3.1 Primitives

Consider a closed, market economy, in the tradition of Diamond (Reference Diamond1965), wherein, at each date t = 1, 2, …, ∞, a continuum of identical two period-lived agents is born. There is no population growth and the size of a cohort of newborns at any date is held fixed at 1.Footnote ¹¹ Agents consume both as young and old but work only as young. When old, they are retired: they consume whatever they have and die. When young, agents work in competitive labor markets at a wage w, consume, and save (s) for old age in perfect capital markets at the gross rate

Assumption 1 (Dynamic efficiency)

$$R_{t + 1} > 1\forall t$$

between t and t + 1. In section 5, we allow for market-determined, endogenous factor prices. There, the single final good is produced using a standard neoclassical production function F(K _t, L _t) where K _t denotes the capital input and L _t denotes the labor input at t. The final good can either be consumed in the period it is produced, or it can be saved to yield capital at the beginning of the following period. Capital is assumed to depreciate 100% between periods. Let k _t ≡ K _t/L _t denote the capital–labor ratio (capital per young agent). Then, output per young agent at time t may be expressed as f(k _t) where f(k _t) ≡ F(K _t/L _t, 1) is the intensive production function. We assume f(0) = 0, f ^′ > 0, and f ^′′ < 0, and that the usual Inada conditions hold. Until further notice though, we focus on exogenously-specified and constant w and R.

Following Chetty (Reference Chetty2015), we draw a distinction between the “true” and “choice” utility of agents. Agents' behavior is dictated by their choice utility, but their actual well-being, our measure of welfare, is governed by their true lifetime utility. Let c ^y denote consumption as young, and c ^o consumption as old. The “true” preferences of the cohort who are young in period t, denoted with a “*”, is the standard, separable

(1)$$\Omega _t^\ast{\equiv} u( {c_t^y } ) + \beta ^\ast u( {c_{t + 1}^o } ) $$

where $\beta ^\ast \in ( {0,1} ]$ is the true discount factor. The felicity function $u( \cdot )$ is assumed to fulfill standard assumptions, including ${u}^{\prime}( \cdot ) > 0$ and ${u}^{\prime \prime}( \cdot ) < 0$ and Inada conditions. At points below, we will use the CES form: $u( c ) = c^{1-\phi }/(1-\phi ).$

Our yardstick for welfare is Ω*. The choice preferences when young are given as $\Omega _t\equiv u( {c_t^y } ) + \beta u( {c_{t + 1}^o } )$ and myopia arises when

Assumption 2 (Myopia)

$$\beta < \beta ^\ast$$

which is assumed, henceforth.

The (marginal rate of substitution (MRS) measures the rate at which an agent wishes to substitute second-period consumption for first-period consumption. In our case, the choice MRS of an agent is given by −u′(c ^y)/βu′(c ^o) and the true MRS is −u′(c ^y)/β*u′(c ^o). A myopic agent places less weight on the future (β < β*), and therefore, cares relatively more about current consumption. Hence, the compensation (in second-period consumption) he seeks for giving up a unit of first-period consumption is higher the more myopic he is. His true indifference curve is flatter than his choice indifference curve.

The government is immune to the myopia of agents and is paternalistic—it decides on policy action using Ω*. All young agents have access to a government-intermediated pension scheme wherein they contribute a lump-sum amount τ _t at date t and receive a pension of b _t+1 at t + 1. A PAYG pension satisfies b _t = τ _t (since the net population growth rate is assumed zero) whereas a fully-funded (FF) pension has b _t = Rτ _t−1.Footnote ¹² Note that a myopic agent perceives the effective return on private saving as Rβ, and that on the PAYG scheme as β. To the government, these returns are higher, Rβ* and β*, respectively.

To get a rough intuitive sense of where we are headed, focus attention on Figure 1. The choice utility is shown by the red indifference curve. Given the initial budget set, the agent chooses point A. The true utility is given by the green indifference curve. The optimal bundle from the point of true utility is A* which has more c ^o and less c ^y than at A. Government intervention via pension schemes, can, in principle, pivot the budget set (the details are fleshed out below) so that the new chosen bundle is B on the dotted budget line. This would have more c ^o and less c ^y than at A, the bundle under laissez faire. Point B has higher true utility than point A does. In fact, we show below that a suitably designed FF scheme can get agents to the bundle A*, something a PAYG scheme cannot.

Figure 1. True vs. choice utility under pension schemes.

3.2 Decision rules

The budget constraints of an agent are

(2)$$c_t^y = w_t-\tau _t-s_t, \;$$

(3)$$c_{t + 1}^o = R_{t + 1}s_t + b_{t + 1}.$$

The private saving decision is determined by

(4)$${u}^{\prime}( {w_t-\tau_t-s_t} ) = \beta R_{t + 1}{u}^{\prime}( {R_{t + 1}s_t + b_{t + 1}} ) {\rm \;for\;}s_t > 0, \;$$

and at the zero private-saving corner by

(5)$${u}^{\prime}( {w_t-\tau_t} ) > \beta R_{t + 1}{u}^{\prime}( {b_{t + 1}} ) \,{\rm for}\,s_t = 0.$$

In line with the pension literature, s _t ≥ 0 is imposed. Agents do not have any wage income as old; all they have is either interest income from prior savings or pension payouts. Allowing negative saving is tantamount to allowing borrowing against future pensions which we disallow; in any case, such borrowing is not possible/allowed in many countries.Footnote ^13, Footnote ¹⁴

For later use, note

(6)$$ \eqalign{\displaystyle{{\partial s_t} \over {\partial R_{t + 1}}} = & -\displaystyle{{\beta {u}^{\prime}( c_{t + 1}^o ) + \beta R_{t + 1}s_t{u}^{\prime \prime}( c_{t + 1}^o ) } \over {{u}^{\prime \prime}( c_t^y ) + \beta R_{t + 1}^2 {u}^{\prime \prime}( c_{t + 1}^o ) }} \vskip1.4pc \cr = & -\displaystyle{{\beta {u}^{\prime}( c_{t + 1}^o ) [ {1 + ( ( c_{t + 1}^o {u}^{\prime \prime}( c_{t + 1}^o ) ) /{u}^{\prime}( c_{t + 1}^o ) ) } ] } \over {{u}^{\prime \prime}( c_t^y ) + \beta R_{t + 1}^2 {u}^{\prime \prime}( c_{t + 1}^o ) }}\,{\rm for}\,s_t > 0.} $$

Hence, ∂s _t/∂R _t+1 > 0 holds when relative risk aversion is less than 1.

Assumption 3 (Risk-aversion)

(7)$$\phi ( {c_{t + 1}^o } ) \equiv{-}\displaystyle{{c_{t + 1}^o {u}^{\prime \prime}( {c_{t + 1}^o } ) } \over {{u}^{\prime}( {c_{t + 1}^o } ) }} < 1, \;$$

a standard condition, well-known in the literature. Henceforth, we assume this is true.

In the absence of a pension scheme (τ = b = 0), the private saving decision satisfies u ^′(w _t − s _t) = R _t+1βu ^′(R _t+1s _t). In this case, the non-negativity constraint on saving is never binding because of Inada conditions. If β = 0, then, of course, s _t = 0 is possible; not otherwise. From the perspective of true utility, the optimal savings level $s_t^\ast$ satisfies

(8)$${u}^{\prime}( {w_t-s_t^\ast } ) = R_{t + 1}\beta ^\ast {u}^{\prime}( {R_{t + 1}s_t^\ast } ).$$

Myopia implies people place less weight on the future (β < β*), and therefore, care relatively more about current consumption—such agents save too little, i.e., $s_t < s_t^\ast.$ Indeed, ∂s _t/∂β > 0 holds implying as β falls (the agent is more myopic), the less he saves, and the gap between his choice and true saving (s _t vs. $s_t^\ast )$ increases.Footnote ¹⁵ It is important to note that, in spite of ∂s _t/∂β > 0, sufficiently high myopia (low-enough but still β > 0) will not drive agents to the zero-saving corner; since b = 0, and the agent earns nothing when old, Inada conditions will prevent that. For future use, note that when b > 0, sufficiently high myopia will drive agents to the own zero-saving corner.

4.1 PAYG

The government is aware that a change in pension benefits affects private saving via changes in the agent's after-tax endowment and his future income. Focus attention on a steady state. The government takes the agent's optimal saving response to its pension into account, s(b) and mandates a pension b by maximizing Ω*(b) = u(w − s(b) − b) + β*u(Rs(b) + b). The optimal level of b, if positive, is defined as the solution to

(9)$$\displaystyle{{d\Omega ^\ast ( b ) } \over {db}} = {-}{u}^{\prime}( {c^y} ) \left[{1 + \displaystyle{{\partial s( b ) } \over {\partial b}}} \right] + \beta ^\ast {u}^{\prime}( {c^o} ) \left[{R\displaystyle{{\partial s( b ) } \over {\partial b}} + 1} \right] = 0, \;$$

Using u′(c ^y) = βRu′(c ^o), dΩ*(b)/db reduces to

(10)$$\displaystyle{{d\Omega ^\ast ( b ) } \over {db}} = {u}^{\prime}( {c^o} ) \left\{{\beta^\ast{-}\beta R + R\displaystyle{{\partial s( b ) } \over {\partial b}}( {\beta^\ast{-}\beta } ) } \right\}$$

How does private saving respond to policy action? For the PAYG scheme, we find, in general,

(11)$$\displaystyle{{\partial s_t} \over {\partial b}} = {-}\displaystyle{{{u}^{\prime \prime}( {c_t^y } ) + \beta R{u}^{\prime \prime}( {c_{t + 1}^o } ) } \over {{u}^{\prime \prime}( {c_t^y } ) + \beta R^2{u}^{\prime \prime}( {c_{t + 1}^o } ) }}\in [ {-1, \;0} ) {\rm \;for\;}s_t > 0$$

leading to

(12)$$\displaystyle{{\partial c_{t + 1}^o } \over {\partial b}} = [ {1-R} ] \displaystyle{{{u}^{\prime \prime}( {c_t^y } ) } \over {{u}^{\prime \prime}( {c_t^y } ) + \beta R^2{u}^{\prime \prime}( {c_{t + 1}^o } ) }} < 0\,{\rm for}\,s_t > 0\,{\rm given}\,R > 1.$$

The PAYG pension is designed to supplement an agent's own saving for retirement. Recognizing that, the agent cuts his own saving as forced saving via the pension increases. If the forced pension and his voluntary saving earned the same return, he would cut his own saving one-for-one in response to an increase in the pension. However, under dynamic efficiency, an extra unit devoted to the pension brings less future income than what private saving would have. As such, he does not reduce his own saving one-for-one; the crowding out—cf. equation (11)—is less than proportionate with the pension increase. Additionally, the present value of lifetime income under the pension, w + ((1 − R)/R)b falls as b rises (since R > 1), the agent's retired consumption falls, cf. equation (12).

Focus attention on equation (10). In the absence of myopia (β = β*), the second term on the r.h.s. of (10) drops out and hence the sign of dΩ*(b)/db is the same as the sign of 1 − R.

In the absence of myopia, then, the optimal PAYG pension is b = 0. The agent dislikes the fact that his total retirement income, given by Rs(b) + b, falls with a rise in b.Footnote ¹⁶ This is clearly not what the government intended. Thankfully, the fall in Rs + b stops once s hits zero. This is so because of the crowding out of private savings (∂s _t/∂b ≤ 0 for s _t > 0); at a sufficiently high level of b, call it $\underline b$, the corresponding level of private saving is zero, s _t = 0. Thereafter, any further raising of b ($b \ge \underline b$) has no effect on s as the non-negativity constraint on s binds—total retirement income is simply b which clearly rises with b!

The question is, how does the presence of myopia help reinstate a role for PAYG pensions? Notice when myopia is absent, the second term on the r.h.s. of (10) drops out implying ∂s(b)/∂b ceases to have any effect on the choice of b. Intuitively, the envelope theorem washes out the effect of b on s. Not so, when myopia is present. In that case, the choice self-views the effect of b on s differently from how the true self does—the true self discounts the effect on future saving at rate β* greater than the rate at which the choice self-discounts the same.

What about the first term on the r.h.s. of (10)? Since (11) tells us that ∂s _t/∂b < 0 for s _t > 0, it follows, that in the presence of myopia, the second term on the r.h.s. of (10) is negative. This means, for dΩ*(b)/db > 0 or PAYG pensions to have a shot at improving true welfare, the first term on the r.h.s. of (10), β* − Rβ, necessarily has to be positive. Equivalently, a necessary condition for a welfare rationale for PAYG pensions is sufficiently-strong myopia, β* > Rβ ⇔ β < β*/R—ordinary myopia, β < β*, is not enough!Footnote ¹⁷ Why? This is for the true self to benefit from the pension, the myopic agent's perceived effective return on private saving, Rβ, must be at least less than his true self's perceived return on the competing PAYG scheme, β*. (In the absence of myopia, this is not possible under dynamic efficiency.) Otherwise, even the true self would prefer no pensions.

Even when myopia is sufficiently strong, how big does b need to be? Recall $s_t < s_t^\ast$—from the standpoint of true utility, a myopic agent is anyway saving too little. From (11), we know the agent cuts s _t in response to the pension when s _t > 0. A PAYG pension crowds out own saving which the true self dislikes, but as b rises beyond $\underline b$, consumption during retirement rises and that makes such a b attractive from the perspective of true utility. Knowing the true self likes $b > \underline b$, what level of b should the government choose? In the present setting with exogenous factor prices, there is no inherent dynamics in the economy. In which case, the pension level may be set, right away, at its long-run optimal value, the one that solves max_b u(w − b) + β*u(b) where

(13)$${u}^{\prime}( {w-b^\ast } ) \equiv \beta ^\ast {u}^{\prime}( {b^\ast } ).$$

Note that b* does not replicate s* (defined in equation (8)). We have

Lemma 1 [Andersen and Bhattacharya (Reference Andersen and Bhattacharya2011)] A necessary condition for the PAYG pension b* to improve true welfare is β* > βR. For CES utility, a sufficient condition for true welfare to increase, i.e., Ω*(b*) > Ω*(0), is

(14)$$\left({\displaystyle{{R + R\beta^{{\ast} 1/\phi }} \over {R + {( {R\beta } ) }^{1/\phi }}}} \right)^\phi > \displaystyle{{R + ( \beta ^\ast{/}\beta ) {( {R\beta } ) }^{1/\phi }} \over {R + {( {R\beta } ) }^{1/\phi }}}.$$

Proof. See Appendix B. ■

For it to be optimal, the PAYG pension has to be large enough to drive voluntary private saving to the corner. Increasing the PAYG beyond that point makes it possible to increase old-age consumption, and thus, counteract the effect of the myopia. However, since the PAYG scheme is return-dominated, myopia (β*/β > 1) alone is not enough to deliver a welfare rationale for a PAYG pension. Sufficiently strong myopia relative to the rate of return (β*/β > R) is required for a welfare improvement to be possible.

Note from (14) that risk-aversion (or intertemporal substitution) plays a role: if ϕ = 1 or ϕ = 0, then Ω*(b*) = Ω*(0) and in either case, b* does not improve welfare. In Appendix B, we present a numerical example of a configuration of parameters satisfying β* > βR that also satisfies (14) for all ϕ ∈ (0, 1).

4.1.1 Fully funded pensions

Consider, next, a mandated FF pension scheme with contribution rate d (τ _t = d and b _t+1 = Rd). The FF-pension also crowds out voluntary saving, and since the returns are the same, the crowding out is one-to-one, i.e., analogous to equation (11), we have

$$\displaystyle{{\partial s} \over {\partial d}} = {-}1{\rm \;for\;}s > 0.$$

Hence, a mandated FF pension contribution only affects total saving if it is sufficiently large, $d \ge \underline d$. The critical contribution level $\underline d$ is defined by ${u}^{\prime}( w-\underline d ) \equiv R\beta {u}^{\prime}( R\underline d )$. The contribution rate maximizing long-run true welfare is determined by (assuming voluntary saving is driven to zero, $d \ge \underline d$) max_d u(w − d) + β*u(Rd) and the optimal level d* is determined by the first-order condition

(15)$${u}^{\prime}( {w-d^\ast } ) \equiv R\beta ^\ast {u}^{\prime}( {Rd^\ast } ).$$

Notice d* = s*. This means a FF program with contribution equal to s* can exactly replicate the desired retirement saving of the true self. Of course, private saving is zero but retirement saving under the FF scheme is exactly what true utility demands. It follows directly that

Lemma 2 A FF pension with contribution rate d*—determined by (15)—generates higher true steady-state utility when compared either to what is possible under laissez faire (τ = b = 0) or an optimal PAYG pension (b*).

Note, the relationship between the optimal PAYG pension, b*, and the FF pension, d*, is, in general, ambiguous $(b^*\displaystyle{ > \over < }d^*)$.Footnote ¹⁸

4.2 Transition from a PAYG to a FF pension system

A PAYG pension has the advantage that it provides, up-front, the current old with a pension, and therefore offers an immediate remedy to their low old-age consumption problem. To that end, suppose the government introduces the long-run optimal PAYG pension at level b*, cf. (13). From Lemma 2, we know that continuing this program is not in the interest of long-run welfare: an optimal FF scheme would do better. The question is: is it possible to make the transition to a FF scheme under the Pareto criterion, the constraint that utility for each cohort remain at least as high had the PAYG pension scheme b* persisted?

To operationalize this question, assume that the long-run optimal PAYG scheme (b*) installed at t and kept in place up to date t + m (m > 0). Recall, this is consistent with every cohort up to t + m being at the zero private-saving corner. Also recall for b* to be welfare-improving, agents must display sufficiently strong myopia, i.e., β*/β > R. At t + m, the government ushers in a FF scheme by mandating the then young to contribute d _t+m > 0 to the scheme. (It is possible that private saving re-emerges, we denote it s(b*, d _t+m)); recall, though, any increase in d is crowded out one-for-one by a decline in s.) Denote the PAYG pension to be received by these retirees by b _t+m+1, i.e., we allow for the possibility that the level of the PAYG scheme is changed after the FF scheme is introduced. True life-time utility for the cohort born at t + m under this policy package is

$$\eqalign{\Omega _{t + m}^\ast{ = } & u( {w-b^\ast{-}d_{t + m}-s( {b^\ast , \;d_{t + m}} ) } ) \cr & + \beta ^\ast u( {Rs( {b^\ast , \;d_{t + m}} ) + Rd_{t + m} + b_{t + m + 1}} ) , \;}$$

which may be rewritten as

$$\eqalign{\Omega _{t + m}^\ast{ = } & u( {w-b^\ast{-}d_{t + m}-s( {b^\ast , \;d_{t + m}} ) } ) \cr & + \beta ^\ast u( {Rs( {b^\ast , \;d_{t + m}} ) + ( {R-1} ) d_{t + m} + d_{t + m} + b_{t + m + 1}} ) .} }}}$$

Note that the term (R − 1)d _t+m captures the return gain to switching from the PAYG to the FF scheme. To foreshadow, this extra income/welfare will be crucial for a successful transition under the Pareto criterion.

The first issue is whether the cohort born at t + m sees a welfare gain from the mandate of a FF pension contribution, d _t+m, on top of their PAYG pension contribution b*?

Lemma 3 True life-time utility of cohort t + m, $\Omega _{t + m}^\ast$, can be improved by mandating them to contribute at the margin to a FF scheme in addition to their PAYG contribution of b*:

$$\left. {\displaystyle{{\partial \Omega_{t + m}^\ast } \over {\partial d_{t + m}}}} \right\vert _{d_{t + m} = 0, b_{t + m + 1} = b^\ast } = [ {R-1} ] \beta ^\ast {u}^{\prime}( {b^\ast } ) > 0\,{\rm given}\,R > 1$$

Proof. See Appendix C. ■

Note that with the PAYG pension at b*, private saving is already at zero; hence adding on an incremental FF pension does not distort saving: ∂s(b*, d _t+m)/∂d _t+m = 0. The marginal unit earns R via the FF scheme which is better than what it would have earned under the PAYG scheme. Hence, on the margin, continuing the initial PAYG scheme and adding a (small) mandated FF pension makes the inaugural set of agents better off.

The welfare gain may be used to phase out the PAYG scheme under the Pareto criterion. Define

$$\Omega _{t + m}^{{\ast} {\rm PAYG}} \equiv u( {w-b^\ast } ) + \beta ^\ast u( {b^\ast } )$$

as the lifetime utility to cohort t + m had the PAYG pension b* continued unchanged. It follows, that for cohort t + m, the Pareto condition is $\Omega _{t + m}^\ast \ge \Omega _{t + m}^{{\ast} {\rm PAYG}}$ where

$$\Omega _{t + m}^\ast{\equiv} u( {w-b^\ast{-}d_{t + m}} ) + \beta ^\ast u( {Rd_{t + m} + b_{t + m + 1}} )$$

where $\Omega _{t + m}^\ast$ captures the changes ushered in by adding a mandated FF contribution, d _t+m, over and above the PAYG contribution of b* as well as the possibility that this cohort will see a PAYG benefit of b _t+m+1 instead of b*. Lemma 3 implies $\exists d_{t + m} > 0$ such that $\Omega _{t + m}^\ast \ge \Omega _{t + m}^{{\ast} {\rm PAYG}}$ holds. Then, there exists a b _t+m+1 < b* such that

(16)$$u( {w-b^\ast{-}d_{t + m}} ) + \beta ^\ast u( {Rd_{t + m} + b_{t + m + 1}} ) = u( {y-b^\ast } ) + \beta ^\ast u( {b^\ast } ) \equiv \Omega _{t + m}^{{\ast} {\rm PAYG}} $$

Hence, the gains from phasing in the FF pension (d _t+m > 0) may be used, under the Pareto criterion, to bring down the level of the return-dominated PAYG pension (b _t+m+1 < b*).

The upshot is that PAYG and mandated FF schemes are both appropriate government interventions for myopic agents. The FF scheme is just better in the long run: a unit of funds taken from the PAYG contribution and shifted to the FF scheme produces a return gain. Yes, in a literal sense, it is true that a transition from a PAYG to a FF scheme requires some cohorts to “pay twice” but, unlike in the classical results (see Proposition 1) derived with time-consistent agents, such cohorts are not worse off.

What happens to generations further down the transition, indexed t + m + j with j > 0? What does the trajectory of d and b look like under the Pareto criterion? Consider, for the sake of argument, a very simple, stylized scheme that sets d _t+m = κb*, κ ∈ (0, 1) so that

$$\eqalign{d_{t + m} + b^\ast = & \; ( {1 + \kappa } ) b^\ast \cr d_{t + m + j} + b_{t + m + j} = & \; ( {1 + \kappa } ) b^\ast {\rm \semicolon \;\;}\;j > 0}$$

i.e., right from the start of the transition, the overall contribution rate (summed across both pensions) is raised relative to the PAYG world. (Many other such schemes can be constructed—see below.) This implies, for example, d _t+m+1 = (1 + κ)b* − b _t+m+1, that is, if we generate a declining sequence for b _t+m+1, we automatically generate an increasing sequence for d _t+m+1—if the PAYG is phased out, the FF is phased in. The equal utility condition for period t + m, $\Omega _{t + m}^\ast{ = } \Omega _{t + m}^{{\ast} {\rm PAYG}}$ now reads

$$u( {w-( {1 + \kappa } ) b^\ast } ) + \beta ^\ast u( {R\kappa b^\ast{ + } b_{t + m + 1}} ) = u( {w-b^\ast } ) + \beta ^\ast u( {b^\ast } ) \semicolon \;$$

apropos Lemma 3, there exists a κ > 0 ensuring b _t+m+1 < b*, the start if the declining sequence for b. Below, we show it is possible to engineer a transition which leaves every cohort along the transition at least as well off as in the counterfactual persistent PAYG scheme. This is our flagship result.

Proposition 2 For an economy with an existing PAYG pension b*, there exists a policy package $\big \{ {b_{t + m + j}, \;d_{t + m + j}} \big \} _{j = 0} ^\infty$ implemented at t + m which satisfies the Pareto condition

(17)$$\eqalign{\Omega _{t + m + j}^{{\ast} {\rm TRANSITION}} \equiv & u( {w-b_{t + m + j}-d_{t + m + j}} ) + \beta ^\ast u ( {Rd_{t + m + j} + b_{t + m + j + 1}} ) \ge \Omega _{t + m + j}^{{\ast} {\rm PAYG}} \; \cr \equiv & u( {w-b^\ast } ) + \beta ^\ast u ( {b^\ast } ) > {\rm for}\,{\rm all}\,j \ge 0} $$

with b _t+m+j following a decreasing sequence and reaching 0 in finite time, and d _t+m+j following an increasing sequence converging to d*, allowing d* to be implemented.

Proof. See Appendix D. ■

Along the transition path, the PAYG-pension is gradually phased out, and the FF pension expanded, ensuring cohorts have the same life-time true utility had the PAYG-world persisted. Eventually, the PAYG pension is fully phased out (b _t+m+j = 0), and replaced by a FF pension (d _t+m+j > 0 for all j ≥ 0). Once that happens, all cohorts are necessarily better off than under the PAYG scheme. Notably, it is possible to implement d* which delivers the optimal level of saving, s*, from the point of view of the true self.

Of course, there may be multiple Pareto-improving transition paths. Proposition 2 outlines a particular path where utility for cohorts is kept at the PAYG level until the pension is fully phased out, after which, subsequent future cohorts get to enjoy higher utility. Other paths, where some of the future gains are distributed up-front such that all cohorts are strictly better off, are possible.

5.1 Equilibrium

Henceforth, we assume factor markets are perfectly competitive, and thus, factors of production are paid their marginal product in each period, i.e.,

(18)$$R_{t + 1} = {\,f}^{\prime}( {k_{t + 1}} ) \equiv R( {k_{t + 1}} ) $$

and

(19)$$w_t = f( {k_t} ) -k_t{\,f}^{\prime}( {k_t} ) \equiv \omega ( {k_t} ) ,$$

where f′(k _t) > 0 and f′′(k _t) < 0 and, hence R′(k _t) = f′′(k _t) < 0 and ω′(k _t) = −k _tf′′(k _t) = −k _tR′(k _t) > 0.

In several places below, we will assume

Assumption 4

$$\eta ( k ) \equiv \displaystyle{{{R}^{\prime}( k ) k} \over {R( k ) }} > -1$$

which holds for a standard production function such as the Cobb–Douglas, $f( k ) = k^\alpha , \;$ α ∈ (0, 1) where η(k) = α − 1. It also holds for a CES production function f(k) = [αk ^(1−σ)/σ + (1 − α)]^σ/(σ−1) with α ∈ (0, 1) and σ ≥ 1.

In passing, note that

$$\displaystyle{{d( {kR( k ) } ) } \over {dk}} = R( k ) + k{R}^{\prime}( k ) > 0\Leftrightarrow \displaystyle{{k{R}^{\prime}( k ) } \over {R( k ) }} > -1\Leftrightarrow \eta ( k ) > -1.$$

This means if k rises, capital income (R(k)*k) also rises if Assumption 4 holds. This fact will be useful in Propositions 5 and 6.

Throughout, a dynamically-efficient economy is assumed.

Assumption 5

$$R( {k_t} ) > 1\;\;\forall t$$

The economy without any government intervention—laissez faire—is identical to that studied in Diamond (Reference Diamond1965). We have from (4), and using (18) and (19),

$${u}^{\prime}( {\omega ( {k_t} ) -k_{t + 1}} ) = R( {k_{t + 1}} ) \beta {u}^{\prime}( {R( {k_{t + 1}} ) k_{t + 1}} )$$

which implicitly defines the equilibrium law of motion for k : k _t+1 = ψ(k _t, β, 0). In the case of a time-invariant PAYG scheme (τ _t = b _t+1 = b > 0), the equilibrium condition in the capital market is k _t+1 = s _t, and hence, we have

(20)$${u}^{\prime}( {\omega ( {k_t} ) -k_{t + 1}-b} ) = R( {k_{t + 1}} ) \beta {u}^{\prime}( {R( {k_{t + 1}} ) k_{t + 1} + b} ) $$

which implicitly defines the equilibrium law of motion for k:

(21)$$k_{t + 1} = \psi ( {k_t, \;\beta , \;b} ).$$

All competitive equilibria with PAYG pensions are characterized by the sequence $\{ {k_{t + 1}} \} _{t = 1}^\infty$ defined by (21) and the government budget constraint. For a FF scheme with contribution rate d (τ _t = d and b _t+1 = Rd), the equilibrium condition in the capital market is k _t+1 = s _t + d _t, and hence, the corresponding equilibrium law of motion for k is

$${u}^{\prime}( {\omega ( {k_t} ) -k_{t + 1}} ) = R( {k_{t + 1}} ) \beta {u}^{\prime}( {R( {k_{t + 1}} ) k_{t + 1}} ).$$

Define

(1) k: the steady state capital–labor ratio in the absence of any government intervention, defined as laissez faire, the solution to k = ψ(k, β, 0).

(2) k ^b: the steady state capital–labor ratio for a given PAYG pension b which solves k ^b = ψ(k ^b, β, b).

(3) k*: the steady state capital–labor ratio in the absence of both myopia and pension which solves k* = ψ(k*, β*, 0).

We make all standard assumptions ensuring the existence of a unique and stable—see Appendix E—steady-state equilibrium, see e.g., De la Croix and Michel (Reference De La Croix and Michel2002):

Assumption 6 (Stability)

(22)$$ {R}^{\prime}( k ) \beta {u}^{\prime}( {c^o} ) + R( k ) \beta {u}^{\prime \prime}( {c^o} ) [ {{R}^{\prime}( k ) k + R( k ) } ] + {u}^{\prime \prime}( {c^y} ) [ {1 + {R}^{\prime}( k ) k} ] < 0 $$

In particular, assume 0 < ψ _k(k ^b, β, b) < 1 (this assumption is necessary for k ^b—see below—to be locally stable).

5.2 PAYG pensions

As before, we start by establishing the impact on private saving (or capital) of myopia and pensions. It is easy to verify (see Appendix E) that

$$\displaystyle{{\partial k_{t + 1}} \over {\partial \beta }} = {-}\displaystyle{{R( {k_{t + 1}} ) {u}^{\prime}( {c_{t + 1}^o } ) } \over {{R}^{\prime}( {k_{t + 1}} ) \beta {u}^{\prime}( {c_{t + 1}^o } ) + R( {k_{t + 1}} ) \beta {u}^{\prime \prime}( {c_{t + 1}^o } ) [ {{R}^{\prime}( {k_{t + 1}} ) k_{t + 1} + R( {k_{t + 1}} ) } ] + {u}^{\prime \prime}( {c_t^y } ) }} > 0$$

implying the bigger the weight (β) agents assign to the future, the larger the saving due to consumption smoothing, and therefore larger the capital–labor ratio at any point in time. Similarly, in Appendix E, we show

(23)$$\hskip-9pt\displaystyle{{\partial k_{t + 1}} \over {\partial b}} = {-}\displaystyle{{{u}^{\prime \prime}( {c_t^y } ) + R( {k_{t + 1}} ) \beta {u}^{\prime \prime}( {c_{t + 1}^o } ) } \over {{R}^{\prime}( {k_{t + 1}} ) \beta {u}^{\prime}( {c_{t + 1}^o } ) + R( {k_{t + 1}} ) \beta {u}^{\prime \prime}( {c_{t + 1}^o } ) [ {{R}^{\prime}( {k_{t + 1}} ) k_{t + 1} + R( {k_{t + 1}} ) } ] + {u}^{\prime \prime}( {c_t^y } ) }} < 0$$

meaning that the PAYG pension crowds out saving and leads to a lower capital–labor ratio. Assume a PAYG scheme is introduced (unanticipated) in period t such that each young pays b > 0 (not too large) to each old in that and all future periods. From equation (23), it follows that upon introduction of the PAYG pension, the capital stock is declining along the equilibrium trajectory and eventually reaches k ^b, defined in section 5.1. Since these results hold in steady state, the next result is immediate.

Lemma 4

$$k^b < k < k^\ast$$

Proof. Since ∂k/∂β > 0 (see Appendix E) and β < β*, k is smaller than the capital stock in a corresponding economy with non-myopic households. Myopia implies agents save too little, and hence, the capital stock is lower. Since ∂k/∂b < 0 (see Appendix E), k ^b is smaller than the capital stock in the absence of a PAYG pensions system, k ^b < k. ■

From Lemma 4, recall k <k* holds, meaning the underlying “undersaving” issue faced by myopic agents persists in the case with endogenous factor prices: myopic agents hold too little capital. Under dynamic efficiency (R(k) > 1), if policy action can incentivize these agents to hold more capital, then steady-state welfare would rise. The problem, as before, is that a higher pension reduces the capital stock. There is, however, one big difference as compared to the case with exogenous factor prices. With endogenous factor prices, as the pension crowds out physical capital, the return to capital would rise (an effect absent earlier), raising the incentive to hold more of it. The equilibrium, therefore, has both voluntary savings and the PAYG pension.

Proposition 3 Suppose Assumptions 1–6 hold. (a) For introduction of a PAYG pension scheme to increase true steady-state welfare over laissez faire, it is necessary that

$$\beta ^\ast > R( k ) \beta$$

holds, (b) it is possible that, upon introduction, true welfare improves both for the inaugural generation and for each and every subsequent cohort, and (c) k* (corresponding to β*) cannot be replicated by a PAYG pension.

Proof. See Appendix F. ■

It is easy to construct numerical examples where (a) voluntary saving (in the form of capital) and public pensions co-exist, and (b) the optimal pension is positive even under dynamic efficiency.

5.3 FF pensions

Under a FF scheme, the decision problem for an individual born in period t is

$$\mathop {\max }\limits_{s_t} \Omega _t = u( {w_t-s_t-d} ) + \beta u( {Rs_t + Rd} )$$

Since s and d have the same return, they are perfect substitutes and only total saving k = s + d matters. It follows that voluntary savings s decreases one-for-one with an increase in d for s > 0. For d so high that s is driven to the zero corner, we have u ^′(ω(k _t) − d) > βR(k _t+1)u ^′(R(k _t+1)d).

The undersaving problem can thus be addressed by a proper choice of the mandated pension contribution. This, however, is a steady-state result, and therefore not of much use in solving the immediate problem for households with inadequate savings.

It is important to note that, in the FF world, even though the entire capital stock is being held by the pension funds, the ownership of these funds lies with the agents (via individual accounts) and not the government.

5.4 Transition to a fully-funded system

Is it possible with endogenous factor prices to make a transition from a PAYG to a FF system under the Pareto criterion so that no cohorts are made worse off along the transition path?

5.4.1 Gain from introducing a FF pension

Let the PAYG pension scheme b be introduced in period t. Is there at some point in time—during adjustment to steady state or in the new steady state—a welfare gain from introducing a FF pension? To answer this question, consider introduction of a mandatory contribution larger than the initial capital stock, d _t+m > k _t+m in period t + m, implying full crowding out of private savings (s _t+m = 0 and k _t+m+1 = d _t+m), where the capital is predetermined at its value at t + m. We have

Proposition 5 At any point in time t + m, m > 0 after the introduction of the PAYG pension scheme, true life-time utility $\Omega _{t + m}^\ast$ can be improved by introducing a FF pension contribution d _t+m > k _t+m, under assumption 4 and

(24)$$\beta ^\ast > \displaystyle{\beta \over {1 + \eta ( k ) }}$$

Proof. See Appendix G. ■

The assumption on η(k _t+m)—see Assumption 4—ensures that the return to capital is not “too sensitive” to the capital stock. Sufficiently-strong myopia (β* > β/(1 + η)) is necessary and sufficient for phasing in of a FF pensions system (d _t+m > 0) to have positive welfare effects on all subsequent cohorts.Footnote ¹⁹ Assuming this holds, it is possible to reduce the PAYG pension while satisfying the Pareto condition, i.e., there exists a b _t+m+1 < b and d _t+m > k _t+m such that

$$\Omega _{t + m}^\ast{\equiv} u( {\omega ( {k_{t + m}} ) -b-d_{t + m}} ) + \beta ^\ast u( {R( {k_{t + m + 1}} ) d_{t + m} + b_{t + m + 1}} ) = \Omega _{t + m}^{{\ast} {\rm PAYG}}.$$

This is the first step in a transition out of the PAYG scheme.

5.4.2 Complete phasing out of PAYG pensions

To work out an explicit case with full phasing out of a PAYG pension under the Pareto criterion, we first analyze an economy that has reached a steady-state with a PAYG pension b > 0 and the associated capital stock, k ^b, and associated life-time utility, Ω*^PAYG.Footnote ²⁰ At t + m > t, an unanticipated announcement is made that a phasing out of the PAYG scheme and a transition to a FF system is underway with a goal of achieving the optimal long-run level of capital d = k*.

We are looking for a policy sequence $\{ {b_{t + m + j}, \;d_{t + m + j}} \} _{j = 0}^\infty$ which satisfies the Pareto condition

$$\eqalign{\Omega _{t + m + j}^{{\ast} {\rm TRANSITION}} \equiv & u( {\omega ( {k_{t + m + j}} ) -b_{t + m + j}-d_{t + m + j}} ) \cr & + \beta ^\ast u( {R( {k_{t + m + j + 1}} ) d_{t + m + j} + b_{t + m + j + 1}} ) \ge \Omega _{t + m + j}^{{\ast} {\rm PAYG}} ( {\,j \ge 0} ) }$$

where k _t+m = k ^b and k _t+m+j+1 = d _t+m+j, implies

$$b_{t + m} = b > b_{t + m + 1} > b_{t + m + 2} > \cdots > \cdots = 0$$

and the introduction of increasing contribution to or phasing in of the FF pensions system:

$$k^b < d_{t + m} < d_{t + m + 1} < d_{t + m + 2} < \cdots < \cdots < d = k^\ast$$

i.e., the pension to the current old (in period t + m) is the pension from the PAYG regime and the current young finance this, where the contribution is at the PAYG steady state level. The current young are also required to contribute to the FF scheme with a contribution d _t+m.

Proposition 5 gives conditions ensuring that cohort t + m are better off when a FF pension is introduced on top of the PAYG pension. Under the Pareto criterion, this welfare gain may be used to bring down the PAYG pension this cohort receives (without hurting them). The next cohort sees an increase in their wage income due to a higher capital stock. That, as well as the reduced contribution to the PAYG pension, enables further increases in FF pension contribution resulting in additional increases in the capital stock and enabling a greater reduction in the PAYG pension they receive. Hence, the first step has more savings and a reduction in the PAYG pension. Downstream the change in savings and thus the capital stock also affects wage. Cohort t + m + 1 will have a higher wage rate because the capital stock has increased (compared to status quo), and this make them better-off. Under the Pareto condition this creates room to decrease the PAYG pension further. Working out this dynamics in detail generates the following result.

Proposition 6 Assume the transition starts from a steady-state equilibrium with a PAYG pension b and associated capital, k ^b. Under the assumptions—see Assumption 4—that η(k ^b) > −1 and β* > β/(1 + η(k ^b)) there exists a trajectory satisfying the Pareto criterion, where the PAYG pension is entirely phased out, and the FF pension expanded so that k* is reached in the long run.

Proof. See Appendix H. ■

The above shows the existence of a transition path assuming that the economy is initially in steady state equilibrium with a PAYG pension b. The result can be considerably generalized.

Proposition 7 Assume that the transition starts from an arbitrary date t with a PAYG pension b _t. Assume

$$\beta ^\ast > \displaystyle{\beta \over {1 + \eta ( {k_{t + j}^b } ) }}\,{\rm for}\,j \ge 0$$

where

$$-1 < \eta ( {k_{t + j}^b } ) \equiv \left. {\displaystyle{{{R}^{\prime}( {k_{t + j}} ) k_{t + j}} \over {R( {k_{t + j}} ) }}} \right\vert _{k_{t + j} = k_{t + j}^b } < 0.$$

Then there exists a trajectory satisfying the Pareto criterion, where the PAYG pension is phased out, and the FF pension is expanded.

Proof. See Appendix I. ■

The bottom line is this. Starting from an initial setting with a PAYG pension in place, it is possible to replace it with another mandated scheme, the FF scheme, which not only preserves (even increases) the benefits of the PAYG in terms of its forced-saving character but also generates a higher return. And along the transition, no one is hurt. Note that once the PAYG scheme is fully phased out, cohorts further downstream are made strictly better off.

5.5 Intuition

We now summarize our intuition about the entire transition. There are many parts, many moving parts, so we approach each one in turn. Suppose a standard Diamond economy is at a laissez faire (LF) steady state with k = k ^LF. Since this was reached under choice preferences and agents are myopic, it is clear that k ^LF < k* where k* is the level of k attainable under true preferences. At any date under LF, all retirees have too low retirement consumption (relative to what they would have had under true preferences) because they saved too little due to their myopia (Figure 2).

Figure 2. Transitions.

Now, suppose the government starts an optimal PAYG scheme, b _t, one derived by maximizing true utility. The scheme, by its very nature, transfers resources from the young to the retired. The young respond by cutting their own saving even further but they end up with more retirement consumption. Their choice utility does not like this but under true preferences, they are made better off. Because saving is increasing in its return, and the PAYG scheme is return-dominated, the latter does not lead to complete crowding out of private saving. Also, the initial old at the point the policy was initiated are made better off because the pension they receive from the initial young is higher than their LF retirement saving. Downstream, all future generations have higher true utility than under the LF.

At some point, the PAYG transition is completed and a new steady state, k ^PAYG is reached. By that time, b _t has converged to its steady state level, b*. To reiterate, this point has higher true utility than at k ^LF, lower personal savings but higher retirement consumption. The retirees are receiving b*. Now, the government initiates a FF scheme, asking the young at that date to not only contribute b* for the current retired but also mandate them to contribute d ₁ into the FF program. Since d and k earn the same return, they are perfect substitutes: a rise in d leads to a one-for-one decrease in k, so that, in fact, k ₁ = d ₁ > k ^PAYG. In other words, d ₁ ensures complete crowding out of personal savings. (Recall, by assumption, they cannot borrow.) In effect, their myopia is rendered impotent. This, recall, is not achieved under the PAYG scheme. It is in this sense that the FF scheme is better at “managing” myopia than the PAYG scheme.

This mandate also does several things. It raises downstream w but lowers downstream R (but even with the reduced R, agents benefit from getting a return R on their contributions vs. 1 in the PAYG scheme). Overall, under the conditions laid out in Proposition 6 (the ones relating to sufficiently strong myopia and low η) there is a welfare gain, and under the Pareto criterion, this can be “taxed away” and a PAYG b < b* can be offered to these young. The transition proceeds with d rising and b falling until a point where b = 0; the PAYG scheme is fully phased out and all that remains are mandated pension contributions, d*. As we have shown in Proposition 6, d* can even replicate k*.

5.6 Numerics

Below, we undertake a short computational analysis to showcase some of the crucial features of the transitional dynamics. The idea is not to conduct a serious calibration exercise but rather to offer some broad brushstrokes and quantitative insights within the confines of our two-period model. The exercise serves two purposes. It allows us to include population growth, and also helps us demonstrate the empirical relevance of the transition we have derived.

Consider a baseline economy where $f( k ) = Ak^\alpha$ and $\Omega _t^\ast{\equiv} ( c_t^y ) ^{1-\phi }/( 1-\phi ) + \beta ^\ast ( ( c_{t + 1}^o ) ^{1-\phi }/( 1-\phi ) )$ and $\Omega _t\equiv ( c_t^y ) ^{1-\phi }/( 1-\phi ) + \beta ( ( c_{t + 1}^o ) ^{1-\phi }/( 1-\phi ) )$. There are five primary parameters to choose, ϕ, β, β*, α, and A. We set them as follows: ϕ = 0.8, β = 0.2, β* = 0.9, α = 0.22, and A = 5. A and α are chosen to deliver a 30-year interest rate of near 2.5 (or an annual real interest rate of around 3%). A discount factor of β = 0.2 implies an annual, one-period discount rate of 6%. We chose a relatively high β* to show that we could implement policies that take the economy close to the Golden rule. The average ratio of public pensions plus old-age cash benefits to GDP across OECD countries in the past three decades has been about 5%. We chose ϕ to come close to that number. In one setting, we allow for population growth 1 + n where n is set to 0.2 (annualized rate of 0.6% close to OECD averages in the past three decades). Below, we present some additional examples.

We report results from three sets of experiments. In the first, the economy is at a steady state with retirees receiving, b*, the optimal PAYG pension. The transition starts with the inaugural young generation being asked to pay b* to the current retirees and contribute d _t to the FF scheme. As explained in the text, we go on to compute the sequence of b _t+j and d _t+j ensuring lifetime utility during the transition is held equal to the lifetime utility at the pre-policy steady state. Once the b _t+j sequence has been fully phased out, we allow the resulting welfare gains to accrue to future generations. The second experiment is the same as just described, except for the transition starts at some date before the steady state under the PAYG scheme has been reached. And the last experiment is like the first except we allow for population growth.

5.6.1 Transitions

In the first experiment, the mandated contribution is introduced and gradually stepped up, see Figure 3a. At first, the PAYG pension can only be reduced marginally under the Pareto condition. Downstream when the gains from having contributed to the FF scheme become larger, the PAYG pension can be reduced more sharply, and eventually fully phased out, and the optimal steady state FF contribution fully phased in.

Figure 3a. Pareto-improving transition starting from and away from a PAYG steady state.

Figure 3b. Transition from a PAYG system to a FF system with population growth.

Figure 3c. Range of myopia satisfying the Pareto condition (24).

Figure 3d. Fast transition.

Figure 3e. Slower transition.

Figure 4. Contribution rates, occupational labor market pensions, and total pension expenditures for public sector and contribution-based pension funds. Note: Contribution rates applies to DI/CO collective agreements. Since 2009 contribution rates have been identical for blue- and white-collar workers. Expenditures: data 2015 onward are projected expenditures. Data source: Ministry of Finance (2017).

Figure 5. (a) LHS and RHS and (b) ϕ*.

From the point where the PAYG pension has been fully phased out, cohorts are strictly better off than in the PAYG steady state, the start of the transition. Implementing the FF pension outside steady state (before the capital stock has converged to the steady state value associated with the PAYG pension b*) is possible but has a longer transition period. FF contributions have to be phased in more gradually, and PAYG pensions phased out more slowly to satisfy the Pareto condition. This is partly because the policy is introduced with retirees receiving the long run PAYG pension b* even though the dynamics under the PAYG pension has not worked itself out fully. Finally, it is seen that population growth makes the transition more slow, see Figure 3b, but still possible. Note that the steady state is different from the case reported in Figure 3a due to the population growth. With population growth, the return difference between the PAYG scheme and the FF scheme is smaller, and this explains why the transition period is longer. Note, in each case, the transition to the FF scheme is completed within four to five generations.

5.6.2 Strength of myopia

Proposition 5 shows that a necessary and sufficient condition (for the phasing out of a PAYG pension system and the phasing in of a FF one) to satisfy the Pareto criterion is the condition (24). For Cobb–Douglas technology, (24) reduces to a simple restriction on parameters:

(25)$$\displaystyle{{\beta ^\ast } \over \beta } > \displaystyle{1 \over \alpha }$$

which may also be viewed as a condition on the strength of myopia needed.

Wang et al. (Reference Wang, Rieger and Hens2016) report estimates of annual discount factors and present bias for 53 countries (see Figure 3 in their paper), where present bias is measured by the value of γ ∈ (0, 1) in the life-time utility: $u( {x_0, \;x_1, \;\ldots , \;x_T} ) = u( {x_0} ) + \gamma \mathop \sum \nolimits^T_{t = 1} \delta ^t u( {x_t} )$ and δ ∈ (0, 1) is a discount factor. Our corresponding measure of myopia (25) isFootnote ²¹

$$\displaystyle{{\beta ^\ast } \over \beta } = \displaystyle{{1 + \gamma ( ( \delta -\delta ^{31}) /( 1-\delta ) ) } \over {\gamma + \gamma ( ( \delta -\delta ^{31}) /( 1-\delta ) ) }}.$$

Wang et al. (Reference Wang, Rieger and Hens2016) report values of γ ranging from approximately 0.01 to 0.99 and δ ranging from approximately 0.78 to 0.9. This implies β*/β ∈ (1, 13.1) in the data. Figure 3c plots 1/α from which it is clear what the minimum β*/β has to be for (25) to be satisfied. For instance, when α ≈ 0.4, (24) is satisfied for β*/β ≈ 2.5 which lies in the range reported in Wang et al. (Reference Wang, Rieger and Hens2016).