# Abstract

We look at the effects of physical activity (PA) recommendation policies by considering a social multiplier model in which individuals differ in their concern for PA. The government can either observe this concern (and implement the First Best) or not (and implement a uniform policy). Whichever the type of policy implemented, while the welfare of individuals the most concerned with PA increases in the social multiplier, the welfare of those the least concerned may decrease in it. For a sufficiently high social multiplier, both government interventions improve the welfare of those most concerned with PA but worsen the welfare of the least concerned individuals if they are not too many. However, compared to the First Best, a uniform recommendation improves the welfare of those most concerned with PA more than it reduces the welfare of those least concerned.

## Appendix A

### Proof of Proposition 1

An LF-equilibrium consists of a pair which solves equations [2*A*], [2*B*] and . Solving this system leads to:

Note that . Therefore, according to Assumption 1 , for all . Consequently, to establish that it is sufficient to show that and . The positivity of is obvious. Having is equivalent to . Therefore, according to Assumption 1, is the LF-equilibrium.

After computations, we obtain . As and , and, consequently, and increase in . We also have . Then, and are convex functions of . □

## Appendix B

### Proof of Proposition 2

According to eq. [3] we have . This value is lower than 1 since, according to Assumption 1, . Therefore, when , and otherwise.

After computations, and . As , and are positive and is increasing and convex in .

Additionally, has the sign of . Since for all we have and , is positive and, consequently, is more convex than and .

Finally note that and, according to Proposition 1, . Therefore, since , for all and there exists a unique such that if and only if . □

## Appendix C

### Proof of Proposition 3

According to eq. [] ( or *B*) we have . Substituting this into eq. [1] allows us to obtain . Consequently, we have and . Both expressions are positive since and are positive. Therefore, and are increasing and convex functions of .

Substituting eq. [3] into eq. [1] allows us to obtain . Then, we have and . Since and are positive, and have the sign of . As and we have . Consequently, is an increasing and convex function of .

We now contrast with . Using Propositions 1 and 2, we obtain and with and .

Then has the sign of with and . Using the facts that and , a sufficient condition to show the positivity of is to establish the positivity of . Rearranging terms leads to with . Let . Since and , we have with . As , and, therefore, . Consequently, meaning that increases in for .

From eq. [1] we obtain . As , we have . Hence, we obtain . Given the definition of (see Proposition 2), . From eq. [1] it follows that . Since, according to Proposition 1, , then . Consequently, since increases in for , there exists a unique such that if and if . □

## Appendix D

### Proof of Proposition 4

In Appendix C, we establish that is an increasing and convex function of and that and have the sign of , i.e., after computations, the sign of . Then and have the sign of with . Consequently, is a decreasing and concave function of if and an increasing and convex function of if .

According to eq. [1], . As , we have . Hence, we obtain:

**Fact 1 – For all**, **when**.

When , increases in for while decreases and, according to Fact 1, we have when . When , complications arise since both and increase in . According to Appendix C, . Then, after computations and using Appendix A and B, has the sign of . The degree of this polynomial function is three and . Moreover, and . As , and consequently and . The fact that , and implies the existence and the uniqueness of root between 0 and 1.^{9} Then, if and if . Consequently, we establish:

**Fact 2 – When**, **decreases in****for****. When****, the function****decreases for****and increases for**.

Note that , and . Then . As , we have . Then, we establish:

**Fact 3 – The function****is a decreasing function of p**.

Given , the maximum of is obtained when , i.e., . Hence:

**Fact 4 – For all**, **if**.

Using Fact 1 to Fact 4, it is straightforward to establish the assertion of our proposition using the (possible) existence of two thresholds and such that:

If , there exists a threshold , such that decreases in for , increases in for and:

If , there exists a unique such that when and when .

If , for all .

If , decreases in and .

The threshold is defined, according to Fact 1 to Fact 4, from the value of *p* such that . This value is given by with and . Then, after computations, is the root of with , , and . Since , , and , we have , and . Then, has a unique root and this latter is positive. Consequently, if then this root corresponds to and if then .

By construction, the threshold exists only when and it corresponds to the unique value such that with which has been defined to prove Fact 2. □

## Appendix E

### Proof of Corollary 1

According to Appendix D, and, consequently, is on the branch of the function which increases in . Then, as decreases in *p* (Fact 3 of Appendix D), the threshold increases in *p*. Moreover, it is straightforward to establish that .

Remark that , , and . As , we have . Then, it is straightforward to show that is a decreasing function of *p*. Consequently, as , for all there exists a threshold such that decreases in *p* if (and only if) . Then, because there exists a unique *p* such that , for a given . Consequently, is a monotonic function of *p*. As , the threshold decreases in *p*.

To summarize, the threshold increases in *p* and while decreases in *p* and . Then, there exists a unique threshold such that is larger (resp: lower) than if and only if *p* is lower (resp: larger) than . Using and Propositions 3 and 4, the assertion of Corollary 1 is straightforward. □

## Appendix F

### Proof of Proposition 5

To determine the FB-equilibrium, the government chooses and so that is maximum given . We first analyze the case of interior solutions (Step 1) and then consider the possibility of corner solutions (Step 2).

**Step 1 – The case of interior solutions****and**.

When FB-equilibrium has interior solutions , the government solves:

After simplifications, the two FOC are given by:

Rearranging terms we obtain:

and

The value *V* decreases (and is concave) when varies from 0 to 1. As and , there exists a unique such that *V* is positive if and only if . As , we have . Thus, is positive and decreases in . As , there exists a unique such that if and only if . Let , then it is straightforward that .

Similarly, it is obvious that is positive and increases in . As , there exists a unique such that if and only if .

After computations, with . As and , increases in .

Moreover, we have with . As and are positive, is positive and is an increasing and convex function of when . Note that and .

We follow by contrasting with and . Note that has the sign of , i.e., the sign of . As , is positive, i.e., .

Moreover, has the sign of , i.e., the sign of . As and we obtain after computations with . Then, the discriminant of the polynomial function is such that . Consequently, the lowest root of is . As and , we have for all and, consequently, .

Note that with . Therefore increases in . Moreover, as and are positive, and is negative, is a convex function of if . Remark that and . As the numerator of is larger than the one of whereas the denominator of is lower than the one of , we have .

Finally, has the sign of , i.e., the sign of . As and we obtain after computations with . Then, the discriminant of the polynomial function is such that . Consequently, the lowest root of is . As and , we have for all and, consequently, .

As regards the comparison between and , when , has the sign of , i.e., the sign of . Then, decreases in . Consequently, if and only if . After computations:

The positivity of this quantity implies that arises for a larger than 1. Hence, and for all .

**Step 2 – The case of corner solutions****and**.

When , and the value which defines solves:

Then, the FOC is and:

which is lower than 1 if and only if .

After computations and . Then, is an increasing and convex function of .

Comparing with , we conclude that has the sign of , i.e., the sign of . As we obtain with . As the discriminant of is , the polynomial function has two positive roots and the product of these roots is given by . As , and , the two roots of are larger than 1 and is positive for all . Consequently, .

**Step 3 – Characterization and properties of the FB-equilibrium**.

Using Steps 1 and 2, it is straightforward to obtain that the FB-equilibrium is given by:

In Step 1, we have established that and are both increasing and convex functions of . In Step 2, we have established that is an increasing and convex function of . Combining Steps 1 and 2, we have also established that , , , and . □

## Appendix G

### Proof of Proposition 6

We separately study the welfare of each type of individual.

**Step 1 – Characterization of type A individuals’ welfare at the FB-equilibrium**.

We first focus on the welfare of type *A* individuals when . Merging eqs. [6] and [7] in Appendix F gives . Then, according to eq. [7] and using the fact that we obtain . Thus, the utility becomes:

Then, with . We obtain with . Then, . As and we have , with . Then . This implies . As , increases in .

Moreover with . Then with . As , we have . Then, is positive. As , is a convex functionof.

We now focus on type *A* individuals’ welfare when . As , then , with . Thus, . As is a positive, increasing and convex function of , it is straightforward to establish that is an increasing and convex function of .

**Step 2 – Characterization of type B individuals’ welfare at the FB-equilibrium**.

We now focus on type *B* individuals’ welfare when . Merging eqs. [6] and [7] in Appendix F gives . Then, according to eq. [6] and using the fact that , we obtain . Then, the utility becomes:

Then, with . After computations, . As , we obtain after simplifying . Therefore, and, by continuity, is an increasing function for sufficiently low values of .

We now focus on type *B* individuals’ welfare when . Using the fact that with we obtain, after computations, with . Thus, . As and we obtain, after computations, . Moreover, it is straightforward that . Then, and and, by continuity, is an increasing function for sufficiently low values of *p* but a decreasing one for sufficiently high values of *p*.

We now compare with for type *A* individuals. It is straightforward to establish that . Moreover, according to Proposition 3, . Then . Similarly, whereas . As we have . By continuity, we have established that when is sufficiently low, whereas for sufficiently large values of .

We finally compare with for type *B* individuals. It is straightforward to establish that . Moreover, according to Proposition 4, . Hence, . By definition of the First Best, we have for all , . As , we necessarily have . By continuity we have established that when is sufficiently low or sufficiently large. □

# Acknowledgment

We are grateful to David Bardey and Pierre Pestieau for their comments and suggestions. We thank participants at the “First workshop IDEI/SCOR/TSE of Long Term Care” (Toulouse, January 6, 2011), and at the “CESifo Workshop on the Economics of Long Term Care” (Venice, July 18–19, 2012) for their discussions. We also thank two referees of this journal for their constructive comments. E. Thibault thanks the Chair Fondation du Risque/SCOR “Marché du risque et création de valeurs” for its financial support.

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- 1
This figure equates to as many deaths as tobacco causes globally.

- 2
See the survey of Blain et al. (2000), and the references therein.

- 3
For more on these campaigns check, respectively, http://www.letsmove.gov/get-active, http://www.mangerbouger.fr, http://www.findthirtyeveryday.com.au, and http://www.nhs.uk/Change4Life, all accessed on September 25, 2012. Similarly, there are uniform campaigns concerning nutrition, in general advising the intake of five portions of fruit and vegetables a day.

- 4
See among others, Warburton, Nicol, and Bredin (2006), regarding PA benefits in reducing the risk of several conditions, Barnett et al. (2003) for reduction in falls and disability, Keysor (2003) for improved independence, McAuley et al. (2005) for improved psychological well-being, and Colcombe and Kramer (2003), and the survey of Vogel et al. (2009) for maintenance of cognitive vitality. Yet note that PA is recommended to all ages since the risk of NCDs starts in childhood (see Warburton, Nicol, and Bredin 2006; PAGAC 2008; and WHO 2010; among others).

- 5
These variables can alternatively be either a time or an amount of wealth devoted to PA.

- 6
is the of Alesina, Glaeser, and Sacerdote (2005, Section 5) when , and .

- 7
Even though we focus on the social multiplier effects associated with PA practice, the present setup is sufficiently general to be used in the analyzes of other goods and services where social multiplier effects occur, such as education or the use of new technologies.

- 8
See for England http://www.telegraph.co.uk/health/healthnews/9777453/Obese-people-may-be-forced-to-exercise-or-lose-benefits.html, and for Japan http://www.nytimes.com/2008/06/13/world/asi a/13fat.html both accessed on February 20, 2013.

- 9
Obviously, and guarantee the existence of a root between 0 and 1. If is not the unique root between 0 and 1, there generally exists three roots , , and such that and , , . Then, as , have at least three roots: however this is impossible since is a polynomial function of degree 2. Consequently is unique.

**Published Online:**2013-11-1

©2014 by Walter de Gruyter Berlin / Boston