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Homework problem:

We have a graph of value:
x coordinate number of widgets made.
y coordinate value in $.

For the first 5 widgets the value greatly increases to $100 and then after 5 gradually flat lines. (diminishing return).

So where the slope equals zero is where the maximum value is found - at 5 widgets.

We solve for the derivative of the equation of the line equal to 0 = 5 widgets.

Ok great.

However, the next step shows a graph of the derivative of the value function and it is labeled the marginal value.

So what does this new simple graph tell me? I am not getting it, I don't know what to deduce from it.

It looks very similar to the original value curve but lower and shorter?

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Graph theory is actually quite a different subject from the topic of this question. –  user12998 Oct 21 '11 at 19:38
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"For the first 5 widgets the value greatly increases to $100 and then after 5 gradually flat lines." What do you notice about the value of the graph of the marginal value for x<5? At x=5? For x>5? –  Grasshopper Oct 21 '11 at 19:54
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If the graph has marginal value, what good is it? Seriously, marginal value is the value of the last widget. So the marginal value of the $7^{\text{th}}$ widget is the value of 7 minus the value of 6. If you have 50 pairs of shoes, are you really that much better off than if you have 49? Certainly the value of the jump from 0 to 1 is greater. –  Ross Millikan Oct 21 '11 at 20:07
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"Marginal value", "Marginal change" is what economists call the change-per-unit, which is (essentially) just the derivative (not quite because, for instance, with "widgets", only an integral number of widgets would make sense). The graph of the marginal value gives you the same information about the value graph as the graph of the derivative of $f$ gives you about the original function in the general case. –  Arturo Magidin Oct 21 '11 at 20:21
    
@Arturo: awesome, thank you..... –  Greg McNulty Oct 21 '11 at 21:20

2 Answers 2

Marginal value", "Marginal change" is what economists call the change-per-unit, which is (essentially) just the derivative (not quite because, for instance, with "widgets", only an integral number of widgets would make sense). The graph of the marginal value gives you the same information about the value graph as the graph of the derivative of f gives you about the original function in the general case. – Arturo Magidin

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Marginal value tells you how much value the production of an additional widget gives you. Since the Value function you described seems to "increase greatly" at first, I take this to mean that the value function increases at an increasing rate. Thus, each additional unit gives you more additional value than the previous unit. Then as the value function increases at a decreasing rate, that means each additional unit gives you a positive value, but lower than the previous rate. Thus, the marginal value curve should be increasing at first in the positive quadrant, then decreasing but still in the positive quadrant, then after the production of the 5th widget, the marginal value curve should be in the bottom right quadrant reflecting that each additional widget actually makes you lose value.

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