DC-1
SEM-II
Lesson: CONVEXITY AND CONCAVITY OF FUNCTIONS
Lesson Developer: S.K. TANEJA
College: Ram Lal Anand College
University of Delhi
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CONVEXITY AND CONCAVITY OF FUNCTIONS
Table of Contents
1. Learning outcomes 2. Introduction 3. Derivative Test for Concavity and Convexity 4. Second Derivative and Concavity and Convexity 4.1 Total Differential Method 4.2 Definitions 4.3 Use of Hession Matrix for the determination of Convexity and Concavity
4.4 Graphical Representation 4.5 Assumption 4.6 Theorem 5. Second derivative test for concavity and convexity. 6. Quasi-concave and Quasi-convex Functions 7. Quasi-convex function 8. Properties of Quasi-concave and Quasi-convex functions 9. Functions of multiple variables 10. Differentiable function 11. Exercises 12. References
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1. Learning outcomes
After you have read this chapter, you should be able to:-
1. Explain concept of concavity and convexity. 2. Explain concept of quasi- concavity and quasi-convexity. 3. Second Derivative Test for Concavity and Convexity. 4. Use of Bordered Hessian Determinant
2. Introduction
A function f is concave if and only if any pair of distinct point p and R in the domain of f and 0 1 f( p (1–)) R f ()(1–)() p f R
0 0 1 1 Where p = (,)x1 x 2 and R = (,)x1 x 2
The definition can be extended to strict concavity by changing the weak inequality ≥ to the strict inequality >.
A function f is convex if and only if any pair of distinct points p and R in domain of f and for 0<θ<1 f( p (1–)) R f ()(1–)() p f R
The right hand side is the height of line segment and the left hand side is the height of the arc AB.
Figure 1
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Till now we have been discussing concavity and convexity of functions of one variable only. The conditions for concavity and convexity, strict and non-strict can be defined for functions of many variables. We shall discuss the concept of concavity and convexity for a two variable function.
ƶ = f(x1, x2)
The function f(x,y) is concave (convex) if and only if for any pair of distinct points A and B on its graph (a-surface) the line segment lies either on or below (above) the surface except at point A and B. Strict concavity requires the line segment AB lies below the arc AB. Imagine a dome-shaped surface. The surface of convex function typically be bowl- shaped. For non-strictly concave and convex function the line segment AB is allowed to lie on the surface itself, some portion of the surface, or even the entire surface may be flat rather than curved
Figure 2
3. Derivative Test for Concavity and Convexity
In the case of functions of two or more than two variable, it becomes difficult to use diagrammatic method or algebraic method to determine the concavity or convexity of function. The functions are such that they require a lot of algebraic manupulation to use the algebraic formula. A way out is to use the derivatives if the functions is differentiable.
A differentiable function f(x) = f(x1, x2,..., xn) is concave if and only if for any given point 0 2 0 1 1 1 p = (x1 , x 2 ,....., xn ) and any other point R = (x1 , x 2 ,...., xn ) in convex domain
n
f()()()(–) R f p fi f R p L1
In the case of function of two variables this can be written as :
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0 0 1 1 given P = (,)x1 x 2 and R = (,)x1 x 2
0 0 1 1 f( R ) f (p) f(x)(x1 i 2 – x 1 ) f(x)(x 2 2 2 – x 1 )
In the case of convex function the inequality will be reversed.
Geometrically it means that for a concave function the tangent plane on point p on the graph of the function lies initially above the graph of the function.
In the case of a convex function graph of the function lies strictly above all the tangent planes or the hyper planes, except the point of tangency.
Example
2 2 ƶ = x1 x 2
The function is convex if for all X = (x1, x2) and Y = (y1, y2)
2 2 2 2 f(Y) – f(X) ≥ fi (X)(y–x)i i (y 1 y 2 )–( x 1 x 2 ) ≥ 2x1 (y1–x1) + 2x2 (y2–x2)
f f 2 2 (where 2x1 and 2 x 2 ) = 2x1 y 1 – 2 x 1 2 x 2 y 2 – 2 x 2 x1 x 2 shifting the right hand side to left hand side we get.
2 2 2 2 2 2 y1 y– x 1 – x 2 – 2 x 1 y 1 2 x 1 2 x 2 y 2 – 2 x 2 ≥ 0
2 2 2 2 y1 x 1 2 x 1 y 1 y 2 x 2 – 2 x 2 y 2 ≥ 0
2 2 (–)(–)y1 x 1 y 2 x 2 ≥ 0
The expression in the brackets will remain positive whatever the value of (x1, x2) and
(y1, y2). This proves that the function is convex
4. Second Derivative and Concavity and Convexity
Till now we have discussed about curvature properties of the function by using algebra or first derivative concavity and convexity of a function is usually discussed using the second derivative. The second derivative shows how the function represented by the first derivative changes. In the case of function of one variable we saw that if f''>0 is convex which means that for f'>0 the function increases more rapidly as x increases while for f'<0 the function values full less quickly. For f''<0 the function is concave which means that for f'>0 the function value increases less quickly as x increases while for f'<0 the function value falls more quickly.
We cannot use the method of determining concavity and convexity for function of two variables (or n variables). Second partial derivatives cannot be used directly because there are infinite number of paths that one can take from same point.
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Example
2 2 ƶ = x1 x 2– 20 x 1 x 2
f1 = 2x1 – 20x2 and f11 = 2 f2 = 2x2 – 20x2 and f22 = 2 and f12 = –20 f11 and f22 are positive, the function is not strictly convex in all directions. Cross partial derivative also plays a role in determining the curvature of the function.
4.1 Total Differential Method
In order to determine the concavity (convexity) of the functions of two variables (this approach can be extended to n-variables also) we shall use the method of total differential.
Let y = f(x) the first order differential at point x = x0 is dy = f (x0) dx dy is a function of both x and dx. Let us regard dx as a given constant. dx is infinitely small change in x. Now we find the total differential of dy which we can write
d() dy d2y = d(dy) = dx dx
d[ f '( x ) dx ] dx dx
d() dx f''( x ) dx dx f '( x ) dx dx
= f''(x) dx2 to (since dx is constant)
= f''(x) dx2
This is called the second total differential of f(x). Since the term dx2 = (dx)2, it is strictly positive for any value of dx ≠ 0. It follows that d2y has some sign as f''(x). Therefore the determination of convexity and concavity which relies on the sign of f''(x) can be presented using the sign of d2y. A function is convex if f''(x) ≥ 0 and concave if f''(x) ≤ 0 then d2y = f''(x) dx2 ≥ 0 for convex function d2y = f''(x) dx2 ≤ 0 for concave functions
The same conditions relating to the sign of d2y to concavity/convexity apply to functions of n-variables. Here we shall explain this method for two variables. y = f(x1, x2)
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Total differential
f f dy dx1 dx 2 x1 x 2
= f1 dx1 + f2 dx2
The second order total differential is the total differential of dy
2 (dy) (dy ) d() dy d y dx1 dx 2 x1 x 2
()()fdx1 1 fdx 2 2 fdx 1 1 fdx 2 2 dx1 dx 2 x1 x 2
= f dx2 f dx dx f dx dx f dx 2 111 211 2 1212 222
2 2 = fdx112 fdxdx 12 1 2 f 22 dx 2 ` (f12 = f21)
2 The expression makes it clear that d y depends on cross partial derivative f12 as well as f11 and f22.
2 A function y = f(x1, x2) is twice continuously differentiable. If d y>0 whenever at least 2 one of the d x1 or dx2 is non-zero is convex. If d y<0 then the function is concave
4.2 Definations
Def: A twice continuously differentiable function y = f(x1, x2) is concave if and only if, d2y is everywhere negative semi definite
f11 ≤ 0, f12 ≤ 0, f11f22 – f12 ≥ 0
2 Def : A twice continuously differentiable function y = f(x1, x2) is convex if and only if d y is everywhere positive semi definite.
If the second order total differential is satisfies the condition d2y ≶ 0 then the function is strictly concave/convex.
The method of determining the sign of d2y directly can involve a lot of algebraic manipulation even when the function is function of two variables. In an earlier topic dealing with maxima and minima we have used quadratic forms and their properties to determine maxima-minima. Here also we can use the same method to determine the sign of d2y.
2 2 2 2 dy fdx11 1 2 fdxdx 12 1 2 fdx 22 2
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We can write the right hand side as
f11 f 12 dx1 []dx1 dx 2 ...... f21 f 22 dx2
We known that f12 = f21 from young's theorem. It follows that 2×2 matrix is symmetric. This matrix whose elements are second order partial derivatives and cross partial derivatives is called the Hession matrix and is denoted by H. Hession matrix can be used to determine the concavity and convexity of the function.
4.3 Use of Hession Matrix for the determination of Convexity and Concavity
Def: For any function y = f(x1, x2, ...., xn) = f(X) where X...... which is twice diffeerntiable with Hession H, the function f is strictly concave on Rn iff H is negative definite for all X in Rn, that is
Let dx = [dx1 dx2]
Then d2y = dxT H dX d2y = dxT H dx < 0
The function f is strictly convex on Rn if and only if H is positive definite for all x € Rn, that is
d2y = dXT H dx > 0.
The Hession H is positive definite if and only if all the leading principal minors of Matrix H are positive.
f11 f 12 f 13 H = f f f 21 22 23 f31 f 32 f 33
For example if y = f(x1, x2, x3) the leading principal minors are
f11 f 12 |H1| = |f1|, |H2| = |H3| = |H| = |f11 f12 f13| f21 f 22
If |H1|<0, |H2|>0 |H3|>0 the f(x1, x2, x3) is strictly convex (the Hession is positive definite.
If |H1|<0, |H2|>0, |H3|<0 then H is negative definite, and the f(x1, x2, x3) is strictly concave.
We now revise the condition for concavity and convexity.
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Suppose H is the Hession matrix associated with twice differentiable function y = f(X) = n f(x1, x2, ...., xn) x ∈ R then
n H is positive definite on R if and only if its leading principle minors are positive |H1|>0, n |H2|>0, .... |Hn| = |H|>0 for X ∈ R
This means that d2y>0 and so f is strictly concave.
H is negative definite on Rn if and only if its leading principle minors alternate in sign begining with a negative value for |H1|. >0 n is even |H1|<0 |H2|>0, ... |Hn| = |H| <0 n is odd
This means that d2y<0 and f is strictly concave.
Note : A leading principle minor of order r of Hession matrix is found by suppressing the last n–r rows and columns.
Example. In the case of a 3×3 matrix |H1| is found by suppressing 2nd and 3rd rows and columns.
|H1| = f11
A leading principle minor of order 2 |H2| is found by suppressing the third row and third column.
So far we have given the conditions for strict concavity and strict convexity. There are functions which are not strictly concave/convex. They are concave or convex.
Example
2 A twice differentiable function y = f(x1, x2...xn) is concave if and only if d y is everywhere negative semidefinite.
In terms of Hession matrix this means H is negative semi definite on Rn if and only if all its principle minors alternate in sign begining with negative or zero value for k=1 (HK).
≥ n is even |Hx1| ≤ 0, |HX2| ≥ 0 ... |HXn| = |H| { ≤ n is odd
2 A twice differentiable function y = f(x1, x2, ... xn) is convex if and only if d y is everywhere positive semi-definite.
In terms of H this means all its principal minors are positive or zero
|HX1| ≥ 0 |HX2| ≥ 0...... , |HXn| = |H| ≥ 0
Note : Principal minors of order are found by suppressing n-k rows and columns of H.
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f11 f 12 f 13 Example H = f f f 21 22 23 f31 f 32 f 33
|H1| = f11, H22, H33
f11 f 12 f 13
|HX2| = f21 f 22 f 23
f31 f 32 f 33
For a 3×3 Hession matrix there are seven principal minor.
For an n×n matrix there are 2n–1 principal minor.
In the case of 2×2 matrix there are three principal minors
f11 f 12 |HX1| f11, f12 |HX2| = f21 f 22
For concave function
f11 f 12 f11 ≤ 0, f22 ≤ 0, ≥ 0 f21 f 22
4.4 Graphical Representation
Let us draw a straight line
If we choose any two points b and d on the line and connect then by a line. The line connecting the points b and d also lies on the straight line which we drew in the beginning. All the point on the line connecting b and d lie on the original line.
Figure 3(a) Figure 3(b)
Now look at the circle choose any two points on or in the circle and connect them by a straight line x1x2. This straight line also lies within the circle.
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Now consider the Fig. 3(b). In this case if we connect the two points x1 and x2 by a straight line we find that there are many points on the x1x2 which do not lie within the figure shown above.
We are now in a position to define the property of straight line and the circle shown above.
A set is convex if the line joining any two points of the set lies entirely within the set. The straight line and the circle which includes the area within the circle is an example of convex set.
Figure 4
In fig. 4, we have drawn the first quadrant of the eucledium space. If we take two points a1 and a2 in the figure and connect them by a straight line then the entire line lies within 1 2 the quadrant. For example if we look at the point ƶ = a a the point ƶ lies on the 31 3 2 straight line connecting the two points a1 and a2. Any point on the straight line connecting points a1 and point a2 can be expressed as :
a1 + (1–)a2 where 0 ≤ ≤ 1 [or ∈ [0, 1]]
Such a point is called a convex combination of pair of points a1 and a2
Def: A set S is convex if for every pair of points x1 ∈ S and x2 ∈ S, point x̄ = x1 + (1–
)x2is also an element of , for every value of when 0 ≤ ≤ 1. A set containing only one point is a convex set. Null set is also considered as a convex set.
Figure 5
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A sold circle is a convex set.
Algeberically x2 + y2 ≤ r defines a convex set.
The equation depicts the circle plus the point in it.
The circle is not hallow. All the three figures given above are examples of convex sets.
Solid figure given below are not convex sets
Figure 6
In these figures there is a feature of reentrance (and also a hole). This is a cause for non-convexity. To qualify as a convex set, the set of points in the figure must contain no holes, and its boundary points must be not be reentered anywhere.
The geometric definition of convexity also applies to points in 3-space as well as n- space. A solid cube is a convex set. A hollow cylender is non convex.
A function which gives rise to a hill over its entire domain is a concave function. A function which gives rise to a valley over its entire domain is convex function.
If the hill (or valley) does not contain any flat surface then the function is suidth be strictly concave (convex) function. In case a function which give rise to hill (or valley) and contains flat surface also, is a concave function (convex function).
4.5 Assumption
The domain of the function is a convex set. This assumption is necessary because we use the combination of x1 and x2 in the domain D to prove whether the f is a concave or convex function.
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Figure 7
In fig. 7, let x ≥ 0 be the domain of the function. This domain is a convex set. If we take two values of x in the domain x1 and x2. The associated values of the function are f(x1) and f(x2) connect these two points by a straight line AB. The graph of the function is also given in the fig1. The graph of f is shown by are AB.
The straight line lies below the arc AB. The value of the function at x̄ between x1 and x2 is f(x̄ ) = C. This is higher than the point (D) on line AB immediately above the value x̄ . It is clear that f(x̄ ) > d. This property can be expressed as strict concavity of the function.
The value of x between x1 and x2 can be written as x̄ = x1 + (1 – )x2 where ∈ [0, 1] or 0 ≤ ≤ 1
The point d on the line AB can be expressed as d = f(x1) + (1 – ) f(x2) 0 ≤ ≤
The point c on the arc AB is c = f(x1 + (1 – )x2)
The function is strictly concave if c = f(x1 + (1 – )x2) > f(x1) + (1 – ) x2
In simple words if we take only two points on the domain where the domain is a convex set then convex combination of these points is also in the domain of the function.
4.6 Theorem
The f is strictly concave if c = f(x1 + (1 – )x2) > f(x1) + (1 – ) f(x2) ∈ [0, 1] and x ∈ Dan interval which is a convex set.
The function f is concave if f(x1 + 1 – )x2) ≥ f(x1) + (1 – ) f(x2) 0 ≤≤1 f(x)
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Figure 8
In fig. 8, line AB lies entirely above the graph of the function except at point A & point B. the f(x) is a convex function. A convex function bends below the line joining points f(x1) and f(x2) (AB).
A function f(x) is strictly convex if f(x1 + (1 – )x2) < f(x1) + (1 – )x2 where ∈ [0, 1]
It is a concave function if f(x1 + (1 – )x2) ≤ f(x1) + (1 – )x2
A linear function is a convex and concave function because it satisfies the conditions of both convex and concave function.
Figure 9a Figure 9b
Figure9c
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Another way of defining concave and convex functions is as discussed below.
Def : A function f is concave on the convex set D where D = [a, b] if {(y, x)| h ≤ f(x), x ∈ D} is a convex set.
For a convex function the inequality is reversed. In fig 9(a) the function is convex and in fig 9(b) the function is concave. In fig 9(a) look at all the points above the graph of the function and below the straight line k parallel to x-axis. The set of points satisfy the above definition and are actually a convex set. Similarly the shaded area in fig b is a convex set. The function depicted in fig a is a convex function on the convex domain [a, b].
Now observe fig 9c. There is a tangent at point x0. The tangent line h on any point on a concave function will lie above the function (except at point f(x0)). For a convex function the tangent line at any point x0 will lie below the graph of the function.
It means if f is concave and differentiable the n f(x) ≤ f(x0) + f'(x0) (x – x0) (1) for any x0 and any other point X.
The right hand side is actually the equation of the tangent at x0 on the function. If we move slightly away from point x0 on either side of x0 the tangent line at point x0 lies above the graph of the function.
For convex function the inequality is reversed. f(x) ≥ f(x0) + f'(x0) (x – x0) (2)
In the case of convex function (if the function is differentiable) the tangent line at (x0, f(x0)) will lie below the graph of the function except at point (x0, f(x0))
5. Second derivative test for concavity and convexity.
If the function is differentiable twice then we can use the second derivative to test the concavity and convexity.
Figure 10a Figure 10b
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In fig. 10(a) tangent at any point on the graph of the function lies below the graph of the function. In fig. 10(b) the tangent line on any point on the graph of the function lies entirely above the graph of the function. We can use the second derivative to determine the concavity and convexity of the function. We know that the sign of the first derivative on an interval (a, b) tells us whether the function is increasing or decreasing. We also know that the first derivative at a point on (a, b) give us the slope of the tangent at that point.
f'(x) > 0 on an interval (a, b) means that the function is increasing on (a, b)
f'(x) < 0 on an interval (a, b) means that the function is decreasing on (a, b)
f''(x) (the second derivative) is the derivative of f'(x)
therefore f''(x) > 0 iff f'(x) is increasing an (a, b)
f''(x) < 0 iff f'(x) is decreasing on (a, b)
If f''(x) > 0 it means the slope of the tangent is increasing as we move from left to right on the graph. In the fig (a) the slope of the tangent is increasing when we move from x1 to x2, where x1 < x2. This happens when the function is convex.
If f''(x) < 0 on (a, b) then tangent becomes flatter when we move to x0 from the left. It means the slope of the tangent is decreasing as we move from left to right on the graph.
In other words when we move from x1 to x2 where x1 < x2 the slope of the tangent is decreasing.
We conclude f is strictly concave on interval I if and only if f''(x) < 0 for all x in the interior of I.
Function f is strictly convex on interval I if and only if f''(x) > 0 for all x on interior of I.
Example
2 Show that x + x is convex on R
The function is convex if
Y X (y + y ) – (x + x ) ≥ (2x1 2x2) Y X
= 2x1y1 – 2x + 2x2y2 – 2x
y + y – x – x2 + 2x + 2x – 2x1y1 – 2x2y2 ≥ 0
y + y + x + x – 2x1y1 – 2x2y2 ≥ 0
2 2 = (y1 – x1) + (y2 – x2) ≥ 0
2 This is true for all (x1 x2) and (y1, y2) in R
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We can also use the second derivative test to check the convexity f1 = 2x1, f2 = 2x2, f11=2, f12 = 0, f22 = 2 f11 < 0 f22 > 0 f11 f22 – (f12)2 > 0 = 4 0 > 0 proved
6. Quasi-concave and Quasi-convex Functions
While discussing concave and convex functions we saw that if the function is concave (convex) there is no need to check the second order condition to determine whether the function achiever maxima (minima) or not. When we are dealing with problem of constrained optimization, it is again possible to dispense with the second order condition. In the case of constrained optimization quasi-concavity of the function obviates need for second order condition for determining the maxima. In a similar manner quasi-convex function removes the need for second order condition when we are trying to find out minimum of the function.
In the beginning we shall discuss the concept of quasi-concavity and quasi-convexity in the case of function of single variable. We are using fig. 11(a).
1) Let f be a function of x [y = f(x)]
2) x and y are non-negative [x, y ≥ 0]
This means that we are in the first quadrant.
3) The domain of the function is convex set.
4) Choose two distinct point xi and xj (or x1 and x2) such that xi < xj in the convex domain of the function.
5) The function f(x) forms an arc between xi and sj such that f (xi) = A and f (xj) =B
Fig 11(a) Fig 11(b)
In fig. 11(a) point B is higher in height than A. In other words f(xj) > f(xi). The function is strictly quasi-concave if all other points on are AB are higher in height than point A.
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In fig. 11 (b).
1) The domain of the is Ist quadrant (xi, xj ≥ 0)
Domain is convex set.
2) Two distinct points are xi and xj such that xi < xj.
3) The function f(x) forms an arc CD between xi and xj such that f(xi) = c and f(xj) =
D. In fig(b) f(xj) < f(xi). The function is strictly quasi-concave if all other points on
the arc are lower in height than f(xi)
Now we shall give an algebric definition of quasi-concavity and quasi-convexity.
Let f be a function of x. Then for any two distinct points xi and xj in the convex domain of the function such that xi < xj and 0 < θ < 1, the function is strictly quasi-concave function if the following inequality is satisfied.
f(xj) ≥ f(xi) f[θxi + (1–θ)xj] > f(xi)
If we replace the strict inequality with weak inequality then the function is quasi- concave. The weak inequality implies that there is some horizontal straight line segment also on the arc AB.
7. Quasi-convex function
Suppose f is a function of x. then for any two points xi and xj and for 0 < θ < 1 the function is strictly quasi-convex if the following inequality is satisfied.
f(xi) > f(xj) f(xi) > f[θxi + (1 – θ)xj]
If we replace the strict inequality with weak inequality the function satisfy the condition is quasi-concave.
f(xi) ≥ f(xj) f(xi)≥ f[θxi + (1 – θ)xj]
Differentiable functions
If f is a function of x and is differentiable then the function is strictly quasi-concave if for the two points xi and xj on their convex domain such that xi < xj, the following inequality is satisfied.
f(xj) > f(xi) f'(xi) (xj – xi) ≥
where f'(xi) = /x=xi
The function is quasi-concave if
f(xj) ≥ f(xi) f'(xi) (xj – xi) ≥ 0
The function is strictly quasi-convex if
f(xi) > f(xj) f'(xi) (xj – xi) < 0
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Quasi-convex if
f(xi) ≥ f(xj) f'(xi) (xj – xi) ≤ 0
8. Properties of Quasi-concave and Quasi-convex functions
1) A linear function is both quasi-convex and quasi-concave.
2) All concave (convex) functions (strict or non strict) arc quasi-concave (quasi- convex). But the opposite is not true.
3) If f(x) quasi convex (strict or non-strict) then –f(x) is quasi-concave.
9. Functions of multiple variables
Let a function z = f(x, y) of two variables
2 The function is defined on R ++ (x, y ≥ 0)
Let u = (x1, y1) and v = (x2, y2) and v > u then f(x, y) is quasi-concave if f(v) > f(x) f[θx + (1 – θ)v] > f(x) strictly quasi-concave f(v) ≥ f(x) f[θx + (1 – θ)v] ≥ f(x) quasi-concave f(v) < f(x) f[θx + (1 – θ)v] < f(x) strictly quasi-convex f(v) ≤ f(x) f[θx + (1 – θ)v] ≤ f(x) quasi-convex
10. Differentiable function
Suppose a function z = f(x1, x2, ...... xn) is twice continuously differentiable. The quasi- concavity and quasi-convexity of the function can be checked with the help of first and second partial derivatives of the function arranged as a bordered determinant.
0 … … … . … … … … . |B| = … … … … . . . . … … … …
The bordered determinant appear to be similar to bordered Hessiam |H| determinant which we use when we deal with constrained maxima (minima). In the case of of bordered Hessian |H| the first row (column) of |H| consists of 0 and fire derivatives of constraint. In this case first row (column) of |B| consists of 0 and first derivatives of the function f (x1, x2,...., fxn) It is because the quasi-concavity (quasi-convexity) depends exclusively on the partial derivatives of function of itself. We use B along with its leading principal minor
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0 0 |B1| = |B2| = |Bn| = |B| = 0
The domain of the function is non-negative or thant (that is x1, x, x3 ...... xn ≥ 0)
z = f(x1, x2,...... xn) is quasi concave on the .
|B1| ≤ 0, |B2| ≥ 0, |Bn| = |B| ≤ 0 if n is odd = |B| ≥ 0 if n is even The function is strictly quasiconcave on the If |B1| < 0, |B2| > 0,..... |Bn| = |B| < if n is odd = |B| > if n is even
Quasi-convex.
|B1| ≤ 0, |B2| ≤ 0, |B3| ≤ 0,...., |Bn| = |B| ≤ 0
Strictly quasiconvex if
|B1| < 0, |B2| < 0,...... |Bn| = |B| < 0.
11. Exercises
1) Are the following function quasiconcave ? Which of them are also concave
a) f(x, y) = x1/2 + y1/4 fx all x, y≥ 0
b) f(x, y) = x1/4 + y3/4 x, y ≥ 0
c) f(x, y) = x2 y3
d) f(x, y) = x y2
2) Which of these function defined on are quasiconvex which are also convex
a) +
b) 3 + 4
c) 2x1 + 3x2 –
3) Are the following functions concave or convex?
a) z = – (x + x ) is it a concave function ?
b) z = xα yβ show that it is concave
if 0 α < , β < 1 and 1 – (α + β) > 0
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c) –3x2 + 2xy – y2 + 3x – 4y + 1 Ans (Concave)
d) x4 + x2 + y2 + y4 – 3x – 8y (Convex)
e) f(x, y) = – xy Neither concave nor convex.
f) x – y – x2
g) –6x2 + (2a + 4)xy – y2 + 4ay
What is the value of a which will make the function concave
h) f(x, y) = –2x2 – y2 + 4x + 4y – 3 for all (x, y)
has a maximum at (x, y) = 1, 2
4) Is this function concave or convex?
/ 2 a) (x1 + x2) defined on R ++
Show that is concave but not strictly concave
2 b) 3x1 + x defined on all R
Show that it is neither strictly concave nor strictly convex.
12. References
Allen, R.G,D, Mathematical Analysis for Economists, London: Macmillan and Co. Ltd
Knut Sydsaeter and Peter J. Hammond, Mathematics for Economic Analysis, Prentice Hall
Carl P. Simon and Lawrence Blume, Mathematics for Economists, London: W .W. Norton & Co.
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