Does Narcan Save Lives?

What can price theory teach us about the ability to reverse an overdose?

Unless you have been living under a rock, you have probably heard about the tens of thousands of people dying each year as a result of opioid use/abuse. In response, policymakers have tried to find appropriate responses to reduce these types of deaths.

In a previous post, for example, Brian discussed Casey Mulligan’s work on the topic. Casey’s works focuses on the idea that there are prescription opioids and illicit opioids and these tend to have different prices. As a result, consumers have a non-convex budget set and a change in the price of prescription opioids can have dramatic effects on consumption since consumers must substitute away from the higher-priced prescriptions to a lower-priced illicit alternative. Rising (prescription) prices can lead to more consumption. This has important implications for policy and highlights the important role that price theory can play in the evaluation of policies designed to reduce opioid use.

In this post, I would like to address a particular type of policy response that requires a different type of analysis. In my home state of Mississippi, the state legislature has approved public schools to keep Narcan in stock. Narcan is a medication that can be used to reverse an opioid overdose. The University of Mississippi, where I work, even offers a training program through the School of Pharmacy that trains people on how to respond to opioid overdoses as well as how to administer Narcan.

News accounts discussing this legislation and these training programs tend to highlight success stories of people saved from opioid overdoses by receiving the medication in a timely fashion. Proponents of this law argue that it will save lives. In fact, to make this argument, a number of stories cite the number of reported uses of Narcan that have saved people’s lives.

But is this true? Does Narcan save lives? There is a sense in which it obviously does. The stories of people whose lives have been saved from receiving the medication are real and these people are alive today due to the existence and use of the medication. However, this is not how an economist would answer the question. The relevant question is not whether we can identify people who have received the medication and survived. Rather, the relevant question is whether fewer people are dying of opioid overdoses relative to the counterfactual in which this medication was unavailable. This is a more difficult question to answer, but price theory can help us sort some things out.

And although it is too early to empirically measure the effects of the policy, price theory can serve as a useful guide for thinking about how to measure the effect of the policy relative to the counterfactual.

A Brief Diversion About Price Theory and Taxes

Although the policy I just described has nothing to do with taxes, I nonetheless think that we can draw lessons from the price theoretic approach to taxes. Please stick with me while I review the price theoretic approach to taxation. I will then apply the same logic of this approach taxes to the market for opioids via a useful analogy. I promise it will be worth it.

When we think about taxes on a particular good, there are two types of costs that are created for participants in this market. There is the cost of paying the tax (the tax revenue that goes to the government) and there are the foregone benefits of the gains from trade that could have been had, but are no longer feasible.

For example, consider a simple example. Suppose that the price of milk is $3 per gallon and people buy 10 gallons of milk. Now suppose that the government levies a tax of $1 per gallon on milk. For typical supply and demand framework, we get the following results. The price paid by consumers goes up. The price received by sellers goes down. The difference between the two prices is $1. The quantity traded declines. Suppose that the quantity that is traded falls to 7 gallons of milk. It follows that the tax revenue generated by the tax is $7. But this isn’t the total cost. There are people who are wiling to pay an amount of money that is above the marginal cost of producing another gallon of milk. However, the difference between their willingness to pay and the marginal cost is less than $1. Thus, without the tax, these buyers and sellers would trade. With the tax, they do not. This is a cost also. We call this additional cost the deadweight loss. Chicago price theorists call this loss the “excess burden” of the tax.

Taxes and the Expenditure Function

Now that we have taken this very brief diversion to think about taxes, I would like to add a little wrinkle to the analysis. A lot of times when we are discussing demand, we are focused on Marshallian demand curves. These Marshallian demand curves can be derived from a utility maximization problem. A consumer chooses how much to consume of each good in order to maximize his or her utility subject to the consumer’s budget constraint. The net result of this approach is to derive the demand for each particular good as a function of its prices, the prices of other goods, and the person’s income. Every point on a Marshallian demand curve is a combination of price and quantity demanded holding other prices and income constant. Every point on an individual’s demand curve is a utility-maximizing point.

As I discussed in a previous post on price theory and the price level, there is an alternative approach to demand. This approach views the consumer’s problem as choosing amount to consume of each good in order to minimize the expenditure required for a given level of utility. This approach derives Hicksian demand curves in which the quantity demanded of a good is a function of its price, the prices of other goods, and the level of utility. It follows that when we plot Hicksian demand curves, we are plotting combinations of price and quantity associated with a particular level of utility, holding all other prices constant.

As I mentioned in my post on the price level, what makes the Hicksian approach useful is that it allows us to utilize something called the expenditure function. The expenditure function can be thought of as follows. Suppose that you know all of the prices of all of the goods. You can plug those prices into the Hicksian demand curve for each good and get the quantity demand for every good, given that set of prices. Then you can take the price of a good multiplied by its quantity demanded, given those prices, to estimate the expenditure on that good. When you add up these expenditures across all goods, you get a measure of expenditures for a given level of utility. What is important about this expenditure function is that you can then consider what happens when things change. In particular, you can think about what happens to total expenditures as the price of one particular good rises and everything else stays the same while simultaneously holding utility constant.

The expenditure function is not only useful for the measurement of the “cost of living,” as I illustrate in my post on the price level, but it has more broad applicability. In fact, the Chicago brand of price theory emphasizes the importance of the use of the expenditure function for understanding things like the cost of taxation.

For example, suppose that we think of our tax example. Consider again the market for milk and let’s suppose for the time being that the supply curve is horizontal, such that milk suppliers are willing to supply unlimited quantities at the going price. Or, put differently, the amount traded is demand-determined. The imposition of the tax on milk will cause the price that consumers pay to rise by the amount of the tax. We thus move along our Hicksian demand curve to a new lower quantity demanded as the price rises from say p to p+t. Now consider the difference in the expenditure function from the imposition of this tax. It must be true that:

Expenditures after the tax - Expenditures before the tax = Tax Revenue + Excess Burden

In other words, when the tax is imposed and the price increases by t, the additional amount of expenditures necessary to keep utility constant would be equal to the amount of taxes paid plus the foregone gains from trade. I will now re-write this equation as follows:

The cost of the tax = Tax revenue + Excess Burden

Why is this useful? Well, it is useful because it allows us to focus on costs while holding utility constant. As a result, we can focus on the composition of the costs associated with taxation. For example, suppose that the government decides to raise the tax on milk from $1 per gallon to $2 per gallon. We know that this will increase the total cost of the tax. We know this because the total cost of the tax change will be measured by the area to the left of the Hicksian demand curve in p-q space between prices p + 1 and p + 2. However, sometimes what we are worried about is the composition of the cost. Is the change in the cost largely in the form of greater tax payments? Or is it largely excess burden?

More importantly, the change in the excess burden can be greater than the cost of the tax! This is not hard to understand why. Consider that tax revenue is measured by the tax per gallon multiplied by the number of gallons purchased. When you increase the tax on milk, the tax per unit goes up. However, the quantity demanded of milk goes down at the the higher after-tax price. Thus, the effect of the increase in taxes on tax revenue is going to depend on the elasticity of demand. If demand is very elastic, a small increase in the tax on milk will result in a larger percentage reduction in the quantity demanded and tax revenue (taxes paid) will decline. But since the cost of the tax must rise, this implies that the increase in the excess burden is even greater than the cost of the tax. On the other hand, when demand is inelastic, the rising cost of the tax will be split between higher taxes paid and a larger excess burden.

Using this Framework to Think About the Narcan Policy

Okay, so what in the world does this have to do with the Narcan policy I mentioned at the beginning of the post?

Consider the market for opioids. There is a Hicksian demand for opioids, just like anything else that people want to consume. Even if we abstract away from all other drug-related public policies, we should recognize that there is an implicit tax associated with opioid use: there is some chance that it will kill you.

Again, let’s think about Hicksian demand and a horizontal supply curve.

We can think of the implicit tax rate on opioids as the number of opioid overdose deaths divided by the quantity consumed. This is the probability of an overdose death per unit of consumption. If people using opioids take this into account when they are making decisions, then this is equivalent to a tax on opioid use equal to the probability of death associated with each unit consumed.

Note that this implies that “tax revenue” in this example is given as

Tax revenue = Deaths/Quantity * Quantity = Deaths

Or, to think about this in terms of the cost of the implicit tax compared to the world in which it was impossible to overdose. This cost is given as:

Cost of the implicit tax = Deaths + Excess Burden

In other words, the cost of this implicit tax to those in the market is the number of deaths caused by opioid overdoses plus the foregone gains from trade that would have occurred if people could use opioids without this implicit tax.

Now, let’s consider the Narcan policy. If there is Narcan on hand, anyone experiencing an opioid overdose can be administered Narcan and the drug user’s death can be prevented (I’m of course abstracting from efficacy rates and timely administration for simplicity of analysis). The increased prevalence of Narcan in public places therefore reduces the probability of death from an opioid overdose. Put differently, the policy lowers the implicit tax associated with opioid use. From our discussion of taxation, we know that the lower tax means a lower price, which necessarily means a lower cost of taxation. What is the composition of this tax cost reduction?

The critical issue here is what happens to deaths when the implicit tax declines. The answer is not obvious. A lower implicit tax implies that the quantity demand of opioids will increase. Whether deaths rise or fall will depend on the elasticity of demand for opioids. If the demand for opioids is relatively inelastic, then deaths will decline since the implicit tax will decline by a greater percentage than the quantity demanded rises. However, if the demand for opioids is elastic, then the percentage increase in quantity demanded will exceed the percentage decline in the implicit tax rate and deaths resulting from an opioid overdose will actually go up.

This is important for thinking about the implications of the policy. For example, if the demand for opioids is elastic, this implies that the benefit of a reduction in excess burden is greater than the overall benefits. This means that the increased benefits of risk-taking behavior exceeds the total benefits of the policy itself and overdose deaths rise. Surely, that is not what policymakers are intending.

But even if the demand for opioids is inelastic, this means that only some of the benefits come in the form of a reduction in the number of opioid-related deaths. Some of the benefits still come from the enjoyment people get from increasing their opoid use.

This is what I meant at the beginning of the post when I said that it is unclear whether Narcan reduces deaths. It reduces the “seen” deaths because we see people’s lives saved by the administration of the medication. However, there are potential “unseen” deaths as a result of the reduced cost of such risky behavior. The number of lives we observe being saved from the availability of the medication overstates the number of total lives saved.

Nonetheless, what this analysis does is give us a way of measuring the extent to which this policy can be successful. The key measurement variable would be to determine what happens to the number of opioid-related deaths after the policy is implemented. If the reduction in the number of deaths is large, then the policy is having its intended impact. On the other hand, if the reduction in the number of deaths is quite small, or if the number of deaths actually rise, then this tells us that the primary beneficiaries of the policy are those who simply enjoy using opioids.

The important point here is that price theory not only gives us a guide to thinking through the effects of the policy, but it also tells us what we should be measuring to judge the effectiveness of the policy in relation to its intended goals.