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Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures

  • Sharmila Gamlath and Radhika Lahiri EMAIL logo
Published/Copyright: November 17, 2022
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 23 Issue 1

Abstract

Empirical evidence suggests a positive correlation between health and educational outcomes at the aggregate level. However, both inter and intra-country data suggest that these variables may not always be monotonically increasing in income, pointing towards household income as a possible mitigating factor in the relationship between health and education expenditures and outcomes. We develop an overlapping generations model where agents spend their childhood studying, undertake expenditures to educate their offspring and health expenditures to improve their own longevity in adult age, and spend old age in retirement. Our model is characterised by two equilibria. In one equilibrium, longevity enhancing health expenditure is an inferior good, resulting in agents substituting health expenditures in favour of education expenditures on offspring as their income increases. In the other equilibrium, health expenditure is a normal good, but for incomes below a certain level, an increase in income causes agents to raise health expenditures whilst lowering education expenditures on offspring, while for incomes above this level all expenditures are increasing in income. These results suggest that the relationship between parental longevity and offspring’s human capital depends on income and whether agents consider longevity enhancing health expenditure to be an inferior or normal good. Dynamics of the model show that the economy could either achieve unbounded growth or converge towards a lower bound of income in the long run.

Keywords: health expenditures; longevity; education expenditures; human capital
JEL Classification: E24; J24; I14; I24
Corresponding author: Radhika Lahiri, School of Economics and Finance, Queensland University of Technology, 2, George Street, 4000, Brisbane, Queensland, Australia, E-mail:
Appendices A

Appendix A.1 Proof of Proposition 1

Consider the left hand side of (2.14) and define:

ψ ( m ) ≡ δ ( φ ( m ) ) − 1 σ + ( 1 − σ ) φ ( m ) φ ′ ( m ) + m .

Differentiating with respect to m and rearranging, we get:

ψ ′ ( m ) = 1 − 1 σ − φ ( m ) φ ″ ( m ) ( φ ′ ( m ) ) 2 δ ( φ ( m ) ) − 1 σ − ( 1 − σ ) φ ( m ) φ ″ ( m ) ( φ ′ ( m ) ) 2 + ( 2 − σ )

Substituting functional forms for ψ′(m) we get:

(A1) ψ ′ ( m ) = 1 + ( 1 − σ ) ( 1 + 2 m ) + δ m 1 + m 1 σ 1 + 2 m − 1 σ .

Since φ″(m) < 0, the term − ( 1 − σ ) φ ( m ) φ ″ ( m ) ( φ ′ ( m ) ) 2 is positive. This means that a sufficient condition for ψ′(m) to be positive is 1 − 1 σ − φ ( m ) φ ″ ( m ) ( φ ′ ( m ) ) 2 > 0 , which simplifies to m > ε * ≡ 1 2 1 σ − 1 .

However, given the continuity of ψ′(m) we would expect it to remain positive for m < ɛ*. Specifically, consider m ∈ [0, ɛ*]. From (A1) we can postulate,

ψ ′ ( m ) > 0 iff 1 + ( 1 − σ ) ( 1 + 2 m ) > − δ m 1 + m 1 σ 1 + 2 m − 1 σ

Rearranging and substituting for φ(m) we can express the above as:

(A2) ψ ′ ( m ) > 0 iff m 1 + m 1 σ > − δ 1 + 2 m − 1 σ 1 + ( 1 − σ ) ( 1 + 2 m ) .

Now, evaluated at m = 0, the right hand side of the above expression is positive, while the left hand side is equal to zero, so the above inequality does not hold. Evaluated at the other end of the range at m = ε*, the right hand side is zero and the left hand side is positive, the inequality holds. We already know this as we derived m > ε* as a sufficient condition for ψ′(m) to be positive. Further, the left hand side and right hand side are both continuous and differentiable in the range in question. We can also show that in this range, the left hand side is increasing in m while the right hand side is decreasing in m. The graph of both functions may be roughly depicted as follows (Figure A.1):

Figure A1: Graphical depiction of functions (A1) and (A2) and Equation (2.14).
Figure A1:

Graphical depiction of functions (A1) and (A2) and Equation (2.14).

To the left of ε ̃ , ψ′(m) < 0, while to the right of ε ̃ , ψ′(m) > 0. We have thus proven the first part of Proposition 1. For the second part, recognise that ε ̃ divides the domain of m into two parts [0, ε ̃ ) and [ ε ̃ , ∞). Corresponding to these domains ψ(m) is respectively monotone decreasing and monotone increasing. As such, partial inverse functions restricted to these domains exist and we denote these by η1(h) and η2(h). Further, there exists an h ̃ corresponding to ε ̃ . Applying the inverse funtion theorem to these partial inverse functions, we can show that these η 1 ′ ( h ) = 1 ψ < 0 and η 1 ′ ( h ) = 1 ψ < 0 . This proves Proposition 1.

Then, there exists an intermediate value m = ε ̃ ∈ [ 0 , ε * ] that the two functions intersect.[3] For future reference, we note that, along η2(h), we have also identified another critical value ε * ≡ 1 2 1 σ − 1 which (given monotonicity) corresponds to a unique threshold level of income h* which is of relevance to subsequent proofs presented below.

Appendix A.2 Proof of Proposition 2

Equations (2.10)–(2.13) imply:

(A3) ∂ d ∂ m ⋅ ∂ m ∂ h = ( 1 − σ ) ( 1 + r ̃ ) 1 − φ ( m ) φ ″ ( m ) ( φ ′ ( m ) ) 2 η 1 ′ ( h )
(A4) ∂ c ∂ m ⋅ ∂ m ∂ h = ( 1 − σ ) ( 1 + r ̃ ) 1 − 1 σ 1 − 1 σ ( φ ( m ) ) − 1 σ φ ′ ( m ) − ( φ ( m ) ) 1 − 1 σ φ ″ ( m ) ( φ ′ ( m ) ) 2 η 1 ′ ( h )
(A5) ∂ e ∂ m ⋅ ∂ m ∂ h = γ ( 1 − σ ) ( 1 + r ̄ ) 1 − 1 σ 1 − 1 σ ( φ ( m ) ) − 1 σ φ ′ ( m ) − ( φ ( m ) ) 1 − 1 σ φ ″ ( m ) ( φ ′ ( m ) ) 2 η 1 ′ ( h )
(A6) ∂ s ∂ m ⋅ ∂ m ∂ h = ( 1 − σ ) 1 − ( φ ( m ) ) φ ″ ( m ) ( φ ′ ( m ) ) 2 η 1 ′ ( h ) .

Recall that φ(m) > 0,φ′(m) > 0 and φ″(m) < 0. Since η 1 ′ ( h ) < 0 , this means that (A3) and (A6) are unambigiously negative, implying that future consumption and thereby savings are decreasing in human capital. The sign of the expressions on the right hand side of (A4) and (A5) depend on the sign of the expressions in curly brackets. Then, we can show that:

1 − 1 σ ( φ ( m ) ) − 1 σ φ ′ ( m ) − ( φ ( m ) ) − 1 σ φ ″ ( m ) ( φ ′ ( m ) ) 2 > 0 iff 1 − 1 σ > φ ( m ) φ ″ ( m ) φ ′ ( m ) .

Substituting for functional forms, this simplifies to:

(A7) 1 − 1 σ ( φ ( m ) ) − 1 σ φ ′ ( m ) − ( φ ( m ) ) − 1 σ φ ″ ( m ) ( φ ′ ( m ) ) 2 > 0 iff m = η ( h ) > 1 2 1 σ − 1

Note however that the expression on the right hand side is the critical value ε * ≡ 1 2 1 σ − 1 , which belongs to the domain for the other partial inverse function η2(h) identified in the proof of Proposition 1. Therefore the term in the curly bracket is negative. From (A4) and (A5), then, we have the result that for η1(h), current consumption and education expenditures are increasing in human capital.

Appendix A.3 Proof of Proposition 3

We now consider the range ( h ̃ , ∞ ) which contains h* implicitly defined by η 2 ( h * ) = 1 σ − 1 2 . For incomes in ( h ̃ , h * ) , we know that the expression in the curly brackets of (A4) and (A5) remain negative, but η 1 ′ ( h ) is replaced by η 2 ′ ( h ) which is positive. This means that current consumption and educational expenditures are decreasing in income. Likewise, η 1 ′ ( h ) is replaced by η 2 ′ ( h ) in (A3) and (A6), and given that the expression in the curly brackets of those equations are unambiguously positive, it is straightforward to show that future consumption and saving are increasing in income.

Finally, examining the range (h*, ∞) we note that the expression in curly brackets of (A4) and (A5) has now become positive. Following through the usual steps, it is evident that all variables are now increasing in income.

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Received: 2021-04-24
Accepted: 2022-10-30
Published Online: 2022-11-17

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. A New General Equilibrium Welfare Measure, with Application to Labor Income Taxes
  4. Labor Share Dynamics and Factor Complementarity
  5. Effect of Monetary Policy on Government Spending Multiplier
  6. News-Driven Housing Booms: Spain Versus Germany
  7. Sovereign Debt Crisis, Fiscal Consolidation, and Active Central Bankers in a Monetary Union
  8. Housing Taxation and Economic Growth: Analysis of a Balanced-Growth Model with Residential Capital
  9. Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures
  10. Contributions
  11. Tolerance of Informality and Occupational Choices in a Large Informal Sector Economy
  12. Uncertainty Shocks, Innovation, and Productivity
  13. Asymmetric Effects of Private Debt on Income Growth
  14. Interpreting Structural Shocks and Assessing Their Historical Importance
  15. Charge-offs, Defaults and the Financial Accelerator
  16. Filtering Persistent and Asymmetric Cycles
  17. Population Aging and Convergence of Household Credit
  18. Robustly Optimal Monetary Policy in a Behavioral Environment
  19. Forward Guidance Effectiveness in a New Keynesian Model with Housing Frictions
  20. The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers
https://doi.org/10.1515/bejm-2021-0095
Keywords for this article
health expenditures; longevity; education expenditures; human capital

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. A New General Equilibrium Welfare Measure, with Application to Labor Income Taxes
  4. Labor Share Dynamics and Factor Complementarity
  5. Effect of Monetary Policy on Government Spending Multiplier
  6. News-Driven Housing Booms: Spain Versus Germany
  7. Sovereign Debt Crisis, Fiscal Consolidation, and Active Central Bankers in a Monetary Union
  8. Housing Taxation and Economic Growth: Analysis of a Balanced-Growth Model with Residential Capital
  9. Intergenerational Linkages, Uncertain Lifetime and Educational and Health Expenditures
  10. Contributions
  11. Tolerance of Informality and Occupational Choices in a Large Informal Sector Economy
  12. Uncertainty Shocks, Innovation, and Productivity
  13. Asymmetric Effects of Private Debt on Income Growth
  14. Interpreting Structural Shocks and Assessing Their Historical Importance
  15. Charge-offs, Defaults and the Financial Accelerator
  16. Filtering Persistent and Asymmetric Cycles
  17. Population Aging and Convergence of Household Credit
  18. Robustly Optimal Monetary Policy in a Behavioral Environment
  19. Forward Guidance Effectiveness in a New Keynesian Model with Housing Frictions
  20. The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers
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