Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2)

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1 Week 3: Sets and Mappings/Calculus and Optimization (Jehle and Reny, Chapter A1/A2) Tsun-Feng Chiang *School of Economics, Henan University, Kaifeng, China September 20, 2015 Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) / 28

2 A Concave Functions For the remainder of this section, we will restrict our attention to real-valued functions whose domains are convex sets. We apply the following assumption to this section: Assumption A1.1 Real-Valued Functions Over Convex Sets Throughout this section, whenever f : D R is a real-valued function, we will assume D R n is a convex set. When we take x 1 D and x 2 D, we will let x t tx 1 + (1 t)x 2, for t [0, 1], denote the convex combination of x 1 and x 2. Because D is a convex set, we know that x t D. In economics, we often encounter concave real-valued functions over convex domains. Concave functions are defined as follows. Definition A1.22 Concave Function f : D R is a concave function if for all x 1, x 2 D, Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 2 / 28

3 Definition A1.22 (Continued) f (x t ) tf (x 1 ) + (1 t)f (x 2 ) for all t [0, 1] Loosely, according to this definition, f is concave if its value at a convex combination of two points is no smaller than the convex combination of the two values. There is a simple case in Figure A1.23. (see the next slide) where y 1 = f (x 1 ) and y 2 = f (x 2 ). We can get a point y t = ty 1 + (1 t)y 2. Additionally, x t = tx 1 + (1 t)x 2. It is clear to see f (x t ) > y t. So f is a concave function. Figure A1.24. (see the next slide) shows a function can have both concave and nonconcave parts. The set of points underneath the graph of the concave regions of both functions are convex sets; the set of points beneath the nonconcave regions is not a convex set. Theorem A1.13 generalizes this. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 3 / 28

4 Figure A1.23(left): A concave function; Figure A1.24(right): Concave and nonconcave regions Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 4 / 28

5 Theorem A1.13 Points on and Below the Graph of a Concave Function Always Form a Convex Set Let A {(x, y) x D, f (x) y} be the set of points "on and below " the graph of f : D R, where D R n is a convex set and R R. Then, f is a concave function A is a convex set. According to the definition of concave function, Figure A1.25 (see the next slide). is concave. Nothing in the defintion, or in Theorem A.13 prohibits linear segments in the graph of the function. The set benethe is still convex. At x t, the value of the function is exactly equal to the convex combination of f (x 1 ) and f (x 2 ), so the inequality f (x t ) tf (x 1 ) + (1 t)f (x 2 ) still holds there. It is sometimes convenient to exclude the possibility of linear segments, particularly flat portions, in the graph of the function. Strict Concavity rules out this kind of thing. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 5 / 28

6 Figure A1.25: f is concave but not strictly concave Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 6 / 28

7 Definition A1.23 Strictly Concave Functions f : D R is a strictly concave function iff, for all x 1 x 2 in D, f (x t )>tf (x 1 ) + (1 t)f (x 2 ) for all t (0, 1). Notice the strict inequality must hold for all t in the open interval (0, 1), rather than the closed interval [0, 1] as before. Thus makes perfect sense because if t were either zero or one, the convex combination x t would coincide with x 2 or x 1, and the strict inequality in the definition could not hold. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 7 / 28

8 A1.4.3 Quasiconcave Functions Concavity, whether strict or not, is a relatively strong restriction to place on a function. Often, one of the objectives in theoretical work is to identify and impose only the weakest possible restrictions needed to guarantee the result sought. Quasiconcavity is a related but weaker property that s often all that is required to get us where we want to go. Definition A1.24 Quasiconcave Functions f : D R is quasiconcave iff, for all x 1 and x 2 in D, f (x t ) min[f (x 1 ), f (x 2 )] for all t [0, 1]. The definition says, if we take any two points in the domain and form any convex combination of them, the value of the function must be no lower than the lowest value it takes at the two points. Another way of describing quasiconcave is in terms of their level sets. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 8 / 28

9 Now consider the functions whose level sets are depicted in Figure A1.26 (see the next slide). We pick two points x 1 and x 2. Then form any convex combination of the these two points to get x t. In each instance, we will assume that f (x 1 ) f (x 2 ). When f (x) is an increasing (decreasing) function, it will be quasiconcave whenever the level set relative to any convex combination of two points, L(x t ), is always on or above (on or below) the lowest (highest) of the two level sets L(x 1 ) and L(x 2 ). The level sets in Figure A1.26 were drawn nicely curved for a good reason. Besides requiring the relative positioning of level sets already noted, quasiconcavity requires very regular behavior in its superior sets. As you may have guessed, these must be convex. Theorem A1.14 Quasiconcavity and the Superior Sets f : D R is a quasiconcave function iff S(y) is a convex set for all y R. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 9 / 28

10 Figure A1.26: Level sets for quasiconcave functions. (a) Left: The function is quasiconcave and increasing. (b) Right: The function is quasiconcave and decreasing. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 10 / 28

11 This theorem establishes an equivalence between quasiconcave functions and convex superior sets. To assume that a function is quasiconcave is therefore exactly the same as assuming that the superior sets are convex, and vice versa. A special case of the quasiconcave functions is depicted in Figure A1.27 (see the next slide) where there is linear segment. There, x 1 and x 2 lie on a flat portion of the same level set. Their convex combination, say x t, will also lie on the linear segment. Here, f (x 1 ) = f (x 2 ) = f (x t ), so the inequality f (x t ) min[f (x 1 ), f (x 2 )] holds, but with equality. Natural enough, this kind of thing is ruled out under strict quasiconcavity. Definition A1.25 Strictly Quasiconcave Functions f : D R is strictly quasiconcave iff, for all x 1 x 2 in D, f (x t )>min[f (x 1 ), f (x 2 )] for all t (0, 1). Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 11 / 28

12 Figure A1.27: Quasiconcavity and linear segments in the level sets Constraining t to the open interval again prevents x 1 and x 2 from being the same point. By requiring the strict inequality, we forbid the convex combination of two points in the same level set from also lying in that level set, as occurs in Figure A1.27. Instead, such convex combinations must lie in strictly higher level sets, as occurs in Figure A1.28 (see the next slide). Thus, strictly quasiconcave functions must have superior sets with no flat segments in the boundary. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 12 / 28

13 Figure A1.28: The function in (a) left: is strictly quasiconcave and increasing. The one in (b) right: is strictly quasiconcave and decreasing. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 13 / 28

14 We begin discussing quasiconcave functions by remarking that quasiconcavity is a weaker restriction than concavity. The follwoing theorem says if a function is concave, it will satisfy all the properties of a quasiconcave function. Theorem A1.15 Concavity Implies Quasiconcavity A concave function is always quasiconcave. A strictly concave function is always strictly quasiconcave. (Note: The converse is not true.) Proof: The theorem follows immediately from the definition that if f is a concave function, f (x t ) = f (tx 1 + (1 t)x 2 ) tf (x 1 ) + (1 t)f (x 2 ) min[f (x 1 ), f (x 2 )] The proof of strictly case is similar. A1 is done. Next chapter (Appendix 2) will review calculus and discuss optimization. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 14 / 28

15 A2.1.1 Functions of a Single Variable A function y = f (x) is differentiable if it is both continuous and "smooth" with no breaks or kink, like the graph in Figure A2.1(b), whereas the one in Figure A2.1(a) is not differentiable at x 0. Figure A2.1: (a) Nondifferentiable and (b) differentiable functions of a single variable Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 15 / 28

16 The derivative of is also a function, giving at each value of x, the slope or instantaneous rate of change in f (x). We therefore sometimes write dy dx = f (x) If the (first) derivative is a differentiable function, we can take its derivative, too, getting the second derivative of the original function d 2 y dx 2 = f (x) If a function possesses continuous derivatives, f, f,, f n, it is call n times continuously differentiable, or a C n. The second derivative, f (x), gives the rate of change of f (x) per unit change in x. That is, f (x) gives the rate at which the slope of f is changing. Consequently, the second derivative is related to the curvature of the function f. For example, Figure A2.3 depicts a concave function. The fact that it is "curved downward" is captured by the fact that the slope of the function Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 16 / 28

17 decreases as x increases, i.e. its second derivative is nonpositive. Figure A2.3: Curvature and the second derivative But something else is also apparent from Figure A2.3. Note that both tangent lines, l 0 and l 1, lie entirely above (sometimes only weakly above) the function f. The equation that describes the straight line is Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 17 / 28

18 l 0 (x) = f (x 0 )(x x 0 ) + f (x 0 ) Now saying that the line l 0 lies above f is just saying that l 0 (x) f (x) for all x. But this then says that f (x) f (x 0 )(x x 0 ) + f (x 0 ) for all x. Thus, this inequality seems to follow from the concavity of f. Theorem A2.1 puts together the preceding observations to characterize concave functions of a single variable. Theorem A2.1 Concavity and First and Second Derivatives Let D be a nondegenerate interval of real numbers on which f is twice continuously differentiable. The following statements 1 to 3 are equivalent: 1. f is concave 2. f (x) 0 x D 3. For all x 0 D : f (x) f (x 0 )(x x 0 ) + f (x 0 ) x D Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 18 / 28

19 Theorem A2.1 (Continued) Moreover, 4. If f (x)<0 x D, then f is strictly concave. (the convese is not true.) Theorem A2.1 also gives a characterization of convex function. Simply replace the word "concave" with "convex", and reverse the sense of all the inequalities. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 19 / 28

20 Functions of Several Variables We will usually be concerned with real-valued functions of several variables, and the ideas just discussed can be easily generalized. Rather than having a single slope, a function of n variable can be thought to have n partial slopes, each giving only the rate at which y would change if one x i, alone, were to change. Each of these partial slopes is called a partial derivative. Formally, Definition A2.1 Partial Derivative Let y = f (x 1,, x n ). The partial derivative of f with respective x i is defined as f (x) f (x 1,, x i + h,, x n ) f (x 1,, x i,, x n ) lim h 0 x i h (eq.a1) Various other notations are sometimes used to denote partial derivatives. Among the most common are y/ x i or just f i (x). Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 20 / 28

21 Like the derivative in the single-variable case, each partial derivative is itself a function. In particular, each partial derivative is function that depends on the value taken by every variable, x 1,, x n. Notice that the partial derivative is defined at every point in the domain to measure how the value of the function changes as one x i changes, leaving the values of the other (n 1) variables unchanged. Consider the following example of a function of two variables. Example A2.1 Let f (x 1, x 2 ) = x x 1x 2 x 2 2. The partial derivative with respect to x 1 and x 2 are f (x 1, x 2 ) x 1 = f 1 = 2x 1 + 3x 2 and f (x 1, x 2 ) x 2 = f 2 = 3x 1 2x 2 respectively. Notice that each partial derivative in this example is a function of both x 1 and x 2. The value taken by each partial derivative will be different at different value of x 1 and x 2. At the point (1,2), their Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 21 / 28

22 Example A2.1 (Continued) values would be f 1 (1, 2) = 8 and f 2 (1, 2) = 1. At the point (2,1), their values would be f 1 (2, 1) = 7 and f 2 (2, 1) = 4. It is easy to see that each partial derivative tells us whether the function is rising or falling as we change one variable alone, holding all others constant. It is sometimes useful to know whether the value of the function is rising or falling as we move in other directions away from a particular point in the domain. So fix a point x = (x 1,, x n ) R n, in the domain of f, and we suppose we wish to know how the value of f changes from f (x) as we move away from x in the direction z = (z 1,, z n ). The function g(t) = f (x + tz) Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 22 / 28

23 defined for t R, in the domain of f. If g(t) increases as t moves from being zero to being positive, then we know that f increases as we move from x in the direction z. Thus we are interested in whether g (0) (where t changes by a very small amount ) is positive, negative, or zero. Note that by definition, g (0) is just the rate at which f changes per unit change in t. Now, the ith coordinate in the domain of f increases at the rate z i per unit change in t. Moreover, the rate at which f changes per unit change in the ith coordinate in the domain is just f i (x), the ith partial derivative of f at x. Consequently, the rate at which f changes per unit change in t due to the change in the ith coordinate is f i (x)z i. The total rate of change of f is then just the sum of all of the changes induced by each of the n coordinates. That is, g (0) = f 1 (x)z f n (x)z n = n f i (x)z i i=1 Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 23 / 28

24 The term on the right-hand side is known as the directional derivative of f at x in the direction z. Now we assemble all preceding n partial derivatives into a row vector f (x) (f 1 (x),, f n (x)) called the gradient of f at x. Assuming all vecotrs are column vectors unless explicitly stated otherwise, the directional derivative of f at x in the direction z can be written as follows g (0) = f (x)z. Note that the partial derivative of f with respect to x i is then just the directional derivative of f in the direction z = (0,, 0, 1, 0,, 0), where the 1 appears in the ith position. The partial derivative (eq. A1) or f i is a function of n variables itself. There is no particular difficulty in calculating the n partial derivatives of the (first-order) partial f i (x). When f i (x) is differentiated with respect to x j, the result is the second-order partial derivatives of f with respect to x i and x j, denoted Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 24 / 28

25 ( ) f (x), or x j x i 2 f (x) x j x i, or f ij (x) Because there are n of these partial derivatives of the partial derivative with respect to x i, they can be arranged into a gradient vector. This time, though, the vector will be the gradient of the partial derivative with respect to x i, f i (x). We can write the gradient vector as f i (x) = ( 2 ) f (x),, 2 f (x) (f i1 (x),, f in (x)) x 1 x i x n x i Because i = 1,, n, we have f 1 (x), f 2 (x),, f n (x). If we arrange all of the f i (x) into a matrix by slacking one on top of the other, we get Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 25 / 28

26 f 1 (x) f 11 (x) f 12 (x) f 1n (x) f 2 (x) H(x) =. = f 21 (x) f 22 (x) f 2n (x) f n (x) f n1 (x) f n2 (x) f nn (x) Notice that H(x) contains all possible second-order partial derivatives of the original function. H(x) is called the Hessian matrix of the function f (x). There is an important theorem on second-order partial derivative to which we will have occasion to refer. It says that the order in which the partial derivative are differentiated makes no difference. Theorem A2.2 Young s Theorem For any twice continuously differentiable function f (x), 2 f (x) x i x j = 2 f (x) x j x i i, j Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 26 / 28

27 Example A2.2 Consider the function f (x 1, x 2 ) = x 1 x x 1x 2. Its two first-order partial derivatives are f 1 (x) = x x 2 and f 2 (x) = 2x 1 x 2 + x 1 It is easy to see that f 12 (x) = f 21 (x) = 2x 2 + 1, just as Young s theorem promised. Young s theorem tells us the Hessian matrix will be symmetric. In the single-variable case, Theorem A2.1 established that the curvature of a concave function was expressed by its second derivative as well as the relation of the function to its tangent lines. The second conclusion also hold in the multivariate case. A simple way to see this is first to understand the following result. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 27 / 28

28 Theorem A2.3 Single-Variable and Multivariable Concavity Let f be a real-valued function defined on the convex subset D of R n. Then f is (strictly) concave if and only if for every x D and every nonzero z R n, the function g(t) = f (x + tz) is (strictly) concave on {t R x + tz D}. Theorem A2.3 says, in effect, that to check that a multivariate function is concave, it is enough to check, for each point x in the domain, and each direction z, that the function of a single variable defined by the values taken on by f on the line through x in the direction z is concave. Week 3: Sets and Mappings/Calculus and Optimization (Jehle September and Reny, 20, 2015 Chapter A1/A2) 28 / 28

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