, Week 2
One Variable Optimisation
Choice between two options a and b with utility u(a. b)
↳ If pa =pis , optimal choice implies that the
marginal utility of each choice must be equal
↳ U'blaI
Utpaabl
=
Pb
Note
µ ,
sometimes the maxima and minima
when the slope are horizontal
É
occur .
I
t
(
dyq=o) such as the
However
graph on the left
other times the maximum
.
i " ,
I i. I :-. and minimum have
points dont a
¥
:
horizontal slope
a ✗0 A
a b
such as on the
right
but other points do .
Convex and Concave Functions
first
limtcctmn-tcclh-oitt.cn
derivative =
t.ca =
yo tcm is
increasing over D
'
it t (a) to tent is decreasing over D
it f ca) > otcx) is strictly 4
'
over D
it 1- (a) { of (a) is
'
strictly * over D
Convexity 1- (a)
"
deal with the functions
and
Concavity second derivative
Definition : Given that it is twice continuously differentiable ,
t
'
f-
"
function is D if (a) 20 cat I
1) A convex over
t (a)
'
f-
"
I
2) A function is concave over D if (R) & O
One Variable Optimisation
Choice between two options a and b with utility u(a. b)
↳ If pa =pis , optimal choice implies that the
marginal utility of each choice must be equal
↳ U'blaI
Utpaabl
=
Pb
Note
µ ,
sometimes the maxima and minima
when the slope are horizontal
É
occur .
I
t
(
dyq=o) such as the
However
graph on the left
other times the maximum
.
i " ,
I i. I :-. and minimum have
points dont a
¥
:
horizontal slope
a ✗0 A
a b
such as on the
right
but other points do .
Convex and Concave Functions
first
limtcctmn-tcclh-oitt.cn
derivative =
t.ca =
yo tcm is
increasing over D
'
it t (a) to tent is decreasing over D
it f ca) > otcx) is strictly 4
'
over D
it 1- (a) { of (a) is
'
strictly * over D
Convexity 1- (a)
"
deal with the functions
and
Concavity second derivative
Definition : Given that it is twice continuously differentiable ,
t
'
f-
"
function is D if (a) 20 cat I
1) A convex over
t (a)
'
f-
"
I
2) A function is concave over D if (R) & O
