Deriving Demand Curves without Utility Functions
By: The Moral Economist
Tags: Economics, Business, Finance, Rationality

Individuals will generally vary from one another in many attributes, not just income. Let us suppose that individuals are generally heterogeneous in their inclinations, complex in their motivations, occasionally whimsical in their choices, and susceptible to a variety of social influences.

— Anwar Shaikh, Rethinking Microeconomics: A Proposed Reconstruction

Typically in an economics course, one starts with a utility maximization individual, then derives a market demand curve by aggregating, sometimes even using an “representative agent”.

However, as Anwar Shaikh points out, “But it does not follow that individual decision making is characterized by the rules of so-called rational choice and rational expectations or by the reductive incentives they embody. Nor does it follow that aggregates can be analyzed in terms of representative agents. The historical, empirical and analytical evidence against this set of hypotheses is overwhelming.”

Utility functions are questionable from a scientific perspective. First, they aren’t directly measurable, without an MRI machine. Second, they are scalar (single) values, instead of multidimensional brain states. Third, they aren’t easily comparable across individuals.

That’s not to say there aren’t valid reasons for studying rational choice based agents. But it would be nice to prove critical economic theory with more robust foundations and less assumptions. In other words, is there a way to derive demand curves without reference to utility functions?

Shaikh believes he’s found a simple way to do so, building off of Gary Becker’s work Irrational Behavior and Economic Theory. For some reason, Becker did not further pursue this line of thinking, but that’s a topic for a later discussion.

Here we only need to make a couple of assumptions. First, assume that all income is partitioned into two parts: necessary and luxury goods, x1 and x2 with relative prices p1 and p2. Since x1 is a necessary good, there is a minimum amount that any consumer must purchase: x1min.

Then, the feasible range of the budget constraint for any individual is the segment between x1min and x1max = y/p1 as shown above (shaded darker).

Then, the relevant equations become:

y = p1*x1 + p2*x2 x1max = y/p1 x2max = (y/p2) -(p1/p2)*x1min

1. is just the general budget constraint, and 2. occurs if the consumer only consumes necessary goods. 3 can be derived as follows.

y = p1*x1min + p2*x2max

y-(p1*x1min) = p2*x2max

y/p2 -(p1/p2)*x1min = x2max

Individuals with income y will generally choose some bundle (x1, x2) within the feasible range, which is shown as point A above. The feasible range is defined by the range between x1min and x1max, since necessary goods are necessary, and luxuries are “just” luxuries.

In the aggregate, consumers will pick a given ratio r of necessary goods.

r is just the chosen ratio x1, relative to x1min and x1max,

r = (x1 -x1min)/(x1max -x1min)

Since we can multiply by p1/p1, r = (p1*x1 -p1*x1min)/(p1*x1max -p1*x1min), and since x1max = y/p1, y = p1*x1max, this expression is equivalent to

r = (p1*x1 -p1*x1min)/(y -p1*x1min)

Therefore, r represents the average propensity to consume, a/b, which is the ratio of discretionary consumption of the necessary good a = (p1*x1 -p1*x1min) to discretionary income b = (y -p1*x1min).

Pause to think about this for a moment. a is the difference between how much necessary good is consumed compared to the minimum, and b is how much income the consumer has left after consuming the minimum bundle.

Finally, we conclude by combining r with equation 2. This gives us the corresponding “per capita consumption demand for the necessary good”.

x1 = (1–r)*x1min + r*xmax

In order to do this, we need to make 2 modest assumptions. We must assume that x1min and r are independent of prices. x1min is the minimum necessary bundle, so that is a safe assumption. Minimum necessary goods are necessary for a reason. They are generally not price sensitive.

However, our assumption that r is independent of prices may initially seem questionable, since that makes the above definition recursive on x1.

Nevertheless, in the aggregate, r is the average propensity to consume, which may depend on relative prices at certain levels, but not on the margin. So, the x1 an individual will choose depends on the average propensity to consume and x1min and x1max.

Since x1max = y/p1, x1 = (1–r)*x1min + r*(y/p1), which proves that x1 is a negatively sloping demand curve with respect to p1. In other words, as p1 increases, x1 decreases.

This might seem almost so simple that it is unimportant. But in fact, it shows that one need not assume hyper-rationality in order to derive negatively sloping demand curves.

They simply occur naturally as the aggregate result of budget constraints and a minimum necessary bundle. Many paths lead to Rome, but some are more robust than others.

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