To those that follow this blog, sorry for not posting anything lately. I have been busy with this project and several others.

Those familiar with the macroeconomic theory of John Maynard Keynes know that his consumption function is defined as follows:

C = a + bY_D

Where C is aggregate consumption, a is autonomous consumption, b is the slope (which in effect measures how much consumption increases with a one unit change in income), and Y_D is income.

As one is able to see, the consumption function closely resembles the basic point-slope equation that many of us learned in middle school mathematics class, and this consumption function is completely identical in how it operates. Thus, those that have expanded their knowledge of mathematics into the Calculus realm know that the derivative of the consumption function is referred to as the slope’s rate of change. In this case, that is called the marginal propensity to consume. The marginal propensity for individuals to consume measures how much of the next dollar in income will be utilized by the owner of the dollar. Real economic factors are largely what dictates the value of the MPC, with income level and general consumer confidence being far and away the most important factors. Because there is no exponents in Keynes’ consumption function, this means that the derivative (the MPC) is simply a constant, and when graphed is a flat line. We can interpret this to be the average MPC at the aggregate level.

Why is this important? What struck me as odd is that we are accepting that the MPC is a single value for all consumers, even despite income difference, when we know that wealthier consumers spend a smaller portion of their income. This means that the MPC must be variable, which is impossible without a quadratic term (a “squared” exponent) in the original consumption function. What I have done is started with the end in mind, i.e. developed a simple equation for an MPC that could be still linear but not a constant, and integrated it back into a consumption function. The variable MPC equation looks like this:

MPC = b – 2xY_D

Where b is the intercept on the Y-axis of the MPC, x is the slope, and Yd is still the original income. When we integrate it back into a new consumption function, we have this (any new constant from integration can be assumed to be lumped into a):

C = a + bY_D – Y_D²

Now that our new consumption function is quadratic, this means that our consumption function is now non-linear, thus the C curve is literally now a curve; and in a regression it could potentially better account for variance and “fit” better.

I found the economic data needed to test this at the Bureau of Labor Statistics (BLS). The BLS administers a Consumer Expenditure Survey every year, in which they survey thousands of average citizens and instruct them to list all of their expenditures and income in fine detail. They make this data available for public use and years worth of survey results are available on their website. I chose 2013 data because it was the most recent. Also it is important to note that every observation in this CE Survey data was measuring a consumer unit, i.e. a family/household.

The first step was to prepare the data set. Because many of the observations had very questionable income/expenditure values, such as negative numbers or extraordinarily high expenditures coupled with zero income, I had to create some rules for inclusion in the data set and in effect, remove outliers. After all, this experiment was to look at data for average households that have  at least someone employed that generates a steady income, thus I removed any observations that had an income under $500/month and over $35,000/month. I also removed any observations that reported expenditures of less than $750/month (as this would indicate they are far under what we accept as autonomous consumption, or the minimal amount that people must spend to have the absolute minimal level of necessities; this would make observations at this level or lower very questionable and unrealistic), and removed expenditure observations of more than $35,000/month.

After using Stata to estimate the coefficients for the new consumption function, we are presented with this:

C = 4389 + 0.332Y_D – 8.9e^-7Y_D²

R² = 0.23

As compared to the consumption function based on Keynes’ original equation, which produces the following results:

C = 4949 + 0.2727Y_D

R² = 0.228

Moving on, the new non-linear consumption function produces the following derivative/MPC equation:

MPC = 0.33218 – 17.8e^-7Y_D

While the original consumption function produces this derivative/MPC equation:

MPC = 0.2727

With the new consumption function and MPC curve plotted on top of each other, we get the following results:

So how can we interpret these results? The new robust and non-linear consumption function accounts for the diminishing utilization of new income across income levels. Also, it intercepts the expenditure-axis at $4389 expenditures/quarter for those hypothetically making $0 income. What this means is that autonomous consumption a, the smallest amount that people must spend to have the absolute minimal level of necessities, is estimated by this model to be $17,556/year/household. The original consumption function, because it was lacking this quadratic term, fit the data in a slightly different manner. This pushed its intercept higher, and would lead us to believe that autonomous consumption is actually $19,796/year/household. Because we are better able to estimate this now, we know that the $19,796 figure is an overestimate. Consumers are actually able to live off of $2,240 less goods and services than the original consumption function would have us believe.

Theory has a necessary confrontation with the mathematics behind the theory here, due to the R²s being essentially identical, 0.23 and 0.228. This means that both models fit the data just as good as the other, but in this case we have an important choice to make. Do we choose the linear model because of its simplicity? Or do we choose the non-linear model because it gives us a variable MPC? I will choose the latter. It is disingenuous for economic textbooks to almost universally teach students that the consumption function is a linear equation because this relegates the MPC equation to only a single constant term, and this cannot be true.

-Tyler