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total-output

Have you ever looked at a graph of total output and wondered to yourself:
“Gee, I wonder which level of output would give me the highest average output?”

Or perhaps you looked at a graph of total profit and asked yourself:
“Hmmmm, which unit of output gives me the highest profit from that unit (marginal profit)?”

If you have tried and run into a wall, then fear not!
These questions can be answered with some simple graphical analysis that is firmly rooted in mathematics.

Okay, you probably don’t ask yourself those questions very often.
Neither do I.
However, the study of economics often requires dipping into the vast pool of data that the world has to offer, gleaning and extracting as much information from it as possible.

This means that it is vital to be comfortable dealing with graphs!

It is for that reason that I’d like to introduce two graph-related skills to you today:
How to look at a graph, and figure out what’s going on with the averages and marginals.

Part 1: Average Values

If you look at a graph of Y against X, the average value of the graph at any one point can be defined as follows:

Average value at point (X,Y) = Y / X
You might be thinking that this is difficult to calculate.
Fortunately, there is a very easy way to represent this on a graph.

Draw a straight line from the origin to a point (X,Y) on the graph.
The value of the gradient of that line is given by Y/X.
This also represents the average value at that point!
In other words, the steeper the line you drew, the higher the average value at that point.
To figure out if the average value is increasing or decreasing along the graph, imagine ‘sliding’ a line along the graph.
If you have the graph on paper in front of you, you can do this by sliding your ruler along the graph! Make sure you remember to hold one point fixed at the origin.
If the line gets steeper, the average value is increasing.
If the line gets shallower, the average value is decreasing.

I’ll demonstrate a few cases below:

a) Straight line graph with positive y-intercept

output_csq1bb The red line that I am sliding along the graph is becoming shallower.
Therefore, the average value is decreasing.

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Mathematical interpretation (intuitive):
The curve has the equation:
$Y = aX + b$
Average value
$= \frac{Y}{X}$
$= \frac{aX + b}{X}$
$= a + \frac{b}{X}$
As the value of X increases, the value of $\frac{b}{X}$ decreases.
We therefore have a constant $a$ plus a decreasing value $\frac{b}{X}$
This gives us a decreasing average!
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Mathematical interpretation (calculus):
As before, we get:
Average value
$= a + \frac{b}{X}$
Differentiate this with respect to X, giving us:
$\frac{\partial AVG}{\partial X} = \frac{-b}{X^2}$ Since this is negative for all values of X, we have a decreasing average.

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b) Straight line graph with negative y-intercept

output_94feip

The red line that I am sliding along the graph is becoming steeper.
Therefore, the average value is increasing.

Mathematical interpretation (intuitive):
The curve has the equation:
$Y = aX - b$
Average value
$= \frac{Y}{X}$
$= \frac{aX - b}{X}$
$= a - \frac{b}{X}$
As the value of X increases, the value of $\frac{b}{X}$ decreases.
We therefore have a constant $a$ minus a decreasing value $\frac{b}{X}$
This gives us an increasing average!

Mathematical interpretation (calculus):
As before, we get:
Average value
$= a - \frac{b}{X}$
Differentiate this with respect to X, giving us:
$\frac{\partial AVG}{\partial X} = \frac{b}{X^2}$ Since this is positive for all values of X, we have an increasing average.

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c) Straight line graph from origin

constant

It doesn’t matter how steep or shallow the graph is.
If it begins at the origin, it has a constant average value!
If you drew a line from the origin to any point on the graph, it would have the same gradient as the graph itself!

Mathematical interpretation (intuitive):
The curve has the equation:
$Y = aX$
Average value
$= \frac{Y}{X}$
$= \frac{aX}{X}$
$= a$
We therefore have a constant average value, $a$ .

Mathematical interpretation (calculus):
As before, we get:
Average value
$= a$
Differentiate this with respect to X, giving us:
$\frac{\partial AVG}{\partial X} = 0$ The average value does not change.

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d) Non-linear graph

output_hin8u8

With this graph, the red line I slide along the graph first gets shallower, then steeper.
The average value first decreases, then increases.
I also show you where you can find the point of minimum average.

VERY IMPORTANT: This point of minimum average will be different on other graphs!
Do not look at a similiarly shaped graph and assume that it has the same feature.

I won’t get into the maths for this one.
Feel free to experiment on your own!

Part 2: Marginal Values

Figuring out the marginal value at any point of the curve is a far easier exercise than figuring out the average value.

In a graph of Y against X, what does the marginal value mean?
It is the amount by which Y increases when we increase X by one unit.
The way to represent this is to take the tangent to the curve at each point.
The slope of the curve at each point represents the marginal value.

Rules:

An upward sloping tangent means the marginal value is positive.
When X increases by one unit, Y will increase as well.
A downward sloping tangent means the marginal value is negative.
When X increases by one unit, Y will decrease.
A steeper tangent means a higher marginal value, whether positive or negative.
ANY straight line curve has a constant marginal value.
This is because the tangent at each point will have the same gradient.

Here’s a quick illustration:

lowest-marginal

I’ve shown you the tangents at two different points.
The tangent on the right represents a higher marginal value than the tangent on the left.

I’ve also indicated where the minimum marginal value would be.
This is the shallowest part of the curve.
You may also know it as the point of inflection in a cubic graph.

Finally, I’d like to show you a contextual example.

profit-data
I’ve given you some figures for total and marginal profit at increasing levels of output.
From the table:
We expect to see the curve being steepest at a quantity of 1.
This is because it gives the highest marginal profit.
Likewise, we expect the curve to be shallowest at a quantity of 6.
This is where we have the lowest marginal profit.

total-and-marginal-profit

And as it turns out, that’s exactly what happens!
The curve is steepest at an output of 1, and shallowest at an output of 6.

Wrapping Up

To figure out the average value at each point on a graph, draw a straight line from the origin to that point.
To figure out the marginal value at each point on a graph, observe the slope of the graph at that point.

This sort of analysis is crucial in all sorts of economic problems.

The next time you are faced with a task requiring this, stare it down with a steely glare.
Your mathematical toolbox just got upgraded!