Monotone Function. Function f is called monotonically increasing, if. x 1 x 2 f (x 1 ) f (x 2 ) x 1 < x 2 f (x 1 ) < f (x 2 ) x 1 x 2

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1 Monotone Function Function f is called monotonically increasing, if Chapter 3 x x 2 f (x ) f (x 2 ) It is called strictly monotonically increasing, if f (x 2) f (x ) Convex and Concave x < x 2 f (x ) < f (x 2 ) x x 2 Function f is called monotonically decreasing, if x x 2 f (x ) f (x 2 ) It is called strictly monotonically decreasing, if f (x ) f (x 2) x < x 2 f (x ) > f (x 2 ) x x 2 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave / 44 Monotone Functions Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 2 / 44 Locally Monotone Functions For differentiable functions we have f monotonically increasing f (x) 0 for all x D f f monotonically decreasing f (x) 0 for all x D f f strictly monotonically increasing f (x) > 0 for all x D f f strictly monotonically decreasing f (x) < 0 for all x D f Function f : (0, ), x ln(x) is strictly monotonically increasing, as A function f can be monotonically increasing in some interval and decreasing in some other interval. For continuously differentiable functions (i.e., when f (x) is continuous) we can use the following procedure:. Compute first derivative f (x). 2. Determine all roots of f (x). 3. We thus obtain intervals where f (x) does not change sign. 4. Select appropriate points x i in each interval and determine the sign of f (x i ). f (x) = (ln(x)) = > 0 for all x > 0 x Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 3 / 44 Example Locally Monotone Functions Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 4 / 44 Monotone and Inverse Function In which region is function f (x) = 2 x 3 2 x x monotonically increasing? We have to solve inequality f (x) 0:. f (x) = 6 x 2 24 x Roots: x 2 4 x + 3 = 0 x =, x 2 = 3 3. Obtain 3 intervals: (, ], [, 3], and [3, ) 4. Sign of f (x) at appropriate points in each interval: f (0) = 3 > 0, f (2) = < 0, and f (4) = 3 > f (x) cannot change sign in each interval: f (x) 0 in (, ] and [3, ). If f is strictly monotonically increasing, then immediately implies That is, f is one-to-one. x < x 2 f (x ) < f (x 2 ) x = x 2 f (x ) = f (x 2 ) So if f is onto and strictly monotonically increasing (or decreasing), then f is invertible. Function f (x) is monotonically increasing in (, ] and in [3, ). Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 5 / 44 Convex Set Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 6 / 44 Intersection of Convex Sets A set D R n is called convex, if for any two points x, y D the straight line between these points also belongs to D, i.e., ( h) x + h y D for all h [0, ], and x, y D. Let S,..., S k be convex subsets of R n. Then their intersection S... S k is also convex. convex: The union of convex sets need not be convex. not convex: Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 7 / 44 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 8 / 44

2 Example Half-Space Convex and Concave Functions Let p R n and m R be fixed, p = 0. Then H = {x R n : p t x = m} is a so called hyper-plane which partitions the R n into two half-spaces H + = {x R n : p t x m}, H = {x R n : p t x m}. Function f is called convex in domain D R n, if D is convex and f (( h) x + h x 2 ) ( h) f (x ) + h f (x 2 ) for all x, x 2 D and all h [0, ]. It is called concave, if f (( h) x + h x 2 ) ( h) f (x ) + h f (x 2 ) Sets H, H + and H are convex. Let x be a vector of goods, p the vector of prices and m the budget. Then the budget set is convex. {x R n : p t x m, x 0} = {x : p t x m} {x : x 0}... {x : x n 0} x x 2 x x 2 convex concave Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 9 / 44 Concave Function Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 0 / 44 Strictly Convex and Concave Functions f ( ( h) x + h x 2 ) ( h) f (x ) + h f (x 2 ) f ( ( h) x + h x 2 ) ( h) f (x ) + h f (x 2) Function f is called strictly convex in domain D R n, if D is convex and f (( h) x + h x 2 ) < ( h) f (x ) + h f (x 2 ) for all x, x 2 D with x = x 2 and all h (0, ). Function f is called strictly concave in domain D R n, if D is convex and f (( h) x + h x 2 ) > ( h) f (x ) + h f (x 2 ) x x 2 ( h) x + h x 2 for all x, x 2 D with x = x 2 and all h (0, ). Secant is below the graph of function f. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave / 44 Example Linear Function Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 2 / 44 Example Quadratic Univariate Function Let a R n be fixed. Then f (x) = a t x is a linear map and we find: f (( h) x + h x 2 ) = a t (( h) x + h x 2 ) = ( h) a t x + h a t x 2 = ( h) f (x ) + h f (x 2 ) That is, every linear function is both concave and convex. However, a linear function is neither strictly concave nor strictly convex, as the inequality is never strict. Function f (x) = x 2 is strictly convex: f (( h) x + h y) [ ( h) f (x) + h f (y) ] = (( h) x + h y) 2 [ ( h) x 2 + h y 2] = ( h) 2 x 2 + 2( h)h xy + h 2 y 2 ( h) x 2 h y 2 = h( h) x 2 + 2( h)h xy h( h) y 2 = h( h) (x y) 2 < 0 for x = y and 0 < h <. Thus f (( h) x + h y) < ( h) f (x) + h f (y) for all x = y and 0 < h <, i.e., f (x) = x 2 is strictly convex, as claimed. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 3 / 44 Properties Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 4 / 44 Properties If f (x) is (strictly) convex, then f (x) is (strictly) concave (and vice versa). For a differentiable functions the following holds: Function f is concave if and only if If f (x),..., f k (x) are convex (concave) functions and α,..., α k > 0, then g(x) = α f (x) + + α k f k (x) f (x) f (x 0 ) f (x 0 ) (x x 0 ) i.e., the function graph is always below the tangent. Function f is strictly concave if and only if x 0 is also convex (concave). f (x) f (x 0 ) < f (x 0 ) (x x 0 ) for all x = x 0 If (at least) one of the functions f i (x) is strictly convex (strictly concave), then g(x) is strictly convex (strictly concave). Function f is convex if and only if f (x) f (x 0 ) f (x 0 ) (x x 0 ) (Analogous for strictly convex functions.) x 0 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 5 / 44 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 6 / 44

3 Univariate Functions Univariate Functions For two times differentiable functions we have f convex f (x) 0 for all x D f f concave f (x) 0 for all x D f For two times differentiable functions we have f strictly convex f (x) > 0 for all x D f f strictly concave f (x) < 0 for all x D f Derivative f (x) is monotonically decreasing, thus f (x) 0. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 7 / 44 Example Convex Function Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 8 / 44 Example Concave Function Exponential function: Logarithm function: (x > 0) f (x) = e x f (x) = e x f (x) = e x > 0 for all x R e f (x) = ln(x) f (x) = x f (x) = x 2 < 0 for all x > 0 e exp(x) is (strictly) convex. ln(x) is (strictly) concave. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 9 / 44 Locally Convex Functions Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 20 / 44 Locally Concave Function A function f may be convex in some interval and concave in some other interval. For two times continuously differentiable functions (i.e., when f (x) is continuous) we can use the following procedure:. Compute second derivative f (x). 2. Determine all roots of f (x). 3. We thus obtain intervals where f (x) does not change sign. 4. Select appropriate points x i in each interval and determine the sign of f (x i ). In which region is f (x) = 2 x 3 2 x x concave? We have to solve inequality f (x) 0.. f (x) = 2 x Roots: 2 x 24 = 0 x = 2 3. Obtain 2 intervals: (, 2] and [2, ) 4. Sign of f (x) at appropriate points in each interval: f (0) = 24 < 0 and f (4) = 24 > f (x) cannot change sign in each interval: f (x) 0 in (, 2] Function f (x) is concave in (, 2]. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 2 / 44 Univariate Restrictions Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 22 / 44 Quadratic Form Notice, that by definition a (multivariate) function is convex if and only if every restriction of its domain to a straight line results in a convex univariate function. That is: Function f : D R n R is convex if and only if g(t) = f (x 0 + t h) is convex for all x 0 D and all non-zero h R n. x 0 h Let A be a symmetric matrix and q A (x) = x t Ax be the corresponding quadratic form. Matrix A can be diagonalized, i.e., if we use an orthonormal basis of its eigenvector, then A becomes a diagonal matrix with the eigenvalues of A as its elements: q A (x) = λ x 2 + λ 2x λ n x 2 n. It is convex if all eigenvalues λ i 0 as it is the sum of convex functions. It is concave if all λ i 0 as it is the negative of a convex function. It is neither convex nor concave if we have eigenvalues with λ i > 0 and λ i < 0. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 23 / 44 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 24 / 44

4 Quadratic Form We find for a quadratic form q A : strictly convex positive definite convex positive semidefinite strictly concave negative definite concave negative semidefinite neither indefinite We can determine the definiteness of A by means of the eigenvalues of A, or the (leading) principle minors of A. Example Quadratic Form 2 0 Let A = 3. Leading principle minors: 0 2 A = 2 > 0 2 A 2 = 3 = 5 > A 3 = A = 3 = 8 > A is thus positive definite. Quadratic form q A is strictly convex. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 25 / 44 Example Quadratic Form 0 Let A = Principle Minors: 2 2 A = A 2 = 4 A 3 = 2 0 A,2 = 0 4 = 4 A,3 = 2 = A 4 2 2,3 = 2 2 = 4 0 A i 0 A,2,3 = = 0 A i,j A,2,3 0 A is thus negative semidefinite. Quadratic form q A is concave (but not strictly concave). Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 26 / 44 Concavity of Differentiable Functions Le f : D R n R with Taylor series expansion f (x 0 + h) = f (x 0 ) + f (x 0 ) h + 2 ht H f (x 0 ) h + O( h 3 ) Hessian matrix H f (x 0 ) determines the concavity or convexity of f around expansion point x 0. H f (x 0 ) positive definite f strictly convex around x 0 H f (x 0 ) negative definite f strictly concave around x 0 H f (x) positive semidefinite for all x D f convex in D H f (x) negative semidefinite for all x D f concave in D Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 27 / 44 Recipe Strictly Convex Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 28 / 44 Recipe Convex. Compute Hessian matrix f x x (x) f x x 2 (x) f x x n (x) f H f (x) = x2x (x) f x2x2 (x) f x2xn (x) f xnx (x) f xnx2 (x) f xnxn (x) 2. Compute all leading principle minors H i. 3. f strictly convex all H k > 0 for (almost) all x D f strictly concave all ( ) k H k > 0 for (almost) all x D [ ( ) k H k > 0 implies: H, H 3,... < 0 and H 2, H 4,... > 0 ] 4. Otherwise f is neither strictly convex nor strictly concave.. Compute Hessian matrix f x x (x) f x x 2 (x) f x x n (x) f H f (x) = x2x (x) f x2x2 (x) f x2xn (x) f xnx (x) f xnx2 (x) f xnxn (x) 2. Check for strict convexity or concavity. 3. Compute all principle minors H i,...,i k. (Only required if det(a) = 0) 4. f convex all H i,...,i k 0 for all x D. f concave all ( ) k H i,...,i k 0 for all x D. 5. Otherwise f is neither convex nor concave. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 29 / 44 Example Strict Convexity Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 30 / 44 Example Cobb-Douglas Function Is function f (strictly) concave or convex? f (x, y) = x 4 + x 2 2 x y + y 2 ( ) 2 x Hessian matrix: H f (x) = Leading principle minors: H = 2 x > 0 H 2 = H f (x) = 24 x 2 > 0 for all x = All leading principle minors > 0 for almost all x f is strictly convex. (and thus convex, too) Let f (x, y) = x α y β with α, β 0 and α + β, and D = {(x, y) : x, y 0}. Hessian matrix at x: H f (x) = Principle Minors: ( α(α ) x α 2 y β H = H 2 = αβ x α y β α β (α ) 0 (β ) 0 x α 2 y β x α y β 2 αβ x α y β β(β ) x α y β ) Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 3 / 44 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 32 / 44

5 Example Cobb-Douglas Function Lower Level Sets of Convex Functions H,2 = H f (x) = α(α ) x α 2 y β β(β ) x α y β 2 (αβ x α y β ) 2 = α(α ) β(β ) x 2α 2 y 2β 2 α 2 β 2 x 2α 2 y 2β 2 = αβ[(α )(β ) αβ]x 2α 2 y 2β 2 = αβ ( α β) x 2α 2 y 2β 2 0 Assume that f is convex. Then the lower level sets of f {x D f : f (x) c} are convex. Let x, x 2 {x D f : f (x) c}, i.e., f (x ), f (x 2 ) c. x y x 2 H 0 and H 2 0, and H,2 0 for all (x, y) D. f (x, y) thus is concave in D. For 0 < α, β < and α + β < we even find: H = H < 0 and H 2 = H f (x) > 0 for almost all (x, y) D. f (x, y) is then strictly concave. Then for y = ( h)x + hx 2 where h [0, ] we find f (y) = f (( h)x + hx 2 ) ( h) f (x ) + h f (x 2 ) ( h)c + hc = c That is, y {x D f : f (x) c}, too. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 33 / 44 Upper Level Sets of Concave Functions Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 34 / 44 Extremum and Monotone Transformation Assume that f is concave. Then the upper level sets of f {x D f : f (x) c} are convex. Let T : R R be a strictly monotonically increasing function. If x is a maximum of f, then x is also a maximum of T f. As x is a maximum of f, we have f (x ) f (x) for all x. As T is strictly monotonically increasing,we have T(x ) > T(x 2 ) falls x > x 2. c c Thus we find (T f )(x ) = T( f (x )) > T( f (x)) = (T f )(x) for all x, i.e., x is a maximum of T f. upper level set lower level set As T is one-to-one we also get the converse statement: If x is a maximum of T f, then it also is a maximum of f. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 35 / 44 Extremum and Monotone Transformation Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 36 / 44 Quasi-Convex and Quasi-Concave A strictly monotonically increasing Transformation T preserves the extrema of f. Transformation T also preserves the level sets of f : 9 4 e 9 e 4 e Function f is called quasi-convex in D R n, if D is convex and every lower level set {x D f : f (x) c} is convex. Function f is called quasi-concave in D R n, if D is convex and every upper level set {x D f : f (x) c} is convex. f (x, y) = x 2 y 2 T( f (x, y)) = exp( x 2 y 2 ) Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 37 / 44 Convex and Quasi-Convex Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 38 / 44 A Weaker Condition Every concave (convex) function also is quasi-concave (and quasi-convex, resp.). However, a quasi-concave function need not be concave. Let T be a strictly monotonically increasing function. If function f (x) is concave (convex), then T f is quasi-concave (and quasi-convex, resp.). Function g(x, y) = e x2 y 2 is quasi-concave, as f (x, y) = x 2 y 2 is concave and T(x) = e x is strictly monotonically increasing. However, g = T f is not concave. The notion of quasi-convex is weaker than that of convex in the sense that every convex function also is quasi-convex but not vice versa. There are much more quasi-convex functions than convex ones. The importance of such a weaker notions is based on the observation that a couple of propositions still hold if convex is replaced by quasi-convex. In this way we get a generalization of a theorem, where a stronger condition is replaced by a weaker one. Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 39 / 44 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 40 / 44

6 Quasi-Convex and Quasi-Concave II Strictly Quasi-Convex and Quasi-Concave Function f is quasi-convex if and only if f (( h)x + hx 2 ) max{ f (x ), f (x 2 )} for all x, x 2 and h [0, ]. Function f is quasi-concave if and only if f (( h)x + hx 2 ) min{ f (x ), f (x 2 )} for all x, x 2 and h [0, ]. Function f is called strictly quasi-convex if f (( h)x + hx 2 ) < max{ f (x ), f (x 2 )} for all x, x 2, with x = x 2, and h (0, ). Function f is called strictly quasi-concave if f (( h)x + hx 2 ) > min{ f (x ), f (x 2 )} for all x, x 2, with x = x 2, and h (0, ). x 2 x x x 2 quasi-convex quasi-concave Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 4 / 44 Quasi-convex and Quasi-Concave III Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 42 / 44 Summary For a differentiable function f we find: Function f is quasi-convex if and only if f (x) f (x 0 ) f (x 0 ) (x x 0 ) 0 Function f is quasi-concave if and only if Monotone function Convex set Convex and concave function Minors of Hessian matrix Quasi-convex and quasi-concave function f (x) f (x 0 ) f (x 0 ) (x x 0 ) 0 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 43 / 44 Josef Leydold Mathematical Methods WS 208/9 3 Convex and Concave 44 / 44