Exercise: Constrained Optimization with One Inequality Constraint
2
2
1. Max ( x 4) ( y 9)
s.t. x 2 y 10
2
2
2. Max x 5 y x y
2
s.t. x y 0.5
2
2
3. Min 4 x 2 y y
2
2
s.t. x y 4
Maxima and Minima
1. Find all critical points of f (x, y) = x2 + y 2 .
Solution. We want fx = fy = 0. We calculate fx = 2x, fy = 2y , and the only way that this can be
true is for x and y to both be 0. Thus, the only critical point is (x, y) = (0, 0) .
2.
Problem 1.
Find the critical points of the function
f (x, y) = x3 + 6xy + 3y 2 9x
and determine their nature.
We calculate
fx = 3x2 + 6y 9, fy = 6x + 6y.
Setting
fy = 0 = y = x.
Using fx = 0 we find
3x2 6x 9 = 0 = x2 2x 3 = 0 = x = 1 or x = 3.
We find the
SOLUTIONS
Problem 1.
Find the critical points of the function
f (x, y) = 2x3 3x2 y 12x2 3y 2
and determine their type i.e. local min/local max/saddle point. Are there any global min/max?
Solution: Partial derivatives
fx = 6x2 6xy 24x, fy = 3x2 6y.
To find
Math 209
Solutions to assignment 3
Due: 12:00 Noon on Thursday, October 6, 2005.
1. Find the minimum of the function f (x, y, z) = x2 + y 2 + z 2 subject to the condition
x + 2y + 3z = 4.
Solution. Lets define g(x, y, z) = x + 2y + 3z, so the problem is t
Chapter 7
Calculus of Several Variables
7.1
Functions of Several
Variables
7.
g(x, y) =
1.
g(1, 2) =
f (x, y) = (x 1)2 + 2xy 3 .
g(2, 3)
The domain consists of all ordered pairs (x, y)
of real numbers. Moreover
=
y
x
+
x y
2 1
5
+ =
1 2
2
3
2
13
+
=
2
3
6
5. Evaluate the following:
a. i% 5+4 c. Vf(x,y, z) where f(:v, y, z) = 80/(1 +902 + 2y2 + 322)
9
_ 22 71. _ a:
b. f/(x) ofy _ (2a:+1) d. f (m) of f(:c) $6 XZX knex
For the given cost and demand functions, nd the production level q that will maximize
pro
Econ 106 Review Problems
1.
9
Find the critical values for minimizing the costs of a firm producing two goods x and y when
the total cost function is c = 8x2 xy + 12y2 and the firm is bound by contract to produce
a combination of goods totaling 42. Solve
Mathematical Economics
Problem Set # 1
Submission Date: January 9, 2012 (Wednesday, at the start of class)
1/ Suppose the soccer team S, the baseball team B, and the table tennis
team T, decide to form a varsity club V, where the number of players in S is
2/22/2013
Exercises
D 1. Examine the following functions for local maxima
and minima. Verify your answers by plotting the
function.
I (a)f(x) = 5x3 - 30:2 + 300
. (b)f(x) = 1:53;: > o
I (c)f(x)=5x+:,x>0
a 2. At what output Q is totalrevenue TR maximized
i
Mathematical Economics
Problem Set #2
(Submission Date: January 25, Friday, 10:00 am.)
@ Let a simple national-income model be dened by the following equations.
Find the effect of every unit of change in investment (Ia) on equilibrium income
(Y), i.e. 6Y/
Exercise: Concavity, Convexity, Inflection
Determine whether the following functions are concave or convex over their entire domain. If
not, identify their inflection point/s and regions of concavity and convexity.
1. y = 3x2 2x + 1 + e-x-3
2. y =
3. y =
X’
\. \
)’
. i
._ -' ' . , “WW”- >
0
Assignment 2&3: Group Quiz (Sets and Spaces; Fiinctionsi)
Econ. 106
X Consider a system of m functionals on X g 82" : f(x) =(f1(x), f2(x), fm(x), ),
which is assumed to be differentiable at, x0 6 X. Write out the matri
Exercise: Elasticity and Single-Variable Optimization
Compute yx (or the elasticity of y with respect to x) of the following functions.
1. y =
2. y = (x3+1)10
3. y =
4. y6 = x5
5.
(
) (
)
Use the first derivative test to obtain the optimum points for the
Economics 106 THW
Elements of Mathematical Economics
Second Semester 2012-2013
Exercise Set 3
G.M. Ducanes
1. (Definiteness) Express each quadratic form as a matrix product involving a symmetric coefficient
matrix and determine its definiteness using eige
Exercise: Profit Maximization, Inventory Control, and Optimization in a closed and
bounded interval
1. What is the maximum profit a firm can make if it faces the demand schedule p = 660 3q and
the total cost schedule TC = 25 + 240q 72q2 + 6q3?
2. If a fir
Exercise: Constrained Optimization with One Equality Constraint
1. The prices of inputs K and L are given as $6 per unit and $9 per unit respectively, and a
2 / 3 1/ 3
firm operates with the production function Q 20K L .
a. What is the maximum output that
Exercise: Profit Maximization, Inventory Control, and Optimization in a closed and
bounded interval
1. What is the maximum profit a firm can make if it faces the demand schedule p = 660 3q and
the total cost schedule TC = 25 + 240q 72q2 + 6q3?
2. If a fir
3/2/2013
Min f(x,y)=x2+xy+y2+x+y
What is the effect on the optimal value of f of
a marginal change in ?
What is the minimum f when =1 ?
What is the minimum f when =2 ?
Max (q1,q2)=100q1-q12+100q2-q22-(q1+q2)
What is the effect on the maximum profit of
Economics 106 THW
Elements of Mathematical Economics
Reading List
Second Semester 2012-2013
Exercise Set 2
G.M. Ducanes
1. Assume that demand for good (Q) depends on its own price (P), income (M) and the price of a
substitute good (S) according to the dem
Exercise: Unconstrained Optimization of Functions of Two Variables
1. Suppose a monopolist is practicing price discrimination in the sale of a product by charging
different prices in two separate markets. Suppose the demand curves are P1=100-Q1 P2=100-Q2
Econ 106
Assignment 1: Matrix Algebra
1.Compute determinants |A| , |Bl , |AB| , IBAI , |A’ I and {B’l for each of the following square
matrices:
—1 0 2 4
A- [5 4:34—19]-
2.Compute inverses A“1,B_1, and (AB)_3l for each of the following matrices:
1 —1 —2 0
1 2 3 100
2 4 6 200
d. 3 6 9 300
100 200 300 10000
6. Find the solutions to the following system of equations using:
3xz = 4
3x+y+z 0
5x+2z = 7
,a) elimination method
b) inverse matrix of coefcients
gCramers rule
. Find the value of :1: if:
, 1 2 5 l 2 3