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 dollars per month let's say per month my the price of chocolate is - is $1 per bar $1 per bar and the price of fruit is $2 per pound $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 quantity of fruit not quantity of four the quantity of fruit if I spent 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 of my money on fruit I could buy 10 pounds per month so this is ten so that's ten pounds per month that would be 20 and so I have a budget line that looks like this I have a budget line that looks like this and the equation of this budget line is going to be well I can 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 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 and I can flip them and I get my quantity if chocolate is equal to 20 minus 2 times my quantity of fruit and I get this budget line right over there well we've also looked at the idea of an indifference curve so for example let's say I'm at 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 you can verify that make sense it's gonna be $18.00 plus 2 which is 20 so let's say I'm at this point on my budget line 18 pounds so 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 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 she can be drive like this so let's say I'm indifferent between any of any of these points any of those points right over there let me draw 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 I could have let's say that is 4 bars of chocolate and let's see 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 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 though a few points below our budget line we would actually save money our preferable so either any of either of these points are not going to maximize our total utility we can maximize our 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 is that that is just that is exactly touches at exactly one point one of our indifference curves we can have an infinite number of indifference curves there could be an indifference curve that looks like that there could be in another indifference 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 is 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 and agenda so I can have an indifference curve that looks like this and because it's tangent it touches in exactly one point and also the slope of my indifference curve which we've learned was a marginal rate of substitution is the exact same as the slope as the slope of our budget line right over there which we learned earlier was the relative price so this right here is the optimal 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 the button every other point on our budget line is to the bottom left of this indifference curve so every other point on our on our budget line is not preferable so remember everything below everything below an indifference curve so all of the shaded area let me actually do it in another color because the indifference curve we are indifferent but everything below an indifference curve so all of this area in green also 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 can say every other point on our budget line is not preferable to the 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 or two go down so the price of fruit were to go from two dollars to one dollar to one dollar per pound so if the price of fruit went from two to one dollar then our actual budget line will look different our new budget line I'll do it in blue would look like this if we spend 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 my new budget line would look something like that would look something like that so that is our new budget line new new budget new budget line so now what would be the optimal allocation of our dollars or the best combination that we would buy we would do the exact same exercise we would assuming that we were we had we had data on all of these 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 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 dropped like a little bit neater so it looks something like that it looks something like that and so based on how the price if we assume we have access to these you know these many many many many indifference curves we can now see based on how a prating all else equal how would a change in the price of fruit changed the quantity of fruit we demanded because now our optimal spend is this point on our new budget line which looks like it's about which looks like it's about well give or take about ten 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 know 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 if assuming we had access to a bunch of indifference curves we can see how a change in price changes our budget line and that how that would change the optimal quantity we would want of a given 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 two quantity is 8 when the price is 2 the quantity demanded is 8 and when the price is actually let me do a little bit different when the price is 2's and these aren't to scale whining demand it is eight and then here is eight and these aren't too scalable and the price is one the quantity demanded is ten so two eight quantity demanded is ten quantity demanded is ten and so our demand curve these are two points on it but we could keep changing it up if soon we had access to a bunch of indifference curves we could keep changing it up and eventually plot our demand curve that looks something that might look something like that