| [21.62s -> 34.42s] Hi, everybody. Hope everybody's having a good day today so far. And I hope everybody is ready to talk about concave and convex functions. Concave, convex functions show up. |
| [34.67s -> 47.28s] all over the place in economics, especially in microeconomics, and we're going to find out about them today. So let's just get started. We're going to be working, as we have been for some time, |
| [47.28s -> 58.61s] in a vector space. So we'll say V is a vector space. And we're going to be working in a domain that is a subset. |
| [58.70s -> 67.66s] of the vector space. And it's going to be a convex domain, a convex set. So x is a convex set. |
| [69.26s -> 83.47s] And we're going to be working with functions, and so let's just say we're working with a specific, some specific function defined on that convex domain, and it's... |
| [83.47s -> 97.20s] going to be a real valued function. So this tells us that it's real valued. So let's say f is real valued. |
| [104.91s -> 117.54s] Okay, so with this setup here, a convex subset of a vector space and a real valued function defined on that convex set. |
| [117.54s -> 130.29s] Here we have our definition of a convex function, a concave function actually to start with, and then a convex function. |
| [130.29s -> 141.58s] This tells us that the function is concave if whenever we take two vectors, two points in the domain, |
| [141.58s -> 153.57s] and we take a convex combination of those two vectors, we have a lambda scalar between 0 and 1, then it's got to be the case that the value of f |
| [153.57s -> 168.19s] at the convex combination on the left of this inequality is at least as large as, if you like, the convex combination of the values at the two points, which is what's on the right-hand side. |
| [168.19s -> 180.72s] of this inequality. And then you'll note that if the inequality goes the other way, if the value at the convex combination is always less than or equal, |
| [180.91s -> 193.15s] less than or equal to the convex combination of the values, then we say the function's convex. And let's also point out that |
| [193.15s -> 202.94s] We say the function is strictly concave or strictly convex if that's a strict inequality always. |
| [202.94s -> 212.61s] Let me just write that a little different way because I think that's useful in terms of thinking about this. So what that inequality says, if we use... |
| [212.61s -> 224.58s] x of lambda, so we've been using x of lambda as our sort of shorthand notation for the convex combination 1 minus lambda x. |
| [224.58s -> 238.18s] plus lambda y. So this inequality could be written f of x of lambda is greater than or equal to 1 minus lambda. |
| [238.18s -> 248.18s] times f of x plus lambda times f of y. So writing the left-hand side of the inequality this way, |
| [248.18s -> 262.45s] We just don't have as much going on in the argument here. And since we kind of know, it's kind of become a habit already, I think, that this represents the convex combination of the two vectors x and y. |
| [262.45s -> 267.15s] To me, this is an easier way to think about this. |
| [267.70s -> 282.19s] So we have our definition, and what we need to do now, as we usually need to do when we have a definition of a new concept, is to look at a few simple examples where the definition |
| [282.19s -> 294.64s] where the definition will actually tell us something. So let's take this off and we will do a couple of examples. |
| [295.38s -> 309.76s] Okay, let's do an example here. Let's take a specific function, and let's let that function be f of x equals 9 minus x squared. |
| [309.76s -> 324.05s] And so this is a real valued function, but it's actually a real function, meaning when we say a real function, we mean that the arguments are real numbers also. So in this case, capital X is some subset of R, and let's just let it be all of R. |
| [324.05s -> 336.98s] So this is a function on the convex domain, all of R. So let's draw a picture here of the graph of this function. |
| [339.06s -> 350.19s] Should look something like that. And let's draw the y-axis in here. Let's draw the horizontal axis here. |
| [354.58s -> 365.62s] Okay, so there's the graph of our function. And let's just note that the value is 9 here at x equals 0. |
| [365.62s -> 379.86s] It's minus 3 and 3 that we're going to have the value of the function being 0. So that's 3. This is minus 3. And let's put the other numbers in here as well. |
| [380.56s -> 395.38s] So minus 2, minus 1, 1, 2. So there's the graph of our function. And let's look at two points, x and y, in the domain. So let's let x. |
| [396.62s -> 407.79s] be 2. And let's let y be minus 1. So this is x. This is y. |
| [409.10s -> 416.18s] Let's let lambda be 1 third. |
| [418.03s -> 432.06s] So in particular, that means that x of lambda, the convex combination of x and y, is one-third of the way from x to y. And so that would be exactly 1. |
| [432.06s -> 443.02s] So two-thirds x plus one-third y is going to be 1. So this is x of lambda. |
| [443.92s -> 449.62s] and that's where lambda equals a third. Well we've already got that down here so we don't really need to do that again. |
| [453.01s -> 466.96s] And so now let's look at the value of f at these points. Well, lambda is third. Let's actually say that x of lambda here is 1. So what's f of x? |
| [468.14s -> 480.98s] at 2. We have a 9 minus 4, that's 5. So let's actually go up here. That's 5. |
| [481.58s -> 494.26s] So this is 5. What about y? f of y. y is minus 1. 9 minus the square of minus 1. 9, that's going to be 8. |
| [497.52s -> 504.02s] So that's 8. And let's look also at f of x of lambda. |
| [507.79s -> 522.45s] x of lambda is 1, so f there, if we've got 1 here, this is going to be 8 again, so let's go over here like this. This is 8. |
| [522.99s -> 533.47s] Okay, so we have the value we have the value of f at the convex combination That's 8 we have the value of f at both x and y |
| [533.47s -> 543.38s] That's 5. And I left out the value of y here. So let's squeeze that in. That is 8. |
| [543.82s -> 557.52s] There we go. Okay. And I didn't even write in f of x of lambda. I drew them up here and forgot to come back down here and put them down here numerically. So f of x of lambda is 8. |
| [559.41s -> 571.15s] And so the only thing left that we have to look at is what's on the right-hand side of this inequality. We want to look at 1 minus lambda. |
| [571.95s -> 584.37s] f of x plus lambda f of y. That's 1 minus lambda is 2 thirds, so that's 2 thirds of 5. |
| [584.72s -> 593.97s] plus 1 third of 8. That's 10 thirds and 8 thirds is 18 thirds, so that's 6. |
| [594.77s -> 606.66s] And we can see that, indeed, in this simple example, the convex combination of the values is 6. |
| [606.66s -> 615.38s] Value at the convex combination is 8, so the left-hand side here is 8, the right-hand side is 6. Our inequality is satisfied. |
| [615.38s -> 628.18s] And so that's consistent. Of course, that doesn't tell us the function's concave. It just says that for this particular x and y and this particular lambda, we have the inequality. |
| [628.18s -> 635.34s] doing the right thing. But of course, to be concave, we have to know the function. |
| [635.34s -> 647.70s] does this for every x and y in the domain and for every convex combination, for every lambda between 0 and 1. So of course, we're not going to be able to check every one of those. |
| [647.70s -> 661.62s] directly, arithmetically, but we could prove that this function's concave. Perhaps I'll give that as an exercise. So let's actually do a little more with the |
| [661.87s -> 669.20s] the diagram here, let's note that if I join up these two points |
| [669.68s -> 679.34s] And what are those two points? Those two points are 1 is x, f of x. |
| [680.37s -> 693.30s] That's this point here. x is the value in the domain, and f of x is the value in the target space here. And y... |
| [693.62s -> 704.30s] f of y, that's this point here, y in the domain and f of y in the target space. So what we're really doing |
| [704.50s -> 718.59s] Here is we're actually looking at the convex combination of these two points. It's just that it doesn't show up that way in the definition. But you can think of it that way. So what's happening here. |
| [718.59s -> 732.11s] when we go one-third of the way from here to here is we go right to there. So, we see that the |
| [732.11s -> 744.56s] convex combination of the two values, that would be the value 8 and the value 5. |
| [745.94s -> 755.58s] on the vertical axis, which is over here, but I've drawn it over here just to put it next to these numbers. |
| [755.58s -> 768.46s] The convex combination of these two values, where lambda is a third, is a third of the way from 5 up to 8. A third of the way from 5. |
| [768.46s -> 782.32s] if you like down here up to eight up here so that it's that convex combination that has to be less than or equal to |
| [783.76s -> 791.89s] the value of f at the convex combination. So another way of thinking about it, of course, and this is perhaps the way you thought about |
| [791.89s -> 805.04s] concave or convex functions before if you've seen them, is that whenever we draw a chord or a line segment joining points on the graph, that line segment has to lie below the graph. |
| [805.17s -> 816.27s] And of course, we can see that with this particular function in this particular graph, that's going to be the case, whatever two points and whatever lambda I choose, I could do this. |
| [816.27s -> 829.95s] I could do this. It is clearly, intuitively, it is always going to be the case that any line segment joining two points on the graph will lie beneath the graph. Moreover, notice that |
| [829.95s -> 844.27s] The line segment always lies strictly beneath the graph. In particular here, 6 is less than 8, not equal to 8. So in fact, again, this is not a proof. This is just... |
| [844.27s -> 850.26s] taking the intuition of the picture here, we can say that this function |
| [850.32s -> 860.98s] 9 minus x squared is actually strictly concave because we can see that it's always going to be the case that the |
| [860.98s -> 874.78s] value at a convex combination is going to be strictly less than the convex combination, sorry, the value at a convex combination, that's here, is going to be |
| [874.78s -> 885.94s] strictly bigger than the convex combination of the two values. So this actually is a strictly concave function. |
| [886.61s -> 897.52s] Let's note one more thing here, and that is that |
| [898.96s -> 906.10s] A function is concave exactly if the negative of the function is convex. |
| [906.10s -> 920.46s] Or you could say it vice versa. You could say a function's convex if it's the negative of a concave function. And the same for being strictly concave and strictly convex. So that tells me that if I take the negative of this, |
| [920.46s -> 934.70s] this function, well the graph in that case is of course going to look like this. And our axis will be here and the horizontal axis will be here. |
| [935.22s -> 950.16s] And so this, of course, would be the value 0, and this would be minus 9. So here what we have is let's call this g of x equals minus. |
| [950.99s -> 964.61s] f of x. So that would be x squared minus 9. Let's just even write that down here. x squared minus 9. And that would look like this. And it's the negative. |
| [964.61s -> 970.86s] of this strictly concave function, so it's strictly convex. |
| [972.24s -> 982.64s] Let's note one other thing here before we move on beyond this example, and that's that a linear function |
| [983.06s -> 996.91s] is both concave and convex. At first, that seems a little bit like, well, how can that be both concave and convex at the same time? But note that a linear function, let's put it a different way. Our function f is linear. |
| [996.91s -> 1011.06s] If this is not an inequality, but an equation. So a function is linear if whenever we have two points in the domain and we have a |
| [1011.79s -> 1024.91s] linear combination of those two vectors, it's got to be the case that what's on the left hand side of this inequality is equal to what's on the right hand side. And actually, that's got to be true for any lambda, not just. |
| [1024.91s -> 1038.72s] lambda is between 0 and 1. But in particular, it will be true for a convex combination. So if I have a linear function so that the left-hand side is always equal to the right-hand side, then obviously |
| [1038.72s -> 1051.76s] The inequality is satisfied, the weak inequality, so the function's concave, but the weak inequality of the other direction is satisfied too, so it's also convex. So indeed, any linear function... |
| [1051.76s -> 1061.98s] is both concave and convex and clearly is not either strictly concave or strictly convex because |
| [1061.98s -> 1071.54s] the inequality here is not going to be a strict inequality. For a linear function, it's always going to be equal. So this is a good starting point. |
| [1071.54s -> 1085.50s] Let's maybe do one more thing before we move on, and that is let's look at the situation if we have the domain being two-dimensional. So let's suppose... |
| [1085.50s -> 1096.27s] that our capital X is a subset now of R2. It could be all of R2. That would be fine. Just like here, it could be all of R. |
| [1096.27s -> 1106.93s] That would be convex. That would be fine. So now our graph isn't going to be like this because it's going to be in three space. The graph is going to be in three dimensions. |
| [1106.93s -> 1117.20s] two dimension, a plane for the domain, and a third dimension for the value of the function. So the graph is going to be some kind of surface in |
| [1117.20s -> 1131.12s] R3. And in particular, if we have a concave function, the graph is going to look like what we have down here. The graph is going to be kind of a |
| [1131.12s -> 1143.89s] It's like a hill like this, sort of the analog of this, but in three dimensions. And if we take the negative of that function, |
| [1143.89s -> 1156.14s] then its graph is going to be the other way. It's going to be like a bowl, sort of the analog in three dimensions of this two-dimensional diagram here. |
| [1156.14s -> 1163.36s] Both of these pictures, the concave diagram, the convex diagram, clearly |
| [1163.36s -> 1176.05s] the functions in those diagrams are actually strictly concave in the one case and strictly convex in the other case because whenever I have two points on the graph, |
| [1176.91s -> 1188.08s] Whenever I have two points in the domain, an x and a y, and then I join up the corresponding points on the graph at x, f of x. |
| [1188.53s -> 1199.71s] And at yf of y on the graph, and I draw the line segment between them, in the concave case, that... |
| [1199.71s -> 1211.47s] line segment will lie strictly below the graph for the graph we have here. And that would be a strictly concave function. And for the convex, |
| [1211.47s -> 1220.05s] function. The line segment will lie strictly above, so it would be a strictly convex function. |
| [1220.05s -> 1233.74s] This gives us a good idea of how concave and convex functions work, not just when we have a one-dimensional domain, but when we have a multi-dimensional domain as well. |
| [1233.74s -> 1246.02s] that in each case, we are talking about a function that is a real valued function, as we said at the outset. We have to be talking about a function f that maps from a |
| [1246.02s -> 1256.85s] convex domain, that's critical, and it has to be the case that the values are real numbers, a real valued function. Okay, so with all of that, |
| [1256.85s -> 1269.90s] Let's now take off some of the stuff we have here on the board and we'll look at another somewhat different example with a little more obvious direct important economic. |
| [1269.90s -> 1273.73s] content to it. Let's take this off. |
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