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Teaching Vaccines Using Internal-to-the-Market Externalities

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Title: Teaching Vaccines Using Internal-to-the-Market Externalities
Language: English
Authors: Chen, Ziyue, Djalalova, Fatima, Rothschild, Casey (ORCID 0000-0002-8960-7997), Hofmann, Annette
Source: Journal of Economic Education. 2023 54(3):289-300.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 12
Publication Date: 2023
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
Descriptors: Teaching Methods, Economics Education, Immunization Programs, Undergraduate Students, Comparative Analysis, Economic Factors, Textbooks, COVID-19, Pandemics, Costs, Well Being, Visual Aids, Correlation
DOI: 10.1080/00220485.2023.2191597
ISSN: 0022-0485
2152-4068
Abstract: Textbook models of externalities tacitly assume that those externalities fall upon individuals "outside" of the market. In many contexts--including common undergraduate examples--externalities fall "inside" the market instead. Positive externalities associated with vaccination, for instance, accrue to other individuals who would potentially demand vaccines and affect their willingness to pay. The authors describe an undergraduate-accessible alternative diagrammatic approach to such internal-to-the-market externalities, using vaccines as their through-running example. They illustrate their approach by applying it in a study of binding mandates for 100-percent-effective vaccines and show how it can be used to depict a striking (known) result that, compared to laissez-faire, such a mandate will "always" lower social welfare. They also discuss important real-world caveats to this result.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1393174
Database: ERIC
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  Value: <anid>AN0164493247;jmd01jul.23;2023Jun27.06:28;v2.2.500</anid> <title id="AN0164493247-1">Teaching vaccines using internal-to-the-market externalities </title> <p>Textbook models of externalities tacitly assume that those externalities fall upon individuals "outside" of the market. In many contexts—including common undergraduate examples—externalities fall "inside" the market instead. Positive externalities associated with vaccination, for instance, accrue to other individuals who would potentially demand vaccines and affect their willingness to pay. The authors describe an undergraduate-accessible alternative diagrammatic approach to such internal-to-the-market externalities, using vaccines as their through-running example. They illustrate their approach by applying it in a study of binding mandates for 100-percent-effective vaccines and show how it can be used to depict a striking (known) result that, compared to laissez-faire, such a mandate will always lower social welfare. They also discuss important real-world caveats to this result.</p> <p>Keywords: Efficiency; undergraduate education; vaccination</p> <p>Vaccination and vaccine mandates are in the news and on our minds as we transition as a society—we hope!—from a pandemic where COVID-19 drives individual, business, and government-level decisions on a day-to-day basis to a world in which COVID-19 is "merely" endemic, and managed largely in the background, like the seasonal flu. Unfortunately, standard undergraduate economics tools are ill-suited to modeling vaccination.</p> <p>Vaccines are different from ordinary goods in two related ways. First, vaccination has positive externalities: when Alice gets vaccinated, the vaccine protects her from infection, and, in so doing, it also protects others that she could potentially infect. In other words, the benefit of Alice's vaccination to Alice is lower than the benefit of Alice's vaccination to society. Second, as discussed in Avery, Heymann, and Zeckhauser ([<reflink idref="bib1" id="ref1">1</reflink>]), the private benefit Alice gets from getting vaccinated—and therefore her willingness to pay for a vaccination—depends on how many other individuals around her have gotten vaccinations. For instance, Alice's private value of vaccination is likely to be low in a society where general vaccination rates are high enough to reduce or eliminate community spread, even if she would have valued the self-protection benefit of receiving a vaccine quite highly in a society with low general vaccination rates.</p> <p>The first feature suggests using a textbook externality graph to understand the effects of vaccines and vaccine mandates. Figure 1 recapitulates this standard textbook approach by way of explaining why it is inappropriate for modeling vaccination. The figure shows a horizontal (e.g., long-run) supply curve and a downward-sloping demand curve whose height measures the private benefit of vaccinations (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mrow><mi>B</mi></mrow><mrow><mi mathvariant="italic">pvt</mi></mrow></msup></math> </ephtml> ). It also shows the marginal social benefit of vaccination</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mrow><mi>B</mi></mrow><mrow><mi mathvariant="italic">soc</mi></mrow></msup></math> </ephtml> as lying a fixed amount</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi></math> </ephtml> above the demand curve. The vertical distance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi></math> </ephtml> is meant to capture the positive externality of vaccination.[<reflink idref="bib1" id="ref2">1</reflink>] The triangular area labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> in the diagram represents the deadweight loss (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">DWL</mi></math> </ephtml> ) associated with such an externality: the market equilibrium quantity</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>Q</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> , where supply hits demand, underprovides the vaccine relative to the efficient quantity</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="italic">eff</mi></mrow></msup></math> </ephtml> , where the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msup><mrow><mi>B</mi></mrow><mrow><mi mathvariant="italic">soc</mi></mrow></msup></math> </ephtml> hits the supply curve. Efficiency can be restored via a Pigouvian subsidy of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi></math> </ephtml> on the purchase or sale of the vaccine.</p> <p>Graph: Figure 1. Vaccines as a "standard" positive externality good.</p> <p>Figure 1 also illustrates the effect of a binding-purchase mandate in the standard textbook analysis. Specifically, the quantity labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">Max</mi></mrow></msup></math> </ephtml> drawn in the diagram represents the size of the population and, hence (assuming each individual needs at most a single course of this vaccine), the maximal quantity demanded. Imposing such a mandate means requiring that all</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">Max</mi></mrow></msup></math> </ephtml> individuals get vaccinated, so</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">Max</mi></mrow></msup></math> </ephtml> units are traded at (we assume) the same price corresponding to the height of the horizontal supply curve. In figure 1, this leads to an <emph>overprovision</emph> of the vaccine by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="normal">Max</mi></mrow></msup><mo>−</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi mathvariant="italic">eff</mi></mrow></msup></math> </ephtml> . Such a mandate would thus eliminate the DWL</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> but replace it with a new DWL labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> in the diagram. Because</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> is larger than</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> in figure 1, the diagram suggests that the mandate improves efficiency relative to the free market, albeit by less than an optimal subsidy would.</p> <p>We suspect that such graphs have been common in microeconomics classrooms in the COVID-19 era. They are not appropriate diagrams for modeling vaccines, however. The reason is that they ignore the second feature of vaccines alluded to above—the internal-to-the-market nature of vaccine externalities. The Setting and Equilibrium section of this article illustrates how to correct the diagram to capture this second feature.</p> <p>Vaccines are far from the only example in which externalities are internal-to-the-market. The use of prophylactics during sexual activity to prevent disease transmission is an obvious example. Another is automobile congestion externalities: one individual's decision not to drive on a congested road has positive externalities on other drivers, so the number of people choosing to work from home affects the willingness of others to choose to work from home.</p> <p>In order to demonstrate that our diagrammatic approach to modeling internal-to-the-market externalities gives insights that differ qualitatively from the standard diagram, we use it in the Application section to prove, graphically, a result about vaccine mandates first derived by Brito, Sheshinski, and Intriligator ([<reflink idref="bib5" id="ref3">5</reflink>]). Specifically, those authors show, when certain conditions are met, that allowing an unfettered free market for vaccines will always be more efficient than—in fact, will Pareto dominate—the imposition of a binding universal mandate for a vaccine. In other words, "standard" externality diagrams like figure 1 are not just conceptually inappropriate for modeling vaccines, but they can give the wrong answer about the effects of policy interventions (such as mandates) in markets for vaccines.[<reflink idref="bib2" id="ref4">2</reflink>]</p> <p>The basic intuition for why universal mandates are undesirable, relative to a first-best policy when vaccines are 100% effective, is straightforward: if vaccination rates are sufficiently high, then society will have herd immunity, and the marginal social benefit of vaccination will be zero. A standard figure like figure 1 can readily be adapted to capture <emph>some</emph> of this intuition.[<reflink idref="bib3" id="ref5">3</reflink>] Our view, however, is that this basic intuition actually points toward more serious shortcomings—both pedagogical and practical—in using standard figures to try and capture internal-to-the-market externalities. Specifically, the height of the demand curve in figure 1 is supposed to capture the private marginal benefit of consumption of the good. In the case of market like vaccines featuring internal-to-the-market externalities, however, these private benefits are socially determined. Conceptually, each individual's willingness to pay for a vaccine, and hence the demand curve, depends on the aggregate vaccination rate. Our main pedagogical contribution, in the Setting and Equilibrium section, is a description of how to adapt supply-demand diagrams to capture such within-market externalities.</p> <p>In the Application section, we apply our modified diagram to Brito, Sheshinski, and Intriligator's setting, where vaccines are 100% effective, and use it to formally derive the result that "universal mandates are always Pareto worse than the unfettered free market." The intuition behind this derivation is that <emph>if</emph> a vaccine is perfectly effective at preventing Alice from becoming ill, then once Alice becomes vaccinated, the vaccination status of others becomes irrelevant to her. It is not just that the externality accrues to the other potential demanders; it accrues <emph>only</emph> to the individuals who remain unvaccinated. So, if all individuals are vaccinated—i.e., if there is a universal mandate—it follows that there are no positive externalities from vaccination anymore.[<reflink idref="bib4" id="ref6">4</reflink>]</p> <p>We are arguing (and providing a graphical tool) for enhancing our undergraduate economics toolbox so as to allow our students to better understand the economics of an important class of externalities. We are not arguing against (or for) vaccine mandates. Indeed, as we discuss in the Discussion and Extensions section, there are several reasons to be wary, even within a narrow economics frame, of applying Brito, Sheshinski, and Intriligator's result to directly inform policy vis-à-vis vaccination. Moreover, our view is that an appropriate policy analysis of vaccine mandates would not rely exclusively on economic reasoning; insights from political science, sociology, and psychology are all important—and arguably more central to the debate. But understanding the economics is important, and our students need the better tools, which we provide here, for understanding these economics.</p> <p>Our article focuses on vaccines and vaccine mandates, both because of the acute and timely relevance for our students and because they were a go-to pedagogical example even before the dawn of the COVID-19 era.[<reflink idref="bib5" id="ref7">5</reflink>] We do, however, see our contribution as being about more than vaccines: in the spirit of Halteman ([<reflink idref="bib10" id="ref8">10</reflink>]) and Duke and Sassoon ([<reflink idref="bib7" id="ref9">7</reflink>]), we aim to provide a new pedagogical tool for enhancing how our undergraduates understand the economics of externalities, specifically the class of "internal-to-the-market" externalities in which the externality accrues to, and influences the willingness-to-pay of, demanders within the externality-causing market. As noted above, congestion externalities, externalities from diligent use of prophylactics, and numerous other pedagogically and policy-relevant examples fall into this class.</p> <hd id="AN0164493247-2">Setting and equilibrium</hd> <p>For expositional clarity and simplicity, we work, in this section and hereafter, with a simple numerical example of a setting with a collection of individuals who differ in their willingness to pay for a single unit of a vaccine for themselves. The insights are readily adapted, e.g., to general willingness-to-pay curves and type distributions.</p> <p>We assume that a unit mass of individuals, indexed by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> , is uniformly distributed on the interval [0, 1]. The willingness of individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> to pay is given by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>10</mn><mo>−</mo><mn>4</mn><mi>i</mi><mo>−</mo><mn>6</mn><mi>α</mi></math> </ephtml> , where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> is the fraction of the population that is vaccinated.</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> represents individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> 's marginal private benefit of being vaccinated, given that the fraction</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> of others are vaccinated.[<reflink idref="bib6" id="ref10">6</reflink>] Note that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> is decreasing both in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> for each</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> —so that higher-indexed types value being vaccinated less, relative to being unvaccinated—and in</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> for each</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> —so that the private benefits of vaccination are lower when society-wide vaccination rates are higher.</p> <p>For any particular aggregate vaccination rate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> , the graph of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> can be understood as a standard demand curve: at a price of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> , individuals with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>></mo><mi>p</mi></math> </ephtml> will demand a vaccine and those with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo><</mo><mi>p</mi><mi mathvariant="normal" /></math> </ephtml> will not. So the quantity demanded at price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> , given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> , is exactly equal to the type</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi><mtext /></mrow><mo stretchy="true">^</mo></mover></mrow></math> </ephtml> such that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mi>α</mi></msub><mo stretchy="false">(</mo><mtext /><mover accent="true"><mrow><mi>i</mi><mtext /></mrow><mo stretchy="true">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></mrow></math> </ephtml> . A higher</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> implies a lower demand curve, as we illustrate in figure 2: it plots, among other things, the demand curve for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>0</mn></math> </ephtml> (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> ) and for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>0.6</mn></math> </ephtml> (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0.6</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> ).[<reflink idref="bib7" id="ref11">7</reflink>] These willingness-to-pay curves differ from ordinary demand curves in one critical way: they are demand curves <emph>conditional</emph> on</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> , and therefore conditional on the aggregate quantity of vaccines consumed. The aggregate quantity is an equilibrium outcome, so the question of which of these standard demand curves will end up being relevant is <emph>endogenous</emph> to the model. It follows that these demand curves cannot be used, in isolation, as a graphical tool for computing or graphically identifying equilibrium.</p> <p>Graph: Figure 2. Equilibrium in the vaccine market.</p> <p>Figure 2 also depicts a third downward-sloping curve labeled "Demand," which <emph>can</emph> be interpreted as a more standard, exogenous demand curve and which can therefore be used for identifying equilibrium graphically. The height of this third demand curve at quantity</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> is given by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> : this is the willingness to pay of the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math> </ephtml> type if all of (and only) the higher-willingness-to-pay types get vaccinated. In other words, it is the willingness to pay of the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math> </ephtml> individual if she is the marginal type.</p> <p>To explain how the graph of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> can be interpreted as a standard, exogenous demand curve, fix any price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≤</mo><mn>8</mn></math> </ephtml> and identify the unique</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>i</mi><mo stretchy="true">^</mo></mover><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> such that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mover accent="true"><mi>i</mi><mo stretchy="true">^</mo></mover><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>i</mi><mo stretchy="true">^</mo></mover><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></mrow></math> </ephtml> . For concreteness, suppose</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> </ephtml> , so that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>i</mi><mo stretchy="true">^</mo></mover><mo>=</mo><mn>0.8</mn></mrow></math> </ephtml> . If all individuals are price takers facing price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> </ephtml> , and all believe that the fraction</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mo>̂</mo></mover></mrow><mo stretchy="true">(</mo><mrow><mi>p</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>0.8</mn></math> </ephtml> of the population is being vaccinated, then exactly 0.8 of the population will, <emph>in fact,</emph> demand a vaccine. Indeed,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0.8</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>5.2</mn><mo>−</mo><mn>4</mn><mi>i</mi></math> </ephtml> , which is greater than</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> </ephtml> for all</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo><</mo><mn>0.8</mn></math> </ephtml> and less than</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> </ephtml> for all</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>></mo><mn>0.8</mn></math> </ephtml> , so precisely the lowest-indexed 0.8 of the population will get vaccinated, consistent with beliefs. Any <emph>other</emph> beliefs</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>≠</mo><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mo>̂</mo></mover></mrow><mo stretchy="true">(</mo><mrow><mn>2</mn></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>0.8</mn></math> </ephtml> would be inconsistent with behavior. For instance, suppose that individuals believed that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math> </ephtml> of the population was getting vaccinated. Then <emph>everyone</emph> would want to get vaccinated at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> </ephtml> (as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msub><mo stretchy="true">(</mo><mrow><mn>1</mn></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>2</mn></math> </ephtml> , so the lowest WTP type is just willing to vaccinate given beliefs</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> </ephtml> ). So actual behavior—100% vaccination—would be inconsistent with beliefs about the aggregate vaccination rate.[<reflink idref="bib8" id="ref12">8</reflink>]</p> <p>Equilibrium is determined by the intersection of the demand curve (i.e., the graph of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> ) and a (standard) supply curve. We assume that the market features a horizontal supply curve. This assumption is not substantively important, but it does simplify the diagrams that follow.[<reflink idref="bib9" id="ref13">9</reflink>]Figure 2 depicts the market equilibrium</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>,</mo><mi /><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></math> </ephtml> where the demand curve intersects a horizontal supply curve. Notice that the demand curve</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> intersects</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>0</mn></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> , respectively, and that for</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>i</mi><mo><</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo><</mo><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo><</mo><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> . The actual demand curve is thus steeper than the "conditional" demand curves. The reason is simple: as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> rises, the intrinsic taste for vaccination of the marginal individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> falls, as with the conditional demand curves, and the willingness to pay of any given individual <emph>also</emph> falls as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> rises along the demand curve, because the fraction of vaccinated individuals rises and the disease environment improves. This second effect, which also can be understood as a shift to lower the conditional demand curve as the vaccination rate increases with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> , makes the market demand curve steeper than each individual conditional demand curve.</p> <p>The preceding equilibrium analysis applies to vaccinations (and other internal-to-the-market externalities) quite generally. Under the additional assumption that vaccination is 100% effective, microeconomics-classroom-style graphical welfare analysis of equilibrium is also straightforward.[<reflink idref="bib10" id="ref14">10</reflink>] This 100% effectiveness assumption directly implies that individuals who are vaccinated are indifferent to the vaccination status of others (they are already fully protected!). More subtly, it also lets us <emph>measure</emph> the positive externality associated with vaccinations via the vertical distance between the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> and the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mi mathvariant="normal" /></math> </ephtml> curves. It therefore allows us to illustrate the positive externality associated with any vaccination rate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> via simple areas on supply-demand graphs like figure 2. To see why</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>−</mo><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> measures the positive externality, observe that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> represents a money-metric difference between (a) the welfare that individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> would get if they received a (free) vaccination and (b) the welfare that the individual would get if they did not get vaccinated and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> others were vaccinated. With 100% effective vaccines, (a) is independent of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> . As such, the vertical distance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>−</mo><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> is the same as the difference in the (b)s respectively associated with vaccination rate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> and a vaccination rate of 0—i.e., the amount by which the vaccination of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> others increases the welfare of the unvaccinated individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> . So, for instance, the vertical distance between the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> and the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> curves can be understood as a positive externality on any</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>></mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> that is caused by equilibrium level</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> of vaccination.</p> <p>The sum of the areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> </ephtml> in figure 3 is the total consumer surplus—and hence, in light of the horizontal supply curve, the total welfare—associated with the equilibrium depicted in figure 2. The areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> </ephtml> are collectively the positive externality associated with vaccination: they sum up the total external benefits that accrue to the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> unvaccinated individuals as a result of the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> individuals who get vaccinated in equilibrium. The trapezoid equal to the combined areas</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> represents the total consumer surplus that accrues to individuals who do get vaccinated. The vertical slice of the trapezoid at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> is the benefit of vaccination (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> </ephtml> ), less the price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> that they pay. This is the net benefit that type</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> would get if they were the <emph>only</emph> individual offered a vaccine at price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> ; it coincides with their consumer surplus in equilibrium because, once they get vaccinated, the vaccination of others provides no additional benefits when vaccines are 100% effective.</p> <p>Graph: Figure 3. Consumer surplus in equilibrium in the vaccine market.</p> <p>Alternatively, we can decompose the consumer surplus of vaccinated individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> into two components: a benefit equal to the vertical slice of the parallelogram consisting of the areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> —which represents the benefit that would accrue to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> <emph>others</emph> got vaccinated—plus a benefit equal to the vertical slice of the triangle consisting of the areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> that represents the additional benefit</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> gets if they get vaccinated knowing that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> others are choosing to get vaccinated. In the trapezoid conceptual approach, we imagine the types</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo><</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> making the decision to vaccinate "before" the others do; then, all the consumer surplus comes from their decision to vaccinate because, having made that decision, they do not get any additional benefit from others who vaccinate. In the triangle-plus-parallelogram conceptual approach, we instead imagine that the types</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo><</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> make the decision "after" the others do, and thus "already" know how many others are getting vaccinated; then, they have already banked the externality benefit (the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>s</mi></math> </ephtml> ) from the others' decision to vaccinate and get an additional benefit (the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi></math> </ephtml> ) from choosing to get vaccinated at price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> . The two approaches are equivalent, but the former is better for foregrounding the idea that vaccination externalities <emph>actually</emph> accrue only to the unvaccinated, while the latter is better for foregrounding the logic of equilibrium for price- and environment-taking agents.</p> <hd id="AN0164493247-3">An application: A welfare analysis of vaccine mandates</hd> <p>Consider imposing a universal vaccine mandate. We imagine this mandate being imposed as a requirement that all individuals purchase a vaccine at the market price,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> , enforced with a sufficiently severe punishment for noncompliance with which all individuals will comply.[<reflink idref="bib11" id="ref15">11</reflink>] Figure 4 illustrates the welfare effects of such a mandate. The total consumer surplus is the sum of the areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> </ephtml> . The collective areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math> </ephtml> represent the total surplus enjoyed by those individuals who would have vaccinated in equilibrium anyway; the collective area <emph>D</emph> coincides exactly with the sum of areas</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> in figure 3 because the extra vaccinations of the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> individuals induced by the mandate do not affect the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo><</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> individuals who were already fully protected by vaccination in equilibrium.</p> <p>Graph: Figure 4. Consumer surplus with a binding mandate in the market for vaccines.</p> <p>Area</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> </ephtml> , which represents the benefit to the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>></mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> individuals, can be understood in two ways (much as areas</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>+</mo><mi>B</mi></math> </ephtml> could be understood in two ways in figure 3). The first way is to imagine each type</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>></mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> getting vaccinated unilaterally "before" the others do. Their unilateral vaccination yields a benefit equal to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> , i.e., equal to the height of area</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> </ephtml> ; and since, once they are vaccinated, they are fully protected, the vaccination status of others is irrelevant, area</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> </ephtml> is the total surplus accruing to the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>></mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> individuals. The second way is to imagine each</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>></mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> individual getting vaccinated "after" the others do. In this case, individual</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> "first" gets a benefit equal to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>−</mo><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> , for a total welfare benefit equal to the parallelogram between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> and the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> curves and between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mi /></math> </ephtml> (the sum of the areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>,</mo><mi /><mi>F</mi></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> ), and "then" is forced to pay</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> for a vaccination that they value only at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo><</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> , resulting in a negative surplus equal to the trapezoid consisting of the areas labeled</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> </ephtml> . Subtracting the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>+</mo><mi>G</mi></math> </ephtml> trapezoid from the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>+</mo><mi>F</mi><mo>+</mo><mi>G</mi></math> </ephtml> parallelogram results in the net surplus of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> </ephtml> , as in the first approach.</p> <p>The total welfare</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>+</mo><mi>B</mi><mo>+</mo><mi>C</mi></math> </ephtml> in figure 3 is larger than the total welfare</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>+</mo><mi>E</mi></math> </ephtml> in figure 4, the difference being the area</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> </ephtml> in figure 4. Imposing a mandate thus unambiguously reduces welfare. Specifically, it reduces the welfare of each type</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>></mo><msup><mrow><mi>i</mi></mrow><mo>*</mo></msup></mrow></math> </ephtml> who would not have chosen to vaccinate absent a mandate (by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>−</mo><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> , the height of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> </ephtml> at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> ) without affecting the welfare of those who would have chosen to vaccinate. Thus, a mandate does not just reduce the aggregate welfare; it causes a Pareto-worsening of welfare.[<reflink idref="bib12" id="ref16">12</reflink>] The argument in Brito, Sheshinski, and Intriligator ([<reflink idref="bib5" id="ref17">5</reflink>]), i.e., that a fully binding mandate of a 100% effective vaccine is Pareto-inferior to the free market equilibrium, can be understood as a more general version of this diagrammatic welfare analysis.</p> <p>The fact that a binding mandate is strictly worse than the free-market equilibrium within this model does not, of course, mean that the free-market equilibrium is optimal. Indeed, just as in standard markets with positive externalities, the surplus maximizing policy involves raising the quantity vaccinated above the equilibrium</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> .[<reflink idref="bib13" id="ref18">13</reflink>] The intuition is simple: in the free market equilibrium, individuals just above</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>i</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> are nearly indifferent to getting vaccinated. But their vaccination would strictly improve the well-being of higher</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> individuals. So inducing them to become vaccinated improves total social welfare.</p> <p>Figure 5 illustrates. The bottom curve in that figure plots the total social welfare</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="true">(</mo><mrow><mi>α</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> as a function of the aggregate level of vaccination</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> , assuming that the highest</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">WTP</mi></math> </ephtml> individuals are the ones who get vaccinated. The figure recapitulates the Brito, Sheshinski, and Intriligator result as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="true">(</mo><mrow><mn>0.6</mn></mrow><mo stretchy="true">)</mo><mo>></mo><mi>W</mi><mo stretchy="true">(</mo><mrow><mn>1</mn></mrow><mo stretchy="true">)</mo><mo>,</mo><mi /></math> </ephtml> i.e., it shows that social welfare is higher at the equilibrium vaccination rate of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.6</mn></math> </ephtml> than it is with universal vaccination. It also shows that surplus is nevertheless increasing in the vaccination rate</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>0.6</mn></math> </ephtml> —i.e., that it is welfare-improving to increase vaccination above the free market equilibrium levels—and that welfare reaches a maximum at a vaccination rate of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>α</mi></mrow><mrow><mi mathvariant="italic">eff</mi></mrow></msup><mo>=</mo><mn>75</mn><mi>%</mi></math> </ephtml> . As is standard, this optimum could be implemented with a Pigouvian subsidy—in this case, a subsidy of $1.50 (or 37.5% of the price), equal to the gap between the price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mn>4</mn></math> </ephtml> and the willingness to pay</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>0.75</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mn>0.75</mn></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>2.5</mn></math> </ephtml> of the marginal consumer at the optimum.[<reflink idref="bib14" id="ref19">14</reflink>]</p> <p>Graph: Figure 5. Total welfare as a function of the fraction vaccinated. All three curves use the same willingness to pay function, WTPα(i)=10−4i−6α. The solid bottom curve plots the welfare with 100% effective vaccines—for which positive externalities accrue only to the unvaccinated. The middle curve and top curves plot the welfare with vaccines that are less than 100% effective and, therefore, also have positive externalities on the vaccinated. In the middle curve, the size of the externality on the vaccinated is 1/16 the size of the baseline externality on the unvaccinated. In the top curve, the externality on the vaccinated is 1/7.5 the size of the baseline externality.</p> <hd id="AN0164493247-4">Discussion and extensions</hd> <p>The standard graphical textbook model of externalities is ill-suited to understanding the externalities associated with vaccination. The reason is that externalities from vaccines accrue to other participants (or potential participants) in the market for vaccines and thereby affect their demand for vaccines. We showed in the Setting and Equilibrium section how to use an undergraduate-accessible graph to capture these sorts of within-market externalities.</p> <p>It is noteworthy that vaccines are far from the only example of products for which there are within-market externalities. Prophylactics (viz., Marks et al. [<reflink idref="bib15" id="ref20">15</reflink>]), theft-prevention devices (viz., Ayres and Levitt [<reflink idref="bib2" id="ref21">2</reflink>]), auto-safety devices (viz., Peltzman [<reflink idref="bib17" id="ref22">17</reflink>]), antivirus software and firewalls (viz., Kshetri [<reflink idref="bib13" id="ref23">13</reflink>]), airline safety (viz., Kunreuther and Heal [<reflink idref="bib14" id="ref24">14</reflink>]), as well as insurance coverage (viz., Hofmann and Rothschild [<reflink idref="bib11" id="ref25">11</reflink>]) are all examples of markets with similar within-market spillovers. As such, the tools we develop here are valuable for studying a broad range of phenomena that our undergraduates may be interested in understanding better.</p> <p>In the Application section, we showed, by example, that appropriately modeling these within-market externalities may be critical for policy analysis: taking them into account implies that, contra the intuition from standard textbook externality models, a binding vaccine mandate for a 100% effective vaccine will always be efficiency- and welfare-reducing relative to a free-market with purely voluntary vaccines.</p> <p>This conclusion about vaccine mandates might suggest a normative inference that vaccine mandates are bad. We believe that the appropriate inference should instead be more nuanced. Specifically, it should be: <emph>if</emph> the simple model is a good model of the real world, <emph>then</emph> the type of vaccine mandates envisioned in this model is bad. Since, in the real world, policymakers and citizens broadly support vaccine mandates, it follows that either their support is misguided or else that the simple model from which we derived those conclusions fails to match up with the real world in one or more important ways. Our view leans toward the latter possibility, and we believe that a discussion of how the model fails to map into the real world is likely to be illuminating for students.</p> <p>A reasonably common anti-mandate view in the United States today is that mandates are unnecessary precisely <emph>because</emph> vaccines work: if vaccines work well, then vaccinated individuals should not care if others do not vaccinate. The analysis illustrated in figure 4 can be understood as a formalization of this view. Asking why the conclusions of Brito, Sheshinski, and Intriligator ([<reflink idref="bib5" id="ref26">5</reflink>]) might be misleading as a guide to policy is thus closely related to asking why someone, in good faith, might disagree with the idea that vaccinated individuals should not care if others choose not to vaccinate.</p> <p>Even within a purely economistic approach to vaccination, we see several reasons. First, vaccines are often imperfect against existing diseases: breakthrough infections are possible. Moreover, as we saw first with the omicron COVID-19 variant, one reason that breakthrough infections happen is the emergence of new vaccine-evading variants. Raising vaccination rates can be expected to lower the probability that a variant will emerge that overcomes the previous vaccine, adding a second conceptual level to the positive externality from vaccination. Consequently, vaccination, <emph>in fact,</emph> has positive externalities on the vaccinated—even if it is 100% effective against existing variants.</p> <p>Second, vaccination may have positive externalities on the vaccinated through other, indirect channels. Many employers, for example, have moved back toward in-person work rather than remote work. If some, but not all, work can still be done remotely, if remote work is desirable, and if employers assign a disproportionate share of the in-person work to the vaccinated, then vaccination will lead to positive externalities on vaccinated coworkers.</p> <p>Third, and critically, vaccines may be unavailable to some individuals. Vaccinated parents, for instance, could quite sensibly favor a vaccine mandate because of the positive externality it would have on their unvaccinatable children.</p> <p>So, models following Brito, Sheshinski, and Intriligator typically understate the benefits of vaccination, particularly the benefits to the vaccinated. It is important to note, however, that the core "mandates decrease total welfare" result still holds for highly but imperfectly effective vaccines. When vaccines are less than perfectly effective, vaccination also has positive externalities on those who vaccinate; consequently, the welfare benefit of additional vaccination is underestimated in figure 4 and the heavy bottom line in figure 5. According to the Centers for Disease Control,[<reflink idref="bib15" id="ref27">15</reflink>] the relative risk of death for unvaccinated versus vaccinated individuals was</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>68</mn><mo>:</mo><mn>1</mn></math> </ephtml> in early 2022, and the relative risk of hospitalization was</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo>:</mo><mn>1</mn></math> </ephtml> . Using the latter as a proxy for the ratio of the magnitude of the externality on the unvaccinated to the magnitude of the externality on the vaccinated, the middle curve in figure 5 plots social welfare as a function of the vaccination rates for the case of imperfectly effective vaccines—which we model by adding a positive externality of vaccination on the vaccinated to the baseline model.[<reflink idref="bib16" id="ref28">16</reflink>]</p> <p>Adding this extra positive externality raises social welfare but does not change the conclusion that the free-market equilibrium yields higher welfare than a binding mandate. Indeed, as the top curve in that diagram indicates, welfare under the free market remains higher than welfare under a mandate as long as the ratio of the baseline externality on the unvaccinated to the externality on the vaccinated is less than</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>7.5</mn></math> </ephtml> (when the welfare levels are exactly equal, as shown in the top curve in figure 5). We are therefore inclined to think that the conclusion that a laissez-faire approach to vaccination is preferable to a binding mandate is at least plausible, even though vaccines are less than fully effective—especially because there are likely substantial enforcement costs of such a mandate from which both Brito, Sheshinski, and Intriligator and we have abstracted.</p> <p>Nevertheless, we think there is an important (if largely semantic) reason—over and above the less than 100% efficacy of vaccines and the associated externalities on the vaccinated—to be wary of applying the lessons of Brito, Sheshinski, and Intriligator and figure 4 directly to policy. A "mandate" in the model outlined above by assumption yields universal compliance. This universality distinguishes it from real-world mandates, which typically include exceptions and which are typically enforced with punishments that are mild enough that compliance with them will be imperfect. A mandate enforced with a modest punishment is behaviorally equivalent to a finite subsidy on vaccination (assuming perfectly rational behavior and abstracting from income effects). Since we know that the welfare-maximizing policy involves a finite subsidy (in our baseline model, 37.5% of the price), it follows that a real-world "mandate" with a finite penalty may, in fact, be an optimal policy. In other words, even if the model itself is a good approximation to the real world, the appropriate conclusion is not that there should be no "mandate." The appropriate conclusion, rather, is that the penalty associated with any mandate should be strictly smaller than the minimum penalty required for inducing full compliance.</p> <hd id="AN0164493247-5">Conclusions—Externalities in the classroom</hd> <p>Standard, graphical undergraduate analyses tacitly assume that all externalities are external-to-the-market. Yet many of the leading real-world examples used in motivating the study of externalities in undergraduate courses involve internal-to-the-market externalities and, as such, are not examples to which the standard graphs apply. The contemporary salience of vaccination and vaccine mandates—as clear an example of an internal-to-the-market externality as there is—exacerbates the tension between analytical and motivational components of externality-related pedagogy. The graphical approach developed here aims to alleviate this tension by providing an undergraduate-accessible way to teach internal-to-the-market externalities.</p> <p>Because the basic graphical tools—supply-demand graphs and welfare analysis via areas thereon—are at the core of most introductory microeconomics courses, and because no mathematics beyond basic algebra is required, we suspect that a dedicated and focused teacher could use the approach developed here to teach internal-to-the-market externalities in an introductory course. Our sense, though, is that for the vast majority of introductory courses, the benefits would not justify the cost of allocating sufficient course time to succeed in such an effort. Moreover, there are some subtleties that would likely make even raising the idea of internal-to-the-market confusing to introductory students. For instance, internal-to-the-market externalities are effects that are external to the <emph>individual transactions</emph> but not external to the market. Instructors who focus on external-to-the-market externalities in their courses can safely elide the potentially confusing distinction between "external-to-the-market" and "external-to-the-transaction."</p> <p>We are therefore inclined to think that teaching internal-to-the-market externalities is instead well-suited to and worth incorporating in intermediate-level microeconomics courses (whether calculus-based or not). Most basically, externalities are a core topic in undergraduate education, and, particularly now, many of the most important examples of externalities involve internal-to-the-market effects; we do a disservice to our majors if we restrict attention to less interesting and topical externality answers—and also if we "get it wrong" for them. Moreover, we believe that teaching internal-to-the-market externalities using a simple example like the through-running one in this article is likely to help students better understand core economic principles, not just about how externalities work but about how welfare analysis works more generally.</p> <p>Teaching internal-to-the-market externalities toward the end of an intermediate microeconomics course also offers an opportunity for instructors to help students understand the relationship between the price-theoretic notions of equilibrium (i.e., partial and general) and the game-theoretic notion of equilibrium (i.e., Nash) that they have typically all seen in their courses. Our preferred interpretation of our model is price-theoretic: individuals take the aggregate environment (prices and aggregate vaccination levels) as given and optimize in that environment; equilibrium is a "stable" aggregate environment, in the precise sense that these optimization decisions generate an outcome consistent with that environment. As we alluded to in note 8, however, we can also interpret the determination of the demand curve in game-theoretic terms. Indeed, fixing any price</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> </ephtml> , the unique quantity</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> for which</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>α</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mi>p</mi></math> </ephtml> associated with the demand curve is, in fact, the unique Nash equilibrium of the simultaneous-move game where each of the players independently decides whether or not to get vaccinated. The socially suboptimal levels of vaccination that emerge in the free-market equilibrium can thus be interpreted in terms of a classic prisoner's dilemma coordination failure or in terms of Pigouvianesque externalities. We thus think that, particularly in light of its current salience, internal-to-the-market externalities provide an excellent summative extension/example at the end of an intermediate micro course.</p> <p>Finally, we think that our analysis adds some value for instructors—e.g., introductory instructors—who opt not to include it in their courses. First, simply being aware that the standard picture is wrong, and why, is both intrinsically valuable and important in guiding instructors to select examples that fit well in the more standard analysis. Moreover, as instructors, we often find ourselves needing to make simplifications that render the analysis formally flawed. Knowing the right way to do it—even if we opt, for pedagogical reasons, not to teach it that way—makes such simplifications more palatable to us, especially (but not only) because it gives us a way to respond to those exceptional students who "notice" and inquire about the flaw.</p> <hd id="AN0164493247-6">Acknowledgments</hd> <p>The authors thank two anonymous referees and the handling co-editor for exceptional suggestions.</p> <ref id="AN0164493247-7"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Avery, C., S. J. Heymann, and R. Zeckhauser. 1995. Risks to selves, risks to others. The American Economic Review 85 (2): 61 – 66.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref4" type="bt">2</bibl> <bibtext> Ayres, I., and S. D. Levitt. 1998. Measuring positive externalities from unobservable victim precaution: An empirical analysis of Lojack. The Quarterly Journal of Economics 113 (1): 43 – 77. doi: 10.1162/003355398555522.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref5" type="bt">3</bibl> <bibtext> Bohanon, C. 1985. Externalities: A note on avoiding confusion. Journal of Economic Education 16 (4): 305 – 7. doi: 10.1080/00220485.1985.10845134.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref6" type="bt">4</bibl> <bibtext> Boulier, B. L., T. S. Datta, and R. S. Goldfarb. 2007. Vaccination externalities. BE Journal of Economic Analysis & Policy 7 (1): Article 23.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref3" type="bt">5</bibl> <bibtext> Brito, D. L., E. Sheshinski, and M. D. Intriligator. 1991. Externalities and compulsory vaccinations. Journal of Public Economics 45 (1): 69 – 90. doi: 10.1016/0047-2727(91)90048-7.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref10" type="bt">6</bibl> <bibtext> Christiansen, V., and S. Smith. 2012. Externality-correcting taxes and regulation. The Scandinavian Journal of Economics 114 (2): 358 – 83. doi: 10.1111/j.1467-9442.2012.01701.x.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref9" type="bt">7</bibl> <bibtext> Duke, J. M., and D. M. Sassoon. 2017. A classroom game on a negative externality correcting tax: Revenue return, regressivity, and the double dividend. Journal of Economic Education 48 (2): 65 – 73. doi: 10.1080/00220485.2017.1285736.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref12" type="bt">8</bibl> <bibtext> Francis, P. J. 1997. Dynamic epidemiology and the market for vaccinations. Journal of Public Economics 63 (3): 383 – 406. doi: 10.1016/S0047-2727(96)01586-1.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref13" type="bt">9</bibl> <bibtext> Goldfarb, R. S. 2013. Shortage, shortage, who's got the shortage? Journal of Economic Education 44 (3): 277 – 97. doi: 10.1080/00220485.2013.795461.</bibtext> </blist> <blist> <bibtext> Halteman, J. 2005. Externalities and the Coase theorem: A diagrammatic presentation. Journal of Economic Education 36 (4): 385 – 90. doi: 10.3200/JECE.36.4.385-390.</bibtext> </blist> <blist> <bibtext> Hofmann, A., and C. Rothschild. 2019. On the efficiency of self-protection with spillovers in risk. The Geneva Risk and Insurance Review 44 (2): 207 – 21. doi: 10.1057/s10713-019-00041-z.</bibtext> </blist> <blist> <bibtext> Hoyt, G. M., P. L. Ryan, and R. G. Houston, Jr. 1999. The paper river: A demonstration of externalities and Coase's theorem. Journal of Economic Education 30 (2): 141 – 47. doi: 10.1080/00220489909595951.</bibtext> </blist> <blist> <bibtext> Kshetri, N. 2009. Positive externality, increasing returns, and the rise in cybercrimes. Communications of the ACM 52 (12): 141 – 44. doi: 10.1145/1610252.1610288.</bibtext> </blist> <blist> <bibtext> Kunreuther, H., and G. Heal. 2003. Interdependent security. Journal of Risk and Uncertainty 26 (2/3): 231 – 49. doi: 10.1023/A:1024119208153.</bibtext> </blist> <blist> <bibtext> Marks, G., N. Crepaz, J. W. Senterfitt, and R. S. Janssen. 2005. Meta-analysis of high-risk sexual behavior in persons aware and unaware they are infected with HIV in the United States: Implications for HIV prevention programs. Journal of Acquired Immune Deficiency Syndromes 39 (4): 446 – 53.</bibtext> </blist> <blist> <bibtext> Mrozek, J. R. 1999. Market failures and efficiency in the principles course. Journal of Economic Education 30 (4): 411 – 19. doi: 10.1080/00220489909596098.</bibtext> </blist> <blist> <bibtext> Peltzman, S. 1975. The effects of automobile safety regulation. Journal of Political Economy 83 (4): 677 – 725. doi: 10.1086/260352.</bibtext> </blist> </ref> <ref id="AN0164493247-8"> <title> Footnotes </title> <blist> <bibtext> The vertical distance could also be—and might naturally be—drawn as a function of</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> </ephtml> ; nothing that follows in this introduction depends in any important way on the assumption of a constant marginal externality.</bibtext> </blist> <blist> <bibtext> We argue that vaccines should be taught differently from standard externalities and their corrections. A broad literature, including corrective taxation (Hoyt, Ryan, and Houston [12]), property rights (Christiansen and Smith [6]), and different types of externalities (Bohanon [3]) discusses teaching those standard externalities; vaccines are nonstandard, given their unique nature.</bibtext> </blist> <blist> <bibtext> E.g., the marginal social benefit for the last few units could be drawn as being negative. And, of course, the figure could be modified to capture the intuition even better by having the size of the positive externality shrink to zero for sufficiently high vaccination levels</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> An analogy may be helpful here. Deciding not to drive perfectly inoculates you from the congestion externalities caused by others who drive. So "not driving" induces positive externalities—but only if there are others who are choosing to drive. So a complete ban on driving will eliminate all of the positive externalities that would plausibly motivate such a plan. In fact, since "don't drive" is always an option available in the laissez-faire outcome, a complete ban on driving cannot make anyone better off than the laissez-faire outcome and will make those who would have chosen to drive strictly worse off.</bibtext> </blist> <blist> <bibtext> Goldfarb ([9]), for instance, uses flu vaccines as an example of demand-deadline shortages. Mrozek ([16]) cites vaccines as a canonical pedagogical example of an externality.</bibtext> </blist> <blist> <bibtext> We make this assumption that the</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">WTP</mi></math> </ephtml> depends only on the aggregate vaccination share</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> for expositional simplicity. This assumption is not without loss of generality. Indeed, as discussed by Boulier, Datta, and Goldfarb ([4]), the pattern of the vaccination-induced externalities tends to be rather complex, and the size of the externality is not necessarily even monotonic in the number of vaccinated individuals, vaccine efficacy, or disease infectiousness. So the real-world externalities of a vaccine mandate are likely to be much more complex than our simple model of it. None of the basic underlying economics in our analysis hinges on these details, however.</bibtext> </blist> <blist> <bibtext> It is not substantively important that these are drawn as parallel.</bibtext> </blist> <blist> <bibtext> As pointed out by a helpful referee, and as we discuss further in the Conclusions section, this "consistency of vaccination behavior with beliefs about vaccination, given prices" can also be understood in game-theoretic terms.</bibtext> </blist> <blist> <bibtext> Moreover, it is likely a reasonable assumption for a vaccine that has already been developed and whose intellectual property lies in the public domain.</bibtext> </blist> <blist> <bibtext> We discuss in the Discussion and Extensions section how algebraic and graphical analysis can be applied when vaccination is imperfectly effective.</bibtext> </blist> <blist> <bibtext> Imposing it while also making it free would be welfare-equivalent but would modestly complicate graphical surplus accounting.</bibtext> </blist> <blist> <bibtext> The experience of one of the authors is that, when faced with this picture, some students will wonder whether there is a way to avoid the reduction in welfare of the non-vaccinators with appropriate transfers. The answer is "no" because total welfare (the size of the pie) is smaller. It is, of course, possible to prevent the non-vaccinators from being made worse off with a sufficiently large transfer targeted at them, but this is just rearranging deck chairs: those transfers have to come from somewhere.</bibtext> </blist> <blist> <bibtext> Note that contra the leading example in Francis ([8]), this means that there <emph>are</emph> externalities in vaccination decisions; the key difference is simply that Francis's leading example involves perfectly homogeneous types with no heterogeneity in willingness to pay.</bibtext> </blist> <blist> <bibtext> In this example, the optimal vaccination level is, in fact, straightforward to compute analytically because the total welfare is the simple quadratic function</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib10" idref="ref8" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo stretchy="true">(</mo><mrow><mi>α</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mi>α</mi><mo stretchy="true">(</mo><mrow><mn>12</mn><mo>−</mo><mn>8</mn><mi>α</mi></mrow><mo stretchy="true">)</mo></math> </ephtml> , i.e., the sum of the trapezoid with height</bibtext> </blist> <blist> <bibl id="bib11" idref="ref15" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib12" idref="ref16" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> and bases</bibtext> </blist> <blist> <bibl id="bib13" idref="ref18" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib14" idref="ref19" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="italic">WTP</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mn>0</mn></mrow><mo stretchy="true">)</mo><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>*</mi></mrow></msup></math> </ephtml> and</bibtext> </blist> <blist> <bibl id="bib15" idref="ref20" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib16" idref="ref28" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mtext /><mi>α</mi><mo>−</mo><msup><mrow><mi>p</mi></mrow><mo>*</mo></msup></mrow></math> </ephtml> (the welfare gain of the vaccinated) and the parallelogram with height</bibtext> </blist> <blist> <bibl id="bib17" idref="ref22" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib18" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><mi>α</mi></math> </ephtml> and base</bibtext> </blist> <blist> <bibl id="bib19" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib20" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mtext /><mi>α</mi><mo>−</mo><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mi>α</mi></msub><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math> </ephtml> (the welfare gain of the unvaccinated).</bibtext> </blist> <blist> <bibtext> https://covid.cdc.gov/covid-data-tracker (accessed February 18, 2022).</bibtext> </blist> <blist> <bibtext> Specifically, we assume that the well-being of a vaccinated individual increases, relative to our baseline model, by</bibtext> </blist> <blist> <bibl id="bib21" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib22" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>k</mi><mi>α</mi></math> </ephtml> when the fraction</bibtext> </blist> <blist> <bibl id="bib23" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib24" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> of the population is vaccinated. The baseline externality is given by</bibtext> </blist> <blist> <bibl id="bib25" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib26" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>−</mo><mi>W</mi><mi>T</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>i</mi></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>6</mn><mi>α</mi></math> </ephtml> , so</bibtext> </blist> <blist> <bibl id="bib27" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib28" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> measures the ratio of the externality on the vaccinated to the baseline externality. We take</bibtext> </blist> <blist> <bibl id="bib29" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib30" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>16</mn></mrow></mfrac></math> </ephtml> to match the relative hospitalization risk—and view this as a plausible upper bound, in light of the much higher relative odds of death. One minor technical note: introducing the externality</bibtext> </blist> <blist> <bibl id="bib31" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib32" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>k</mi><mi>α</mi></math> </ephtml> on the vaccinated <emph>while holding the</emph></bibtext> </blist> <blist> <bibl id="bib33" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib34" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">WTP</mi></math> </ephtml> <emph>functions constant</emph> actually increases the well-being of the unvaccinated as well as the vaccinated and the unvaccinated. This has two implications. First, it implies that social welfare increases by</bibtext> </blist> <blist> <bibl id="bib35" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib36" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>k</mi><mi>α</mi></math> </ephtml> relative to the baseline model; this is what we use to compute the higher lines in figure 5. Second, it implies that the externality on the unvaccinated is actually</bibtext> </blist> <blist> <bibl id="bib37" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib38" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mn>6</mn></math> </ephtml> , so the ratio of externality magnitudes is technically</bibtext> </blist> <blist> <bibl id="bib39" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib40" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msup><mrow><mo stretchy="true">(</mo><mrow><mn>1</mn><mo>+</mo><mi>k</mi></mrow><mo stretchy="true">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> rather than</bibtext> </blist> <blist> <bibl id="bib41" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib42" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> —but for</bibtext> </blist> <blist> <bibl id="bib43" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib44" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>16</mn></mrow></mfrac></math> </ephtml> this is not a quantitatively important distinction.</bibtext> </blist> </ref> <aug> <p>By Ziyue Chen; Fatima Djalalova; Casey Rothschild and Annette Hofmann</p> <p>Reported by Author; Author; Author; Author</p> </aug>
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  Group: Ti
  Data: Teaching Vaccines Using Internal-to-the-Market Externalities
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  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Chen%2C+Ziyue%22">Chen, Ziyue</searchLink><br /><searchLink fieldCode="AR" term="%22Djalalova%2C+Fatima%22">Djalalova, Fatima</searchLink><br /><searchLink fieldCode="AR" term="%22Rothschild%2C+Casey%22">Rothschild, Casey</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-8960-7997">0000-0002-8960-7997</externalLink>)<br /><searchLink fieldCode="AR" term="%22Hofmann%2C+Annette%22">Hofmann, Annette</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Journal+of+Economic+Education%22"><i>Journal of Economic Education</i></searchLink>. 2023 54(3):289-300.
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  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Label: Peer Reviewed
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  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 12
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2023
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
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  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
– Name: Subject
  Label: Descriptors
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  Data: <searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Economics+Education%22">Economics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Immunization+Programs%22">Immunization Programs</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Students%22">Undergraduate Students</searchLink><br /><searchLink fieldCode="DE" term="%22Comparative+Analysis%22">Comparative Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Economic+Factors%22">Economic Factors</searchLink><br /><searchLink fieldCode="DE" term="%22Textbooks%22">Textbooks</searchLink><br /><searchLink fieldCode="DE" term="%22COVID-19%22">COVID-19</searchLink><br /><searchLink fieldCode="DE" term="%22Pandemics%22">Pandemics</searchLink><br /><searchLink fieldCode="DE" term="%22Costs%22">Costs</searchLink><br /><searchLink fieldCode="DE" term="%22Well+Being%22">Well Being</searchLink><br /><searchLink fieldCode="DE" term="%22Visual+Aids%22">Visual Aids</searchLink><br /><searchLink fieldCode="DE" term="%22Correlation%22">Correlation</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/00220485.2023.2191597
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0022-0485<br />2152-4068
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Textbook models of externalities tacitly assume that those externalities fall upon individuals "outside" of the market. In many contexts--including common undergraduate examples--externalities fall "inside" the market instead. Positive externalities associated with vaccination, for instance, accrue to other individuals who would potentially demand vaccines and affect their willingness to pay. The authors describe an undergraduate-accessible alternative diagrammatic approach to such internal-to-the-market externalities, using vaccines as their through-running example. They illustrate their approach by applying it in a study of binding mandates for 100-percent-effective vaccines and show how it can be used to depict a striking (known) result that, compared to laissez-faire, such a mandate will "always" lower social welfare. They also discuss important real-world caveats to this result.
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  Data: 2023
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  Data: EJ1393174
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    Identifiers:
      – Type: doi
        Value: 10.1080/00220485.2023.2191597
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 12
        StartPage: 289
    Subjects:
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Economics Education
        Type: general
      – SubjectFull: Immunization Programs
        Type: general
      – SubjectFull: Undergraduate Students
        Type: general
      – SubjectFull: Comparative Analysis
        Type: general
      – SubjectFull: Economic Factors
        Type: general
      – SubjectFull: Textbooks
        Type: general
      – SubjectFull: COVID-19
        Type: general
      – SubjectFull: Pandemics
        Type: general
      – SubjectFull: Costs
        Type: general
      – SubjectFull: Well Being
        Type: general
      – SubjectFull: Visual Aids
        Type: general
      – SubjectFull: Correlation
        Type: general
    Titles:
      – TitleFull: Teaching Vaccines Using Internal-to-the-Market Externalities
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Chen, Ziyue
      – PersonEntity:
          Name:
            NameFull: Djalalova, Fatima
      – PersonEntity:
          Name:
            NameFull: Rothschild, Casey
      – PersonEntity:
          Name:
            NameFull: Hofmann, Annette
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2023
          Identifiers:
            – Type: issn-print
              Value: 0022-0485
            – Type: issn-electronic
              Value: 2152-4068
          Numbering:
            – Type: volume
              Value: 54
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Journal of Economic Education
              Type: main
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