I've been drawing my indifference curves to look something like this. That's the vertical axis, that's one good. So this is the quantity of Good A. This is the quantity of Good B. And I've been drawing the indifference curves like this. So it might look like that. That's one indifference curve. Then another indifference curve would look like that. And I could keep drawing indifference curves. And this is what an indifference curve would look like for two normal goods. Let me write that down. These are normal goods. And the reason why normal goods indifference curves would look like that are when I'm trying to figure out the combinations of two normal goods. Because if I have a lot of one good -- so at this point right over here I have a lot of Good A and I have very little of Good B -- I would be willing to trade off a lot of A to get one extra of B. But if all of a sudden I have a lot of B and less A, I would be willing to trade off very little A to get an incremental B. So that's why we have kind of this inward bow-shaped curve right over here. Or you could -- mathematically it looks like it's part of a hyperbola. And so that's what "normal goods" - indifference curves, if we were trading off between normal goods, would look like. Now let's think about the indifference curve - so it would be this kind of curved thing. The marginal rate of substitution would constantly be changing. Now let's think about different types of goods. Let's say that this is the quantity of $5 bills. And let's say this is the quantity of $10 bills. $10 bills. And we're talking about the actual -- the good now is actually the dollar bills. So let's say that this right over here is ten $5 bills. Well, that's 50 bucks. I'd be indifferent between that and five $10 bills. This is five right over here. And any point in between I would be indifferent because I'm always willing to trade off two $5 bills for one $10 bill. So it would look -- my indifference curve would be linear in this case. My indifference curve would be linear. So, no matter what, on this indifference curve I'm always willing. If I want to get one extra $10 bill I'm always willing to give up two $5 bills, which makes complete sense. Because two $5 bills are completely equivalent to one $10 bill. Now we could take it to another extreme: let's say I have an indifference -- let me draw the quantity of, I don't know, the quantity of M&M's. Let's say red M&M's. And I should have done that in red, but I won't. And let's say this is the quantity of blue M&M's. And let's say that I actually am indifferent between red and blue M&M's. Some people aren't. Red M&M's and blue M&M's. So having 10 red M&M's to me is completely equivalent to having 10 blue M&M's. I'm willing to trade them off one for one. I don't care. I get the same total utility. I get the same total utility. So in this case, assuming that I really don't care the color of my M&M, I am completely indifferent as I swap them out. And so this is a case of perfect substitutes. Perfect substitutes. Now, I'd always be happy to have more M&M's. So another indifference curve might look something like this. But it's always going to have a slope of negative one. I'm always giving up one red M&M to get one blue M&M, then I would be indifferent. And likewise over here you could have another indifference curve between $5 bills and $10 bills that looks like this. But the slope would be the exact same thing. Now, the last situation I want to think about is what we'll call perfect complements. So, goods that if you have one of them you really need the other one. Otherwise one of the two is somewhat useful. And maybe the most pure version of perfect complements - let me write it over here. So let's say this is the quantity of Right shoes and this is the quantity of Left shoes. So obviously, if we're just talking about just one pair, you have one of each. Now, do you really care if you get more left shoes? No. You have the exact same preference. It doesn't really change your life. You have the same total utility. In fact it might even be negative, because you have to store them all. But let's just assume you have the same total utility and you don't get any benefit of having those spare shoes, in case your shoe gets destroyed or anything like that. In terms of what you can get out of it, what you can wear, you get the same utility. And so you're really indifferent, no matter how many extra left shoes someone gives you. And you'd also be indifferent no matter how many extra right shoes someone gives you. Now, you would be happier if you had maybe two right shoes and two left shoes, because now you have two pairs. So this would be another indifference curve. And once again, if you have two right shoes you don't really care how many more than two left shoes you get. And if you have two left shoes you really don't care how many more than two right shoes you get. This indifference curve in green is clearly preferable to the one in white, but along each indifference curve it doesn't benefit you to have three left shoes and only two right shoes. So this is what perfect complements would look like. This is perfect substitutes. And this is normal goods.