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[數學,M1&M2] Prove the law of diminishing marginal return (LDMR)



Prove the law of diminishing marginal return (LDMR)

[隱藏]
Given the following curve:
Total
Prove that Total Product becomes concave since the onset of LDMR.

[ 本帖最後由 SC3U 於 2019-10-13 05:28 AM 編輯 ]

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Denote:


TP = Total Product = P x Qty = Price  x Quantity Supplied = Total revenue

MP = Marginal Product = (TP)'



How to show that (MP)' = (TP)" = 0 when LDMR effectuates?

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Solution

Note that Qty is labour supplied.
MP = P(Qty)' + Qty(P)'
With respect to Qty, then we have
MP = P + Qty(P)'
Then
MP' = P' + Qty(P)" + (P)' = 2(P)' + Qty(P)"
MP' = 0 shows that Qty = -2(P)'/(P)" =  -P/(P)'
What's next?

[ 本帖最後由 SC3U 於 2019-10-11 07:47 AM 編輯 ]




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Solution (Continued)

Note that from the graph,

When MP' = 0, MP >0

We have MP = P + Qty(P)' > 0

which is equivalent to

Qty > -P/(P)' >0

This also implies that

(P)' < 0

Note that MP = 0 when P = 0

So

(P)" >0

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What does P stands for?

[隱藏]
P = cost

We have (P)" > 0
such that total cost is a convex function

Also

(P)' < 0

When Qty is great enough, MP = 0

When MP' < 0 , set up

2(P)' + Qty(P)"  < 0  

0 < Qty < -2(P)'/(P)"

When Qty is great enough, MP' < 0

Note that (P)'/(P)" < -P/(P)'

So LDMR must be onset before MP = 0

[ 本帖最後由 SC3U 於 2019-10-11 08:01 AM 編輯 ]

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Note

When TP < 0, MP < 0 and thus P < 0

So we have MP = P + Qty(P)' < 0

So Qty is great enough to have:

Qty = -P/(P)' > 0

Since P is negative, (P)' becomes positive!

Below is the curve for total cost:

main-qimg-1d42e4e426e27a27ac705c09360b0001.png

Note that 1 x ATC = 1 x P = TC

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