Pension reform and labor market incentives

Abstract

This paper investigates how parametric reform in a pay-as-you-go pension system with a tax–benefit link affects retirement and work incentives of prime-age workers. We find that postponed retirement tends to harm incentives of prime-age workers in the presence of a tax–benefit link, thereby creating a policy trade-off in stimulating aggregate labor supply. We show how several popular reform scenarios are geared either towards young or old workers or, indeed, both groups under appropriate conditions. We characterize the excess burden of pension insurance and show how it depends on the supply elasticities of both decision margins and the effective tax rates.

Appendix

1.1 A Comparative statics

In part A of the Appendix, we calculate the log-linearized versions of the retirement condition Eq. 12 and the PAYG pension budget Eq. 19. The resulting expressions take into account, subject to pension earnings Eq. 14, the intensive labor supply decision Eq. 11, and solve—in terms of percentage changes relative to an initial equilibrium—for the equilibrium retirement date x and flat pension b. They serve as the basis for our comparative statics analysis in Sections 2, 3, and 4.

To begin, we note that pension earnings are a complex function of the parameters of the PAYG system: $p\left( x,b;\tau ,m_{0},\alpha \right) \!=\!m\left( x;m_{0},\alpha \right) \!\tau z\!\left( x;\tau ,m_{0},\alpha \right)\!+\!b$. Obviously, $p^{\prime }\equiv \partial p/\partial x$ is independent of the flat pension b. We derive how the relative change $\hat{\tau}_{R}\equiv d\tau _{R}/\left( 1-\tau _{R}\right)$ of the participation tax rate depends on changes in retirement behavior, x, and pension parameters τ, m ₀, α, and b. The effective tax rate $\tau _{R}\equiv \tau +p-(1-x)p^{\prime }$ is defined in Eq. 12. Defining the elasticity $\varepsilon \equiv \tau _{R}^{\prime }x/\left( 1-\tau _{R}\right)$, where $\tau _{R}^{\prime }$ is given in Eq. 16, and noting that pension parameters affect the participation tax rate by their impact on p and p ^′, we obtain:

$$ \hat{\tau}_{R}=\varepsilon \cdot \hat{x}+\frac{db}{1-\tau _{R}}+\frac{ \partial \tau _{R}}{\partial \tau }\frac{d\tau }{1-\tau _{R}}+\frac{\partial \tau _{R}}{\partial m_{0}}\frac{dm_{0}}{1-\tau _{R}}+\frac{\partial \tau _{R} }{\partial \alpha }\frac{d\alpha }{1-\tau _{R}}. $$

(49)

The derivatives of τ _R will be derived for specific policy scenarios in part B of the Appendix. Substituting Eq. 49 into the retirement response noted in Eq. 13, we derive, after rearranging, the following equation for the impact on retirement in terms of parametric shifts and the change db in the endogenous level of flat pensions:

$$ \hat{x}=-\frac{\eta }{1+\eta \varepsilon }\frac{1}{1-\tau _{R}}\left[ db+ \frac{\partial \tau _{R}}{\partial \tau }d\tau +\frac{\partial \tau _{R}}{ \partial m_{0}}dm_{0}+\frac{\partial \tau _{R}}{\partial \alpha }d\alpha +\left( 1-\tau _{R}\right) \hat{\gamma}\right]. $$

(50)

This equation corresponds to the retirement locus in Figs. 1 and 2. It is downward sloping, since a higher flat pension induces, holding τ,m ₀,α constant, earlier retirement.

The other constraint that pins down the equilibrium is the condition for budget balance in Eq. 19: $\tau \cdot \left( 1+x\right) =\left( 1-x\right) p$. Taking the differential of revenues and spending yields:

$$ \left( 1+x\right) d\tau +\tau dx=\left( 1-x\right) \left[ p^{\prime }dx+ \frac{\partial p}{\partial \tau }d\tau +\frac{\partial p}{\partial m_{0}} dm_{0}+\frac{\partial p}{\partial \alpha }d\alpha +db\right] -pdx. $$

(51)

Using the fact that $\tau _{R}=\tau +p-\left( 1-x\right) p^{\prime }$, we solve for db in terms of $\hat{x}$ and the shifts in the pension parameters:

$$ db=\frac{\tau _{R}x}{1-x}\cdot \hat{x}+\left[ \frac{1+x}{1-x}-\frac{\partial p}{\partial \tau }\right] \cdot d\tau -\frac{\partial p}{\partial m_{0}} \cdot dm_{0}-\frac{\partial p}{\partial \alpha }\cdot d\alpha . $$

(52)

This equation corresponds to the PAYG budget locus in Figs. 1 and 2. It is upward sloping since an increase in retirement age relaxes the pension budget and allows for a larger flat pension as long as the participation tax rate τ _R is positive. This is intuitive, since the participation tax measures the net fiscal loss to households and, thus, the net gain to the system, if retirement is marginally postponed. The tax rate τ _R captures the extra tax paid plus the pension earnings foregone minus the increase in pensions over the remaining life-time 1 − x, corresponding to the number of pensioners in the cross-section of the population.

The solution of Eqs. 50 and 52 determines the reduced-form, equilibrium expressions for the retirement response and the size of the flat, lump-sum pension payments in terms of the changes in the system parameters $\left( \tau ,m_{0},\alpha \right)$ and the preference parameter γ. This solution yields, in turn, the reactions of the other variables of interest, e.g., the response of intensive labor supply $\hat{L}$ of young workers, the latter due, as discussed above in Eqs. 10 and 11, to the impact of the participation decision $\hat{x}$ on the implicit tax $\hat{\tau}_{L}$. We can also infer the impact on the participation tax $\hat{\tau}_{R}$ of the old, which yields the welfare change according to Eq. 27.

1.2 B Effects on the participation tax rate

In Appendix B, we derive the responses of the participation tax to changes in, respectively, the contribution rate, the tax–benefit link, and the degree of actuarial fairness. Combined with Eqs. 50 and 52, the resulting partial derivatives determine the equilibrium shifts in old-age participation and flat pensions for the policy reforms considered in Sections 3 and 4. All the subsequent partial derivatives are calculated for a given retirement date.

Statutory tax rate

Using $\tau _{R}=\tau +p-\left( 1-x\right) p^{\prime }$ from Eq. 12, we compute first the effect of an increase in the contribution rate:

$$ \frac{\partial \tau _{R}}{\partial \tau }=1+\frac{\partial p}{\partial \tau } -\left( 1-x\right) \frac{\partial p^{\prime }}{\partial \tau }. $$

(53)

The impact on earnings-linked pensions depends on the reaction of the assessment base, z = x + L, which, in turn, is driven by first-period labor supply in Eq. 9. Using τ _L as given in Eq. 6 and holding x constant, we find that a higher contribution rate discourages intensive labor supply and thereby erodes the assessment base:

$$ \tau \frac{\partial z}{\partial \tau }=\tau \frac{\partial L}{\partial \tau _{L}}\frac{\partial \tau _{L}}{\partial \tau }=-\frac{\tau _{L}}{1-\tau _{L}} \cdot \sigma L<0. $$

(54)

In calculating the effect on pensions p = mτ z + b, we note that the conversion factor $m=\frac{\alpha }{1-x}+m_{0}$ and its derivative $m^{\prime }=\frac{\alpha }{\left( 1-x\right) ^{2}}=\frac{m-m_{0}}{1-x}$ are independent of τ. A higher contribution rate thus affects the pension level and the pension increment $p^{\prime }=\tau \cdot \left[ zm^{\prime }+mz^{\prime }\right]$ that is offered if retirement is marginally postponed:

$$ \frac{\partial p}{\partial \tau }=m\left[ z+\tau \frac{\partial z}{\partial \tau }\right] ,\quad \frac{\partial p^{\prime }}{\partial \tau }=m\left[ z^{\prime }+\tau \frac{\partial z^{\prime }}{\partial \tau }\right] +m^{\prime }\left[ z+\tau \frac{\partial z}{\partial \tau }\right] . $$

(55)

The term z ^′ = 1 + L ^′ = 1 − Ψ, with $\Psi \equiv \frac{\sigma L}{1-\tau _{L}}\cdot \frac{m_{0}\tau }{R}$, follows from Eq. 11.²⁸ Assuming a fixed wage elasticity of labor supply σ, we obtain:

$$ \tau \cdot \frac{dz^{\prime }}{d\tau }=-\Psi \cdot \left[ 1+\left( 1-\sigma \right) \frac{\tau _{L}}{1-\tau _{L}}\right] . $$

(56)

Using the relationships $\left( 1-x\right) m^{\prime }=m-m_{0}=\alpha /\left( 1-x\right)$ and substituting the relevant derivatives into Eq. 53, we find:

$$ \begin{array}{cl} \displaystyle\frac{\partial \tau _{R}}{\partial \tau }= & 1-\alpha +m_{0}\cdot \left[ z-\left( 1-x\right) -\displaystyle\frac{\tau _{L}}{1-\tau _{L}}\sigma L\right] \\[12pt] & +\ \left( 1-x\right) m\cdot \Psi \left( 2+\left( 1-\sigma \right) \displaystyle\frac{ \tau _{L}}{1-\tau _{L}}\right) . \end{array} $$

(57)

From this general expression, we deduce several cases: full actuarial fairness: α = 1, m ₀ = Ψ = 0, (with b = 0) and hence $\partial \tau _{R}/\partial \tau =0$. While the participation tax rate is zero in this case, there remains a positive implicit tax on young workers, $\tau _{L}=\tau \cdot \left[ 1-1/R\right]$, which is smaller than the statutory rate because PAYG contributions earn no interest. The other extreme case is no tax–benefit link, α = m ₀ = 0, so that $\partial \tau _{R}/\partial \tau =1$.

The case with a fixed conversion factor independent of retirement behavior, m = m ₀ and α = 0, yields an intermediate case. The square bracket in Eq. 57 can safely be assumed positive, at least if the labor supply elasticity is not too large. In our simple model, the worker-retiree ratio is $\left( 1+x\right) /\left( 1-x\right)$, which exceeds unity in a realistic setting. If, instead, considering the effective number of workers, L + x, and realistically assuming that hours worked of young and older workers are not too different, i.e., L close to 1, we also have $z=L+x>\left( 1-x\right)$. Therefore, the first two terms in the square bracket are clearly positive. A natural assumption, which is actually stronger than required, is that the erosion of the assessment base will not be so large as to exceed the net effect of the first two terms in the square bracket.

It will also be instructive to consider the case of fixed first-period labor supply, given by σ = Ψ = 0, which again leads to an increase in the participation tax rate if the statutory tax rate is raised, $\frac{\partial \tau _{R}}{\partial \tau }=1-\alpha +m_{0}\left[ z-\left( 1-x\right) \right] >0$, $\alpha \in \left[ 0,1\right]$. By continuity, the total effect on $\frac{\partial \tau _{R}}{\partial \tau }$ remains positive at least for small values of σ. In any case, the influence of L is likely to be small, given the econometric evidence on the labor supply response of young workers.

Tax–benefit link

Consider the effect of a tighter tax–benefit link $m=\alpha /\left( 1-x\right) +m_{0}$, through a rise in m ₀, starting from m ₀ = 0. The parameter $\alpha \in \left[ 0,1\right]$ can take arbitrary values, with α = 0 being one special case. The partial effect on $\tau _{R}=\tau +p-\left( 1-x\right) p^{\prime }$ is:

$$ \frac{\partial \tau _{R}}{\partial m_{0}}=\frac{\partial p}{\partial m_{0}} -\left( 1-x\right) \frac{\partial p^{\prime }}{\partial m_{0}}.$$

(58)

Using τ _L as given in Eq. 6, and holding x constant, we find that a tax–benefit link encourages intensive labor supply and thereby expands the assessment base z = x + L:

$$ m\frac{\partial z}{\partial m_{0}}=m\frac{\partial L}{\partial \tau _{L}} \frac{\partial \tau _{L}}{\partial m_{0}}=\frac{\tau -\tau _{L}}{1-\tau _{L}} \cdot \sigma L>0.$$

(59)

Raising the conversion factor m ₀ affects the pension level, p = τmz + b , and the pension increment $p^{\prime }=\tau \left[ mz^{\prime }+m^{\prime }z\right]$ in Eq. 15 by:

$$ \frac{\partial p}{\partial m_{0}}=\tau \left[ z+m\frac{\partial z}{\partial m_{0}}\right] ,\quad \frac{\partial p^{\prime }}{\partial m_{0}}=\tau \left[ z^{\prime }+m\frac{\partial z^{\prime }}{\partial m_{0}}+m^{\prime }\frac{ \partial z}{\partial m_{0}}\right] .$$

(60)

Since we evaluate the policy change starting from m ₀ = 0 in the initial equilibrium, the marginal effect of later retirement on the assessment base is unity, z ^′ = 1 + L ^′ = 1. Given m ₀ = 0 initially, the term $\left( 1-x\right) m=\alpha$ and the effective tax rate $\tau _{L}=\tau \left[ 1-\left( 1-x\right) m/R\right]$ remain constant, and therefore, first-period labor supply, is independent of retirement age. Consequently, we obtain:

$$ \frac{\partial p^{\prime }}{\partial m_{0}}=\tau \left[ 1+m^{\prime }\frac{ \partial z}{\partial m_{0}}\right] .$$

(61)

Combining Eqs. 58–61 and noting that m ₀ = 0 implies $\left( 1-x\right) m^{\prime }=m$, we find:

$$ \frac{\partial \tau _{R}}{\partial m_{0}}=\tau \left[ z-\left( 1-x\right) \right] >0.$$

(62)

Thus, a strengthening of the tax–benefit link (from an initial value of m ₀ = 0) results in a partial effect on the participation tax rate corresponding to $\partial \tau _{R}/\partial m_{0}=\tau \left[ z-\left( 1-x\right) \right] >0$, where the term in square brackets can safely assumed, as before, to be positive.

More actuarial fairness

Raising the parameter α not only introduces a tighter tax–benefit link, but also makes it fairer. Again, we assume m ₀ = 0 initially. The partial impact on the participation tax rate, $\tau _{R}=\tau +p-\left( 1-x\right) p^{\prime }$, is:

$$ \frac{\partial \tau _{R}}{\partial \alpha }=\frac{\partial p}{\partial \alpha }-\left( 1-x\right) \frac{\partial p^{\prime }}{\partial \alpha }.$$

(63)

Using τ _L in Eq. 6, and holding x constant, we find the pension base z = x + L grows by:

$$ \frac{\partial z}{\partial \alpha }=\frac{\partial L}{\partial \tau _{L}} \frac{\partial \tau _{L}}{\partial \alpha }=\frac{\sigma L}{1-\tau _{L}} \cdot \frac{\tau }{R}>0.$$

(64)

The conversion factor changes by $\partial m/\partial \alpha =1/\left( 1-x\right)$ and $\partial m^{\prime }/\partial \alpha =1/\left( 1-x\right) ^{2}$. The tax–benefit link thus affects the pension level p = mτ z + b and the pension increment $p^{\prime }=\tau \left[ m^{\prime }z+mz^{\prime } \right]$ in Eq. 15 according to:

$$ \frac{\partial p}{\partial \alpha }=\tau \left[ \frac{z}{1-x}+m\frac{ \partial z}{\partial \alpha }\right] ,\quad \frac{\partial p^{\prime }}{ \partial \alpha }=\tau \left[ \frac{z}{\left( 1-x\right) ^{2}}+m^{\prime } \frac{\partial z}{\partial \alpha }+\frac{1}{1-x}\right] .$$

(65)

Our assumption of m ₀ = 0 initially implies that $\left( 1-x\right) m=\alpha$ does not vary with x. Later retirement thus expands the assessment base z = L + x by z ^′ = 1, with $\partial z^{\prime }/\partial \alpha =0$. Combining Eqs. 63–65 yields, upon using $m=\left( 1-x\right) m^{\prime }$:

$$ \frac{\partial \tau _{R}}{\partial \alpha }=\frac{\partial p}{\partial \alpha }-\left( 1-x\right) \frac{\partial p^{\prime }}{\partial \alpha } =-\tau .$$

(66)

Consequently, introducing more actuarially fairness reduces the participation tax rate.

1.3 C Preferences

To relate our assumptions on preferences in Section 2.1 to the existing literature, we consider the general life-cycle problem in which we only impose time separability, as is usual in macroeconomics. To be concise, we abstract from the pension system:

$$ V=\max_{c_{1},l_{1},l_{2},x}u\left( c_{1},l_{1}\right) +\beta u\left( c_{2},l_{2},x\right) \quad s.t.\quad c_{2}=\left( w_{1}l_{1}-c_{1}\right) R+w_{2}l_{2}x. $$

(67)

The modified notation refers to wages w _t , intensive labor supply l _t , t = 1, 2, and the subjective discount factor β (≡ 1/R in the main text). Defining partial derivatives as $u_{c}^{1}\equiv \partial u\left( c_{1},l_{1}\right) /\partial c_{1}$, $u_{l}^{1}\equiv \partial u\left( c_{1},l_{1}\right) /\partial l_{1}$, $u_{c}^{2}\equiv \partial u\left( c_{2},l_{2},x\right) /\partial c_{2}$, $u_{l}^{2}\equiv \partial u\left( c_{2},l_{2},x\right) /\partial l_{2}$, and $u_{x}^{2}\equiv du\left( c_{2},l_{2},x\right) /dx$, the resulting optimality conditions are:

$$ \frac{u_{c}^{1}}{\beta u_{c}^{2}}=R,\quad -\frac{u_{l}^{1}}{u_{c}^{1}} =w_{1},\quad -\frac{u_{l}^{2}}{u_{c}^{2}}=xw_{2},\quad \frac{u_{l}^{1}}{ \beta u_{x}^{2}}=\frac{w_{1}R}{w_{2}l_{2}}. $$

(68)

These conditions express the usual tangency conditions, where the marginal rates of substitution are equal to relative prices. For example, the last condition in Eq. 68 refers to the trade-off between the old-age participation rate (retirement date) x and first-period labor supply l ₁ . The marginal rate of substitution measures by how much old-age participation must decline to compensate for a marginal increase in first-period labor supply, $\left. dx/dl_{1}\right\vert _{dV=0}=-u_{l}^{1}/\left( \beta u_{x}^{2}\right)$. The right hand side gives the marginal rate of transformation, i.e., the reduction of old-age participation in exchange for a marginal increase in first-period labor supply such that life-time income is unchanged.

Income effects substantially complicate the comparative static analysis of Eq. 68. To focus on incentive effects, a large part of the literature imposes separability between consumption and labor market activities in each period. Two recent examples are Cigno (2008) and Fenge and Pestieau (2005). Assuming fixed labor supply during the economically active part of the second period, l ₂ = 1, preferences specialize to:

$$ V=\max_{c_{1},l_{1},x}u\left( c_{1}-\varphi \left( l_{1}\right) \right) +\beta u\left( c_{2}-\gamma \phi \left( x\right) \right) . $$

(69)

The analysis in Cigno (2008) corresponds to this case, except that he suppresses second period participation and, instead, analyzes individual decisions in the presence of credit constraints. In the absence of the latter, the savings condition is $u_{c}^{1}=\beta Ru_{c}^{2}$. Noting $u_{l}^{1}=-u_{c}^{1}\varphi ^{\prime }\left( l_{1}\right)$ and $u_{x}^{2}=-u_{c}^{2}\gamma \phi ^{\prime }\left( x\right)$, optimal labor market activities are given by $w_{1}=\varphi ^{\prime }\left( l_{1}\right)$ and $w_{2}=\gamma \phi ^{\prime }\left( x\right)$. These conditions are identical to Eqs. 6 and 12 in the main text, where the pension system determines consumer prices w ₁ = 1 − τ _L and w ₂ = 1 − τ _R and, thereby, the intertemporal trade-off in labor market behavior. In particular, the trade-off between first and second period labor supply still corresponds to the last condition in Eq. 68, except that the marginal rate of substitution reduces to:

$$ \left. dx/dl_{1}\right\vert _{dV=0}=-\frac{R\varphi ^{\prime }\left( l_{1}\right) }{\gamma \phi ^{\prime }\left( x\right) }. $$

(70)

To reiterate, since this paper focusses on labor market behavior rather than savings, we further specialize preferences in Eq. 2 to be linear in consumption. Given $u_{c}^{t}=1$, t = 1, 2, the savings condition above requires 1 = βR. Aside from fixing the interest rate, this simplification leads to no further restrictions on labor market behavior.

Given factor prices, the impact of the pension system on l ₁, x and s follows, in general, from the comparative statics of Eq. 68, where the third condition is omitted if l ₂ = 1 is fixed. As long as income effects are ‘sufficiently small’ compared to substitution effects, our qualitative results are not affected.