Comments on the supply/demand curve.
Generally when I read a discussion of the supply and demand curve I see something that goes like this:
The thing that has always bothered me about this representation, however, is that the curves representing supply and demand are always these vague representations, arbitrarily drawn on the chalkboard. There is never an effort to quantify what those curves should look like before we launch into discussions of things like increasing demand for lower-priced goods or what happens when price controls are put into place.
But we can quantify these things to some extent.
In all of the graphs below, I preserve the convention of price along the Y axis and quantity along the X axis, so some of this may seem “sideways.”
First, let’s look at the demand curve.
Let’s suppose we’re selling a thing, and people want one of these things. The question we ask of that population of people is “how much should that thing cost?”
Of course the question is a complex one, but to first order we may expect that overall the answer to that question becomes a bell curve centered around the most expected price for that thing:
Again, note the convention: price is along the Y axis, quantity is along the X axis. This is a bell curve centered around the price B with about 1/3rd of B being the standard deviation in the price people are willing to pay. The equation representing this curve is given by:
The demand curve would then be the integral from +∞ to y; that is, as the price declines more people buy–and the total amount of product sold would be the area under the curve from price y to infinity.
(I’ve scaled the X axis to fit in our graph.)
What this graph represents is what we intuitively already know: that as the price gets cheaper–as we drop down on the Y axis–the amount of product we can sell–that is, the number of people who are willing to buy our product–increases. But also notice a few features at the extreme ends of the curve: there is a point where we won’t sell very much product at all: if our product is priced way out of our audience’s reach, dropping the price a little bit just won’t move stuff off the shelves. And at the other extreme, if we’ve reached market saturation, dropping our price more just won’t move more product–because we’ve reached market saturation.
The equation for this curve is given by:
Where the function erf is the error function.
Now let’s add the supply curve. We can use a similar argument as with the demand curve: if we have a whole bunch of suppliers and they believe they can bring in a price at a given price point, our supply bell curve looks very similar to the demand curve. However, the number of people who are willing to sell at a given price winds up being the integral above, except from -∞ to y. Plotting this on our graph:
There are some interesting things to note here.
First, let’s suppose that our suppliers are able to bring the cost of production down. We get the classic plot showing the supply curve shifting to stimulate demand:
As prices fall because suppliers determine a cheaper way to make their products, demand rises, and in this case they rise considerably.
But, as was noted before, if we are at either extreme: if, for example, we’ve reached market saturation, then the curve simply does not bend that much. We’ve reached saturation–and the incremental effort to sell to the remaining few who haven’t purchased a product can be rather considerable.
The same thing happens at the other end of the curve: if a product is just too far out of the price range of the vast majority of people, dropping the price a little bit just isn’t going to move the needle. You’re not going to sell a lot more $300,000 Ferraris if you drop the price to $290,000.
We can also use this graph to describe price ceilings and price floors.
Suppose, for example, we’re describing raising minimum wages. In this case, “supply” are the workers who are willing to work a given job at a given price, and “demand” is the willingness of employers to employ people at a given price.
So what happens at a given minimum wage?
Well, that depends on if the minimum wage is raised above the natural price given by the supply/demand curve.
Suppose our proposed minimum wage is below the crossing point of our supply and demand curves:
Then our price floor–minimum wage–doesn’t affect things very much at all. If currently workers for a given job are being paid more than the proposed floor, everyone’s going to ignore that floor because already prevailing wages are above the floor.
This is, in fact, the situation in most areas of software development: for me personally you could raise minimum wage to $30/hour and it won’t make a significant impact on my own take-home pay. (It may make some change, because remember: the above graph really represents a narrow slice of the supply/demand curve in a particular profession for a particular market. And as the minimum wage is raised, the shape of the overall market changes, which then affects people’s budgets and that may make them reallocate where they spend money–just as if a sudden sale on steak may make you rethink buying chicken.)
It’s not to say that at this level the minimum wage doesn’t serve a useful social function. In fact, it does: there are undoubtedly employers who are assholes who are trying to convince people to work for nothing–and a minimum wage floor gives those employees another legal tool to prevent abuse. This social function, however, is not really relevant to the economic discussion.
Now, on the other hand, suppose the price floor is raised above the natural supply/demand curve. (I’ve taken the liberty to label the supply and demand curves for what they represent: the willingness of employers to hire at a given wage verses the willingness of people to work at a given wage.)
What happens is simple: as wages go up, more people want to make money at that given wage. However, as wages go up, employers are less likely to want to hire at that wage. And that leads to a gap–a gap between the number of people who want to work and the number of employers willing to hire.
And that is, by definition, unemployment.
Similar arguments can be said about price floors and price ceilings in other economic arenas. For example, the long lines and rationing that had to take place in the 1970’s during the gas shortage came from the government imposing price ceilings. A ceiling then creates a gap between the amount of stuff someone can sell, and the amount of stuff people want to buy–which creates shortages:
And people react to shortages by being willing to stand in long lines; the line becomes an added “price” people are willing to pay.
Now let’s do something completely different.
Suppose I’m a company which makes widgets. I mass manufacture those widgets.
Mass manufacturing is interesting in that it entails a setup cost and a per-product cost. Meaning that, for example, if I’m making chairs using plastic injected moulding, then I have to pay a large setup cost to set up the mould, and then I pay a per-chair charge for the plastic for each chair.
That is, the total price it costs me to make n chairs is:
where n is the number of chairs I’m making, s is my setup cost (such as the cost of the mould), and c is the per-chair cost (such as the cost of the plastic).
Now if I make x chairs and sell them at a price y, this means that I have xy amount of money to make chairs. Subtract out the setup price s, divide by the per-chair cost and this gives me the number of chairs I can make at a given price:
We can solve for the quantity x and the price y to give me the number of chairs I am, as a manufacturer, able to sell at a given price:
Plugging our final formula into our graph for given values of s and c, and we get a completely different curve than we’re used to seeing when drawing supply/demand curves:
Note that our supply curve, instead of sloping upwards from lower-left to upper-right, instead, curves downwards; as we ship more product we can lower our per-product price.
And notice our supply curve intersects our demand curve in two places–at a high price/low quantity point, and at a low price/high quantity point.
In many ways this models very well something many manufacturers do, when they take a high-price premium “luxury” item (at the upper-left) and move it downstream towards the low-price mass-produced point (at the lower-right). As the setup charge pays for itself in the luxury market arena, it becomes easier to then make the product in larger quantities for the mass market arena by reusing the same mould or jigs or manufacturing tools, and cranking the assembly line wide open for the mass market.
Of course, the problem with consumers is that if you sell too many luxury items as mass-manufactured items, it becomes tougher to sell upscale products.
You can tackle this problem one of two ways: either (a) through branding (by having a luxury brand and a mass-manufactured brand, such as Toyota and Lexus), or (b) by simply deciding to sell all your products at the upper-left corner (as Bang-Olufsen does) or the lower-right corner of the curve (as Apple does).
Now we can also use this to explain why certain technologies suddenly “appear”; that is, we can easily describe what happens when some technology waiting in the wings suddenly explodes on the scene:
Suppose we have a new technology, and our setup costs are high and our per-item charge is high–because it’s an all new technology. Perhaps tinkerers play with the technology or it’s something that floats around in the labs–but to get it out there would cost a lot of money.
Then we may see a supply curve that looks like this:
Note the curves never cross.
At no price point does the supply curve ever cross the demand curve, because the item simply cannot be made at a price that anyone would want it at.
This is far different than the traditional supply/demand curve we looked at where we model the supply of something as a gaussian curve; that gaussian assumed a large number of suppliers who could supply a good, and a few could create a handful of items at a high price point.
Using the old supply/demand curve, what happens when the price drops a little bit is what we would expect:
We get a tiny gain in the population willing to buy a product.
But if we use our equation describing the actual manufacturing costs we don’t just get an incremental number of adopters buying a luxury product. Instead, for a small drop in manufacturing price we get an explosion of demand:
This explains how we went from just a handful of mobile smart phones in 2006 to nearly everyone carrying an iPhone less than a decade later. As manufacturing costs dropped, it intersected the demand curve–and suddenly we went to a world were nearly everyone had one. And it wasn’t like the price to manufacture smart phones radically dropped in order to stimulate demand, as would be suggested if we used the traditional supply curve.
I really don’t have a point to all of the above, except to note that some rather interesting things come out of the traditional supply/demand curve if you attempt to actually model what the supply curve actually looks like and what the demand curve actually looks like, rather than just gesturing at the chalk board and drawing an upward and downward sloping line.
I’m sure if a more generalized treatment of the demand curve was done–for example, if we have a population where half are willing to jump on at one price, and the other at a second price, we’d see some other interesting things jump out of the graph. Just as I’m sure if we could model the additional dimensions behind the supply/demand curve, such as the different values different employees have for a corporation, or we could model businesses which use loss leaders to make a profit–we’d get a much deeper insight into how the economy works.
Fuzzy little things that I find interesting. 
