_{1}

Population in developed countries has dramatically aged in recent years. At the same time, most of these countries finance their old-age public pension benefits on a pay-as-you-go (PAYGO) basis. Such a pension scheme combined with an ageing population, represent a financial challenge for the governmental budgets. Accordingly, several studies have been done on the effects of population ageing. In the present paper, the purpose is to study how population ageing will affect capital accumulation. A simple OLG model with PAYGO pension is applied. The formulation of the model makes it possible to deduce an explicit expression for this relation. It is shown that an increase in the elderly dependency ratio will increase the long-run capital stock.

In the vast majority of countries around the world, the demographic structure is expected to change in the com- ing decades. The demographic change is due to declining fertility rates and higher life expectancies. This de- mographic transition process results in a general ageing of the population and a reduction in the population growth rate. Some countries are even expected to experience negative growth rates (Börsch-Supan, et al., [

The ageing process can be illustrated by studying the old-age dependency ratio, as in

Country | 2000 | 2015 | 2030 | 2045 |
---|---|---|---|---|

USA | 20.9 | 24.7 | 36.8 | 39.1 |

Finland | 24.7 | 35.4 | 47.8 | 48 |

Germany | 26.2 | 35.3 | 52.1 | 63 |

Norway | 25.8 | 27.9 | 36.1 | 41.8 |

Japan | 27.6 | 47.2 | 58.2 | 75.3 |

Source: United Nations, 2012.

It is clear from this table that all the included countries are characterized by a transition to an older popu- lation.

Most of the countries that experience population ageing also mainly rely on pay-as-you-go (PAYGO) pension systems. At the time these systems where implemented, the first old generation did not contribute.

The combination of ageing and PAYGO may cause severe pressure on public finances. The number of con- tributors decrease, while the number of retirees increase. That implies a smaller tax base, and higher pension liabilities.

Motivated by these issues the current paper explores how population ageing affects capital accumulation, when the social security system is run as a PAYGO pension program.

The literature considering population ageing and macroeconomic implications is vast and characterized by many different approaches. Cutler et al. [

In contrast to these papers the current one will focus on a simple model structure. This enables more tractable solutions with respect to the underlying mechanisms. Narrowing the literature to theoretical approaches, central contributions are Meijdam and Verbon [

This paper proceeds as follows. Section 2 present the model. In Section 3, the effect of population ageing on capital accumulation is analyzed. Section 4 concludes.

Let us consider an overlapping generations economy in the spirit of Diamond [

Each household live for two periods, works in the first period and are retired in the second. Population grows exogenously according to:

where the exogenous variable ^{1} Notice that the entire population size has a certain growth rate. This simplification is convenient to derive explicit solutions. Popu- lation ageing is within this framework interpreted as a permanent reduction in

where a permanent drop in fertility at time t corresponds to an increase in the dependency ratio, i.e. the number of young workers becomes a smaller fraction of total population, relatively to retired individuals in period t,

A representative individual born in period t, endowed with perfect foresight, maximize a lifetime utility function defined over consumption when young

Young agents in period t inelastically offer one unit of labor at wage

where

Upon the Cobb-Douglas utility function along with the budget constraints in period t, the expression for optimal individual savings becomes:

Savings are thus increasing in net income and the interest rate, while the social security benefit reduces individual savings. By excluding the government and considering a Laissez-Faire economy, all contributions and benefits are set to zero. Hence:

Which simply gives savings as an increasing function of the wage. The interest rate do not affect optimal savings in this case due to the logarithmic utility function, which yields an intertemporal substitution elasticity equal to 1.

Let us turn to the production side of the economy. There are two input factors, capital K, and labor L. The de- preciation rate of capital is assumed to be zero. Output is given by a the Cobb-Douglas production function

where the last assumption imply constant returns to scale. Each young individual supplies one unit of labor, and production per head of the young is thus expressed by:

Firms are maximizing profits under perfect competition and their competitive behavior leads to the equali- zation of factor input prices and marginal products:

The government enter the economy in a simple way and is not involved in neither consumption nor production activities. In each period t, the government collects taxes

With a PAYGO transfer scheme, the budget constraint of the government is given by

where

There are three markets in this economy. Equilibrium in the labor market is given by

The equilibrium condition of the capital market equates the supply of capital due to savings, and firms demand for capital. The next period’s capital is therefore given as

Inserting the savings function (5) and constant taxes into the equilibrium condition in (11), gives the dynamics of capital per worker as:

The above expression describes the equilibrium in the capital market. Notice that the first order conditions in (8) and (9), imply that

Accordingly, by inserting the first order conditions in (8) and (9), and the government’s budget restriction in (10), into Equation (12), one obtains the dynamics of capital intensity as a trajectory

where

To consider the dynamical behavior of capital and stability issues, it is necessary to define a dynamic equi- librium and a steady-state equilibrium for the capital market. The dynamic equilibrium is given by Equation (13). To study this dynamic equilibrium it is appropriate to define a fixed point of capital per worker

Stability is scrutinized by conducting two simplifications, that subsequently are relaxed. The first simplifi- cation is to assume that the population growth rate is constant. The second is to exclude the government and thereby the public pension scheme.

Assumption 1 The population growth rate is assumed to be time-invariant, i.e.

Assumption 2 Governmental interventions are absent, i.e.

The rationale for this step-wise strategy is to present the necessary background for the following analysis. Assumptions 1 and 2 imply that the savings function is augmented to

Equation (14) is a homogenous and nonlinear difference equation of the first order. To evaluate stability around a steady-state equilibrium it is useful to define the state transition function:

where

There are two steady-states. As

Hence, as

The next step in the stability analysis is to relax Assumption 1 and 2. The dynamic behavior of the capital intensity is then given by the first order, nonlinear difference Equation in (13). The state transition function is now given by:

where

To study stability conditions of the interior steady-state, it is convenient to consider

where

Evaluating Equation (19) demonstrates that:

i.e. the steady-state, satisfying Equation (18), is unique and locally asymptotically stable.

A PAYGO social security imply that the capital market equilibrium given in Equation (13) must hold in every period. Inspection of this intertemporal equation reveals that capital intensity is affected by demo- graphic changes in the economy. Endowed with perfect foresight with respect to demographic trends, each individual in the model economy is capable of internalizing future changes in the population in their consumption and saving decisions. Hence, if the population growth rate changes in period

To study the consequences on the economy of population ageing it is convenient to reformulate capital in- tensity given in Equation (13) to:

where

In which

In the analysis of

and one can state the following proposition:

Proposition 1 A permanent increase in the elderly dependency ratio gives rise to the long-run capital stock, i.e.

Proof. The proposition is proved by contradiction. Inserting for

Inspection of the above expression reveals that the denominator is unambiguously positive. Since the proof is showed by contradiction, it is assumed that

But, as the LHS equals the wage^{2}, and the wage obviously must be higher than the lump-sum tax, the above inequality yields a contradiction. Therefore one has that:

Hence, population ageing increase the long-run capital intensity. QED

The result reflects that old individuals, who own assets due to earlier saving, are becoming a larger proportion of the population, while the proportion of young individuals, who own no assets, are becoming less. Subse- quently, saving increases and boosts capital accumulation. Moreover, the transition path is characterized by a gradual increase in the wage rate and a fall in the interest rate. This is seen by differentiating the first order conditions in (8) and (9), while exploiting the relation in (1) and the definition in (7):

These qualitative results correspond well to the results given by several more detailed and complicated mo- dels.

This paper is motivated by the ongoing debate regarding the consequences of population ageing. Since several countries rely on PAYGO pension systems, the current paper investigates how ageing affects capital accumu- lation with such a pension scheme. Thus, a standard OLG model is extended with a PAYGO pension system, and an exogenous variable for the population growth rate. In order to achieve explicit solutions, specific func- tional forms has been chosen on production and utility behavior.

The main result in the paper is interesting. It is shown that an increase in the elderly dependency ratio will increase the long-run capital stock. Thus, although the demographic transition along the PAYGO scheme may cause a financial challenge for the governmental budgets, capital accumulation will increase. This may stimulate economic growth led by physical capital accumulation. As far as such a pattern increase income in the society, this may cause an indirect improvement in the governments budgets. Thus, we can have two effects on the financial budgets working in opposite directions. To scrutinize such effects is a natural extension of the current paper.

The contribution of the current paper is accordingly threefold. Firstly, the formulation allows explicit solu- tions for the impact of population ageing on capital accumulation. Secondly, it is shown that population ageing will increase the capital stock. Thirdly, the formulation opens up for different extensions to shed light on several aspects of pension economics. It is also important to note that the simplified formulation limits the possibility of country-specific studies of the details in the social security systems. To conduct such studies require large-scale models that needs to be solved numerically.

I would like to thank Steinar Holden, Øystein Thøgersen, Hans Fehr, participants at the 2015 FIBE conference in Bergen, Norway, and an anonymous referee for helpful comments.