Marginal Cost
The Big Idea: Marginal Cost- Derivatives in Economics
In economics, derivatives are applied when determining the quantity of the good or service that a company should produce.
For example: You can model cost as a function of quantity: $C(x)=(.000001x^3)-(.003x^2)+5x+1000$
You can take the first derivative of this equation to understand the rate of change in cost for each additional product produced:
$C'(x)=(.000003x^2)-.006x+5$→ This is called the Marginal Cost.
As expected, the marginal cost, is positive for all X values. This makes sense because no matter what the quantity is, producing more will result in higher costs. In other words, more production never costs less.
What is its importance??
While considering the average cost of production is important, it is not as relevant as marginal cost. For instance, if it costs a toothbrush company \$300,000 to make 500,000 toothbrushes, each one costs the company \$0.60. However, this doesn’t tell the company how much each new additional toothbrush is costing them to make. It could be \$0.25, or it could be \$0.75. The average cost doesn’t capture this idea while the marginal cost does. This analysis is crucial if a company needs to reduce its cost, while still trying to make as much money as possible.
The marginal cost does increase and decrease. For instance, C'(6) > C'(500), illustrating that when producing 6 units, the cost of the next additional unit is higher than if you were to produce 500 units and wanted to produce the 501st.
$C'(6)=(.000003(6)^2)-.006(6)+5=4.85$
$C'(500)=(.000003(500)^2)-.006(500)+5=2.75$
This concept is called economies of scale; in this case, economies of production. When you are producing more, producing an additional unit is cheaper than when you are producing less. However, this pattern doesn’t continue forever, since C’ increases and decreases there is a point where it has a minimum. In other words, there is a certain level of production (x) at which the cost of producing one additional product is as low as possible. In the example above, this happens when
$C”(x)=0.$
$C”(x)= .000006x-.006$
$0= .000006x-.006$
$x=1000$
At a production level of 1000 units, the marginal costs is at its minimum. Meaning that producing one additional product costs more than it did previously. This ultimately results in less profit. Given that X= 1000 is an inflection point on the graph of y=c(x), it is concave down for when x> 1000 meaning that the marginal cost is decreasing. The marginal cost is increasing when the function is concave up (x >1000).
Sources:
http://math.hawaii.edu/~mchyba/documents/syllabus/Math499/extracredit.pdf
http://tutorial.math.lamar.edu/Classes/CalcI/BusinessApps.aspx
https://www.economicsonline.co.uk/Definitions/Marginal_cost.html
https://www.investopedia.com/terms/m/marginalcostofproduction.asp
Great Post! It’s interesting to have a little foray into the world of economics with respect to calculus in the way you provided. It never really occurred to me that the cost of making a product would increase as more units are made. That being said, if it’s cheaper to produce in bulk that probably explains why it is cheaper to buy in bulk. Also, perhaps since there is a marginal cost that can be found in the derivative of average cost, I surmise that there is a similar function to be found in the derivative of the average sales price of a product that would be beneficial to consumers and merchants looking to save money?
I enjoyed this post. You do a great job presenting the marginal cost concept. You explain its meaning by analyzing hypothetical situations. You then apply its formulaic expression, with the derivative, to an example. This post was easy to follow, well-thought out, and interesting.
I specifically appreciate how you differentiate between average cost and marginal cost. As you explain, marginal cost can be understood as “the rate of change in cost for each additional product produced.” Average cost does not accurately measure this rate of change. I like how you illustrate this idea by presenting “C'(6) > C'(500)”. Since the derivative of producing 6 units is greater than that of 500 units, the cost to produce the 7th unit is higher than the 501st unit. Since the derivative is the rate of change, we know that the company who produces 6 units has a higher increasing cost (slope of price) as compared with the company that produces 500 units. It is interesting to apply this concept to economics.
Next, you do a great job of explaining the term “economies of scale/production”. I appreciate how you explain this term: “when you are producing more, producing an additional unit is cheaper than when you are producing less. In a way, this concept can also be thought of as diminishing returns. In diminishing returns, the more input you put in, the less output you receive over time. So, I get less out of studying calculus for my 20th hour than I do for my 1st hour.
Also, what would a marginal cost graph look like if X=1000 was instead a root or a critical point. Would there be 0 marginal cost at y=0? What would it mean, in terms of marginal cost, to have a local maxima or minima at a certain point? Is that point the maximum or minimum marginal cost in a company’s production. Can the cost of producing another unit be limited? Will it ever be free?
Overall, this was an excellent post. I really got a great lesson in economics! Awesome job!