# Relationship between utility function and indifference curve In Section 3 we analyse the agent's indifference curves and ask how she makes In turn, a utility function tells us the utility associated with each good x ∈ X, and is .. curve defines an implicit relationship between x1 and x2, u(x1,x2(x1)) = k. In utility theory, the utility function of an agent is a that any set of three or more bundles forms a transitive relation. From a mathematical point of view, the indifference curve is an equation Now you have a one-dimensional function, and for it to be convex, its second derivative with From this you derive your relationship to the MRS. . Relationship between convexity and a perfect complements type utility function.

That is, if each point on I2 is strictly preferred to each point on I1, and each point on I3 is preferred to each point on I2, each point on I3 is preferred to each point on I1. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.

With 2convex preferences [ clarification needed ] imply that the indifference curves cannot be concave to the origin, i. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged. Assumptions of consumer preference theory[ edit ] Preferences are complete.

The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him. Assume that there are two consumption bundles A and B each containing two commodities x and y. A consumer can unambiguously determine that one and only one of the following is the case: A is preferred to B, formally written as A p B  B is preferred to A, formally written as B p A  A is indifferent to B, formally written as A I B  This axiom precludes the possibility that the consumer cannot decide,  It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.

This assumption makes indifference curves continuous. Preferences exhibit strong monotonicity If A has more of both x and y than B, then A is preferred to B. This assumption is commonly called the "more is better" assumption. An alternative version of this assumption requires that if A and B have the same quantity of one good, but A has more of the other, then A is preferred to B. It also implies that the commodities are good rather than bad.

## Indifference curves and marginal rate of substitution

Examples of bad commodities can be disease, pollution etc. Indifference curves exhibit diminishing marginal rates of substitution The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.

This assumption also set the stage for using techniques of constrained optimization because the shape of the curve assures that the first derivative is negative and the second is positive. Another name for this assumption is the substitution assumption. It is the most critical assumption of consumer theory: Consumers are willing to give up or trade-off some of one good to get more of another.

The fundamental assertion is that there is a maximum amount that "a consumer will give up, of one commodity, to get one unit of another good, in that amount which will leave the consumer indifferent between the new and old situations"  The negative slope of the indifference curves represents the willingness of the consumer to make a trade off. Assuming it does, a full demand schedule can be deduced as the price of one good fluctuates.

Consumer theory uses indifference curves and budget constraints to generate consumer demand curves. For a single consumer, this is a relatively simple process.

First, let one good be an example market e. Budget constraints give a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line illustrated. That is my indifference curve. Now let's think about it. Obviously, if I go all over here, 20 pounds of fruit and I don't know, that looks like about 2 bars of chocolate. To me, the same utility, based on my preferences, as where I started off with. If someone just swapped everything out, I would kind of just shrug my shoulders and say, "No big deal. I wouldn't be sad. What about points down here? What about a point like this? That is clearly not preferable, because, for example, that point I just showed, I can show a point on the indifference curve where I am better off.

For example, that point that I just did, that's 5 pounds of fruit and about 5 bars of chocolate. Assuming that the marginal benefit of more chocolate is positive, in the way I've drawn this, the assumption is that it is, then I'm obviously getting more benefit if I get even more chocolate per month. Anything down here, below the indifference curve, is not preferred.

Using the same exact logic, anything out here, well that would be good, because we're neutral between all of these points on the curve. This green point right over here, I have the same number of bars as a point on the curve, but I have a lot more pounds of fruit. It looks like I have 11 or 12 pounds of fruit.

Assuming that I'm getting marginal benefit from those incremental pounds of fruit, we will make that assumption, then this right over here, anything out here is going to be preferred. This whole area is going to be preferred to everything on the curve. The whole area down here is obviously we've not preferred to anything on the curve.

Just to show you that it's not those points. Let me do that in a different color, actually, because our curve is purple. Everything in blue, is not preferred. The last thing I want to think about in this video, is what the slope of this of this indifference curve tells us. When I talk about the slope, and this is really kind of an idea out of calculus. We're used to thinking about slopes of lines. If you give me a line like that, the slope is, how much does my vertical axis change for every change in my horizontal axis.

The typical algebra class, that axis is your Y axis, that is your X axis. When we think about slope we say, "Okay, when I have a certain change in Y" "when I change in X by 1. When I change, I get a certain change in Y.

The triangle means change in delta. Delta Y, the change in Y over change in X, is equal to the slope. This is when it's aligned and the slope isn't changing. At any point on this line, if I do the same ratio between the change in Y and the change in X, I'm going to get the same value.

On a curve like this, the slope is constantly changing. What we really do, to figure out the slope exactly to point, you can imagine it's really the slope of the tangent line at that point, a line that would just touch at that point. For example, let's say that I draw a tangent line. I'm going to draw my best attempt at drawing a tangent line. I'll do it in pink. Let's say I have a tangent line right from out starting predicament, just like that.

It looks something like that. Right where we are now, exactly at this point.

## Indifference curve

If we veer away it seems like our slope is changing. When in fact it definitely is changing. It's becoming less deep as we go forward to the right. It's becoming more deep as we go to the left. Right there, the slope of the tangent line looks like that, or you can view that as the instantaneous slope right there.

### Indifference curve - Wikipedia

We can measure the slope of the tangent line. We could say, "Look, if we want an extra …" This looks like about This is 5 and this is 2. What is your change in What is the slope here?

The slope here is going to be your change in bars. I should actually say this is a negative right over there. It's going to be your change in bars, your change in chocolate bars over your change in fruit. Over your change in fruit. In this situation, it is negative 5 bars for every 2 fruit that you get. So, bars per fruit. Or, you can say this is equal to negative 2. It's essentially saying, exactly at that point, how are you willing to trade off bars for fruit. Exactly at that point it's going to change as things change along this curve.