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
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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
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
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
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
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
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
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: 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
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 =