WEBVTT 00:00:01.000 --> 00:00:08.902 In business, there is often a need to increase revenue, and so the problem comes up, how do we do this? 00:00:08.904 --> 00:00:18.501 Should we lower prices to increase demand or should we raise prices? We may sell fewer items, but 00:00:18.503 --> 00:00:24.502 we would get more dollars per item. What we have here is a problem, and what we need to do is to 00:00:24.504 --> 00:00:33.699 develop a model for it. This is topic 16, price elasticity of demand. Alright, let's take a look at what 00:00:33.701 --> 00:00:40.801 we have so far. We know that we're going to be discussing the relationship between price of an object 00:00:40.803 --> 00:00:56.901 and quantity sold. So on the axis that we normally think of as the x-axis we're going to plot price, 00:00:56.903 --> 00:01:05.299 and on the axis we normally think of as the y-axis, we're going to use, this is price p, 00:01:05.301 --> 00:01:15.901 and on this axis, we're going to use x, which is a function of p, and I use capital D to represent demand. 00:01:15.903 --> 00:01:23.901 So the first thing you have to get adjusted to here is that p is the variable that you usually think of as x, 00:01:23.903 --> 00:01:28.800 and x is what you usually think of as y. Alright, now why would we do that? 00:01:28.802 --> 00:01:33.201 Well because when you develop a model you want to think about what happens. 00:01:33.203 --> 00:01:46.101 So when the price is high the demand, or the quantity sold, might be low. If you lowered the price, 00:01:46.103 --> 00:01:54.001 so say the price was here, you might find that the demand or the quantity sold would be higher. 00:01:54.003 --> 00:02:02.603 Now what you can do over time is keep records that relate price and demand, and what you get usually 00:02:02.605 --> 00:02:12.402 is some line, whether it's linear or some piece of a higher level polynomial, you get a function that seems 00:02:12.404 --> 00:02:24.402 to be decreasing. Now what we want to do is determine whether we should raise or lower the price 00:02:24.404 --> 00:02:32.905 in order to increase revenue. So we begin to create a model. What we're interested in is the relative 00:02:32.907 --> 00:02:47.000 rate of change of the quantity sold, and I'm going to use D of p, just like f of x, in relation to the price. 00:02:47.002 --> 00:02:54.407 Now when you hear the words rate of change, you think of the derivative. Well what we want to know is 00:02:54.407 --> 00:03:03.801 what is the relationshop, or how fast is the sales, that's x the quantity sold, changing. So we would take 00:03:03.803 --> 00:03:12.402 the derivative of that, and then we want that as a ratio to the rate of change of price. Now the problem 00:03:12.404 --> 00:03:19.604 comes up here that the price is going to be a constant number. Let's say for example, a knit shirt. 00:03:19.606 --> 00:03:26.503 It might sell for 15 dollars. Well if you take the derivative of that you're going to get 0, and you're 00:03:26.505 --> 00:03:32.500 going to get nothing from this because division by 0 is undefined. So we have to be creative here. 00:03:32.502 --> 00:03:38.503 We know about natural logarithms. We know that the natural log of a number is the power you raise e to 00:03:38.505 --> 00:03:46.503 to get that number. So instead of setting up our relationship this way, we're going to take the natural log 00:03:46.505 --> 00:03:54.206 of the demand function, and then we're going to take the derivative and find the rate of change of the 00:03:54.208 --> 00:04:03.207 natural log of the demand function. Then we're going to find the ratio of that number to the natural log of 00:04:03.209 --> 00:04:15.104 the price. And then we're going to take the derivative of this number, and we'll see if maybe that will give 00:04:15.106 --> 00:04:21.601 us a model. Now let's just go over why this is going to work and how we're going to interpret this answer. 00:04:21.603 --> 00:04:32.604 We want to know whether the rate of change of the quantity sold is larger or faster than the rate of change 00:04:32.606 --> 00:04:42.101 of the price. So if this fraction that we eventually get, way down the line here, is greater than 1 that would 00:04:42.103 --> 00:04:50.804 mean that the quantity sold is changing faster than the price. In that case, we should lower the price 00:04:50.806 --> 00:04:58.104 to increase revenue. So if this number turns out to be greater than 1, we should lower the price of the 00:04:58.106 --> 00:05:06.403 object to increase revenue. If this number equals 1, we'd better stick with what we have because we are 00:05:06.405 --> 00:05:14.203 at a situation where this relative rate of change is equal, so we are at maximum revenue for that item. 00:05:14.205 --> 00:05:22.003 If the denominator is larger, you would have a number that would tell you that the rate of change of 00:05:22.005 --> 00:05:28.508 price is faster and the rate of change of demand is not keeping up, and in that case you should 00:05:28.510 --> 00:05:34.200 raise the price to increase revenue. Alright, let's develop. 00:05:34.202 --> 00:05:47.208 We have the natural log of the demand function, and we want to take the derivative of that function. 00:05:47.210 --> 00:05:52.307 So I'm going to write this function with prime, and that's a shortcut to saying take the derivative. 00:05:52.309 --> 00:06:03.708 Alright, the derivative would be 1 over D of p and then times D prime of p. Now this is a lot of 00:06:03.710 --> 00:06:12.307 different notation so let's look at it. This is 1 over u, du/dx. Then the denominator of that relationship 00:06:12.309 --> 00:06:25.908 was the natural log of the price. We're taking the derivative of this function, and we need to think of this 00:06:25.910 --> 00:06:37.508 as our variable, so this is like l-n-x, so the derivative is 1 over x, except we're using the letter p for price. 00:06:37.510 --> 00:06:47.407 So our relationship becomes this term, divided by this term, and it forms a ratio that we're going to analyze. 00:06:47.409 --> 00:07:00.808 Well, this is a complex fraction. It's kind of messy, so let's clean it up. We have D prime of p, over D of p, 00:07:00.810 --> 00:07:09.805 and then divided by 1 over p, which is the same thing as times p over 1. We multiply across. I'm going to 00:07:09.807 --> 00:07:23.507 bring this p out to the front because it's easier to handle. So I have p times D prime of p over D of p. 00:07:23.509 --> 00:07:32.410 Now this number in the form that I have here is going to actually be a negative number. That's because 00:07:32.412 --> 00:07:39.511 the demand function, the one I drew earlier, is a decreasing function. So when you take this derivative, 00:07:39.513 --> 00:07:46.507 this number is going to be negative. So what economists do, and they can just do this when they're 00:07:46.509 --> 00:07:53.305 developing a model. We prefer to work with positive numbers, so we take this expression, and you've 00:07:53.307 --> 00:07:58.708 probably always wanted to do this. You just put a negative sign in front of it. And what that does is 00:07:58.710 --> 00:08:12.309 that makes the final result positive and then our interpretation is easier. This relationship is called elasticity, 00:08:12.311 --> 00:08:23.808 and the letter capital E is used to represent elasticity. Formally it's called Price Elasticity of Demand. 00:08:23.810 --> 00:08:31.810 We can use this to measure the elasticity of the price of a product. Remember that we got here by using 00:08:31.812 --> 00:08:43.010 derivatives of natural logarithms. Let's take a look at the interpretation. This formula is exactly what 00:08:43.012 --> 00:08:52.108 we developed right here. If E is greater than 1, we say the price is elastic, and we should decrease the 00:08:52.110 --> 00:09:01.507 price to increase revenue because the decrease in price will be smaller than the increase in sales. 00:09:01.509 --> 00:09:10.910 So we actually end up with an increase in revenue. If the elasticity turns out to be less than 1, then we say 00:09:10.912 --> 00:09:17.608 the price is inelastic, and we should increase the price of the object if we want to increase revenue 00:09:17.610 --> 00:09:25.611 because the demand is not going to keep up with the price change. If elasticity turns out to be 1, 00:09:25.613 --> 00:09:33.809 that means the rate of change of the two quantities is, we call it unitary elasticity, and what we should 00:09:33.811 --> 00:09:39.704 do is simply maintain the same price. We are at maximum revenue, and there isn't much we are going 00:09:39.706 --> 00:09:46.010 to be able to do about that product. Alright, let's take a look at an example. 00:09:46.012 --> 00:09:56.007 The marketing department of Madison's has determined by analyzing sales data. This is similar 00:09:56.009 --> 00:10:01.709 to the graph I drew in the begining where you just have to establish a price, see how many you sell, 00:10:01.711 --> 00:10:08.111 establish another price, see how many you sell there, and then get a function to model this. Ok. 00:10:08.113 --> 00:10:14.708 They've come up with this model for their knit tops, something like this. It says x, the quantity sold, is 00:10:14.710 --> 00:10:26.806 the demand in terms of the price, and it would be 50 dollars minus 2 times p, the price. Backup, 50 objects, 00:10:26.810 --> 00:10:35.409 50 in this case, what are they? Knit tops. Because this is quantity. Ok, we have to know the current price 00:10:35.411 --> 00:10:43.501 in order to evaluate the price and see if it's elastic. The current price is 15 dollars. So what's the question? 00:10:43.503 --> 00:10:48.702 Well they want to increase revenue, and so they want to know if they should take this 15 dollar price 00:10:48.704 --> 00:10:58.502 and raise it or lower it. So let's test to see if the 15 dollar price is elastic or inelastic or unitary. We have 00:10:58.504 --> 00:11:08.400 this relationshop for elasticity. It's the opposite of the price, then the derivative of the demand function, 00:11:08.402 --> 00:11:15.600 divided by the demand function. This will all be evaluated at the 15 dollar price for the knit tops. 00:11:15.602 --> 00:11:29.601 So we have for this problem elasticity is negative 15 times the derivative of the demand function. 00:11:29.603 --> 00:11:45.601 Ok, from the example, the demand function was 50 minus 2p. That needs to be right here where we can see it. 00:11:45.603 --> 00:11:54.400 So we're going to take the derivative of this function, and we get negative 2. Now at this point we would 00:11:54.402 --> 00:12:04.501 normally evaluate this function at 15, but because it's a constant function, we still have negative 2. 00:12:04.503 --> 00:12:14.001 So the second substitution that goes here is negative 2. This is our work tablet. 00:12:14.003 --> 00:12:22.902 And then this would be divided by the demand function evaluated at 15. Here's the demand function, 00:12:22.904 --> 00:12:34.702 go back to this one, not this one. The demand function evaluated at 15 would be 50 minus 2 times 15, or 00:12:34.704 --> 00:12:46.401 50 minus 30 which is 20. So I put 20 here. So what do I have for elasticity? Negative 15 times negative 2 00:12:46.403 --> 00:13:05.901 is 30, over 20 which is 3/2. Let's interpret. 3/2 is greater than 1. So the price of the knit tops is elastic, 00:13:05.903 --> 00:13:20.301 which means that if we want to increase revenue from the sale of these tops we should decrease the price. 00:13:20.303 --> 00:13:36.800 Decrease price to increase revenue. Now it occurs to me that you might wonder. That's nice, how much 00:13:36.802 --> 00:13:44.901 should I decrease the price? I mean it's 15 dollars. Should I sell them for 3 dollars, or is there some way 00:13:44.903 --> 00:13:52.301 here to quickly figure out how far to go? Well this is not part of the first example, but let's see if we can 00:13:52.303 --> 00:14:02.100 figure it out. Let's see. If we're at maximum revenue, then that means elasticity is 1. So we're going to form 00:14:02.102 --> 00:14:10.901 the elasticity again, but this time instead of evaluating at a particular price we're just going to use p, and 00:14:10.903 --> 00:14:24.901 then we're going to solve for p. Watch. We will have negative p times the derivative of this function. 00:14:24.903 --> 00:14:38.401 So right here I put negative 2, over the demand function itself. So that's 50 minus 2p. Now this is a little 00:14:38.403 --> 00:14:50.602 different so let's go back. I used the elasticity formula, but instead of using a particular price, I used p. 00:14:50.604 --> 00:14:59.502 And now I'm going to set this equal to 1. This will tell me what price gives me unitary elasticity. 00:14:59.504 --> 00:15:12.700 Alright, this is 2p here. Multiply both sides by 50 minus 2p. I get 50 equals 4p, and from some very basic algebra 00:15:12.702 --> 00:15:24.301 I am able to get 12 dollars and 50 cents, and that is the price that I should sell these knit tops for to increase 00:15:24.303 --> 00:15:32.501 revenue. Sales will go up to accommodate, and this would increase and bring us maximum revenue. 00:15:32.503 --> 00:15:40.601 Alright, let's take a look at an example that uses something a little more complicated. Here's example 2. 00:15:40.603 --> 00:15:47.101 It starts out as the same type of problem. The marketing department of a newspaper, so you have to 00:15:47.103 --> 00:15:53.801 keep in mind now we're thinking about newspapers. Ok, so here's our newspaper. They've analyzed 00:15:53.803 --> 00:15:59.600 the sales data, and they've established the following demand relationship. Now this is a pretty 00:15:59.602 --> 00:16:06.000 standard demand relationship for newspapers, the quantity of newspapers in terms of the price. 00:16:06.002 --> 00:16:13.101 This number right here is usually the average distribution of the newspaper, and then it's times 00:16:13.103 --> 00:16:19.500 the square root of, and then this is in cents, 75 cents. It's going to be different for different papers 00:16:19.503 --> 00:16:26.100 in different cities and countries and then minus the price. The current price is 50 cents. 00:16:26.102 --> 00:16:32.598 We want to increase revenue. Should we raise the price of the paper, bringing in more money? 00:16:32.600 --> 00:16:40.701 But maybe sales would drop off. Or should we do the reverse? Well let's test the elasticity of this. 00:16:40.703 --> 00:16:50.502 Let's try to do this by remembering the formula. The elasticity is the opposite of the price times the 00:16:50.504 --> 00:17:04.601 derivative evaluated at the current price. D of p is 80,000, then we're going to write this as 75 minus p, 00:17:04.603 --> 00:17:19.200 raised to the 1/2 power. Ok, that's the function, the demand function. Let's take the derivative, 00:17:19.202 --> 00:17:33.401 and let's be really careful here. 1/2 times 80,000 is 40,000 times, this is the chain rule, 75 minus p, 00:17:33.403 --> 00:17:45.300 to the negative 1/2, times the derivative of the inner function, which is negative 1. Now I've seen 00:17:45.302 --> 00:17:50.500 a lot of people do this problem and forget the negative 1, and of course the answer comes out ridiculously, 00:17:50.502 --> 00:17:54.701 and you eventually realize you need to go back and check your work. But I thought it was good to 00:17:54.702 --> 00:18:00.600 point out here that you have to watch this. Ok, so what do we have for this derivative? Wow. 00:18:00.602 --> 00:18:12.901 We have negative 40,000, and then we have the square root of 75 minus p, and I'll put that in the 00:18:12.903 --> 00:18:24.199 denominator because it's raised to the negative 1/2. We want to now evaluate at the current price 00:18:24.201 --> 00:18:40.201 of 50 cents. Alright, so D prime of 50 is negative 40,000 divided by, 50 into here would be 00:18:40.203 --> 00:18:53.000 75 minus 50 which is 25. The square root of that is 5. Simplifying we get negative [8,000]. 00:18:53.002 --> 00:19:10.501 That substitution is our second quantity. 8,000. Let's see 5 into 40, yes, 8,000. And now we 00:19:10.503 --> 00:19:24.202 have to divide by this function, evaluated at 50 cents. So D at the current price of 50 cents would be 00:19:24.204 --> 00:19:38.801 80,000 newspapers times 75 minus 50, which is 25, to the 1/2, which is 5. That's the square root of 25. 00:19:38.803 --> 00:19:57.202 So it's 80,000 times 5. So that's 400,000. That is our denominator. Now we've done so much work here 00:19:57.204 --> 00:20:02.500 we probably need to stop and look at what we're doing. We are trying to figure out the price elasticity 00:20:02.502 --> 00:20:08.801 at a current price of 50 cents per newspaper to determine how we can increase revenue. 00:20:08.803 --> 00:20:20.401 So have you looked ahead to see what we have? Well, let's see. We have positive 4, 40, 00:20:20.403 --> 00:20:31.301 and multiplying here, divided by... So we have 400,000 divided by 400,000 which is 1. 00:20:31.303 --> 00:20:38.700 So what did we discover? Let's go back and look at our interpretation. We discovered that we have 00:20:38.702 --> 00:20:45.701 unitary elasticity, and if we either raise or lower the price of the newspaper, we would actually 00:20:45.703 --> 00:21:02.802 decrease revenue. So what we should do here is maintain the current price of 50 cents. 00:21:02.804 --> 00:21:09.101 If we need to increase revenue of the newspaper, we're going to have to think of something else 00:21:09.103 --> 00:21:16.802 more clever than adjusting the price. Now what we have been able to do here involves the derivative, 00:21:16.804 --> 00:21:24.499 but remember that it's even a little deeper than that. Where did this come from? This relationship that 00:21:24.501 --> 00:21:31.801 we modeled and developed came from using the derivative of natural log functions in order to avoid 00:21:31.803 --> 00:21:36.701 having a zero denominator when we take the derivative of the price in the denominator. This is just 00:21:36.703 --> 00:21:48.401 one of the many, many useful applications of natural logarithms. It makes analyzing elasticity quite simple.