Optimal point on budget line

So let's just review what we've seen with budget lines. Let's say I'm making $20 a month. So my income is $20 per month. Let's say per month. The price of chocolate is $1 per bar. And the price of fruit is $2 per pound. And we've already done this before, but I'll just redraw a budget line. So this axis, let's say this is the quantity of chocolate. I could have picked it either way. And that is the quantity of fruit. If I spend all my money on chocolate, I could buy 20 bars of chocolate a month. So that is 20. This is 10 right over here. At these prices, if I spent all my money on fruit I could buy 10 pounds per month. So this is 10. So that's 10 pounds per month. That would be 20. And so I have a budget line that looks like this. And the equation of this budget line is going to be-- well, I could write it like this. My budget, 20, is going to be equal to the price of chocolate, which is 1, times the quantity of chocolate. So this is 1 times the quantity of chocolate, plus the price of fruit, which is 2 times the quantity of fruit. And if I want to write this explicitly in terms of my quantity of chocolate, since I put that on my vertical axis and that tends to be the more dependent axis, I can just subtract 2 times the quantity of fruit from both sides. And I can flip them. And I get my quantity of chocolate is equal to 20 minus 2 times my quantity of fruit. And I get this budget line right over there. We've also looked at the idea of an indifference curve. So for example, let's say I'm sitting at some point on my budget line where I have-- let's say I am consuming 18 bars of chocolate and 1 pound of fruit. 18-- and you can verify that make sense, it's going to be $18 plus $2, which is $20. So let's say I'm at this point on my budget line. 18 bars of chocolate, so this is in bars, and 1 pound of fruit per month. So that is 1. And this is in pounds. And this is chocolate, and this is fruit right over here. Well, we know we have this idea of an indifference curve. There's different combinations of chocolate and fruit to which we are indifferent, to which we would get the same exact total utility. And so we can plot all of those points. I'll do it in white. It could look something like this. I'll do it as a dotted line, it makes it a little bit easier. So let me draw it like this. So let's say I'm indifferent between any of these points, any of those points right over there. Let me draw it a little bit better. So between any of these points right over there. So for example, I could have 18 bars of chocolate and 1 pound of fruit, or I could have-- let's say that is 4 bars of chocolate and roughly 8 pounds of fruit. I'm indifferent. I get the same exact total utility. Now, am I maximizing my total utility at either of those points? Well, we've already seen that anything to the top right of our indifference curve of this white curve right over here-- let me label this. This is our indifference curve. Everything to the top right of our indifference curve is preferable. We're going to get more total utility. So let me color that in. So everything to the top right of our indifference curve is going to be preferable. So all of these other points on our budget line, even a few points below or budget line, where we would actually save money, are preferable. So either of these points are not going to maximize our total utility. We can maximize or total utility at all of these other points in between, along our budget line. So to actually maximize our total utility what we want to do is find a point on our budget line that is just tangent, that exactly touches at exactly one point one of our indifference curves. We could have an infinite number of indifference curves. There could be another indifference curve that looks like that. There could be another indifferent curve that looks like that. All that says is that we are indifferent between any points on this curve. And so there is an indifference curve that touches exactly this budget line, or exactly touches the line at one point. And so I might have an indifference curve that looks like this. Let me do this in a vibrant color, in magenta. So I could have an indifference curve that looks like this. And because it's tangent, it touches at exactly one point. And also the slope of my indifference curve, which we've learned was the marginal rate of substitution, is the exact same as the slope of our budget line right over there, which we learned earlier was the relative price. So this right about here is the optimal allocation on our budget line. That right here is optimal. And how do we know it is optimal? Well, there is no other point on the budget line that is to the top right. In fact, every other point on our budget line is to the bottom left of this indifference curve. So every other point on our budget line is not preferable. So remember, everything below an indifference curve-- so all of this shaded area. Let me actually do it in another color. Because indifference curve, we are different. But everything below an indifference curve, so all of this area in green, is not preferable. And every other point on the budget line is not preferable to that point right over there. Because that's the only point-- or I guess you could say, every other point on our budget line is not preferable to the points on the indifference curve. So they're also not preferable to that point right over there which actually is on the indifference curve. Now, let's think about what happens. Let's think about what happens if the price of fruit were to go down. So the price of fruit were to go from $2 to $1 per pound. So if the price of fruit went from $2 to $1, then our actual budget line will look different. Our new budget line. I'll do it in blue, would look like this. If we spent all our money on chocolate, we could buy 20 bars. If we spent all of our money on fruit at the new price, we could buy 20 pounds of fruit. So our new budget line would look something like that. So that is our new budget line. So now what would be the optimal allocation of our dollars or the best combination that we would buy? Well, we would do the exact same exercise. We would, assuming that we had data on all of these indifference curves, we would find the indifference curve that is exactly tangent to our new budget line. So let's say that this point right over here is exactly tangent to another indifference curve. So just like that. So there's another indifference curve that looks like that. Let me draw it a little bit neater. So it looks something like that. And so based on how the price-- if we assume we have access to these many, many, many, many, many indifference curves, we can now see based on, all else equal, how a change in the price of fruit changed the quantity of fruit we demanded. Because now our optimal spent is this point on our new budget line which looks like it's about, well, give or take, about 10 pounds of fruit. So all of a sudden, when we were-- so let's think about just the fruit. Everything else we're holding equal. So just the fruit, let's do, when the price was $2, the quantity demanded was 8 pounds. And now when the price is $1, the quantity demanded is 10 pounds. And so what we're actually doing, and once again, we're kind of looking at the exact same ideas from different directions. Before we looked at it in terms of marginal utility per dollar and we thought about how you maximize it. And we were able to change the prices and then figure out and derive a demand curve from that. Here we're just looking at it from a slightly different lens, but they really are all of the same ideas. But by-- assuming if we had access to a bunch of indifference curves, we can see how a change in price changes our budget line. And how that would change the optimal quantity we would want of a given product. So for example, we could keep doing this and we could plot our new demand curve. So I could do a demand curve now for fruit. At least I have two points on that demand curve. So if this is the price of fruit and this is the quantity demanded of fruit, when the price is $2, the quantity demanded is 8. And when the price is-- actually, let me do it a little bit different. When the price is $2-- these aren't to scale-- the quantity demanded is 8. Actually let me do it here-- is 8. And these aren't to scale. But when the price is $1, the quantity demanded is 10. So $2, 8, the quantity demanded is 10. And so our demand curve, these are two points on it. But we could keep changing it up assuming we had access to a bunch of indifference curves. We could keep changing it up and eventually plot our demand curve, that might look something like that.