Physical Activity and Policy Recommendations: A Social Multiplier Approach
-
Catarina Goulão
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 [2A], [2B] 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:
![[6]](/document/doi/10.1515/bejeap-2012-0075/asset/graphic/bejeap-2012-0075_eq23.png)
![[7]](/document/doi/10.1515/bejeap-2012-0075/asset/graphic/bejeap-2012-0075_eq24.png)
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.
©2014 by Walter de Gruyter Berlin / Boston
Artikel in diesem Heft
- Frontmatter
- Editorial
- The Economics of Long-Term Care: An Introduction
- Advances
- The Economics of Long-Term Care
- Informal Care and Inter-vivos Transfers: Results from the National Longitudinal Survey of Mature Women
- Long-Term Care Insurance and Family Norms
- Long-Term Care, Altruism and Socialization
- Working Mums and Informal Care Givers: The Anticipation Effect
- Endogenous Altruism, Redistribution, and Long-Term Care
- Do Local Governments Respond to (Perverse) Financial Incentives in Long-Term Care Funding Schemes?
- Behavioral Biases and Long-Term Care Insurance: A Political Economy Approach
- Physical Activity and Policy Recommendations: A Social Multiplier Approach
Artikel in diesem Heft
- Frontmatter
- Editorial
- The Economics of Long-Term Care: An Introduction
- Advances
- The Economics of Long-Term Care
- Informal Care and Inter-vivos Transfers: Results from the National Longitudinal Survey of Mature Women
- Long-Term Care Insurance and Family Norms
- Long-Term Care, Altruism and Socialization
- Working Mums and Informal Care Givers: The Anticipation Effect
- Endogenous Altruism, Redistribution, and Long-Term Care
- Do Local Governments Respond to (Perverse) Financial Incentives in Long-Term Care Funding Schemes?
- Behavioral Biases and Long-Term Care Insurance: A Political Economy Approach
- Physical Activity and Policy Recommendations: A Social Multiplier Approach
