un constrained optimisation

Question 1

what is the unconstrained optimal ?

Answer 1

the absolute max of a function

Question 2

how do you find a critical point of a function ?

Answer 2

you set the F.O.C ( first derivatives ) to equal 0 and solve

Question 3

what is the marginal rate of substitution ( MRS ) ?

Answer 3

it is the ration of the input prices or the value of one interms of another

MRS = - dk/dl = dq/dl . dk/dq = w/p / p/r = w/r

the MRS is equal to the slope of the isoquant

Question 4

how do you find the optimal output when you are selling in two different countries ( have two different critical points) ?

Answer 4

you put the Pa, Pb, Qa and Qb into the profit function

find the F.O.C ( 1st derivatives, by both variables )

solve to find the optimal Qa and Qb ( solving simultaneous equations )

and substitute to find the profit :)

Question 5

name the four methods you could use to solve simultaneous equations ?

Answer 5

1. slope intercept method
2. method of substitution
3. method of elimination
4. cramers rule

Question 6

explain the slope intercept method ?

Answer 6

set both equations equal to one of the variables
equal them to each other
and solve

Question 7

explain the method of substitution ?

Answer 7

rearrange one of the equations to make one of the variables the subject
then substitute the equation into the other equation
and solve

Question 8

explain the method of elimination ?

Answer 8

same method we always used in school

you manipulate and then sub or add one equation to the other to get rid of now of the variables and then solve

Question 9

explain cramer’s rule ?

Answer 9

this is when you make matrices out of the equations and then calculate the determinant of A ( with the matrix B slotted into the first column for x* and 2nd for y* divided by the determinant of A

Question 10

how do you determine whether the equations have a unique solution or 0/infinite solutions ?

Answer 10

when the determinant is non zero, there is a unique solution

when the determinant is zero, there are 0 or infinite solutions

Question 11

how do you find the equation of the tangent plane/isoquant ? with one variable

Answer 11

y = f(x0) + df/dx0 . x - df/dx0 . x0

Question 12

how do you find the equation of the tangent plane/isoquant ? with two variables

Answer 12

z = z0 + df/dx . dx - df/dy . dy

when on an isoquant all qs are the same so z = z0
therefore :

0 = df/dx . dx + df/dy . dy

dx/dy = -df/dy / df/dx slope of iso / level curve

Question 13

how do you find the absolute maximum of a function that wants you to find a minimum ?

Answer 13

when f(x,y) min of f = max of -f

Question 14

what are the first order conditions ?

Answer 14

df/dy = 0

df/dx = 0

Question 15

when z = f(x,y) , what is the total differential ?

Answer 15

dz = df/dx . dx + df/dy . dy

Question 16

what is the chain rule of differentiation ?

Answer 16

when y = f(x)^t

dy/dx = f’(x).f(x)^t-1.t

Question 17

how do you ensure that you have found or are finding the correct critical point ?

Answer 17

first you must use the f.o.c but you must also look at the s.o.c because with out this you may be calculating a saddle point rather than the absolute max or min.

Question 18

what are the conditions for a minimum point ?

Answer 18

F.O.C S.O.C

1st dervs = 0 2nd dervs > 0, the product of 2nd derivatives > cross second derivative squared

Question 19

what are the conditions for a maximum point ?

Answer 19

F.O.C S.O.C

1st dervs = 0 2nd dervs < 0, the product of 2nd derivatives > cross second derivative squared

Question 20

what are the conditions for a saddle point ?

Answer 20

F.O.C S.O.C

1st dervs = 0 2nd dervs = any value , the product of 2nd derivatives < cross second derivative squared

Question 21

what are the conditions for a un determined point ?

Answer 21

F.O.C S.O.C

1st dervs = 0 the product of 2nd derivatives = cross second derivative squared