Optimisation

Question 1

What does ‘only if’ mean?

Answer 1

Necessary but not sufficient

Question 2

What does ‘if’ mean?

Answer 2

Sufficient but not necessary

Question 3

What does ‘iff’ mean?

Answer 3

Both necessary and sufficient

Question 4

Describe the interval (a,b)

Answer 4

It is an open interval, that does not include a or b

Question 5

Describe the interval [a,b]

Answer 5

closed bound interval aka. including a and b

Question 6

What is a strict inequality?

Answer 6

<>

Question 7

What is a weak inequality?

Answer 7

≤≥

Question 8

How do you denote a maximising solution?

Answer 8

*

Question 9

What does a convex function look like?

Answer 9

U

Question 10

What does a concave function look like?

Answer 10

∩

Question 11

How can you determine convexity or concavity of a 1-variable function?

Answer 11

1. Using second order derivatives
   Convex (min point) → f’‘(x)≥0
   Concave (max point) → f’‘(x)≤0
2. Line segment
   If the line segment joining any 2 points on the graph f(x) is below (above) the graph, or on the graph∴f(x) is concave (convex) if for any two x1,x2 , and t∈[0,1]

tf(x1)+(1-t)f(x2 )≤f(tx1+(1-t)x2)

If the inequality holds as ≥ the function in convex

Question 12

How can you determine convexity or concavity of a 2-variable function?

Answer 12

1. ABC:
   If A<0 and AC-B2>0 → concave (max)

If A>0 and AC-B2>0 → convex (min)

2. Line segment
   A function f(x_1,x_2) defined over a convex set S is concave if…

λf(x)+(1-λ)f(y)≤f(λx+(1-λ)y) If the inequality holds as ≥ the function in convex

Question 13

What is a local extreme point?

Answer 13

An extreme point over a given domain of the function

Question 14

What is a global extreme point?

Answer 14

An extreme point for the entire function

Question 15

What are the sufficient conditions for extreme points?

Answer 15

If f(x) or f(x,y) or L(x,y,λ) is a convex/concave continuous function over an interval I/concave set S, and c is an interior stationary point, then c in a min/max point for f(x)/f(x,y) in I/S

Question 16

What are the first order conditions for 1 and 2 variable functions?

Answer 16

1-varible → f(x) is differentiable over an interval I and that c is an interior point of I. Then c is an extreme point in I only if f’(x)=0

2-varibles → f(x,y) is differentiable over an interval I and that (x0,y0) is an interior point of I. Then c is an extreme point in I only if f’1 (x0,y0)=0 and f’2(x0,y0)=0

Question 17

Can you maximise a transform function?

Answer 17

Maximising/minimising a transformed function F(f(x)) is equivalent to maximising f(x), where F is strictly an increasing function

Question 18

What is the method of find + determining stationary points for 1-variable functions?

Answer 18

Finding an extreme point:

1. Check interior extreme points using FOC → f’(x)=0
2. Check end point values, and where f(x) isn’t defined for non-interior extreme points

Defining a stationary point, c:
1. Sufficient conditions → show if concave/convex to show point is a max/min

2. Let f^j(x) be the jth derivative and n the smallest no. such that f^n (x)≠0. Then c is:
   - A local max. if n is even and f^n (c)<0
   - A local min. if n is even and f^n (c)>0
   - An inflection point if n is odd

Question 19

What is the method of find + determining stationary points for 2-variable functions?

Answer 19

Finding an extreme point:

1. Check interior extreme pointing using FOC) → f’1 (x0,y0)=0 and f’2(x0,y0)=0
2. Evaluate (solve simultaneous) the function at stationary points and compare
3. Check end point values, and where f(x,y) isn’t defined to see if extreme point isn’t interior

Defining a stationary point, (x0,y0):
1. Determine if graph is concave/convex → shows if point is a max/min
2. Evaluate the function at stationary points and compare
Let: A=f’‘11 B=f’‘22 C=f’‘12 Then (x_0,y_0)is:
If A<0: AC-B2>0 → strict max AC-B2<0 → saddle point
If A>0: AC-B2>0 → strict min AC-B2=0 → test is inconclusive

Question 20

Define a saddle point

Answer 20

Saddle point (x0,y0) is a stationary point with the property that arbitrarily close to (x0,y0 ) there exists point (x,y) with f(x,y)

Question 21

What is a binding constraint?

Answer 21

A constraint that stops you from achieving the actual max

Question 22

How would you solve a non-linear constraint?

Answer 22

Solve normally, if constraints are violated + look for corner solutions

Question 23

How would you solve a non-negativity constraint?

Answer 23

Solve normally if x^≥0 and y^≥0 you are done. If x^<0 and y^<0 the constraints are violated + you must look for corner solutions

Question 24

What are the 2 methods of solving linear constrained optimisation?

Answer 24

1. Substituting the constraint → sub constraint in
2. Lagrange multiplier method → L(x,y,λ)=f(x,y)-λ[g(x,y)-c]
   maxf(x,y) is subject to g(x,y)=c,& λ is an unknown constant called the Lagrange multiplier
   → Find FOCs of the Lagrange equation; use simulations equations to find x, y and λ
   → Sufficient + necessary conditions are the same, find if L(x,y,λ) is concave etc.
   → For n variable cases L(x1…xn,λ1…λn )=f(x1…xn )-∑[(j=1)^m]λj [gj(x1…xn )-cj]

Question 25

What is the interpretation of the Lagrange multiplier, λ?

Answer 25

x^* and y^*  solve maxf(x,y) subject to g(x,y)=c  →  f* (c)=f* (x*(c),y*(c))    
then  df(c)= λdc⟺ df*(c) / dc=λ   

∴ λ represents the rate at which the value of f(x,y) changes when the constant c changes
(λ is called the shadow price / marginal value of the resource represented by the constraint)

Question 26

State the extreme value theory

Answer 26

Extreme value theorem (EVT) → if f is continuous over a closed bounded interval [x0,x1], then there exists a point min. point c in [x0,x1] and a max. point c’ so that f(c)≤f(x)≤f(c^’)

Question 27

States the mean value theorem

Answer 27

Mean value theorem (MVT) → if f is continuous over [x0,x1] and differentiable in (x0,x1), then there exists at least 1 interior point c with in (x0,x1) such that f’(c)=(f(x1)-f(x0))/(x1-x0 )

(aka. A point where the gradient of the sectant line = the gradient of the function at that point) → If f’(c)=0 there is a stationary point (Rolle’s theorem FOC)

Question 28

Define elasticity

Answer 28

A unit-free measure of the Δ in the function’s value when one of its arguments changes

Question 29

What is the equation for elasticity of f(x,y) in regards to x?

Answer 29

El(x)f= %∆f(x,y) / %∆x 
= ∆f(x,y) / f(x,y) ÷ ∆x/x
= ∆f(x,y) / f(x,y)∙x/∆x
= x/f(x,y)∙∂f(x,y)/∂x
=x/f∙∆f/∆x

Question 30

What is the equation for the function y=f(x,z) partial elasticities of f with respects to x & y?

Answer 30

El(x)f=x/f(x,y)∙∂f(x,y)/∂x = ∂lnf/∂lnx

El(y)f=y/(f(x,y))∙∂f(x,y)/∂y = ∂lnf/∂lnz