Marginal utility and budget lines
Optimal point on budget line Using indifference curves to think about the point on the budget line that maximizes total utility
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.
- The price of chocolate is one dollar per bar.
- And at the price of fruit is two dollars 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've picked it either way.
- And that is the quantity of fruit.
- Not quantity of four. 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 lbs. per month.
- So this is 10. That's 10 pounds per month.
- So I have a budget line that looks like this. 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 want to write this explicitly in terms of my quantity of chocolate, since I put that on my vertical axis,
- since 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 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 one pound of fruit.
- 18. And you can verify that and it makes sense. It's going to be $18 plus 2 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 one pound of fruit per month. 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 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. I should draw it like this.
- So let's say I'm indifferent between any of these points - those points 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 one pound of fruit, or I could have - let's say that is four bars of chocolate
- and, let's see, and roughly eight 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 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.
- 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 our 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 our total utility at all these other points in between along our budget line.
- So to actually maximize out total utility, what we want to do is find a point on our budget line
- that is just tangent, that just touches at exactly one point one of our indifference curves.
- We can have an infinite number of indifference curves. There could be one that looks like that.
- There could be 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 touches exactly this budget line,
- or exactly touches the line at one point.
- So I might have an indifference curve that looks like this.
- Let's 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 in exactly one point - and also the slope of my indifference curve
- which we've learned is 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 here is the optimal allocation on our budget line.
- That right here is optimal. And how do we know it is optimal?
- Well, there's 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, all this shaded area -
- we should actually do it in another color.
- Because indifference curve we are indifferent: 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 is 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 are also not preferable to that point right over there, which acutally is on the indifference curve.
- Now 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. $2 to $1 per pound.
- So if the price went from 2 to 1 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 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. Would look something like that.
- So that is our new budget line. 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. Assuming 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. There's another indifference curve that looks like that.
- Try and make it a little bit neater, so it looks something like that.
- And so based on how the price -- assume we have access to these many, many indifference curves
- we can now see based on how a price, with 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, give or take, about 10 pounds of fruit.
- So all of a sudden, when we were -- let's think about just the fruit.
- Everything else we're holding equal. So just the fruit.
- When the price was two, the quantity demanded was eight pounds.
- And now when the price is one, the quantity demanded is 10 pounds.
- 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 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 assuming 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 two the quantity is eight.
- And when the price is -- actually let me do it a little bit different.
- When the price is two, and these aren't to scale, the quantity demanded is eight.
- And then -- actually I should do it here. And these aren't to scale.
- When the price is one the quantity demanded is 10. Two, eight, quantity demanded is 10.
- And so our demand curve -- these are two points on it -- 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.
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