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A Work Project presented as part of the requirements for the Award of a Master Degree in Economics from the NOVA – School of Business and Economics.

The Health Production Function Revisited:

The Role of Social Networks and Liquid Wealth

Technical Appendix

Carolina Borges da Cunha Santos Student number 793

A Project carried out on the Master in Economics Program, under the supervision of: Professor Pedro Pita Barros

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Technical Appendix 1

Adjustment after a negative health shock, for 0 < 𝜃 < 1

𝑀𝑎𝑥{𝑡𝐻,𝑡𝑁,𝐶,𝑀,𝜃,𝑁} 𝑈 = 𝐻(𝑀, 𝑡𝐻, 𝑁) + 𝑈2( 𝐶) + 𝑈3[𝑊𝜃 + 𝛼(1 − 𝜃)𝑊], for α>1 (1)

s.t. 𝐻 = 𝐻(𝑀, 𝑡𝑁); 𝑟𝜃𝑊 + 𝑌 = 𝑝𝑀 + 𝐶; 𝑇̅ = 𝑡̅ + 𝑡𝐿 𝐻+ 𝑡𝑁; 𝑁 = 𝑁(𝑡𝑁), N’>0; 𝑌 = 𝜔𝑡̅𝐿

Which, after the appropriate substitutions have been made, can be reduced to:

𝑀𝑎𝑥{𝑡𝑁,𝜃,𝑀} 𝑈 = 𝐻(𝑀, 𝑡𝑁) + 𝑈2(𝑟𝜃𝑊 + 𝜔𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3[𝜃𝑊 + 𝛼(1 − 𝜃)𝑊] (2)

And the first order conditions are:

𝜕𝑈

𝜕𝑡𝑁=

𝜕𝐻

𝜕𝑡𝑁 = 0 (3)

𝜕𝑈

𝜕𝜃 = 0 ⬄

𝜕𝑈2

𝜕𝐶 𝜕𝐶

𝜕𝜃+

𝜕𝑈3

𝜕𝑍 𝜕𝑍

𝜕𝜃= 0⬄

𝜕𝑈2

𝜕𝐶 (𝑟𝑊) +

𝜕𝑈3

𝜕𝑍 𝑊(1 − 𝛼) = 0 (4)

Where 𝑍 = 𝑊𝜃 + 𝛼(1 − 𝜃)𝑊.

𝜕𝑈

𝜕𝑀= 0⬄

𝜕𝐻

𝜕𝑀− 𝑝

𝜕𝑈2

𝜕𝐶 = 0 (5)

And the associated second order conditions: 𝑈11= 𝜕

2𝐻

𝜕𝑡𝑁2< 0 (6)

𝑈22 =𝜕 2𝑈2

𝜕𝐶2 (𝑟𝑊)2+𝜕

2𝑈3

𝜕𝑧2 (1 − 𝛼)2𝑊2< 0 (7)

𝑈33 =𝜕 2𝐻

𝜕𝑀2+ 𝑝2 𝜕

2𝑈

𝜕𝐶2 < 0 (8)

𝑈12= 𝑈21 = 0 (9)

𝑈13= 𝑈31 = 𝜕 2𝐻 𝜕𝑡𝑁2

𝜕𝑡𝑁

𝜕𝑀 > 0 (10)

𝑈23= 𝑈32= −𝑝𝑟𝑊𝜕 2𝑈2

𝜕𝐶2 > 0 (11)

Therefore, the hessian matrix is H = [

𝑈11 0 𝑈13

0 𝑈22 𝑈23

𝑈31 𝑈32 𝑈33

] and its determinant is:

|𝐻| = 𝑈11𝑈22𝑈33− (𝑈13)2𝑈22− (𝑈23)2𝑈11 < 0. (12)

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𝑈22 < 0, 𝑈33< 0 and |𝐻| < 0. Additionally, we assume the utility function evidences diminishing

marginal utility and that both the health production function as well as the social network function display decreasing marginal returns on their respective inputs. Lastly, we assume that the isoquants associated with the utility function are convex, which implies that the second-order cross-derivatives must be positive or null.

Let k be a negative shock in the stock of health (𝑘 < 0), so that the model can be represented by

𝑈 = 𝐻(𝑀, 𝑡𝑁, 𝒌) + 𝑈2(𝑟𝜃𝑊 + 𝜔𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3[𝜃𝑊 + 𝛼(1 − 𝜃)𝑊] (13)

Totally differentiating the first order conditions in respect to k:

{

𝑈11𝜕𝑡𝑁 ∗

𝜕𝑘 + 𝑈13

𝜕𝑀∗

𝜕𝑘 = −𝑈1𝑘

𝑈22𝜕𝜃 ∗

𝜕𝑘 + 𝑈23

𝜕𝑀∗

𝜕𝑘 = 0

𝑈31𝜕𝑡𝑁 ∗

𝜕𝑘 + 𝑈32

𝜕𝜃∗

𝜕𝑘 + 𝑈33

𝜕𝑀∗

𝜕𝑘 = −𝑈3𝑘

(14)

Which in matrix form can be written as: [

𝑈11 0 𝑈13

0 𝑈22 𝑈23

𝑈31 𝑈32 𝑈33 ]

[ 𝜕𝑡𝑁∗

𝜕𝑘 𝜕𝜃∗ 𝜕𝑘

𝜕𝑀∗

𝜕𝑘 ]

= [−𝑈01𝑘 −𝑈3𝑘

] (15)

By applying the Cramer rule we can determine the impact that a negative shock in the stock of health (k) will have in the equilibrium levels of 𝑡𝑁, 𝜃 and 𝑀. Nonetheless, we first must take into consideration that two different situations may arise. Firstly, if there are no social networks (or if these are not responsive to the needs of the individual), then 𝑈1𝑘 < 0. In fact, in the absence (or weakness) of social networks the effect that 𝑡𝑁 has on health via 𝑡𝐻, which is negative, dominates over the positive effect that 𝑡𝑁 has on health through N. Therefore, the larger the magnitude of the negative shock in health (𝑘) the greater will be the amount of time allocated to health-enhancing activities and, since 𝑇̅ = 𝑡̅ + 𝑡𝐿 𝐻+ 𝑡𝑁, the lower will be the time devoted to social networks. Additionally, in the absence of social networks, in order to ensure the convexity of isoquants, it must be assumed that 𝑈13= 𝑈31< 0. Indeed, in the absence of social networks, if we assume that the marginal utility of medical goods and services increases with the time

devoted to health, then 𝜕( 𝜕𝐻 𝜕𝑀)

𝜕𝑡𝐻 > 0, which in turn implies that 𝜕(𝜕𝑀𝜕𝐻)

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4

Secondly, if there exist social networks and if these are responsive to the needs of the individual, then it is plausible to assume that the positive effect that hat 𝑡𝑁 has on health through

N will prevail over the negative impact it exerts through 𝑡𝐻. Hence, the greater the amplitude of the negative shock in the stock of health, the greater the amount of time that will be channelled to social networks, which implies 𝑈1𝑘 > 0.

In both situations, however, 𝑈3𝑘> 0. Indeed, regardless of the existence of social networks or the degree of support provided by them, the larger the health shock the greater will be the marginal utility of consumption of medical services.

Technical Appendix 1.1.

Joint effect of 𝑡𝑁 in the stock of health through 𝑡𝐻 and 𝐻: 𝑈1𝑘 > 0 and 𝑈3𝑘 > 0.

𝜕𝑡𝑁∗

𝜕𝑘 =

|𝐴|

|𝐻|=

|−𝑈01𝑘 𝑈022 𝑈𝑈1323

−𝑈3𝑘 𝑈32 𝑈33

|

|𝐻| (16)

Where, |𝐴| = 𝑈3𝑘𝑈22𝑈13+ 𝑈1𝑘[(𝑈32)2− 𝑈22𝑈33]. Since the first term of the determinant is negative and 𝑈1𝑘 > 0, we are interested in determining whether (𝑈32)2< 𝑈22𝑈33. If so, then |𝐴| < 0, otherwise it is not possible to conclude anything about the sign of |𝐴|.

(𝑈32)2< 𝑈22𝑈33⬄(𝑝𝑟𝑊)2(𝜕 2𝑈2

𝜕𝐶2)

2

< [𝜕2𝑈2

𝜕𝐶2 (𝑟𝑊)2+𝜕

2𝑈3

𝜕𝑍2 (1 − 𝛼)2𝑊2][𝜕 2𝐻

𝜕𝑀2+ 𝑝2 𝜕

2𝑈2 𝜕𝐶2]

⬄(𝑝𝑟𝑊)2(𝜕2𝑈2

𝜕𝐶2)

2 <𝜕𝑀𝜕2𝐻2

𝜕2𝑈2

𝜕𝐶2 (𝑟𝑊)2+ (𝑝𝑟𝑊)2( 𝜕2𝑈2

𝜕𝐶2)

2 +𝜕2𝑈3

𝜕𝑍2 (1 − 𝛼)2𝑊2[ 𝜕2𝐻

𝜕𝑀2+

𝜕2𝑈2

𝜕𝐶2] (17)

⬄0 <𝜕𝜕𝑀2𝐻2𝜕 2𝑈2

𝜕𝐶2 (𝑟𝑊)2+𝜕

2𝑈3

𝜕𝑍2 (1 − 𝛼)2𝑊2[𝜕 2𝐻

𝜕𝑀2+𝜕

2𝑈2 𝜕𝐶2]

And, since the right-hand side of the equation is positive, we can conclude that |𝐴| < 0. Hence,

𝜕𝑡𝑁∗

𝜕𝑘 =

|𝐴| |𝐻|> 0.

Likewise, 𝜕𝜃∗

𝜕𝑘 =

|𝐵|

|𝐻|=

|𝑈011 −𝑈01𝑘 𝑈𝑈3𝑘23

𝑈31 −𝑈3𝑘 𝑈33

|

|𝐻| (18)

Where |𝐵| = −𝑈1𝑘𝑈23𝑈31+ 𝑈3𝑘𝑈23𝑈11< 0. Therefore, 𝜕𝜃∗

𝜕𝑘 =

(5)

5

Lastly, 𝜕𝑀∗

𝜕𝑘 =

|𝐶|

|𝐻|=

|𝑈011 𝑈022 −𝑈01𝑘

𝑈31 𝑈32 −𝑈3𝑘

|

|𝐻| (19)

Where |𝐶| = −𝑈3𝑘𝑈11𝑈22+ 𝑈31𝑈22𝑈1𝑘 < 0. Hence, 𝜕𝑀∗

𝜕𝑘 =

|𝐶| |𝐻|> 0.

Technical Appendix 1.2.

Exclusive effect of 𝑡𝑁 in the stock of health through 𝑡𝐻 (no effect via social networks): 𝑈1𝑘 < 0; 𝑈3𝑘 > 0 and 𝑈13 = 𝑈31< 0.

𝜕𝑡𝑁∗

𝜕𝑘 =

|𝐴|

|𝐻|=

|−𝑈01𝑘 𝑈022 𝑈𝑈1323

−𝑈3𝑘 𝑈32 𝑈33

|

|𝐻| (20)

Where |𝐴| = 𝑈3𝑘𝑈22𝑈13+ 𝑈1𝑘[(𝑈32)2− 𝑈22𝑈33] > 0. Thus, 𝜕𝑡𝑁∗

𝜕𝑘 =

|𝐴| |𝐻|< 0.

Similarly, 𝜕𝜃∗

𝜕𝑘 =

|𝐵|

|𝐻|=

|𝑈011 −𝑈01𝑘 𝑈𝑈3𝑘23

𝑈31 −𝑈3𝑘 𝑈33

|

|𝐻| (21)

Where |𝐵| = −𝑈1𝑘𝑈23𝑈31+ 𝑈3𝑘𝑈23𝑈11 < 0. Thus, 𝜕𝜃∗

𝜕𝑘 =

|𝐵| |𝐻|> 0

Lastly, 𝜕𝑀∗

𝜕𝑘 =

|𝐶|

|𝐻|=

|𝑈011 𝑈022 −𝑈01𝑘

𝑈31 𝑈32 −𝑈3𝑘

|

|𝐻| (22)

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Technical Appendix 2

Adjustment after a negative health shock, for 𝜃 = 0 and 𝜃 = 1

We now extend the free maximization problem (appendix 1), in order to determine the outcome for individuals who have a share of liquid wealth equal to one or zero.

𝑀𝑎𝑥{𝑡𝑁,𝜃,𝑀} 𝑈 = 𝐻(𝑀, 𝑡𝑁) + 𝑈2(𝑟𝜃𝑊 + 𝜔𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3[𝜃𝑊 + 𝛼(1 − 𝜃)𝑊] (23)

s.t. {

𝑡𝑁 ≥ 0 0 ≤ 𝜃 ≤ 1

𝑀 ≥ 0 ⇔ {

𝑡𝑁 ≥ 0 𝜃 ≥ 0 𝜃 ≤ 1 𝑀 ≥ 0

(24)

Which corresponds to three non-negativity restrictions and one inequality restriction:

L= 𝑈 = 𝐻(𝑀, 𝑡𝑁) + 𝑈2(𝑟𝜃𝑊 + 𝜔𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3[𝜃𝑊 + 𝛼(1 − 𝜃)𝑊] + 𝜆(1 − 𝜃) (25)

The associated Karush-Kuhn-Tucker conditions are:

{ 𝜕𝑈

𝜕𝑡𝑁≤ 0; 𝑡𝑁≥ 0; 𝑡𝑁 𝜕𝑈

𝜕𝑡𝑁 = 0

𝜕𝑈

𝜕𝜃≤ 0; 𝜃 ≥ 0; 𝜃 𝜕𝑈

𝜕𝜃 = 0

𝜕𝑈

𝜕𝑀≤ 0; 𝑀 ≥ 0; 𝑀 𝜕𝑈

𝜕𝑀= 0

𝜕𝑈

𝜕𝜆 ≥ 0; 𝜆 ≥ 0; 𝜆 𝜕𝑈

𝜕𝜆 = 0

(26)

Technical Appendix 2.1.

Adjustment after a negative health shock, for 𝜃 = 1

Assuming that 𝑡𝑁 > 0, 𝜃 = 1 and 𝑀 > 0, then the system can be simplified to:

{

𝜕𝐻

𝜕𝑡𝑁= 0

𝜕𝑈2

𝜕𝐶 𝑟𝑊 +

𝜕𝑈3

𝜕𝑍 𝑊(1 − 𝛼) − 𝜆 = 0 𝜕𝐻

𝜕𝑀− 𝑝

𝜕𝑈2

𝜕𝐶 = 0

1 − 𝜃 = 0; 𝜆 > 0

(27)

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7

𝐻 = [

𝑈11 0 𝑈13 0

0 𝑈22 𝑈23 −1

𝑈13 𝑈23 𝑈33 0

0 −1 0 0

] (28)

Applying the Laplace rule to the last column, we can obtain the determinant of the hessian matrix.

|𝐻| = (−1) ∗ (−1)2+4|𝑈𝑈11 0 𝑈13

13 𝑈23 𝑈33

0 −1 0 | = 𝑈13

2− 𝑈

33𝑈11, which we assume to be positive since

the hessian matrix must be positive definite.

Let k be a negative shock on the stock of health, so that the Lagrangian function becomes: L= 𝑈 = 𝐻(𝑀, 𝑡𝑁, 𝒌) + 𝑈2(𝑟𝜃𝑊 + 𝜔𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3[𝜃𝑊 + 𝛼(1 − 𝜃)𝑊] + 𝜆(1 − 𝜃) (29)

Then, by totally differentiating the first-order conditions in respect to k, we obtain:

{

𝑈11𝜕𝑡𝑁 ∗

𝜕𝑘 + 𝑈13

𝜕𝑀∗

𝜕𝑘 = −𝑈1𝑘

𝑈22𝜕𝜃 ∗

𝜕𝑘 + 𝑈23

𝜕𝑀∗

𝜕𝑘 −

𝜕𝜆∗

𝜕𝑘 = 0

𝑈31𝜕𝑡𝑁 ∗

𝜕𝑘 + 𝑈32

𝜕𝜃∗

𝜕𝑘 + 𝑈33

𝜕𝑀∗

𝜕𝑘 = −𝑈3𝑘

−𝜕𝜃𝜕𝑘∗= 0

(30)

Note that, from the last equation of the system, we can conclude that in face of a negative health shock the share of liquid wealth will not change: 𝜕𝜃∗

𝜕𝑘 = 0.

Writing the system in matrix form, we obtain:

[

𝑈11 0 𝑈13 0

0 𝑈22 𝑈23 −1

𝑈31 𝑈32 𝑈33 0

0 −1 0 0

]

[ 𝜕𝑡𝑁∗

𝜕𝑘 𝜕𝜃∗ 𝜕𝑘 𝜕𝑀∗ 𝜕𝑘 𝜕𝜆∗ 𝜕𝑘 ] = [ −𝑈1𝑘 0 −𝑈3𝑘 0 ] (31)

Lastly, by applying the Cramer rule we can determine 𝜕𝑡𝑁∗

𝜕𝑘 and

𝜕𝑀∗

𝜕𝑘.

𝜕𝑡𝑁∗

𝜕𝑘 =

|𝐴|

|𝐻|=

|

−𝑈1𝑘 0 𝑈13 0

0 𝑈22 𝑈23 −1

−𝑈3𝑘 𝑈32 𝑈33 0

0 −1 0 0

|

|𝐻| (32)

Where |𝐴| = − 𝑈1𝑘𝑈33+ 𝑈3𝑘𝑈13> 0. Hence, 𝜕𝑡𝑁∗

(8)

8

Similarly,

𝜕𝑀∗

𝜕𝑘 =

|𝐶|

|𝐻|=

|

𝑈11 0 −𝑈1𝑘 0

0 𝑈22 0 −1

𝑈13 𝑈32 −𝑈3𝑘 0

0 −1 0 0

|

|𝐻| (33)

Where|𝐶| = −𝑈11𝑈3𝑘+ 𝑈13𝑈1𝑘 > 0 and so 𝜕𝑀∗

𝜕𝑘 > 0.

Technical Appendix 2.2.

Adjustment after a negative health shock, for 𝜃 = 0 For 𝑡𝑁> 0, 𝜃 = 0, 𝑀 > 0 we know that 𝜕𝑈

𝜕𝜃< 0, given that the marginal benefit of additional

liquidity is lower than the associated marginal cost. Moreover, since the share of liquid wealth is null, then the restriction 𝜃 ≤ 1 is not active, which implies that 𝜆 = 0. Thus, the Lagrangian function simplifies to:

L= 𝑈 = 𝐻(𝑀, 𝑡𝑁) + 𝑈2(𝑟𝜃𝑊 + 𝜔𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3[𝜃𝑊 + 𝛼(1 − 𝜃)𝑊] (34)

Which is precisely the same as in the case of free optimization. Therefore, in face of a negative health shock 𝜕𝑡𝑁∗

𝜕𝑘 > 0, 𝜕𝜃∗

𝜕𝑘 > 0 and

𝜕𝑀∗

(9)

9

Technical Appendix 3

Marginal rate of substitution between social network interactions and the share of liquid wealth

𝑈 = 𝐻(𝑀, 𝑡𝑁) + 𝑈2(𝑟𝜃𝑊 + 𝑤𝑡̅ − 𝑝𝑀) + 𝑈𝐿 3 (𝜃𝑊 + 𝛼(1 − 𝜃)𝑊) (35)

The marginal rate of substitution between social network interactions and the share of liquid wealth is given by:

𝑑𝑡𝑁

𝑑𝜃 = −

𝑀𝑈𝜃

𝑀𝑈𝑡𝑁 (36)

Where, 𝑀𝑈𝑡𝑁 =

𝜕𝐻

𝜕𝑡𝑁> 0, by construction (37)

𝑀𝑈𝜃 =𝜕𝑈𝜕𝐶2(𝑟𝑊) +𝜕𝑈𝜕𝑍3𝑊(1 − 𝛼) (38)

Which implies,

𝑑𝑡𝑁

𝑑𝜃 = −

𝜕𝑈2

𝜕𝐶(𝑟𝑊)+𝜕𝑈3𝜕𝑍𝑊(1−𝛼)

𝜕𝐻 𝜕𝑡𝑁

(39)

Note that, by assumption 𝜕𝑈2

𝜕𝐶 > 0 and so

𝜕𝑈2

𝜕𝐶 (𝑟𝑊) > 0. Moreover, we also assumed from the

onset that 𝜕𝑈3

𝜕𝑍 > 0 and that 𝛼 > 1, with the last condition expressing the greater weight that

illiquid assets have on utility, due to the services they provide (i.e. housing services). Therefore, the sign of the expression greatly depends on how greater 𝛼 is relative to 1 and, consequently, how the absolute values of𝜕𝑈2

𝜕𝐶 (𝑟𝑊) and

𝜕𝑈3

𝜕𝑍 𝑊(1 − 𝛼) relate.

For 𝛼 > 1 assume that 𝛼 → 1, so that liquid wealth and illiquid wealth become perfect substitutes in the wealth utility function (𝑈3). Under these conditions,

lim𝛼→1𝑑𝑡𝑑𝜃𝑁= lim𝛼→1(− 𝜕𝑈2

𝜕𝐶(𝑟𝑊)+𝜕𝑈3𝜕𝑍𝑊(1−𝛼)

𝜕𝐻 𝜕𝑡𝑁

) = −𝜕𝑈2𝜕𝐶(𝑟𝑊)+lim𝛼→1𝜕𝑈3𝜕𝑍𝑊(1−𝛼) 𝜕𝐻

𝜕𝑡𝑁

= −𝜕𝑈2𝜕𝐶(𝑟𝑊) 𝜕𝐻 𝜕𝑡𝑁

(10)

10

(11)

11

Technical Appendix 4

Relation between the theoretical and empirical models

If the system of first order conditions (derived in technical appendix 1) of the maximization problem at hand is linearized with a Taylor series approximation, then it can be written as:

{𝑎𝑎1121𝑡𝑡𝑁𝑁+ 𝑎+ 𝑎2212𝜃 + 𝑎𝜃 + 𝑎1323𝑀 = 𝑏𝑀 = 𝑏12 𝑎31𝑡𝑁+ 𝑎32𝜃 + 𝑎33𝑀 = 𝑏3

(41)

Where 𝑎𝑖𝑗 and 𝑏𝑖 are combinations of derivatives evaluated at a reference point for the Taylor expansion. The demand functions of 𝑡𝑁,𝜃 and 𝑀 result from the resolution of the system with respect to 𝑡𝑁, 𝜃 and 𝑀, respectively.

However, for the corner cases 𝜃 = 0 and 𝜃 = 1, the condition associated with the share of liquid wealth is not relevant, and so the system can be simplified to:

{𝑎̃11𝑡𝑁+ 𝑎̃13𝑀 = 𝑏̃1 𝑎̃31𝑡𝑁+ 𝑎̃33𝑀 = 𝑏̃3

(42)

If the system is evaluated at the same extension point and knowing that 𝑏̃1 = 𝑏1− 𝑎12𝜃 and 𝑏̃3= 𝑏3− 𝑎32𝜃, then solving the system we obtain:

{𝑡𝑀 = 𝑚0− 𝜃𝑚1

𝑁 = 𝑡𝑁0− 𝜃𝑡𝑁1 (43)

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