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Lecture-82 MRTS: Few Examples - YouTube
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right now let us
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try to calculate the mrts for three
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different cases one when the production
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function is cobb douglas production
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function
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so now you should understand that cobb
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douglas is the name of the form
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particular form that we use it can be
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used for utility it can be used for
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production function
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now let us take a particular one
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q is the power a
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l to the power b
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what will be the mrts how can we
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calculate the m r t s
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d q by d l by d k d by d k so before we
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do that
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let us take a general case what is
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basically happening
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we are using let us say to produce q
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naught label we are using k naught and l
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naught
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and now what we are talking about we are
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changing
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k naught 2
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a naught plus delta k naught and let us
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leave it
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leave it as positive because we know if
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one is in one input is increased second
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input has to be reduced to come to the
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same level but expression will bring
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that minus sign so do not worry about it
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l naught plus
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delta l naught so basically what we have
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done
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remember in this graph we start with
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here
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and we are trying to move
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we are tr we we will change
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l such that that we come to the so one
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so we will come to here so earlier point
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is
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k earlier point here is and the new
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point is
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of course what we are assuming that they
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are on the same isoquant here we have l
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naught plus delta l naught
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comma k naught plus
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delta k naught
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fine
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ok
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now
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in both case this
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should also give us the same level of
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output
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fine
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so if you know the taylor series
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expansion then it is very simple
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what we can do what we can write this is
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equal to
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f k naught
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comma l naught
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plus
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of course we will take the approximate
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because taylor's expansion we will have
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infinite term that we cannot you know we
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will not use so just first terms
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ok
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and here what we have
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ok this will get cancelled and what we
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will get
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d
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is equal to
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zero
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or in other word what we will get
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mpl by npk
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fine
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and what is this
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mpl
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mpl
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here there will be a minus sign here
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mp k and of course we need to take limit
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so basically what we are getting
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the slope is equal to minus mpl divided
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by
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mpk divided by
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sorry
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mpl divided by mpgas sorry just a minute
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let me check it
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ha m p l divided by m p k
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is it clear
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and this is how we can calculate this is
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what we had just discussed that this is
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m r t s so this is m r t s there is
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another way to do it also just
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mathematics you should be familiar with
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what we are saying is
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that
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what we are saying is that with the new
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label
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again the same technique we can use
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we are producing the same amount
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we are producing the same amount or in
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other word by changing the del
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k in capital and del l in
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labor we are not able to we are not
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changing the
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label of production so by
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totally differentiating it what we can
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get
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this is
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or let us say
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just leave it like this
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this is q and by changing k and l we are
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not
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aim is not to change the q but aim is to
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keep the q fixed so by if we are
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changing k and l what we can write
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that by d k
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delta k delta k
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plus
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delta l this would be equal to 0 and we
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will again reach to the same level and
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there are several other techniques also
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the other technique that also this is
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called total differentiation other here
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we use taylor series just you should be
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familiar with the technique this is
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taylor
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sort and this is called total
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differentiation
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ok
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and third is
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again we will reach to the same point
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implicit
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function
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theorem
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again i am not getting into detail but
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you should know look at it the graph
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this is l this is f and value is q
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naught so basically what we are saying q
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naught is equal to
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function of capital and labour
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but q naught is fixed q naught is a
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number so what we are saying that to
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obtain q naught k if we are changing
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that l then k should be a function of l
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fine so basically this is what we are
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saying
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k as a function of l and this is l this
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is your independent variable now
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differentiate it with respect to l again
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you will reach to the same point
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but that is immaterial what is important
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is that the marginal rate of technical
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substitution is in this particular case
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minus mpl divided by mpk
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and now we have a cobb douglas function
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crop douglas function is k to the power
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a l to the power b
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let us calculate the marginal rate of
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technical substitution how can we
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calculate first we have to calculate mpl
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and how much is mpl
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b
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k to the power l
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l b minus 1
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and what is mpk
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a
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k to the power b minus 1 a minus 1
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l to the power b and now it is very
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simple marginal rate of technical
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substitution is equal to minus
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or in other word minus b k divided by a
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l
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is it clear
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let us take another example earlier we
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talked about
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a production function where
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the both inputs are perfect substitute
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of one another it means what we have
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is
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the production function is linear in knl
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k and l
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okay here we have k here we have l
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q is equal to
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a k
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plus
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b l
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and what would be the marginal rate of
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technical substitution
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minus b by a minus b by
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a
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in many books you will see in many
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places you will see that there is no
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minus but you know for yourself that you
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know that it has to be
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negative so you can omit this negative
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sign as long as it is clear in your mind
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that marginal rate of technical
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substitution cannot be positive
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ok
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fine
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in some book they define that it is
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negative of this
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mpl by
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mpk in some book they just divide
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sorry
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its its
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described as it is simply a mpl divided
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by mpk but does not matter
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fine its clear how about when these two
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factors of production are
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complement
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of each other
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perfect substitute
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think about it
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so what we can say at corner point we
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cannot
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we cannot define how about in this zone
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where it is horizontal
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it is zero
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so sometime you know just looking at
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graph you can figure out here in this
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zone it is zero and where it is
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vertical it is infinite infinite and at
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corner point it is not defined its clear
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okay
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do
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you
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