# Modelling the Supply Curve and Minimum Wage

David Card recently won the Economics Nobel Prize (along with Joshua Angrist and Guido Imbens). His work on labor economics (with Alan Krueger) caused quite a stir in the 90s when they showed a case where the minimum wage actually increased employment. Economists freaked out because this finding supposedly showed the impossible. Supposedly there’s no way a wage increase also increase employment; economics is about trade-offs, there is no such thing as a free lunch, etc. Card winning the Nobel has exposed me once again to all the same rhetoric.

It’s a bit strange though, since it had already been previously shown (at least mathematically) that a minimum wage could increase employment. I discussed in a previously post here. Both Joan Robinson and Milton Friedman were aware of the possibility. However Friedman called it a “theoretical curiosum”, something akin to negative mass matter, only occurring in the case of monopsony (my previous post goes a bit into the strange misuse of the term “monopsony” throughout economics too).

Friedman comes to this conclusion because he has an incorrect mental model of what constitutes the supply curve. Indeed, he is most well known for advocating for free market and minimal government policies in order to achieve greater market competition. By the end of this post I hope to show something that may seem initially contradictory, that you can achieve perfect competition through government intervention such as the minimum wage.

This post will be split into two parts. The first is a model for generating the supply curve. Hopefully the model is intuitive enough so that you don’t fall into the same trap as Friedman. The second part will use this curve to show that a minimum wage can increase employment.

The Supply Curve

Before starting the model, it’s important to take a step back and think about what the supply curve represents. It’s not just some pure abstraction, not just lines on a graph. It should be connected to real phenomena in the economy. For the labor supply curve, we should be thinking about the labor force/ the actual people that make up the labor force. How does this set of people create our supply curves?

Perhaps it’s easier to start with the case of perfect competition. What does this look like in terms of the prospective employees? Perfect competition in the labor market implies that firms will be price takers. That is, a firm will have no power over setting the wage given to workers; the wage is *constant*.

Let’s imagine the labor force: it is filled with people, each with a wage that they are willing to work at. Now if the wage is constant, say $16, then everybody is thinking of that same number. We can represent this as a set:

{16,16,16,…,16}. We can plot this set via a histogram and it would look like this (in this case I have a set of 2000 people):

Not surprisingly it’s just a single bar at $16.

Now let’s begin testing possible modifications of this distribution. What if some people would be willing to work at a lower wage (imperfect competition)? So the distribution has some spread rather being fixed at a single point.

I’m going to to use the simplest model of spread, the Normal distribution (and I will only be focusing on the left side, i.e. Half-Normal distribution). This is what our histogram now looks like:

So though the majority are asking for about $16, but there’s quite a few willing to work for much less. The socialist reasoning for why somebody would be willing to work for less is that society is constructed in such a way that a person will need to take what they can get in order to survive. Having a low paying job is better than no job when your life is on the line.

Of course this is all a simplification of the complicated real world, but the question is whether having a spread is a better representation of the real world compared to perfect competition. Note that you can re-obtain the perfect competition case by setting the standard deviation of the distribution to 0.

Okay, so let’s say a firm pays out a wage of $13, then the firm will be able to hire everyone at $13 and everyone less than $13, i.e. the sum of people to the left of $13 in our histogram. If you’re familiar with probability, then you’ll recognize that this defines the cumulative distribution function (the integral of the probability density function). And since econ convention has Price on the y-axis, then we need to flip our CDF, i.e. use the Inverse CDF (quantile function). Here is the inverse CDF of our Half-Normal distribution:

It is a function giving the price of labor for the amount of labor employed; it is the supply curve.

The shape may be a bit unfamiliar, but look what happens in the case of the perfect competition distribution (0 standard deviation):

As expected the perfect competition curve is a flat line, i.e. the labor price is constant.

So there we have it, we have a method of of creating the supply curve. It is important to recognize that it’s the probability distribution itself that forms the basis of this analysis. All the upcoming figures and supply-related curves will be generated from the probability distribution and *not* from an arbitrarily defined supply curve. You should be thinking about the distribution of people first, rather than the curves first. I often see economists making claims about how certain policies will affect supply curves, which are clearly nonsensical once you think about the distribution. (Of course similar mistakes can happen with distributions as well, but at least it’s a step in the right direction)

Minimum Wage and Employment

What follows is mostly a rehash of the math that I’ve gone through in previous posts, though this time I’ll be using the probability distribution model for the supply curve.

A firm can compute it’s profit the following equation:

The max L(labor) means that the firm will want to hire exactly the right amount of labor to maximize profits. Calculation of the max involves taking the derivative with respect to L and setting it to zero.

Taking a look at the right hand side you have three terms:

- pQ: this represents the total revenue obtained from selling whatever product is created by the firm with the employed labor. I will not be going into detail about this term (it will be arbitrarily generated, but you could perform similar analysis as I’m doing with supply), you will just see it referenced as the Total Revenue curve, or it’s derivative, the Marginal Revenue of Product curve (MRP).
- wL: this represents the total cost incurred by the firm for hiring however number of employees. L is the number of employees hired. ‘w’ is the wage function ( the supply curve), it is a function of L so it is more appropriately written w(L). Thus the total cost is w(L)*L. For example (using the half-normal supply curve provided above), if the firm wanted to hire 0.3 of the total labor force, then it would need to pay a wage of ~$14. Thus the total cost is 14*0.3*N= 4.2*N, where N is the size of the total labor force (e.g. if the total labor force was 2,000 people, then the total cost would be $8,400).

Since a specific N is not important, I will be omitting N from now on.

The derivative of the total cost is called the Marginal Factor Cost curve (MFC) - FC: this represents fixed costs, it is not a function of L and thus its derivative is 0 and has no bearing on the maximization of profits here.

Since FC plays no role, then we can see that the derivative is 0 when MRP = MFC, i.e. when the curves intersect.

Let’s first plot the MFC and Supply/Wage curve for the half Normal distribution used above (these get automatically generated through the power of software magic):

The MFC is always above (or equal to) the S curve, and the amount is related to the slope of the supply curve. The greater the slope of the supply curve, the more the MFC curve deviates from the Supply curve.

A non-0 slope means that if you hire more workers, you also need to raise wages for everyone, and in the reverse direction, if you decrease the amount of workers employed, you also save money by decreasing the wages for everyone. What this means to a firm is that if there is a separation between S and MFC, then the firm can reduce wages without a commensurate loss in labor willing to work at the new wage (at least compared to perfect competition).

When the Supply curve is constant (perfect competition, i.e. just a horizontal line and 0 slope), then the MFC curve and S curve lie exactly on top of each other.

Here’s what it looks like for perfect competition:

Now let’s plot the MRP curve.

Computation of the optimal level of employment is straight forward; as mentioned above, you find the spot where MFC = MRP. The x-location will gives you the level of employment. To find the wage, just check the S curve at that level of employment. In the above example the optimal labor employed is 0.4, and thus the optimal wage is the value of the blue line at 0.4, a bit less than $14.5.

Now the question then is whether there is some operation that lowers the MFC curve such that the MRP curve crosses the MFC curve farther to the right of the graph (i.e. more employed), while also ensuring that the wage obtained is equal or greater? The MFC curve is effectively a function of the Supply curve (both are functions of our base probability distribution), so you can’t modify S and MFC independently. Also shifting the curves does not help since the MRP curve is a straight downward sloping line.

But we do know that if you reduce the slope of the Supply curve, then the deviation of the MFC from the Supply curve will also be reduced. Hmm…

Let’s see what happens when we add a minimum wage (set at $15). I won’t be doing this by modifying the curves directly, rather I will be modifying the base probability distribution that we generated in the first section. Thus I will be making no assumptions about what the new supply curve will or will not look like. I will be imposing the min wage by simply raising all values less than $15 to $15.

The new histogram looks like this:

Notice that the x-axis range now only stretches from 15 to 16. All of the people that were willing to work for less than $15 now end up in the huge $15 bar.

Here is what the new S and MFC curves look like:

The left 50% of the graph shows effectively a horizontal line. This is because, as shown in the histogram, about 50% of the total population asks for $15. If you hire one person for $15, and the next person asks for $15, then there is no change in the wage when you hire the next person. Thus both S and MFC are horizontal and lie on top of each other.

This means that (within this region) the firm is a price-taker, it is effectively facing perfect competition. Of course this is not generally what economists *imply* when they talk about perfect competition, but it does fit the mathematical definition of perfect competition. The disconnect between the implied and the math is more evidence for economists’ faulty mental models.

Now I’ll overlay the graphs with and without minimum wage:

For the dotted lines, the MFC curve intersects with the MRP curve at about 0.5, and dotted blue line value at 0.5 is $15. This is both a **higher wage and more people employed** when using a minimum wage compared to no minimum wage.

What if you don’t believe me, or don’t believe in calculus. Maybe my explanations have been unclear and there’s some sleight of hand occurring. So what if you were to just use the profit equation directly. This would involve brute force computing the amount of profit for every amount of labor employed. Fortunately we can just let the computer do that.

Here is what the curves for Total Revenue and Total Cost look like as labor employed varies:

Since the profit is the Total Revenue minus the Total Cost, then you just need to find the widest vertical gap between the Total Revenue curve and the appropriate Total Cost curve. The computed profits generate these curves below:

And here are the maximum profit results:

That’s a 25% increase in employment (9 percentage points), and the value of a minimum wage only becomes starker if you increase the standard deviation of the initial Half-Normal probability distribution.

This matches results obtained by the derivative method. In fact all of the above MFC curve figures were obtained by just plugging the probability distribution into the profit equation and using automatic differentiation.

The minimum wage, if set correctly, will increase employment. This is basic theory, but people have done some mental gymnastics to avoid coming to terms with it. The same analysis can be done for things like price controls or rent control. Obviously it’s a simplistic model, but it’s this simplified model that economists misuse to argue against any benefits to the working class.

Once more complicated economic models such as effective demand get introduced, it only makes it likelier that a minimum wage is useful. Or you could introduce a suitable level of unemployment insurance, or have the government become a larger employer within the labor market. There’s quite a few different ways the government could go about changing that distribution of the labor force such that we achieve a less exploitative and more productive economy.

Footnotes:

- The code for generating the figures is here. Sorry the code is a mess since I really just used it to create the figures rather than be a tutorial.

2. Throughout the post I use the Half-Normal distribution but you could of course use the full Normal Distribution. The supply curve would then be the inverse CDF of the Normal distribution. It would look like this:

All the same math used throughout the post would still apply. However, I do not like the idea of using the right half of the Normal distribution in the context of the minimum wage. Notice how on the right side of the curve the slope actually increases as you move from left to right. This might actually be more familiar to economists as textbooks will generally have either a straight supply curve or one that does have increasing slope as you move left to right. It does provide the correct insight if you are talking about what happens as you begin using up a finite resource. For example, the marginal cost of obtaining the last barrel of oil will be more expensive than obtaining the first.

However, this is not the dynamic that we would should expect for modelling the minimum wage. It’s the left hand side of the distribution which is more realistic.

For example, I recall seeing a textbook claim that low-wage work should be more elastic than high-wage work. Their reasoning was Friedman-esque; fewer barriers to entry for low-wage work, more people could enter the low-wage work market.

This is contrary to how I think of the market. Low-wage work involves the some of the most exploited people in society. They need to accept terrible wages in order to survive, thus low-wage work should be more inelastic, they take what they can get.

I can’t find any studies comparing low-wage and high wage elasticities (to be honest I wouldn’t know where to look), but there has been tons of studies showing that low wage work is extremely inelastic.

Footnote to the footnote: Elasticity is not exactly equal to the slope, and its possible for the elasticity to trend differently from the slope. You need to crank up the standard deviation of the Half-Normal to show lower elasticity at the low end; even though it is not necessary for discussions surrounding minimum wage and low wage work in general. Here are the elasticity graphs for our Half-Normal distribution with standard deviation 2.0 and 5.0

This is mostly because using elasticity as a measure implies an expected shape of the distribution that is non-Normal. In either case both distributions end up showing that the minimum wage works. This is why I mostly refer to the slope rather than elasticity throughout the post, since it’s the term that actually matters.

Actually, I probably should’ve used an Exponential distribution throughout the post since it’s elasticity and inverse CDF are similar. Though of course who knows if it’s any more realistic. Here’s the inverse CDF and Elasticity graphs for an Exponential distribution.