Monotone Functions Function f is called monotonically increasing, if ( ) x x f (x ) f (x ) f x2 Chapter 8 1 ≤ 2 ⇒ 1 ≤ 2 It is called strictly monotonically increasing, if f (x1) x < x f (x ) < f (x ) x1 x2 Monotone, Convex and 1 2 ⇔ 1 2 Extrema Function f is called monotonically decreasing, if x x2 f (x ) f (x2) 1 ≤ ⇒ 1 ≥ f (x1) It is called strictly monotonically decreasing, if f (x2) x < x f (x ) > f (x ) x1 x2 1 2 ⇔ 1 2 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 1 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 2 / 47 Monotone Functions Locally Monotone Functions For differentiable functions we have A function f can be monotonically increasing in some interval and decreasing in some other interval. f monotonically increasing f (x) 0 for all x D ⇔ 0 ≥ ∈ f For continuously differentiable functions (i.e., when f 0(x) is continuous) f monotonically decreasing f 0(x) 0 for all x D f ⇔ ≤ ∈ we can use the following procedure: 1. Compute first derivative f 0(x). f strictly monotonically increasing f 0(x) > 0 for all x D f ⇐ ∈ 2. Determine all roots of f (x). f strictly monotonically decreasing f (x) < 0 for all x D 0 ⇐ 0 ∈ f 3. We thus obtain intervals where f 0(x) does not change sign. 4. Select appropriate points xi in each interval and determine the Function f : (0, ∞), x ln(x) is strictly monotonically increasing, as 7→ sign of f 0(xi). 1 f 0(x) = (ln(x))0 = > 0 for all x > 0 x Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 3 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 4 / 47 Locally Monotone Functions Monotone and Inverse Function In which region is function f (x) = 2 x3 12 x2 + 18 x 1 If f is strictly monotonically increasing, then − − monotonically increasing? x1 < x2 f (x1) < f (x2) We have to solve inequality f (x) 0: ⇔ 0 ≥ 1. f (x) = 6 x2 24 x + 18 immediately implies 0 − 2 2. Roots: x 4 x + 3 = 0 x1 = 1, x2 = 3 x = x f (x ) = f (x ) − ⇒ 1 6 2 ⇔ 1 6 2 3. Obtain 3 intervals: ( ∞, 1], [1, 3], and [3, ∞) − That is, f is one-to-one. 4. Sign of f (x) at appropriate points in each interval: 0 So if f is onto and strictly monotonically increasing (or decreasing), f (0) = 3 > 0, f (2) = 1 < 0, and f (4) = 3 > 0. 0 0 − 0 then f is invertible. 5. f 0(x) cannot change sign in each interval: f (x) 0 in ( ∞, 1] and [3, ∞). 0 ≥ − Function f (x) is monotonically increasing in ( ∞, 1] and in [3, ∞). − Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 5 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 6 / 47 Convex and Concave Functions Concave Function Function f is called convex, if its domain D is an interval and f f (1 h) x + h x (1 h) f (x ) + h f (x ) − 1 2 ≥ − 1 2 f ((1 h) x + h x ) (1 h) f (x ) + h f (x ) − 1 2 ≤ − 1 2 for all x1, x2 D f and all h [0, 1]. It is called concave, if ∈ ∈ f (1 h) x + h x − 1 2 f ((1 h) x + h x ) (1 h) f (x ) + h f (x ) − 1 2 ≥ − 1 2 (1 h) f (x ) + h f (x ) − 1 2 x1 x2 (1 h) x + h x − 1 2 x1 x2 x1 x2 convex concave Secant below graph of function Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 7 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 8 / 47 Convex and Concave Functions Strictly Convex and Concave Functions For two times differentiable functions we have Function f is called strictly convex, if its domain D f is an interval and f convex f 00(x) 0 for all x D f f ((1 h) x + h x ) < (1 h) f (x ) + h f (x ) ⇔ ≥ ∈ − 1 2 − 1 2 f concave f 00(x) 0 for all x D f ⇔ ≤ ∈ for all x , x D , x = x and all h (0, 1). 1 2 ∈ f 1 6 2 ∈ It is called strictly concave, if its domain D f is an interval and f ((1 h) x + h x ) > (1 h) f (x ) + h f (x ) − 1 2 − 1 2 f 0(x) is monotonically decreasing, For two times differentiable functions we have thus f (x) 0 00 ≤ f strictly convex f (x) > 0 for all x D ⇐ 00 ∈ f f strictly concave f (x) < 0 for all x D ⇐ 00 ∈ f Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 9 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 10 / 47 Convex Function Concave Function Logarithm function: (x > 0) Exponential function: f (x) = ln(x) 1 f (x) = ex 1 e f (x) = f (x) = ex 0 x 0 1 x f 00(x) = 2 < 0 for all x > 0 1 e f 00(x) = e > 0 for all x R − x ∈ ln(x) is (strictly) concave. exp(x) is (strictly) convex. 1 1 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 11 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 12 / 47 Locally Convex Functions Locally Concave Function A function f can be convex in some interval and concave in some other In which region is f (x) = 2 x3 12 x2 + 18 x 1 concave? − − interval. We have to solve inequality f 00(x) 0. ( ) ≤ For two times continuously differentiable functions (i.e., when f 00 x is 1. f 00(x) = 12 x 24 continuous) we can use the following procedure: − 2. Roots: 12 x 24 = 0 x = 2 1. Compute second derivative f 00(x). − ⇒ 3. Obtain 2 intervals: ( ∞, 2] and [2, ∞) 2. Determine all roots of f 00(x). − 4. Sign of f 00(x) at appropriate points in each interval: 3. We thus obtain intervals where f 00(x) does not change sign. f (0) = 24 < 0 and f (4) = 24 > 0. 00 − 00 4. Select appropriate points xi in each interval and determine the 5. f 00(x) cannot change sign in each interval: f 00(x) 0 in ( ∞, 2] sign of f 00(xi). ≤ − Function f (x) is concave in ( ∞, 2]. − Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 13 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 14 / 47 Problem 8.1 Problem 8.2 Determine whether the following functions are concave or convex (or In which region is function neither). f (x) = x3 3x2 9x + 19 (a) exp(x) − − monotonically increasing or decreasing? (b) ln(x) In which region is it convex or concave? (c) log10(x) (d) xα for x > 0 for an α R. ∈ Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 15 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 16 / 47 Problem 8.3 Problem 8.4 In which region the following functions monotonically increasing or Function decreasing? 1 a f (x) = b x − , 0 < a < 1, b > 0, x 0 In which region is it convex or concave? ≥ 2 (a) f (x) = x ex is an example of a production function. Production functions usually have the following properties: x2 (b) f (x) = e− (1) f (0) = 0, lim f (x) = ∞ x ∞ 1 → (2) f 0(x) > 0, lim f 0(x) = 0 (c) f (x) = x ∞ x2 + 1 → (3) f 00(x) < 0 (a) Verify these properties for the given function. (b) Draw (sketch) the graphs of f (x), f 0(x), and f 0(x). (Use appropriate values for a and b.) (c) What is the economic interpretation of these properties? Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 17 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 18 / 47 Problem 8.5 Problem 8.6 Function Use the definition of convexity and show that f (x) = x2 is strictly f (x) = b ln(ax + 1) , a, b > 0, x 0 convex. ≥ 2 is an example of a utility function. Hint: Show that inequality 1 x + 1 y 1 x2 + 1 y2 < 0 holds for all 2 2 − 2 2 Utility functions have the same properties as production functions. x = y. 6 (a) Verify the properties from Problem 8.4. (b) Draw (sketch) the graphs of f (x), f 0(x), and f 0(x). (Use appropriate values for a and b.) (c) What is the economic interpretation of these properties? Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 19 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 20 / 47 Problem 8.7 Problem 8.8 Show: Show: If f (x) is a two times differentiable concave function, then If f (x) is a concave function, then g(x) = f (x) convex. − g(x) = f (x) convex. You may not assume that f is differentiable. − Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 21 / 47 Josef Leydold – Bridging Course Mathematics – WS 2021/22 8– Monotone, Convex and Extrema – 22 / 47 Problem 8.9 Problem 8.10 Let f (x) and g(x) be two differentiable concave functions.