Unit 9

Question 1

Geometric point about convex sets?

Answer 1

Have no gaps, holes or depressions

Question 2

See and practice top of my notes U9 table on convex and not convex in different dimensions

Answer 2

Now

Question 3

Note regarding convex combinations and linear combinations?

Answer 3

All convex combinations are linear combinations, but not vice versa

Question 4

Normal English, what makes a set convex?

Answer 4

A set is convex if, given any 2 points that belong to it, ALL convex combinations of those 2 points also belong

Question 5

See example page 4 notes at bottom

Answer 5

Now

Question 6

Key point about 2 convex sets, and why it is important?

Answer 6

The intersection of 2 convex sets is also convex

Useful since if there are several constraints, then their intersection (the feasible set) is also convex

Question 7

Can you have a concave set?

Answer 7

NO, only a concave function

Question 8

Finish the sentences:

1) if f is concave…
2) if f is convex…

Answer 8

1) if f is concave the set of points lying BELOW its’ graph is convex
2) if f is convex, the set of points lying ABOVE its’ graph is convex

Question 9

What is a bundle?

Answer 9

Some vector (eg. a_) of goods

Question 10

What is a level set?

Answer 10

A set of all bundles that are indifferent to some given bundle (eg. If only 2 goods in question it is an indifference curve)

Question 11

What is an upper level set?

Answer 11

A set of bundle that are valued at least as highly as a given bundle

Question 12

What is a lower level set?

Answer 12

A set of bundles that are valued less than or equal to a given bundle

Question 13

Note

Answer 13

When n>2 is incorrect to talk about curves/contours tf use level set terminology

Question 14

See general definition of level and upper set in my notes

Answer 14

Now

Question 15

What is a quasiconcave function?

Answer 15

A function that doesn’t have local maxima that aren’t global maxima

Question 16

2 examples of quasiconcave functions?

Answer 16

-x^2, e^(-x^2)

Question 17

Sufficient condition for a critical point to be a global maximum?

Answer 17

Objective function is concave (NOTE: it is not a necessary condition)

Question 18

All (1) functions are also (2), but not the other way round.

Answer 18

1) concave

2) quasiconcave

Question 19

How many maximum points do quasiconcave functions have?

Answer 19

<=1 (although this isn’t how they are defined)

Question 20

Definition of quasiconcave function?

Answer 20

If f:R^n->R, and the upper level sets of f are all convex sets, then we say f is a quasiconcave function

Question 21

What is the lower level set of a quasiconcave function?

Answer 21

Convex

Question 22

Note

Answer 22

See and read example pages 10&11