Final Solution - Economics 2P30 Foundations of Economic...

Economics 2P30
Foundations of Economic Analysis
Lester M.K. Kwong
Department of Economics
Brock University
Winter 2013
Final Examination - Suggested Solutions
Time: 3 Hours
This examination will not be deposited in the library reserve.
Section A: Definitions
∗ ∗ ∗ ∗ ∗ ∗ ∗ Define 6 of the following 8 terms in two sentences or less. ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ This Section of the exam is worth 10%. ∗ ∗ ∗ ∗ ∗∗
1. Continuous function at some point x
2. Strictly concave function
3. Homogeneous function of degree k
4. Mean Value Theorem
5. Complement of a set X
6. Contrapositive of P ⇒ R
7. Compact set
8. Diameter of a set X
Solution :
1. A function f is said to be continuous at some point x if ∀ δ > 0, ∃ ✏ > 0 so that if d ( x,y ) < ✏
then d ( f ( x ) , f ( y )) < δ .
2. A function f is said to be strictly concave if for all x ≠ y , f ( ↵ x + ( 1 − ↵ ) y ) > ↵ f ( x ) + ( 1 −
↵ ) f ( y ) for all ↵ ∈ ( 0 , 1 ) .
3. A function f is said to be homogeneous of degree k if for all λ > 0 and for all x > 0,
f ( λ x ) = λ
k
f ( x ) .
4. If f ∈ C
1
over [ a,b ] then there exists some c ∈ ( a,b ) so that:
f
′
( c )( b − a ) = f ( b ) − f ( a )
5. The complement of a set X is X
c
= { x ∶ x ∉ X } .
1

6. The contrapositive of P ⇒ R is ∼ R ⇒ ∼ P .
7. A set is said to be compact if it is both closed and bounded.
8. The diameter of a set X is diam ( X ) = sup { d ( x,y ) ∶ x,y ∈ X } .
Section B: Proofs
∗ ∗ ∗ ∗ ∗ ∗ ∗ Choose 6 of the following 8 questions. ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗∗ This section of the exam is worth 40%. ∗ ∗ ∗ ∗ ∗∗
For each of the following, answer true or false. If true, prove. If false, derive a counterexample
1. If f ∈ C
1
is homogeneous of degree k then f
′
is homogeneous of degree k − 1.
Solution : True. If f is homogeneous of degree k then:
f ( λ x ) = λ
k
f ( x )
Taking the derivative of the above identity with respect to x yields:
f
′
( λ x ) λ = λ
k
f
′
( x ) ⇒ f
′
( λ x ) = λ
k − 1
f
′
( x )
and hence f
′
is homogeneous of degree k − 1.
2. The function f ( x ) = 20 + x is weakly concave and weakly convex over R .
Solution : True. f
′′
( x ) = 0 and so f
′′
≥ 0 and f
′′
≤ 0 for all x ∈ R . Hence, f is weakly
concave and weakly convex over R .