📈 economics

The Design of Optimally Balanced Pay-as-you-go Social Security Systems

This paper proposes a framework for designing optimally balanced pay-as-you-go social security systems that resemble notional accounts by applying a backward calculation algorithm to find optimal monetary equilibria in non-stationary overlapping generations economies, illustrated through demographic and productivity projections for five major countries from 1950 to 2070.

Original authors: Leandro Lyra Braga Dognini

Published 2026-04-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Leandro Lyra Braga Dognini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, never-ending relay race called Social Security. In this race, the current generation of workers (the "young") hands a baton of money to the next generation of retirees (the "old"). The rule is simple: what the young put in today pays for the old's benefits today. This is called a Pay-As-You-Go system.

For a long time, this race worked smoothly because there were always plenty of young runners to carry the baton for a few older runners. But recently, the world has changed. People are having fewer babies, and people are living longer. Suddenly, there are fewer young runners and many more old ones. The baton is getting heavy, and the system is starting to wobble.

This paper, written by Leandro Lyra Braga Dognini, proposes a new way to design this relay race so it never collapses, even when the number of runners changes drastically.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Problem: The "Demographic Tsunami"

The paper starts by pointing out a global crisis. Countries like Brazil, China, India, Italy, and the US are all facing a "demographic transition."

The Analogy: Imagine a family dinner where, for decades, there were 20 kids for every 1 grandparent. The kids could easily buy the grandparent a slice of cake. Now, the kids are having fewer children, and the grandparent is living longer. Soon, there might be only 2 kids for every grandparent. If the kids still try to buy the same huge cake, they will go broke, or the grandparent won't get enough cake.
The Result: Governments are forced to either raise taxes on the few young workers, cut benefits for the many old retirees, or print money (which causes inflation). None of these are good solutions.

2. The Solution: The "Fictitious Bank Account"

The author suggests a clever trick. Instead of thinking of Social Security as a government tax-and-spend program, imagine it as a giant, invisible bank account for every single person.

The "Notional Account": Think of this like a savings account where you can't actually withdraw the money until you retire, but the government tracks your balance.
The Twist: In a normal bank, your money grows because of interest rates set by the market. In this new system, your "interest rate" is determined by how many young people are working next year.
- If there are lots of young workers, your account grows fast (high return).
- If there are few young workers, your account grows slower (low return).
The Magic: By linking your benefits directly to how many people are working, the system automatically balances itself. If the population shrinks, your benefits adjust down slightly, but you also pay in less. No one is left holding the bag.

3. The Secret Sauce: "Looking Backward to Move Forward"

This is the most complex part of the paper, but here is the simple version.

Usually, when governments plan for the future, they guess what will happen in 2050 and set rules today. The author says this is backward. Instead, he uses a mathematical method called "Backward Calculation."

The Analogy: Imagine you are trying to solve a maze. Most people start at the entrance and try to find the exit. The author says, "Let's start at the exit (the very distant future, say the year 2100) and work our way backward to today."
How it works:
1. The Tail: First, the author imagines a "Tail Economy"—a simplified version of the world far in the future where things are stable. He figures out the perfect rules for that future.
2. The Step-Back: Then, he takes one step back in time. He asks: "Given the perfect rules for the future, what must the rules be today to make sure we arrive at that perfect future?"
3. The Chain: He repeats this step-by-step, moving backward from 2100 all the way to 1950 (or today).
The Result: This process reveals the perfect contribution rate (how much you pay) and the perfect replacement rate (how much you get) for every single generation, ensuring that no generation is unfairly treated compared to the next.

4. Why This is "Optimally Balanced"

The paper argues that this method creates a system that is Pareto Optimal.

The Analogy: Imagine a pie. A "Pareto Optimal" solution is one where you cannot give a bigger slice to one person without taking a slice away from someone else.
In this system, the government isn't just trying to balance the budget (making sure taxes equal benefits). It is trying to balance fairness. It ensures that the young aren't paying too much for the old, and the old aren't getting too little for the young. It finds the "Goldilocks" zone where everyone is as happy as possible given the demographic reality.

5. Real-World Application

The author tested this idea using data from five countries: Brazil, China, India, Italy, and the US.

The Findings: The math showed that as these countries age, the "interest rate" on the notional accounts will naturally drop.
The Lesson: Instead of panicking and suddenly cutting pensions, these countries should adjust their systems gradually and automatically. If the population shrinks, the system automatically adjusts the math so that everyone's "account" reflects the new reality.

Summary: The Big Takeaway

This paper is a blueprint for fixing Social Security before it breaks.

Instead of treating Social Security as a political battle between generations, the author treats it like a mathematical relay race. By using a "backward calculation" method, we can design a system where:

Your benefits are linked to your contributions (like a real bank account).
The system automatically adjusts to population changes (fewer workers = lower growth, but also lower costs).
No generation is left behind, and the system remains fair and balanced for centuries to come.

It's a way to turn a looming demographic crisis into a manageable, mathematically perfect plan.

1. Problem Statement

The paper addresses the critical challenge of sustaining Pay-As-You-Go (PAYG) social security systems amidst global demographic transitions.

The Core Issue: Plummeting fertility rates and rising life expectancy are causing a rapid decline in the ratio of working-age population to the elderly. This creates fiscal imbalances where outflows (benefits) exceed inflows (contributions), leading to rising public debt, tax hikes, or benefit cuts.
Limitations of Current Models: Existing literature often relies on stationary models, short time horizons (2-period households), or assumes specific demographic patterns. There is a lack of a general theoretical framework capable of designing optimal PAYG systems for non-stationary economies with heterogeneous households and arbitrary lifespans that can adapt to dynamic demographic shifts.
Goal: To construct a theoretical basis and an algorithmic design for an "optimally balanced" PAYG system that is fiscally sustainable, Pareto optimal, and adaptable to future demographic and productivity changes.

2. Methodology

The paper utilizes General Equilibrium Theory, specifically Overlapping Generations (OLG) models, combined with a backward calculation algorithm derived from Dognini (2025).

A. Theoretical Framework

Fictitious Economy: The author constructs a fictitious competitive infinite-horizon economy ( $E$ ) composed of the insured population.
Walras' Law & Notional Accounts: Under equilibrium prices, the system satisfies Walras' Law. This implies that the optimal contribution and replacement rates can be interpreted as the balances of a Notional Account (or Non-Financial Defined Contribution - NDC) system. Contributions and benefits are explicitly linked, mimicking a market mechanism where households maximize utility subject to inter-temporal budget constraints.
Optimality Criteria: A system is "optimally balanced" if:
1. It is fiscally balanced (total contributions = total benefits).
2. It is Pareto optimal (no household is made worse off, and no further Pareto improvements are possible).
3. It is forward-looking: Parameters adjust based on future demographic and productivity dynamics.

B. The Backward Calculation Algorithm

The core methodological contribution is the application of a backward calculation algorithm to find the unique Pareto optimal monetary equilibrium in a non-stationary OLG economy.

Tail Economies: The infinite horizon problem is approximated by a sequence of finite economies. The algorithm requires defining a "well-behaved" tail economy (a stationary or predictable economy starting at a future time $T$ ) where the set of Pareto optimal equilibria is known.
Relabeling: For economies with lifespan $S \geq 3$ , the model uses the Balasko-Cass-Shell relabeling algorithm to transform the economy into an equivalent 2-period model with multiple goods.
Backward Solving: Starting from the boundary conditions of the tail economy (Pareto optimal prices), the algorithm solves the market-clearing equations backward in time to determine the equilibrium prices (and thus return rates) for the current period.
Selection Criterion: Since multiple Pareto optimal equilibria may exist, the paper proposes selecting the sequence that minimizes the variance of return rates (or minimizes the distance to current parameters) to ensure stability.

3. Key Contributions

General Algorithm for Non-Stationary OLG: The paper provides a rigorous algorithm to calculate optimal contribution and replacement rates for economies with arbitrary lifespans and non-stationary demographic dynamics (e.g., rapid aging), moving beyond the standard 2-period stationary models.
Theoretical Justification for Notional Accounts: It demonstrates that an optimally balanced PAYG system is mathematically equivalent to a Notional Account system where the "interest rate" is the equilibrium return rate of the fictitious economy. This provides a micro-founded justification for NDC systems (used in Sweden and Italy).
Forward-Looking Design: The design explicitly incorporates future demographic shifts. The paper proves that changes in future demographic growth rates reverberate backward to the present, necessitating immediate adjustments in current contribution and benefit rates to maintain Pareto optimality.
Handling Heterogeneity: The framework accommodates heterogeneous households and varying endowments (productivity), allowing for the integration of real-world data (GDP per capita, population counts).

4. Results and Empirical Illustration

The author applies the methodology to a simplified 3-period OLG model (representing 20-year blocks) using demographic and productivity data from 1950 to 2070 for five countries: Brazil, China, India, Italy, and the United States.

Data Inputs:
- Demographics ( $H_t$ ): Derived from UN World Population Prospects (working-age vs. elderly).
- Productivity ( $e_t$ ): Derived from World Bank GDP per capita data.
- Growth Rates: Demographic growth ( $\alpha_t$ ) and productivity growth ( $\beta_t$ ) are calculated to form aggregate growth rates ( $\gamma_t$ ).
Simulation Findings:
- Return Rates: The calculated optimal return rates generally track the aggregate endowment growth rate (the "biological interest rate"). However, during periods of sharp demographic transition (e.g., Brazil and China), the optimal rates deviate significantly from simple growth trends to smooth consumption across generations.
- Contribution/Replacement Rates: The algorithm generates specific contribution ( $\sigma_t$ $σ_{t}$ ) and replacement rates for each generation.
  - Example: In countries facing rapid aging (like China), the algorithm suggests that future demographic declines must be priced in today, leading to immediate adjustments in contribution rates to prevent future fiscal collapse.
- Minimum Variance Selection: Among the multiple possible equilibrium paths, the "minimum variance" path was selected to minimize volatility in returns, providing a stable benchmark for policy.
Key Insight: The design is highly sensitive to long-term forecasts. A change in the projected demographic growth rate for 2050 requires an immediate reform of the contribution rates for the current generation (2026).

5. Significance and Implications

Policy Relevance: The paper offers a concrete, theoretically sound method for governments to reform pension systems. It moves away from ad-hoc political adjustments toward a systematic, forward-looking approach based on general equilibrium principles.
Notional Accounts Validation: It strengthens the case for Notional Defined Contribution (NDC) systems, showing they are not just accounting tricks but can represent the unique Pareto optimal allocation in a PAYG system.
Addressing the "Time Inconsistency" Problem: By making the system "forward-looking," the design prevents the common political failure where current administrations delay reforms, passing the burden to future generations. The algorithm mathematically proves that delaying adjustment leads to sub-optimal (non-Pareto) outcomes.
Limitations & Future Work: The current model assumes exogenous demographics and productivity. Future work needs to incorporate endogenous labor supply, stochastic shocks, and more complex preference structures. Additionally, the reliance on long-term forecasts introduces uncertainty, suggesting a need for "buffer funds" or automatic stabilizers in real-world applications.

In summary, Dognini bridges the gap between abstract general equilibrium theory and practical pension reform, providing a mathematical tool to design social security systems that are robust against the inevitable demographic shifts of the 21st century.