to multi-decision problems.

We have 3 choices

A:2 Pizza slices, 1 Cookie

B:1 Pizza slice, 2 Cookies

C:2 Pizza slices, 2 Cookies

I will assume you are indifferent with A and B meaning you are equally as happy with 2 pizzas and 1 cookie or 1 pizza and 2 cookies. This isn't from a property. I just assumed it.

I will assume that you prefer C the most because the more goods, the better from the property of non-satiation.

X-axis details the number of cookies.

Y-axis represents the number of pizzas.

Here are the indifference curves where A and B are on the same curve because you are indifferent to both A and B.

So, modeling the indifference curve can show the combination of goods that you are indifferent in.

4 Properties of Indifference Curves

1)Consumers prefer high indifference Curves because more goods are better. So, C is the most preferable option because it is higher on the graph.

2)Indifference curves are always downward sloping. We will understand by that is by pointing out the fault with an upwards indifference curve. Down below is an upward-sloping Indifference Curve. We can see that point(1,1) and Point (2,2) lie on the same curve, which would mean they are indifferent to each other. Obviously, that is wrong because it violates the property that more is better(Non-Satiation). We will understand more later in the article.

3)Indifference Curves never intersect. We will understand by that is by pointing out the fault with two indifference curves intersecting. Down below are two curves intersecting. According to the graph A and B are indifferent, and A and C are indifferent. So, all points are indifferent. However, B is preferable than C because it is higher (more goods). So, this violates transitivity since B and C are not indifferent.

4) There is one indifference curve through every consumption bundle(point). If you have two curves for one bundle (one point), it is creating a problem for consumers because they won't know how they feel. This is related to completeness.

There is another property that will be discussed later in the article because you need to understand another concept before understanding the property.

Let's think of a real-life example of using indifference curves to help make the right decisions. Let's say you got accepted to 3 job offers, and you care about two things: location and faculty. The 3 jobs are working in Princeton university, Santa Cruz university, and IMF, a research institution. Princeton had the best faculty then IMF then last Santa Cruz. Santa Cruz had the best location then IMF then Princeton. Let's model the choices on a graph and see which job you should take.

As you can see you are roughly indifferent between Harvard and Santa Cruz. Harvard has a good faculty but bad location. Santa Cruz has a wonderful location but worse faculty than Harvard. So, it balances out. IMF would be the best choice because it has the best of both factors. It has a better faculty than Santa Cruz but not as good as Harvard. It has a better location than Harvard but not as good of a location as Santa Cruz. It has a higher indifference curve too.

Utility Functions

Next let's learn how to mathematically know what choice is best to take.

We've modeled graphically utility functions. Now, we do it mathematically.

Let's take the same example before of pizzas and cookies.

Let's say that

Utility =

I'm not saying it's true for you, but it's just a way to represent Utility. We need to know that utility is an ordinal concept meaning it only answers which is a better option, but not how much better it is.

So, let's see the utility for A (2 pizzas and 1 cookie),B(1 pizza and 2 cookies), and C(2 cookies and 2 pizzas) from our previous example.

As we can see A and B are indifferent matching the graph we did before, and C is the one that gives the most happiness/utility. Therefore, C is the best choice to take that will maximize utility.

Margin Utility

It is the utility that a consumer gets from the next unit of a good or service. The important thing that margin unit is diminishing, meaning as you have more of a good, you have less satisfaction from the next unit. It's important to know that you'll get more happiness overall with more of a good, but everytime you'll get the next unit you get less utility/satisfaction.

Let's take an example. When you eat cookies, the first and second cookie will give you the most happiness, but as you eat the third, fourth and so on, you'll get less and less happiness from it.

When you are super thirsty, the first bottle you drink will give you more happiness than the second one.

Here is a graph representing the total utility in the y-axis and number of cookies in the x-axis while pizza constant at 2. As you can see with more goods, your total utility is increasing. In other words, you are getting happier. However, every time you eat the next unit, you get less happiness. In the first cookie, your utility is 1.4 since √2×1 =1.4 from the equation. If you have two cookies, it's 2 since

√2×2 =2 and so on. You get more happiness with 2 cookies than only 1, but you get less happiness from the second cookie than the first, so it's diminishing and that's the margin utility. It's important to know that as I said before the total utility number doesn't mean anything specifically. it's just to rank which is better or which gives more utility. You can measure margin utility in some cases.

This graph represents the margin utility in the y-axis and number of cookies in the x-axis. It gives us the utility you get from the first unit of cookie, the second, the third, and so on. It's diminishing.

Marginal Rate of Substitution (MRS)

It is the slope of the indifference curve between two specific points.

It is the rate at which you are willing to substitute one good for another.

So, as I said it's the slope and, in this case, p is pizzas and c are cookies.

In the graph, A, B and C are indifferent because they are in the same curve. So, you can get the rate at which you are willing to substitute one good for another. So, get MRS from point A to B. MRS=2-4/2-1 = -2. So, -2 pizzas/1 cookie,

meaning you are willing to substitute 2 pizzas for one cookie. Let's get from B to C. MRS=1-2/4-2=-1 pizza/2 cookies, meaning you are willing to substitute 1 pizza for 2 cookies. What's interesting is that MRS diminishes from left to right because of diminishing margin utility. For example, at point A, you want more cookies than pizzas at point A because you have 4 pizzas(less margin utility) and 1 cookie(high margin utility. So, you would trade 2 pizzas to get one cookie.

We can make a formula in how MRS related to margin utility

Another way to think of MRS is that tells you how relative margin utilities change as you move down the indifference curve. At point A, you have 4 pizzas but 1 cookie. So, pizza has a small margin utility while cookie has a large margin utility. So, as you have more of a thing, the less you want the next unit of it (Margin utility is low), which explains why we put cookies in the numerator while pizza in the denominator. Since if cookie has a larger margin utility than pizza, you would be willing to trade more pizzas for cookies. Example: at point A, since we have 4 pizza and 1 cookie. So, the numerator would be big and the denominator would be small. Therefore, you would trade more pizzas for cookies. At point B to C, the margin utility of pizza is increasing (since number of pizza decrease, but margin utility of cookies is decreasing (since number cookies increase). So, as you move down an indifference curve, you're more willing to give up the good on the x-axis to get the good in the y-axis and vice versa(as you move up an indifference curve, you're more willing to give up the good on the y-axis to get the good in the x-axis).

Last property of Indifference Curves

The last property you need to know is that indifference curves can't be concave (like curved inwards) to the origin because they would violate diminishing margin utilities. Let me explain what I mean.

In the concave graph, at point A, you have 4 pizzas (low margin utility) and 1 cookie(high margin utility). So, substituting 1 cookie for 1 pizza does not make sense in that case the margin utilities are not equal. so, you should be willing to substitute more pizzas for cookies.

How companies use this in real-life

Let's take an example of what we took and see how it impacts the prices of different sizes and goods. Let's take Starbucks.

A small coffee is at a certain price but a large coffee (double amount of the small) isn't double the prize of the small. It only increases by a small amount because your margin utility is diminishing. As you have more coffee available, you're willing to pay less for it. If you get a small coffee, you probably won't get another small, so they make a larger size so you would buy even though you most likely don't need the extra amount. So, the prices are the market's reaction to margin utility.

Citations:

https://ocw.mit.edu/courses/economics/14-01-principles-of-microeconomics-fall-2018/

emanuelhest2000.blogspot.com

Source: https://www.major-potential.com/post/preferences-and-utility-functions