Mathematics in the principles of economics: concave functions and convex functions concave and convex functions of a single variable_ economics

Mathematical methods for Economic theory Martin J. Osborne    contents text exercises     3 .1 concave and convex functions of a single variable definitions the twin notions of concavity and convexity are used Ly in economic theory, and are also the "to Optimization theory." a  function of a single variable is  concave if every line segment joining two points on its graph Doe s not lie above the graph at any point. Symmetrically, a function of a single variable is  convex if every line segment joining two points in its graph Does not lie below the graph at any point. These concepts are are illustrated in the following figures.

A Concave Function:no line segment Joiningtwo points on the graphlies above the Graphat any Point   &nbs P A Convex function:no Line segment Joiningtwo points on the graphlies below the Graphat any Point    a function is Neitherconcave nor convex:the line segment shown liesabove the graph at somepoints and below it at others

Here is a precise definition. Definition let F is a function of a single variable defined on a interval. Then F is concave if every line segment joining two points on it graph is never above the graph convex if every line seg Ment joining two points on it graph is never below the graph. To make this definition is more useful and we can translate it into a algebraic condition. Let f is a function defined on the interval [x 1, x 2].  This function is concave according to the definition if, for every pair of numbers a and B with X 1≤a≤x 2 and X 1 ≤b≤x 2, the line segment from (A, f (a)) to (b, f (b)) lies on or below the graph of the function, as illustrate D in the following figure.

X1A (1−λ) a +λbbx2f (a) (1−λ) f (a) +λf (b) f ((1−λ) a +λb) F (b)

Denote the "line segment" (A, f (a)) to (B, f (b)) at the Point x by ha,b (x). Then according to the definition, the function f is concave if and only if for every pair of numbers a&nbsp ; And b with x1 ≤ a ≤ x2 andx1 ≤ b ≤ x2 we have F (x)  ≥  ha,b (x) for all x with a ≤ x ≤ b.         (*) Now, every point  x with  a ≤  x ≤  B may is written as& nbsp x = (1 −λ) a +λb, whereλis a real number from 0 to 1. (whenλ = 0, we have  x =  A; whenλ = 1 we have  x =  B.) The fact that  H a,b is linear means that ha,b ((1 − λ) a + λb)  =  (1 −  λ) Ha,b (a) +λha,b (b) for any value Ofλwith 0 ≤λ ≤1. Further, we have  H a,b (a)  =  f (a) and  h A,b (b)  =  F (b) (the line segment coincides with the function in its endpoints), and so Ha,b ((1 −&n bsp;λ) a + λb  =  (1 − λ) f (a)  +λf (b). Thus the condition (*) is equivalent to F ((1−λ) a + λb)  ≥  (1 −λ) f (a)  +λf (b) for Allλwit H 0 ≤λ ≤1. We can make a symmetric argument for a convex function. Thus the definition of concave and convex functions may is rewritten as follows. Definition let  f be a  function of a single variable defined on the  I. then  F is concave if for All a ∈ i, All b ∈ i, and allλ ∈[0,  1] We have f ((1−λ) a +λb)  ≥  (1 − λ) f (a) +λf (b) convex if for All a ∈ i, AL L b ∈ i, and allλ ∈[0, 1] we have f ((1−λ) a +λb)  ≤  (1 − λ) f (a) +λf ( b). In a exercise you are asked to show that&nbsp F is convex if and only if−f is concave.

The

Note this is a function may be both concave and convex. Let f be such a function. Then for all values Of a and bwe have f ((1−λ) a + λb)  ≥  (1 − λ) f (a)  +λf (b) for allλ ∈ [0, 1] and F (1−λ) a + λb)  ≤  (1 − λ) f (a)   +λf (b) for allλ ∈ [0, 1]. Equivalently, for all values of  a and  b we have f ((1−λ) a + λb)  =  (1 −& nbsp;λ) F (a)  +λf (b) for allλ ∈ [0, 1]. This is, the a function is both concave and convex if and only if it is  linear  (or, more properly,  affine), T Aking the form  f (x)  =α +βx for all  x, for some constantsαandβ.

Economists often assume that a firm ' s production the function is increasing and concave. Examples of such a function for a firm this uses a single input are shown in the next two. The fact that such a production function is increasing means, more input, generates more output. The fact, it is concave means, the increase in output generated by each one-unit increase in the input does Crease as more input is used. In economic jargon, there are ' nonincreasing returns ' to the ' input ', or, given that ' firm a single input, ' uses Asing returns to scale ". In the example in the ' the ' following two figures, the increase in output generated by each one-unit increase in th E input not only does not increase as more of the "input is used", but in fact decreases, so this in economic jargon a Re "diminishing returns", not merely "nonincreasing returns" and to the input.

Z→0F (z) concave production function (z = input, f (z) = output) z→0f (z) concave production function (z = input, f (z) = out Put

The notions of concavity and convexity are important in optimization theory because, as we shall you, a simple condition I s sufficient (as OK as necessary) for a maximizer of a differentiable concave function and for a minimizer of a differen Tiable convex function. (precisely, every point at which the derivative of a concave differentiable function are zero is a maximizer of the Functio N, and every point at which the derivative of a convex differentiable function are zero is a minimizer of the function.

The next result shows this a nondecreasing concave transformation of a concave function is concave. Proposition let  u be a  concave  function of a single variable and  g a nondecreasing and concave function of a single variable. Define the function  f by  f (x)  =  g (U (x)) for all  X. then  f is Concave. Proof We need to show that  F ((1−λ) a +λb)  ≥ (1−λ) f (a)  +λf (b) for all values of  A an d  b with  a ≤  B.

By the definition of f we have f ((1−λ) a + λb)  =  g (U (1−λ) a +λb)). Now, because  u is concave we have U (1−λ) a + λb)  ≥  (1 −λ) u (a) +λu (b). Further, because  g is nondecreasing,  r ≥  s implies  g (r)  ≥  g (s). Hence g (U (1−λ) a + λb)  ≥  g ((1−λ) U (a)  +λu (b)). But now by the concavity of  G we have g ((1−λ) U (a)  +λu (b))  ≥ (1−λ) G (U (A))  +λg (U (b)) &N Bsp;= (1−λ) f (a)  +λf (b). so  f is Concave. Jensen ' s inequality:another characterization of concave and convex functions If we letλ1 = ; =λin the  earlier definition of a concave function and replace  a by  x 1 and  b&nbs p;by  x 2, the definition becomes:  f is concave on the interval  i if for all  x 1 ∈&N Bsp I, all  x 2 ∈  I, and all Λ1 ≥0 andλ2 ≥0 withλ1 +λ2 = 1 we have F (λ1x1 + λ2x2)  ≥ λ1f (x1)  + Λ2F (x2). The following result, due to  Johan jensen  (1859–1925), shows so this characterization can be generalized. (The J in each of Jensen ' s names are, incidentally, pronounced the way an 中文版 speaker pronounces a Y.) Proposition (Jensen ' s inequality)   SOURCE a  function f of A single variable defined on The&nbs P interval  i is  concave if and only if for all  n ≥2 F (λ1x1 + ... +  ΛNXN)  ≥ λ1f (x1)  + ...  +λnf (xn) for all  x 1 ∈  I, ...,  x n ∈  I&nbsp ; and allλ1 ≥0, ..., λn ≥0 with∑n
I=1λi = 1.

The function f of a single variable defined on the interval I are convex if and only if to all n≥2 f (λ1x1 + ... +λnxn) ≤Λ1F (x1) + ... +λnf (xn) for all x 1∈i, ..., x n∈i and Allλ1≥0, ..., λn≥0 with∑n
I=1λi = 1. Differentiable functions The following diagram of a differentiable concave function should convince you ' graph of Such a function lies on or below every tangent to the function. In the figure, the red line is the graph of the function and the tangent of the "the" is the "x*, which has slop E F ' (x*).

X*XF (x*) F ' (x*) (x−x*) f (x) f (z) slope = f ' (x*)

The fact that graph's function lies below this tangent are equivalent to F (x)  −  F (x*)  ≤  F ' (x*) (x −  x*) for all  x.

The next result states this observation, and the similar one for convex functions, precisely. It is used to showthe important the for a concave differentiable function f every point  X for which f ' (x)  = 0 is a global maximizer, with for a convex differentiable The point is a global minimizer. Proposition   PROOF the  differentiable  function f of A single variable defined on an open   interval  i is  concave on  i if and only if f (x)  − f (x*)  ≤  F ' (x*) (x − x*) for all x ∈ i and x* ∈ i and is convex on  I if And only if f (x)  − f (x*)  ≥  F ' (x*) (x − x*) for all x ∈ i and  X* ∈ i. twice-differentiable functions We often assume the functions in economic (models. A e.g ' s firm Uction function, a consumer ' s utility function) are  twice-differentiable. We may determine the concavity or convexity of such a function by examining its second derivative:a function whose second Derivative is nonpositive everywhere is concave, and a function whose second derivative are nonnegative everywhere is conv Ex. Proposition   SOURCE a  twice-differentiable  function f of A single variable defined On the  interval  i is concave if and only If f ' (x)  ≤0 for All x in The int Erior of i convex if and only If f ' (x)  ≥0 for All x in the interior of& Nbsp;i. Example is  x 2 −2 x + 2 concave or convex on any interval? Its second derivative are 2 ≥0, so it is convex to all values of  x. Example is  x 3 −  x 2&nbsp ; concave or convex on any interval? Its second derivative are 6 x −2, so it are convex on the interval [1/3, ∞) and concave the interval (−∞, &NBSP;1/3]. The next result shows how the characterization of concave twice-differentiable functions can is used to prove a earlier R Esult when The functions involved are twice-differentiable. The earlier is true for  all functions, so the next result proves something we already know to be true. I include it only as a example of the usefulness of the characterization of concavity in the previous proposition. Proposition let  u be a  concave  function of a single variable and  g a nondecreasing and concave function of a single variable. Assume that  u and  g are  twice-differentiable. Define the function  f by  f (x)  =  g (U (x)) for all  X. Then f is Concave. Proof We have  F ' (x)  =  G ' (U (x)) U ' (x), so which F "(x)  =  G" (U (x)) · U ' (x) · U ' (x)  + g ' (U (x)) U "(x). since  g "(x)  ≤0 (g is concave),  g ' (x)  ≥0 (G&NBsp;is nondecreasing), and  U "(x)  ≤0 (u is concave), we have F" (x)  ≤ 0. That is,  f is concave. A point at which a twice-differentiable function changes from being convex to concave, or vice versa, are an inflection poi NT. Definition the point  c is an  inflection point of a  twice-differentiable  function& Nbsp;f of a single variable if  F "(c)  = 0 and for some values of  a and  Bsp a <  c <  B we have either f "(x)  > 0 if a < x  < c and f "(x)  < 0 if c < x < b or f" (x)  < 0 If a < x < c and f "(x)  > 0 if c < x <  b. The function  f in the following figure has a inflection point at  c. for  x between  a and  C, the value of&nbsp F "(x) is negative, and for  x between  c and  B, it is positive.

X→ACBF (x) concave production function (z = input, f (z) = output)

Note this some authors, including Sydsæter and Hammond (1995) (p. 308), give a slightly different definition, in which the Conditions F "(x) > 0 and F" (X) < 0 are replaced by F "(X) ≥0 and F" (x) ≤0. According to the alternative definition, F "does not have to change sign at C. For example, for a linear function, every point satisfies the alternative definition. Strict convexity and concavity The inequalities in the definition of of concave and convex functions are weak:such functions may have linear parts, as does the ' function in ' following for x > A.

X→AF (x) A function, concavebut not strictly concave

A concave function that has no linear parts are said to be strictly concave. Definition the function f of a single variable defined on the interval I am strictly concave if for all a∈i, all b∈ I with A≠b, and allλ∈ (0,1) we have

F ((1−λ) a +λb) > (1−λ) F (a) +λf (b). Strictly convex if for all a∈i, all b∈i with A≠b, and allλ∈ (0,1) we have

F ((1−λ) a + λb)  <  (1 − λ) f (a) +λf (b). an  earlier result states that if  f is twice differentiable then f is concave on [a,  b] I F and only if  F "(x)  ≤0 for all  x ∈ (a,  b). Does this result have a analogue for  strictly concave functions? Not exactly. if  F "(x)  < 0 for all  x ∈ (A, B) then f is strictly concave on [a,  b],  but the Converse is not true:if  f is strictly concave then its second derivative are not necessarily negative at All points. (Consider the function  f (x)  =−x 4. It is concave, but its second derivative at 0 is zero,  not negative.) It, f is strictly concave on [a,  b] if  F ' (x)  < 0 for all  x ∈ (a,  b), &nbs P but if  F is strictly concave on [a,  b] then  F ' (x) is  not necessarily negative for all  x ∈ (a,  b). Analogous observations apply to the case of convex and strictly convex functions, with the conditions  F "(x)  ≥ 0 and  F "(x) > 0 replacing the conditions  F" (x)  ≤0 and  F "(x) < 0.
from:http://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/cv1/t