key: cord-0780296-jveuxnuj
authors: Lee, Kangoh
title: Pandemics and support for mitigation measures
date: 2021-11-25
journal: J Econ (Vienna)
DOI: 10.1007/s00712-021-00765-5
sha: b5b981e6b9e26147f44a420a3abc3cde9d6af7cd
doc_id: 780296
cord_uid: jveuxnuj

Measures reduce health risk but limit economic activities and affect disproportionately the contact-intensive sectors whose economic activities involve more person-to-person interactions. The analysis shows that the size of the contact-intensive sectors shapes the stringency of measures due to the economic interactions between the contact-intensive sectors and other sectors although they constitute a minority of the labor force. Exploiting variation in measures and economic conditions across states, an empirical analysis shows that the number of contact-intensive workers has a negative effect on the stringency of measures.

During the Covid-19 pandemic, policymakers have implemented various containment measures. Such measures are intended to reduce health risks but also result in economic losses. A question concerns the severity of a measure that balances the health benefit and the economic loss, the topic that has recently received a good deal of attention (for example, Alvarez et al. 2020; Eichenbaum et al. 2020; Jones et al. 2020) . This paper attempts to contribute to this literature by studying the heterogeneous utility effects of a measure between the contact-intensive sectors and other sectors, a topic that has been rarely considered to the best of my knowledge, although heterogeneous job-loss effects have been studied (Barrot et al. 2020; Mongey et al. 2020 ).

In the model, a jurisdiction/state consists of two sectors, the contact-intensive sector such as the leisure and hospitality industry, called sector a, whose economic activities involve more person-to-person interactions and are affected more by a mitigation measure, and the other sector such as online businesses, called sector b, that is less affected. A worker produces in one sector but consumes both good (representing goods and service) A, produced by sector a, and good B, produced by sector b. A measure such as a partial business closure (or partial opening with 25% capacity) and social distancing limits consumption of good A. The measure also decreases production efficiency in both sectors, for example, as production may require coordination among workers and the measure may make such coordination difficult. A measure is assumed to affect sector-a's production more. The economic loss from limited consumption affects both sector-a workers and sector-b workers, as they consume both goods. The economic loss, from limited production due to the lower consumption of good A and the decrease in production efficiency, affects disproportionately sector-a workers. The measure mitigates health risks workers are exposed to and benefits them, but benefits sector-a workers more. Sector-a workers and sector-b workers thus differ in their preferred stringency.

Taking the argument above one step further, the size of sector a can be related to the stringency of a measure. Sector-b workers benefit from a larger share of sector a, as it decreases the price of good A they consume but does not directly affect their income. As a result, when the share of sector a is larger, sector-b workers want to reduce the stringency to enjoy the benefit of the lower price. This does not say that sector-b workers prefer a less-stringent measure than sector-a workers. Rather, it is about the change in the preferences for the stringency of each sector in response to a change in the size of sector a, which helps in establishing the empirical hypothesis below. Thus, if a majority chooses a measure or a policymaker reflects the preferences of the majority when choosing it, the measure becomes less stringent as the size of sector a increases although sector-a workers constitute a minority of the labor force, a reasonable assumption in modern developed economies, as discussed below. That is, measures tend to be less stringent in states with a larger share of sector a. This result serves as the hypothesis in the empirical analysis. This paper is related to a large number of recent papers on the Covid-19 pandemic. First, much of the literature has considered the infection externalities that individuals impose on others. The key result is that a severe and early measure reduces the mortality rate at the cost of a severe economic downturn (Alvarez et al. 2020; Atkeson 2020; Bethune and Korinek 2020; Jones et al. 2020; Eichenbaum et al. 2020 ). This literature, like this paper, studies the health-benefit and economicloss trade-off, but does not consider the heterogeneous utility effects between two sectors and political support for measures.

Second, a number of papers consider multi-sector models. Barrot et al. (2020) estimate the effects of social distancing on GDP by taking into account the relationships between sectors. Koren and Petö (2020) study which jobs are most affected by measures such as social distancing. Mongey et al. (2020) find that lowwork-from-home and high-physical-contacts jobs are more vulnerable to various measures. These papers focus on the heterogeneous job-loss effects across sectors, rather than comparing the utilities of workers across sectors.

Another literature considers the effects of political leaning on the response to measures. Allcott et al. (2020) find that Democrats and Republicans differ in social distancing behavior. Likewise, Democrats and Republicans differ in their views of the importance of health risks and economic losses (Beauchamp 2020; Coppins (2020) ). This literature, like this paper, studies a political aspect of measures, but does not consider the heterogeneous utility effects of measures.

The next section considers a simple model. Section 3 analyzes the heterogeneous utility effects and the determination of the stringency of a measure. Section 4 provides an empirical analysis, and the last section offers a conclusion.

There are two goods (representing both goods and services), A and B. Good A ðBÞ is produced by sector a ðbÞ. There is a continuum of workers with mass 1. A worker produces in one sector but consumes both good A and good B. Sector i employs n i identical workers, i ¼ a; b; so n i is the proportion of workers in sector i.

h denotes the severity of a (mitigation) measure under consideration. The question concerns how the choice of the severity h in a jurisdiction depends on the economic and health conditions of the jurisdiction, enabling an empirical analysis of the relationship between the severity and the conditions across jurisdictions. As such, it should be stressed from the outset that this paper does not study the effect of introducing a measure.

The utility of a worker is written as v i ðhÞ u i ðhÞ À h i ðhÞ; i ¼ a; b;

u i represents the economic utility from a sector-i worker's economic activities, consumption and labor supply. h i ðhÞ represents the health risk for a sector-i worker.

The worker enjoys the utility from the consumption of A and B, denoted by A i and B i , respectively. A represents goods and services whose consumption involves more person-to-person interactions and is affected more by a measure, and B represents goods whose consumption is less affected. For instance, A represents dine-in services in a restaurant, and B may represent online purchase of goods. The last term of u i , À' i , is the disutiliy of labor supply and will be discussed below.

Consumption of A poses more health risk due to more person-to-person contacts than that of B, and a measure is assumed to limit consumption of A to AðhÞ, so

In reality, almost all sectors are affected by a measure to some degrees, but (2) captures the idea that some sectors are more affected than others. A more severe measure is assumed to limit consumption of A more, so

Health risk from consumption of A should be included in the utility function, but it does not affect the analysis qualitatively and is relegated to the appendix to avoid cluttering up notations. An increase in the stringency of a measure is assumed to decrease the health risk h i ðhÞ, but more for sector-a workers, so

A sector-i worker supplies ' i units of labor. With measure h, the effective units of labor supply become 1

(5) says that a measure makes labor less productive, as in Jones et al. (2020) . For instance, production may require coordination and interaction among workers, and a measure such as social distancing or partial business opening makes it more difficult to coordinate and interact, reducing production efficiency (Koren and Petö 2020) . With a measure, some tasks can be completed online but may not be done as efficiently as in a physical workplace. The loss of production efficiency, measured by /ðhÞ, is assumed to increase in h. Given the difference between two types of goods, the subsequent analysis assumes

Thus, a measure or an increase in its stringency affects production more in sector a. The sector-i worker earns w i L i ðhÞ with w i denoting the wage in sector i, and the budget constraint reads as

The model above is simple enough to generate closed-form solutions but has limitations. First, in (2), consumption of A is assumed to be the same as AðhÞ. In reality, the constraint may not be binding and A\AðhÞ for some individuals. However, the model is restricted to identical individuals and assumes that the constraint is binding, because otherwise the severity h plays no role. Second, constraint (7) assumes that income is completely spent on A and B. In practice, individuals may save out of their income. However, the analysis is restricted to a one-period model, and saving is not considered.

Third, u i in (1) assumes disutility of labor. More generally, workers value leisure, and u i could be written as

where z i denotes leisure and ' is the maximum amount of time available for leisure or production. (1) essentially assumes a linear leisure function, so that Wðz i Þ ¼ z i ¼ ' À ' i to obtain closed-form solutions.

A sector-i worker chooses A i ; B i and ' i , subject to constraints (2) and (7), to maximize u i ðhÞ. As shown in the appendix, 2

One unit of effective labor is assumed to produce one unit of the good in each sector. Since sector i employs n i workers and each supplies L i ðhÞ units of effective labor, the supply of a good equals n i L i ðhÞ. In equilibrium,

In addition, in equilibrium,

Only the relative price can be determined, so

where P B is set at one. Using (8) through (11),

3 Policy choice of a measure

Substituting (8) and (12) into (1), it can be shown again in the appendix that v a ðhÞ ¼ ½ ð1 À bÞ bs À 1AðhÞ À / a ðhÞ À h a ðhÞ;

The first term of v a ðhÞ, ½ ð1ÀbÞ bs À 1AðhÞ, shows the utility of consumption and disutility of supplying effective units of labor L a ðhÞ: That is, ðAðhÞÞ a ðB a Þ b À L a ðhÞ ¼ ½ ð1ÀbÞ bs À 1AðhÞ once consumption and labor supply are chosen to maximize the utility. The sign of the expression inside the pair of square brackets, ð1ÀbÞ bs À 1, plays an important role and depends on the magnitude of s ¼ n a =n b , the (relative) size of sector a in (11). Thus, an increase in the stringency of a measure, that limits AðhÞ more, may increase or decrease the utility, depending on s. The second term of v a ðhÞ; namely À/ a ðhÞ, represents the disutility of supplying non-productive labor. That is, to supply effective units of labor LðhÞ, a sector-a worker has to supply ' a ðhÞ ¼ L a ðhÞ þ / a ðhÞ. / b ðhÞ in v b ðhÞ can be interpreted analogously.

Given the difference in the utilities in (13) between sector-a workers and sector-b workers, they prefer different policies. Letting h i maximize v i ðhÞ, h i satisfies the FOC (first-order condition) for an interior maximum of

The SOC (second-order condition) is assumed satisfied and v 00 i ðhÞ; so that the comparative statics results below can be signed. Nevertheless, sufficient conditions for the satisfaction of the SOC are briefly discussed.

Since the benefit of an increase in the severity of the measure is the reduction in health risk h i ðhÞ while the cost is the loss of production efficiency and consumption of A, it appears to be reasonable to assume h 00 i ðhÞ ! 0; / 00 i ðhÞ ! 0 and A 00 ðhÞ 0: That is, the cost, / i ðhÞ or ÀAðhÞ, is increasing at an increasing rate, 3 and the benefit, ÀhðhÞ, is increasing at a decreasing rate. Since v 00 i ðhÞ is identical to v 0 i ðhÞ, except that A 00 ðhÞ; / 00 i ðhÞ and h 00 i ðhÞ replace A 0 ðhÞ; / 0 i ðhÞ and h 0 i ðhÞ; respectively, v 00 i ðhÞ still cannot be signed with these assumptions, because the term involving A 00 ðhÞ can be positive or negative. To have a sense of sufficient conditions, consider an example:

; denote the expression inside the pair of square brackets in (14) and (15), v 00 a ðhÞ and v 00 b ðhÞ are, respectively, v 00

Given d [ 2 and [ 0, the second term and the third term of v 00 a ðhÞ are negative. Q a in (14) cannot be signed in general. If Q a ! 0; the first term is also negative, given c 2 ð1; 2Þ, so v 00 a ðhÞ: If Q a ; the first term is positive and the sign of v 00 a ðhÞ cannot be determined. A sufficient condition for v 00 a ðhÞ is a [ a à with a à denoting a critical value of a, because v 00 a ðhÞ decreases in a. The same comment applies to v 00 b ðhÞ; and a sufficient condition

The question concerns the relationship between s and the preferred choice h i :

where 'ffi' means 'the same sign as.' These results can be stated as:

Proposition 1 oh a =os [ 0 and oh b =os (as the number of sector-a workers increases relative to that of sector-b workers, sector-a workers prefer a morestringent measure while sector-b workers prefer a less-stringent measure).

The intuition of the result is simple. As in (14), the utility-maximizing h a balances the marginal benefit of an increase in the stringency, resulting from the reduction in the health risk Àh 0 a ðhÞ, and the marginal cost from the reduction in the economic utility, the remaining terms in (14). An increase in the ratio s ¼ n a =n b increases the supply of labor in sector a, relative to sector b, decreasing the price of good A, P A : Sector-b workers gain from a lower price P A as they consume it. Sectora workers lose from the lower price even though they also consume A, as their wages decrease. 4 Thus, if s is larger, the economic utility of a sector-a worker is lower, and so is the marginal cost of the increase in the stringency. Sector-a workers thus prefer to increase the stringency when s is larger, as in (16). The decrease in P A , due to an increase in s, increases the economic utility of a sector-b worker and hence increases the marginal cost of the increase in the stringency. Sector-b workers thus prefer to reduce the stringency in response to an increase in s.

The proposition does not say that sector-b workers gain and sector-a workers lose from an increase in the severity. Rather, both lose (and enjoy the health gain), but sector-a workers lose less from an increase in the severity when s increases. The proposition does not say that sector-a workers prefer a more-stringent measure than sector-b workers, either. Rather, the proposition is about the response of a worker in each sector to an increase in s. A policy implication of the proposition is stated as:

Corollary 1 Assume that n b [ n a ; and a majority of voters chooses the stringency of a measure or a policymaker chooses the stringency preferred by the majority. Then, oh b =os (if sector-b workers constitute a majority, the measure is less stringent in a jurisdiction with a larger number of sector-a workers relative to that of sectorb workers).

With the assumption, the corollary follows from proposition 1. The question concerns which of two sectors constitutes a majority. The relationship between n a and n b is an empirical issue. However, as noted in the Introduction, a measure appears to affect disproportionately a minority of workers. For instance, workers in the leisure and hospitality industry such as restaurants and hotels (Barrot et al. 2020; Suneson 2020) have been hit hardest during the Covid-19 pandemic, and those workers account for about 11% of the labor force in the U.S., as discussed in the next section (Bureau of Labor Statistics 2020). Other contact-intensive sectors may include the trade and transportation industry such as retail stores, and cruises and airlines. The sum of these sectors still constitutes a minority of the labor force. In fact, Koren and Petö (2020) find that about 49 million workers are in the contactintensive sectors in the U.S., about 30% of the labor force. The assumption of n b ! n a thus appears to be reasonable. The corollary serves as the empirical hypothesis in Sect. 4, as the size of sector a and the stringency vary across states.

The dependent variable is duration, which equals the duration (the number of days) of a statewide stay-at-home order, compiled by Tolbert et al. (2020) . A longer duration is interpreted as a more stringent measure. The hypothesis concerns the effects of the size of sector a on the stringency of a measure, duration, at the time of making policies (issuing the orders). That is, given information available at that time, the question is how the size of sector a affected the stringency of the measure. States had imposed statewide stay-at-home orders once between March 2020 and April 2020. California first issued the order on March 19, 2020. By April 6, 2020, 42 states and D.C. had issued the orders, but eight states did not issue. 5 The key independent variable is the size of the contact-intensive sectors. Two sector-a variables are considered. sector-a1 equals the percentage of employment in the leisure and hospitality industry such as hotels and restaurants. sector-a2 equals the percentage of employment in the leisure and hospitality industry plus that in the trade and transportation industry such as retail stores, and cruises and airlines. These two industries have been greatly affected by the pandemic and are consistent with the notion of the contact-intensive industries in recent research findings (Barrot et al. 2020; Mongey et al. 2020) . Employment data is extracted from the U.S. Bureau of Labor Statistics (2020), and employment equals the seasonally-adjusted number of employees in February 2020, the most recent month before the pandemic, but the percentage of employment in sector-a1 or sector-a2 had been stable at least for the past five years. All data below is from the most recent available sources before the pandemic.

The regressions include other variables. First, there are three health variables. old equals the percentage of the population 65 years or older, and this variable is included as the old are known to be at risk. vulnerability measures the extent to which a state is vulnerable to the Covid-19 pandemic and is based on 20 factors such as social and physical environment and high risk population. It is a score, ranging from 0 to 100, and a higher score means more vulnerability. deathrate is the number of confirmed deaths resulting from Covid-19 per 100,000 population and is expected to affect the stringency. deathrate had evolved over time, and it is assumed that states used information as of the order-issuing dates, and states with no order did as of March 26, 2020, the average date among those states with the orders. 6 Second, political ideologies often shape policies, especially pandemic-related policies. For instance, Democrats and Republicans differ in their beliefs about the health risks associated with the virus and behave differently (Allcott et al. 2020). To control for the difference, the governor's party affiliation and the partisan composition of the legislature of each state are considered. In particular, partisan equals 1 if a state is controlled by Democrats, and -1 if controlled by Republicans, and zero if divided. In addition, to see if measures in swing states differ from others, swing is added, a dummy variable equal to 1 if a state is a swing state, and 0 otherwise.

Third, there are other controls. density equals the number of residents per square mile, and the pandemic is expected to affect denser areas more adversely. percapitagdp equals per capita gross state product and may be viewed as a broad control. These seven variables come from Index Mundi (2020) , Barclay and Rodriguez (2020) , University of Oxford (2020), the National Conference of State Legislatures (2020), Gilbert (2020) , State Key Data (2020) and U.S. Bureau of Economic Analysis (2020), respectively. Table 1 presents summary statistics. duration ranges from 0 day in the eight states with no order to 90 days. In California, Kentucky, Maryland, New Jersey, Oregon and West Virginia, the duration is indefinite ('until revoked') at the time of issuing the orders and is assumed to be 90 days, 7 as the longest duration among the states with a definite time line is 72 days. If it is set at 80 days or 100 days, it changes the regression results little. sector-a1 and sector-a2 are on average 11.2% and 29.4%, respectively. About 30% of jobs in the U.S. 'rely heavily on face-to-face communication or require close physical proximity to other workers' (Koren and Petö, 2020) , so sector-a2 is close to their estimates and considered a broader notion of the contact-intensive industries. old, vulnerability, deathrate, density and percapitagdp are on average 16.4%, 45, 0.3, 424, and $59,267, respectively. As for partisan, 15 and 22 states are controlled by Democrats and Republicans, respectively, and 13 states are divided. There are six swing states, Arizona, Florida, Michigan, North Carolina, Pennsylvania and Wisconsin, so the mean of swing equals 0.12. sector-a1iv and sector-a2iv serve as instruments for sector-a1 and sector-a2 for IV estimation results and will be discussed more. Data on vulnerability, partisan and percapitagdp is not available for D.C., so there are 50 observations for the three variables. Table 2 shows regression results with duration as the dependent variable and sector-a1 as the size of the contact-intensive sector. It appears that a health variable and the political variable, partisan, along with a sector-a variable, play an important role in explaining duration. Thus, specifications (i) through (iii) include one of the three health variables, with sector-a1 and partisan in all three specifications. Specification (iv) includes all variables.

The coefficients of sector-a1 are negative in all specifications. To relate this result to the model, note that sector-a workers prefer to increase the duration of the stay-at-home order in (16) and sector-b workers prefer to decrease it in (17), as the size of the contact-intensive sectors increases. Thus, the negative coefficients of sector-a1 are consistent with (17), but not with (16). However, under the maintained assumption in corollary 1, sector-b workers constitute a majority, and the policymaker of a jurisdiction determines the stringency to maximize the utility of sector-b workers. As a result, the model predicts (17), and the negative coefficients of sector-a1 are consistent with model predictions in corollary 1.

The significance of the coefficient of sector-a1 depends on the health variable included. The coefficient of sector-a1 is significant at the 1% level with vulnerability in (i), but it is significant at the 5% level with old in (ii) or deathrate in (iii). It is significant at the 5% level with all variables in (iv). The coefficient of sector-a1 ranges from -267 in (iii) to -346 in (i), so a one-percentage point increase in the size of the intensive sector decreases the duration by 2.67 days in (iii) and by 3.46 days in (i), given that sector-a1 is expressed as a decimal number such as 0.23 rather than 23%. To have a sense of the explanatory power of sector-a1, note that the standard deviation of sector-a1 is 0.0255, and that of duration is 27.23 in Table 1 . Thus, in (iii), a one-standard-deviation increase in sector-a1 decreases duration by 267 x 0.0255 = 6.81, explaining about 25% (= 6.81/27.23) of the standard deviation of duration. Likewise, in (i), a one-standarddeviation increase in sector-a1 explains about 32.4% (= 346 x 0.0255/27.23) of the standard deviation of duration.

As for health variables, the coefficients of vulnerability, old and deathrate are positive, as expected, although the coefficients of old and deathrate become negative in (iv). The coefficient of vulnerability in (i) is significant at the 5% level, but that of old in (ii) or deathrate in (iii) is not significant. 8 When all variables are included in (iv), the coefficient of vulnerability becomes significant at the 10% level.

The positive coefficients of partisan in all specifications show that the duration tends to be longer in Democrat-controlled states. This result is consistent with findings in the literature that Democrats are more likely to adhere to mitigation measures and place more emphasis on health risks than Republicans (Allcott et al. 2020; Beauchamp 2020; Coppins 2020) . The coefficients of partisan are significant at the 1% level in all specifications. The coefficient of swing is positive, but not significant, so swing states do not appear to differ from others. Regarding other controls, the coefficient of density is positive, as expected. The coefficient of percapitagdp is negative. There is no expected sign of the coefficient of percapitagdp, but the negative sign may mean that a larger economy may lose more from an order and may shorten the duration. Neither coefficient is significant.

duration was set to 0 for 8 states with no order and set to 90 for 6 states with no definite time line. Tobit regressions were considered, as 0 and 90 can be viewed as corner solutions to the determination of an optimal duration that balances the health benefits and the economic losses. Although the results are not reported to conserve space, the results remain little affected. Table 3 is identical to Table 2, except that sector-a2 replaces sector-a1. The results in Table 3 are qualitatively the same as those in Table 2 in the sense that the signs of all coefficients in Table 3 coincide with those in Table 2 . The coefficients of sector-a2 are less significant than those of sector-a1. In particular, it is significant at the 10% level in (ii) and (iii), but at the 5% level in (i) and (iv). The coefficients of sector-a2 range from -199 in (iii) to -280 in (i). The coefficients mean that a one-percentage point increase in the size of the intensive sector, measured by sector-a2, decreases the duration by 1.99 days in (iii) and by 2.8 days in (i). To see the explanatory power of sector-a2, note that the standard deviation of sector-a2 is 0.0358, so a one-standard-deviation increase in sector-a2 decreases duration by 199 x 0.0358 = 7.12, explaining about 26.1% (= 7.12/27.23) of the standard deviation of duration in (iii). Likewise, in (i), a one-standard-deviation increase in sector-a2 explains about 36.8% (= 280 x 0.0358/27.23) of the standard deviation of duration. The explanatory power of sector-a2 is thus about the same as that of sector-a1 in Table 2 .

To take into account the possibility of omitted variables bias, instrumental variables for sector-a variables are considered, and IV estimation results are presented in Table 4 . The average of the past five years of percentage of employment is used as an instrument. 9 That is, sector-a1iv equals the five-year average of sector-a1 for 2015 through 2019, and similarly for sector-a2iv. Firststage weak instrument statistics, the Cragg-Donald Wald F Stat and the Kleibergen-Paap Wald F stat, in the table show that sector-a1iv or sector-a2iv does not appear to be a weak instrument. Table 4 includes regression results with both sector-a1 and sector-a2. (i) and (ii) use sector-a1, and (iii) and (iv) use sector-a2. Rather than each of the three health variables, only one health variable, vulnerability, is considered in (i) and (iii) again in order to conserve space. (ii) and (iv) include all variables. Comparing  Tables 2 and 3 with Table 4 , IV estimation results are qualitatively the same as those in Tables 2 and 3 in the sense that the signs and explanatory power of the key variables, sector-a1 and sector-a2, remain largely unaffected. The coefficients of the two variables are significant at the 1% level in all specifications. Although not reported, with old or deathrate replacing vulnerability in (i) and (iii), the coefficients of sector-a1 and sector-a2 become significant at the 5% level.

The coefficients of other variables change little. In particular, the coefficients of vulnerability are positive and significant at the 5% level, as in Tables 2 and 3 . partisan continues to have a positive significant effect on duration. The coefficient of density is again positive while that of percapitagdp is negative, but they are not significant.

The paper has studied the heterogeneous welfare effects of measures intended to mitigate health risk during a pandemic. The analysis has shown that the size of the contact-intensive sector plays an important role in the stringency of a measure and hence in political support for the measure. Available empirical evidence confirms the analysis and shows that the duration of stay-at-home orders tends to be shorter in states with a larger contact-intensive sector.

The analysis highlights the importance of multisectors in studying the economic effects of measures. In particular, the size of the contact-intensive sectors shapes the stringency of measures, not because they constitute a majority, but because the majority or the non-contact-intensive sectors benefit from a larger share of the contact-intensive sectors. This benefit effect occurs as production in a sector is related to another through consumption and illustrates the value of a multisector model. The role of the interaction between sectors in the determination of mitigation measures has received little attention but appears to deserve more research. 

Substitution of P A in (11) into the last expression above and subtraction of h a ðhÞ give v a ðhÞ in (13). The expression of v b ðhÞ can be obtained in an analogous manner. h

The utility function in (1) is modified as v i ðhÞ ¼ ðA i Þ a ðB i Þ b À lðhÞA i À ' i À h i ðhÞ:

lðhÞ represents the risk involved in consumption of A. A measure is assumed to reduce the risk, so l 0 ðhÞ. The utilities in (13) are modified as v a ðhÞ ¼ ð1 À bÞ bs À 1 À l

The FOCs in (14) and (15) become v 0 a ðh a Þ ¼ ð1 À bÞ bs À 1

A 0 À ðlAÞ 0 À / 0 a À h 0 a ¼ 0;

Although (19) and (20) include the additional term ÀðlAÞ 0 , relative to (14) and (15), it does not affect the key result in proposition 1 and corollary 1, namely the sign of oh a =os and oh b =os:

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Acknowledgments I am grateful to Jan Brueckner and an anonymous referee for their helpful comments.

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Proof of (8) A sector-i worker chooses ðA i ; B i ; ' i ; g i1 ; g i2 Þ to maximizewhere g i1 and g i2 are the multipliers. The FOCs for an interior maximum of M i are (8). Substituting A i ¼ AðhÞ and the expression of B i into oM i =og i2 ¼ 0 and solving the resulting equation for ' i , the expression of ' i in (8) 

Proof of (11) Only one equilibrium condition matters, given two goods. The first one in (9) leads towhere A ¼ AðhÞ; and the expression of ' a in (8) is used. The condition can be solved for P A , and the solution is in (11). h Proof of (12) Substitution of the expression of P A in (11) and that of ' i in (8) into the expression of B i in (8) leads to (12). h