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Healthy Grands, Fertility and Pensions

  • Giam Pietro Cipriani and Tamara Fioroni EMAIL logo
Published/Copyright: May 22, 2025
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 25 Issue 1

Abstract

In this paper, we build a simple overlapping generations model with endogenous fertility and a PAYG pension system. Children are costly, but healthy grandparents can reduce this cost by providing grandparental care. However, if grandparents are unhealthy, their adult children have to spend time caring for them. The results show how longevity and grandparenting affect equilibrium fertility and pensions, depending on the elderly’s health, thus adding two important dimensions – grandparenting and elderly health – to the stylized textbook OLG model with longevity and endogenous fertility.

Keywords: OLG model; PAYG pension; endogenous fertility; grandparenting
JEL Classification: H55; J11; J13
Corresponding author: Tamara Fioroni, Department of Economics, University of Verona, Polo Universitario Santa Marta, Via Cantarane 24, 37129 Verona, Italy, E-mail: 
Appendix A

From Eq. (9) the steady state level of fertility can be written:

(13) n * = γ ( 1 − τ − π m z ) C [ q − g π ( 1 − m ) ] { ( 1 + γ ) C + δ π α }

where C = α + τ(1 − α).

Some calculations show that ∂n/∂π can be simplified as follows:

(14) − δ π 2 α m z ( 1 − m ) g + 2 ( 1 − m ) ( 1 − τ ) g π δ α + ( 1 − m ) g ( 1 + γ ) ( 1 − τ ) C − q [ m z ( 1 + γ ) C + ( 1 − τ ) α δ ] .

Thus ∂n/∂π = 0, if:

(15) π 1,2 = ( 1 − τ ) ( 1 − m ) α δ g ± ( 1 − m ) α δ g [ m z ( 1 + γ ) C + α δ ( 1 − τ ) ] [ ( 1 − τ ) ( 1 − m ) g − m q z ] ( 1 − m ) α δ g m z .

Let’s define:

(16) g ̄ = m q z ( 1 − τ ) ( 1 − m )

(17) g ̃ = q [ α δ ( 1 − τ ) + ( 1 + γ ) C m z ] α δ ( 1 − τ − m z ) + α δ ( 1 − τ ) + ( 1 + γ ) C ( 1 − τ )

(18) g ̂ = q [ α δ ( 1 − τ ) + ( 1 + γ ) C m z ] ( 1 + γ ) C ( 1 − τ ) ( 1 − m )

where simple calculations show that g ̃ < g ̂ . From eq. (15) if g < g ̄ the determinant is negative, g < g ̂ ensures that π 2 > 0 and when g < g ̃ π 2 > 1 . If g > g ̄ , Assumption 1 implies π < q/g(1 − m), moreover simple calculations show that π 1 < q/g(1 − m) if g > g ̄ .

Thus:

  1. If g < g ̃ then ∂n/∂π < 0

  2. If g ̃ < g < g ̂ then ∂n/∂π < 0 if 0 < π < π 1 and ∂n/∂π > 0 if π 2 < π < q/g(1 − m)

  3. If g > g ̂ then ∂n/∂π > 0.

From Eq. (10), some calculations show that ∂b*/∂π < 0 if:

(19) ( 1 − 2 α ) [ 2 π ( 1 − m ) g − q ] [ ( 1 + γ ) C + α δ π ] ( 1 − τ − π m z ) < ( 1 − α ) [ m z ( 1 + γ ) C + ( 1 − τ ) α δ ] [ q π − π 2 ( 1 − m ) g ]

where given Assumption 1, it is easy to see that when π < q/2(1 − m)g eq. (19) always holds. To study Eq. (19) for q/2(1 − m)g < π < q/(1 − m) for the sake of simplicity let’s define the left hand side of Eq. (19) LHS(π) and the right hand side RHS(π).

Thus:

(20) L H S π = q 2 ( 1 − m ) g = 0

(21) L H S π = q ( 1 − m ) g > 0   if     g > g ̃ 1

and:

(22) R H S π = q 2 ( 1 − m ) g > 0

(23) R H S π = q 2 ( 1 − m ) g > 0

(24) ∂ R H S ( π ) / ∂ π < 0

thus if g < g 1 the LHS is below the RHS for each q/2(1 − m)g < π < q/(1 − m) and therefore ∂b*/∂π < 0. If g < g 1 there exists a threshold level of π, i.e. π*, such that if π < π*, then ∂b*/∂π < 0, above this threshold ∂b*/∂π > 0.

Appendix B

Optimal fertility is now given by:

(25) n t = w t R t + 1 γ n t − 1 ( 1 − τ ) − m π z n t − 1 w t R t + 1 ( 1 + γ + π δ ) [ q − g ( 1 − m ) π ] − γ τ w t + 1 .

At the steady-state:

(26) w t = w t + 1 = w * , R t + 1 = R * , n t = n t − 1 = n * .

where the steady state level of capital and factor prices are the same as in the baseline model. This yields:

(27) A ( n * ) 2 − B n * + C = 0 ,

where:

A = q − g ( 1 − m ) π R * ( 1 + γ + π δ ) − γ τ , B = R * γ ( 1 − τ ) , C = R * γ m π z .

The steady-state equation is quadratic, so there are two possible solutions for n*. For real solutions, the discriminant must be non-negative:

(28) Δ = B 2 − 4 A C

Since economic feasibility requires selecting a positive and stable solution, let us rewrite the mapping:

(29) n t = f ( n t − 1 ) = N ( 1 − τ ) − m π z n t − 1 ,

where:

(30) N = R * γ A .

The parameter selection in our simulation ensures the existence of two positive steady-state solutions for n; of which only one is locally stable – namely, the solution given by:

(31) n * = B + Δ 2 A .

This result stems from the fact that the function f(n t−1) which defines the fertility dynamics, is concave in n t−1, i.e. f′(n t−1) > 0 and f″(n t−1) < 0. The analytical study of Equation (31) is rather complex. However, numerical analysis shows that for a range of parameter values fertility behaves as in the model without siblings. In particular, it always decreases with respect to the probability of being unhealthy (see Figure 9), whereas the impact of life expectancy depends on the level of grandparenting. If grandparenting is sufficiently low, an increase in life expectancy always negatively affects fertility (see Figure 10a). If grandparenting lies within a certain intermediate range, fertility first decreases and then increases as longevity rises (see Figure 10b). Finally, if grandparenting is sufficiently high, fertility always increases with respect to longevity (see Figure 10c).

Figure 9: Equilibrium fertility versus probability of being unhealthy. Parameter values: τ = 0.12, z = 0.2, α = 0.3, q = 0.4, δ = 0.74, γ = 0.9, g = 0.2, m = 0.6, A = 1.
Figure 9:

Equilibrium fertility versus probability of being unhealthy. Parameter values: τ = 0.12, z = 0.2, α = 0.3, q = 0.4, δ = 0.74, γ = 0.9, g = 0.2, m = 0.6, A = 1.

Figure 10: Equilibrium fertility versus life expectancy. Parameter values: τ = 0.12, z = 0.2, α = 0.3, q = 0.4, δ = 0.74, γ = 0.9, g = 0.2, m = 0.6, A = 1.
Figure 10:

Equilibrium fertility versus life expectancy. Parameter values: τ = 0.12, z = 0.2, α = 0.3, q = 0.4, δ = 0.74, γ = 0.9, g = 0.2, m = 0.6, A = 1.

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Received: 2024-07-12
Accepted: 2025-04-21
Published Online: 2025-05-22

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/bejm-2024-0112
Keywords for this article
OLG model; PAYG pension; endogenous fertility; grandparenting

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. A Matter of Minutes: Unexpected FOMC Communication and Fed Credibility
  4. Do Firm-Level Shocks Generate Aggregate Fluctuations? A Cross-Country Analysis
  5. Financial Frictions at Entry, Average Firm Size, and Productivity
  6. Healthy Grands, Fertility and Pensions
  7. Contributions
  8. Robust Optimal Monetary Policies in Behavioral New Keynesian DSGE Models
  9. On the Impact of Fiscal Policy on Inflation: The Case of Fiscal Rules
  10. Gini in the Taylor Rule: Should the Fed Care About Inequality?
  11. Housing Booms and Busts: Dissecting Housing and MBS Markets Linkages
  12. Guardians of (In)Equality: Unmasking the Role of Military Spending in Shaping Income Inequality
  13. Natural Disasters and Capital Accumulation: The Role of Precautionary Saving and Capital Market Openness
  14. Short-Run Impacts of Floods: A Case Study from India
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