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Deriving demand curve from tweaking marginal utility per dollar

Where does a buyer's demand curve come from? A rational buyer wants to get as much "bang per buck" from their consumption as possible. In economics, that's called marginal utility per dollar spent. When the price of a good decreases, the "bang per buck" on that good increases, which incentivizes consuming more of it. In this video, we derive the individual's demand curve for a good by tweaking the marginal utility per dollar spent. Created by Sal Khan.

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Video transcript

A few videos ago we saw that we could maximize total utility given our $5 spending by calculating the marginal utility per dollar for each incremental dollar we could spend on each of these goods and then just for each dollar maximizing it. Our 1st dollar, we got 100 utility points per dollar for that first chocolate bar and that was more than any than the first fruits, so that's where we spent it. Then we got more for the 2nd chocolate bar for that 1st pound of fruit, so we spend it there. Then for the 3rd pound or for the 3rd bar of chocolate. Then it became equal to spend for the 1st pound of fruit so then we spent the next $2 on that 1st pound of fruit because the price of fruit were $2. What I want to do in this video is explore what happens when I change the price of the chocolate bars. What happens to our marginal utility per dollar over here? In particular, what happens to the quantity demanded? If you think about what we're doing it, we figured out with 1 price what was the quantity demanded, we demanded 3 bars. If we change the price and we get another quantity demanded, we're essentially starting to plot our our demand curve and we can actually derive our demand curve from this information right over here. Let's see how we could do that. Let's now assume that our chocolate bars are $2. Now we're going to calculate. The marginal utility per dollar, this applies to both of these columns. This is for what it was $1 per bar, this is now when it's $2 per bar. Well, for that first bar, I'm still getting 100 points of marginal utility, but now it's $2. So 100 divided 2 is going to give me 50 marginal utility points per dollar. Then for that next bar, I get 80 marginal utility points. I'm still enjoying it but enjoying it a little bit less but I'm paying $2 for it. I'm getting 80 divided by 2 is 40 points, you know, and I'm just giving these arbitrary units, 40 points per dollar. Then the 3rd bar is 30 points per dollar. Then the 4th bar is 20 points per dollar. Now, how would I spend my $5? Let me do this a little bit, let me do it over here. How would I spend my $5 now? My first dollar, where would I get the most marginal utility per dollar? Where would I get the most bang for my buck? My very 1st dollar, I can either buy half a bar here, I could buy half a here and I'm assuming that, for the sake of simplicity let's assume that I get the same marginal utility per dollar for the 1st half a bar and for the 1st bar. That is constant until I get to one entire bar. That's also true for, and even if I buy a fraction of the pound here. My 1st dollar, I can't use these numbers, this is when the bars were a dollar per bar, now they're $2 per bar. This is the reality. Now actually it makes sense for me to, at least, for that 1st dollar I can buy a half pound of my fruit at a marginal utility per dollar of 60. My 1st dollar will go towards .5 pounds of fruit and I'm getting a marginal utility per dollar of 60. Where is my 2nd dollar going to go? Well, I can still get another half pound at a marginal utility of 60. Remember, we have to ignore these right here for the sake of this argument or for the sake of this scenario right now. I could still get another half pound for marginal utility per dollar of 60, so now I buy another half pound of fruit and my marginal utility per dollar is 60. Now, where is my 3rd dollar, my 3rd dollar going to be spent? Well, I could spend it now at a rate of a dollar per half bar, or $2 per bar for chocolate or a dollar per half bar, $2 per bar for fruit over here. I'm actually neutral. I could spend it. Let me just, for the sake of fun, say list on half a bar of chocolate and my marginal utility per dollar is 50. Then my 4th dollar, once again, I could do a couple of different things here. I could buy another half bar because I can buy up to a whole bar at this marginal utility per dollar up to a whole bar. So why not do that? I'll buy another half chocolate bar, so now I have a whole chocolate bar. Once again, I'm able to continue buying that at 50 utility units per dollar. Then my 5th dollar over here, what would I do with that? Well, I don't want to buy any more chocolate bars because my marginal utility per dollar of the chocolate bar because I've exhausted what I can buy at this utility, this utility per dollar. My marginal utility per dollar has gone down now, but now I could still buy fruit at that same 50. Now, with that dollar, since the fruit is $2 per pound, I can buy another half pound of fruit at a marginal utility per dollar rate of 50. Now I buy another half pound of fruit at a marginal utility per dollar of 50. You can calculate the total marginal utility I got, this is the marginal utility per dollar and this is a dollar spent at that marginal utility per dollar. My total utility I should say, the marginal utility is the increment, but my total utility now is 60 + 60 is 120 plus 50 + 50 + 50. So it's 120 + 150 = 270 total utility. Even more interesting here, let's think about the quantity of chocolate bars that I have now bought once the price is gone up. I have now bought exactly 1 chocolate bar. You could say my 3rd and 4th dollars were spent on 1 bar right over here, I bought 1 bar. Let's think about it, all else equal. Remember, ceteris paribus. We haven't changed the price of fruit, we haven't changed consumer preferences which would have changed your marginal utility numbers right over here. All else equal. What happened just when we changed the price of chocolate bars? Let me write it down. Just think about chocolate. If we just think about chocolate bars. Let me write price and quantity. When the price was $1 the quantity demanded was 3 bars. That was the 1st video we saw on marginal utility. We demanded 3 bars. Now when the price has gone up to $2, the quantity demanded is exactly 1 bar. We could do everything in between, we could see what happens if the price was a dollar 50 or if the price was 50 cents, if we actually lower the price. We would see how the, and there might actually be a situation where you would have to have higher quantities here especially when you lower the price. But by doing that, assuming you have enough rows here and we might not have it, if you lower the price. Assuming you have the marginal utility at different quantities for the two goods, you can figure out exactly how much chocolates someone would buy given different changes in price. We at least have 2 points for the demand curve now. If we assume that this is price and this is quantity right over here, when the price was $1, the quantity demanded was 3, and when the price is $2, the quantity demanded is 1. There, we have 2 points for our demand curve. Our demand curve might look something like that. If it was linear, it would go straight. It would go something straight like that. But we at least have 2 points on the curve and we could keep trying different prices out using these information to figure out the exact shape of that curve.