Microsoft Word - ECON2016 Unit 4 and 5 Practice Questions.docx
1
Units 4 and 5 Additional Practice Questions
Find additional practice questions for Units 4 and 5 in preparation for the graded quiz.
Solutions will not be posted but you can email me if you have any specific queries.
Unit 4 – Convexity and Concavity
Question 1
Classify the stationary point(s) of each of the following functions as convex, concave
(strictly or weakly) or neither.
(i)
(ii)
(iii)
(iv)
Question 2
Show that the Co
– Douglas production function , with
and is strictly concave for .
Unit 5 - Unconstrained and Constrained Optimization
Question 1
Find the global maximum and minimum values of each function and indicate where
they occur.
(i) subject to .
(ii) subject to .
(iii) subject to .
(iv) subject to .
Question 2
Find the stationary points and the values of the Lagrange multipliers for the
following problem.
(i) Min subject to
(ii) Max subject to
(iii) Max subject to and
Question 3
Determine whether the stationary point represents a maximum/minimum
subject to
Question 4
Dell, a manufacturer of computers and laptops, finds that it takes units of labour
and units of capital to produce units of the product. If a unit
of labour costs $100, unit of capital $200, and $200, 000 is budgeted for production,
2
2
2
12121 ),( xxxxxxf --=
21
2
22
2
XXXXXXXXXX),( xxxxxxxxf -+++=
21
2
2
2
121 32),( xxxxxxf ++=
12333),( 221
23
22
2
121 +--+= xxxxxxxf
( ) ba 2121, xAxxxf =
1,0
ba 1<+ ba 0,0 21
xx
2121 43),( xxxxf += 11,10 21 ££-££ xx
523),( 22
2
121 ++= xxxxf 102,100 21 ££££ xx
2
2
2
121 ),( xxxxf += 41,31 21 ££-££- xx
21
2
2
2
XXXXXXXXXX),( xxxxxxxxf +--+= 10,10 21 ££££ xx
f (x1, x2 ) = 4x1x2 + 7x1x3 + 9x2x3 x1x2x3 = 2016
f (x1, x2, x3) = 3x
2
1x2x3 x1 + x2 + x3 = 32
XXXXXXXXXX),,( xxxxxxf ++= 2
2
2
2
1 =+ xx 132 =+ xx
f (x1, x2 ) = 20x1 − 4x1
2 + 9x1x2 − 6x
2
2 +18x2 x1 +3x2 = 64
1x
2x f (x1, x2 ) =100x1
0.75x2
0.25
2
determine how many units should be expended on labour and how many units
should be expended on capital in order to maximize production. (You MUST justify
that the stationary point you found does indeed maximize the given function).
Question 5
Find the amount of capital (K) and labour (L) that should be employed to maximize
output (q) given the following Co
-Douglas production functions and constraints
set by PK, PL and budget B:
(You MUST justify that the stationary point you found does indeed maximize
the given function).
Question 6
Show that following problem is a convex programming problem and use the KKT
conditions to find the optimal solution.
Question 7
Consider the following nonlinear programming problem
Determine whether can be optimal by applying the KKT
conditions.
q(K,L) = K 0.4L0.6
PK = 8,PL = 6,B = 300
Maximize f (x1,x2 ) = 24x1 − x1
2 +10x2 − x2
2
subject to XXXXXXXXXXx1 ≤ 8
XXXXXXXXXXx2 ≤ 4
and XXXXXXXXXXx1 ≥ 0, x2 ≥ 0
0,0 XXXXXXXXXXand
3 XXXXXXXXXXsubject to
XXXXXXXXXXMaximize
21
21
3
22
3
1
2
11
³³
£+
-+-+=
xx
xx
xxxxxZ
( ) )2,1(, 21 =xx
ECON2016_u XXXXXXXXXXv1rec
UNIT 4
Convexity and Concavity
Overview
In this unit, we will use the Hessian matrix to solve determine the convexity and concavity
properties of a function of several variables. These properties are critical to the optimization
process.
Learning Objectives
By the end of this unit, you will be able to:
1. Identify the graphs of concave and convex functions of one variable.
2. Compute the Hessian matrix of functions of several variables.
3. Use the Hessian matrix to determine the convexity and concavity properties of a
given function of more than one variable.
This unit has one session as follows:
Session 4.1: Hessian Matrix
Readings & Resources
Note to Students: Sometimes hyperlinks to resources may not open when
clicked. If any link fails to open, please copy and paste the link in your
owser to view/download the resource.
Required Reading
Hoy, M., Livernois, J., McKenna, C., Rees, R., & Stengos, T XXXXXXXXXXMathematics for
economics. MIT Press.
Required Resources
Khan Academy: Concavity
https:
www.khanacademy.org/math/ap-calculus-a
ab-derivatives-analyze-
functions/ab-concavity/e
ecognizing_concavity
Khan Academy: Hessian Matrix
https:
www.khanacademy.org/math/multivariable-calculus/applications-of-
multivariable-derivatives/quadratic-approximations/a/the-hessian
Session 4.1
Concavity and Convexity
Introduction
This session focuses on computing the Hessian matrix of a function of several variables. We will
also use the Hessian Matrix to determine the convexity and concavity properties of a variety of
multivariate functions.
Preliminaries
Recall the types of functions:
One variable: ?(?) = ?' + 6? + 7
Two variables: ?(?, ?) = 6?? + ?' − 2?/
Three variables: ?( ?, ?, ?) = ?/ + 6???' + 7?? a
N variables: ?2 ?3 ,?' , ?/ ……… . ?67 or ?( ? ) where x = 2 ?3 ,?' , … . . ?67
Notation: ?;< =
= >? (@A,…….@B)
=@C
Example:
Let ? 2 ?3,?'7 = ?3' + 6?3?' + ?'/ . Then
?3 = 2?3 + 6?' , ?' = 6?3 + 3?''
?33 = 2 , ?3' = 6 ?'' = 6?' , ?'3 = 6
If partial derivatives are continuous ?3' = ?'3
In general, ?;< = ?<;
The Gradient Vector
The gradient vector ∆? =
⎝
⎜
⎛
?3
?'
⋮
⋮
?/⎠
⎟
⎞
, consists of first order partial derivatives.
Example:
Find ∆? for ?2?3,?'7 = 4 + 5?3 + 6?'
First ?3 = 5 ?' = 6
So ∆? = O56P
For 2nd order derivatives, we have the following notation:
∆' ? = R
?33 ?3'
?'3 ?''
S for n = 2
∆' ? = T
?33 ?3' ?3
?'3 ?'' ?'
?/3 ?/' ?
U for n = 3
∆' ? = V
?33 ?3' ?36
⋮ ⋮ ⋮
?63 ?6' …?66
W for n variables.
Concavity and Convexity
Recall what you learnt in your introductory calculus course. In Figure 4.1 below, we have a
polynomial graph with a concave and a convex portion as well as a point of inflection. These give
you an idea of what is meant by a function that is convex and one that is concave.
Figure 4.1
Now here in Figure 4.2 we have a fully concave function.
Figure 4.2
Any point ?Y between ?? and ?? (inclusive) can be written as
�̅� = ??3 + ( 1 − ?) ?' , ? ? [0,1] where 0 ≤ ? ≤ 1
Now ? Y is between ?(?3) and ?(?')
So ?̅ = ? ?(?3 ) + ( 1 − ?) ?(?') , ? ? [0,1]
In terms of formal definitions:
The function ? is concave if:
?(�̅�) ≥ ? ?(?3 ) + ( 1 − ?) ?(?')
Where ? Y = ??3 + ( 1 − ?) ?' and ? ? [0,1]
On the other hand, the function ? is convex if:
?(�̅�) ≤ ? ?(?3 ) + ( 1 − ?) ?(?')
It is strictly concave if strict inequality holds when ? ? [0,1]
Figure 4.3
As you will recall, to show convexity and concavity, we use the first and second derivatives
subject to specific conditions as shown below.
Figure 4.4
?f(?) > ? ?f(?) < ?
?ff(?) < ? ?ff(?) < ?
Figure 4.5
Strictly convex functions
?f(?) > ?, ?′′(?) < ? ?f(?) < ?, ?′′(?) < ?
Figure 4.6
Strictly concave functions
Proving convexity and concavity can also be shown with functions requiring total differentials.
Let us recap functions of two variables to show this.
Definition
The first order total differential for the function ? = ?2?3,?'7 is
?? = ?3 ??3, + ?' ??'
The second order total differential for the function ? = ?2?3,?'7 is
?'? = ?33 ??3' + 2 ?3' ??3 ??' + ?'' ??''
where ??'' = (??;)'
Theorem
Given the function ? = ?2?3,?'7 . If ?'? > 0 whenever at least one of ??3 ?? ??' is non-zero then
? = ?2?3,?'7 is a strictly convex function.
Let us now apply what we have reviewed to matrices, specifically the Hessian Matrix.
Recall:
Given ? = ?2?3,?'7
?'? = ?33 ??3' + 2 ?3' ??3 ??' + ?'' ??''
A function is strictly convex iff ?'? > 0 whenever at least one of ??3 ?? ??' is non-zero.
Example:
Show that ?2?3,?'7 = ?3' + ?'' is strictly convex.
First
?3 = 2?3
?'y 2?'
?33 = 2
?3' = 0
?'' = 2
?'3 = 0
Now ?'? = 2 ??3' + 2?'' > 0 if one of ??3 ??? ??' is non-zero so ? is strictly convex.
Theorem
The function ? = ?2?3,?'7 is
(Weakly) convex iff ?'? ≥ 0
(Weakly) concave iff ?'? ≤ 0
Example
Show that 2?3,?'7 = 5 − (?3 + ?')' is concave
Now
?3 = −2(?3 + ?')
?' = −2(?3 + ?')
?33 = −2
?'' = −2
?3' = −2
So ?'? = −2 ??3' − 4 ??3 ??' − 2??''
= −2 (??3' + 2??3 ??' + ??'')
= −2 ( ??3 + ??')' ≤ 0, therefore ? is concave
We want to extend the idea of concavity and convexity to ? (?3 , ?' … . ?6)
Now ?? = ?3 ??3, + ?' ??' + ⋯+ ?6 ??6 = ∑ ?;6;y3 ??;
And ?'? = ∑ ∑ ?;< 6;y3 6
Or ?'? = ??~ ? ?? where. ? = V
?33 ⋯ ?36
⋮ ⋱ ⋮
?63 … ?66
W , is
the Hessian matrix of ?