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Subsidy Policy and Elderly Labor

Abstract

In economically developed countries, aging societies with fewer children are progressing. Increased longevity has necessitated postponement of the working retirement age. Our paper presents an examination of how subsidies for an elderly labor supply and a decrease in the population growth rate affect the elderly labor supply and intergenerational wage inequality between younger people and elderly people. Our paper presents the derivation that this subsidy raises the elderly labor supply. Based on the contribution rate for pension and the elasticity of substitution between young labor and elderly labor, the effects of a decrease in population growth on the elderly labor supply are ascertained. Moreover, the study described in this paper derives the positive subsidy rate for elderly labor to maximize social welfare.

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Fig. 1

Data: OECD Statistics

Notes

  1. 1.

    Data: OECD Data “Labour force participation rate”. In the OECD average, the labor participation of 25–54 years was 80.5% in 2001, rising to 81.59% in 2015. The labor participation of 55–64-year-olds was 50.45% in 2001, rising to 61.08% in 2015.

  2. 2.

    We consider labor time \(1-l_{t+1} \) as retirement timing. If labor time \(1-l_{t+1} \) is small, then we consider that retirement year old is earlier. This paper assumes that the younger people provide labor supply inelastically. Therefore, in this model, the effect of an income taxation for young labor income is the same as lump-sum taxation because of the lack of distortion. Considering that the younger people supply labor inelastically as the regular employees in the real world, this assumption is appropriate.

  3. 3.

    If we consider the case of \(K=1\) as a simple economic model, the result obtained by our paper does not change substantially. However, our paper shows to obtain the result for any \(K>0.\)

  4. 4.

    With \(\rho =1\), we obtain the production function \(Y_t =K_t^\theta \left( {\gamma L_t +\delta L_{t-1} } \right) ^{1-\theta }\) . This production function is considered as a perfectly substitutive form. Then, the wage rates of young workers and old workers are constant in a small open economy. With \(\rho =0\), we obtain the product function \(Y_t =K_t^\theta L_t^{\gamma \left( {1-\theta } \right) } L_{t-1}^{\left( {1-\gamma } \right) \left( {1-\theta } \right) } \). Then, the wage rates of young workers and old workers depend on the labor supply of older people and the population growth rate, as shown by (11) and (12). Our paper presents consideration of different cases including a substitutive case \(\left( {0<\rho <1} \right) \) and a complementary case \(\left( {-\infty<\rho <0} \right) \), except for perfect substitution and perfect complementarity. The elasticity of substitution between \(L_t \) and \(L_{t-1} \) is given as \(\frac{1}{1-\rho }\). With \(\rho <1\), we consider the case of \(0<\frac{1}{1-\rho }<\infty \).

  5. 5.

    This paper does not present consideration of the human capital accumulation. Lin (1998) sets the human capital accumulation model and considers the intergenerational difference of the human capital accumulation between two generations. However, Lin (1998) does not consider the labor supply by the older people and the people of the first generation have no human capital. Our paper can consider the human capital accumulation. However, even if we consider the human capital accumulation, the results obtained by this paper do not change because human capital accumulation brings about the only effect of an increase in the effective labor supply. Our paper presents consideration of \({\gamma }\) and \(1-{\gamma }\), respectively, as the productivity of younger people and older people.

  6. 6.

    Then, we obtain \(L_t =N_t \) because younger people provide labor inelastically. However, older people provide \(1-l_t \) of labor time. We obtain \(L_{t-1} =\left( {1-l_t } \right) N_{t-1} \).

  7. 7.

    As the other taxation, we consider lump-sum taxation. However, because of the inelastic labor supply of young labor in this model, the wage income proportional is substantially equal to the lump-sum taxation. Our paper assumes that the pension benefit and the employment subsidy are financed by the wage income of younger people. In the real economy, pension benefits and employment subsidies are financed mainly by the wage income of younger people.

  8. 8.

    We can consider the government budget constraint including (14) and (15). However, an increase in the subsidy for labor reduces the allocations for pension benefit. This additional effect complicates the analysis of this paper. Then, we consider the separate government budget constraints.

  9. 9.

    See Appendix for a detailed proof.

  10. 10.

    Although footnote 4 explains that the production function (8) includes the substitutive case \((0<\rho <1)\), we consider that the production function (8) has mutual complementarity because the marginal productivity of the labor input of younger people can be pulled up by an increase in the labor input of older people, vice versa.

  11. 11.

    This result can be obtained by the log-utility function. Proposition 1 might not be obtained because of the income effect by which the taxation for the subsidy decreases the labor supply and the substitution effect that the subsidy raises the labor supply if we consider constant elasticity of substitution form (CES utility function). In CES utility function, the total effect on the labor supply of the older people is ambiguous.

  12. 12.

    See Appendix for a detailed proof.

  13. 13.

    See Appendix for a detailed proof.

  14. 14.

    See Appendix for a detailed proof of socially optimal allocations and socially optimal policies.

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Correspondence to Masaya Yasuoka.

Additional information

We would like to thank Shin Imoto, Koji Kitaura, Naomi Kodama, Kazutoshi Miyazawa, Keisuke Osumi, seminar participants of the 73rd Annual Congress of the International Institute of Public Finance and the journal editor and anonymous referees for helpful comments. Research for this paper was supported financially by JSPS KAKENHI Grant numbers 26380253, 17K03791, and 17K03746. Nevertheless, any remaining errors are the authors’ responsibility.

Appendices

Appendix

Derivation of (18) \(\frac{dl_{t+1} }{dl_t }\)

With \(\tau =\delta =\sigma =0\), (16) changes to

$$\begin{aligned} l_{t+1} =\left( {1+r} \right) \left( {1-\alpha -\beta } \right) \left( {\frac{w_{1t} }{w_{2t+1} }+\frac{1}{1+r}} \right) , \end{aligned}$$
(A.1)

From total differentiation of (A.1) with respect to \(l_{t+1} \) and \(\frac{w_{1t} }{w_{2t+1} }\) at the steady state, we obtain the following:

$$\begin{aligned} dl_{t+1} =\left( {1+r} \right) \left( {1-\alpha -\beta } \right) d\frac{w_{1t} }{w_{2t+1} }. \end{aligned}$$
(A.2)

\(\frac{w_{1t} }{w_{2t+1} }\) is given by \(\frac{w_{1t} }{w_{2t+1} }=\frac{\gamma }{1-\gamma }\left( {\frac{\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l_t }{n}} \right) ^{\rho }}{\gamma \left( {\frac{n}{1-l_{t+1} }} \right) ^{\rho }+1-\gamma }} \right) ^{\frac{1}{\rho }-1}\). Then, \(d\frac{w_{1t} }{w_{2t+1} }\) are shown as presented below:

$$\begin{aligned} d\frac{w_{1t} }{w_{2t+1} }= & {} -\frac{\gamma \left( {1-\rho } \right) }{n\left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l}{n}} \right) ^{\rho }} \right) }dl_t \nonumber \\&-\frac{\gamma ^{2}\left( {1-\rho } \right) \left( {\frac{n}{1-l}} \right) ^{\rho -1}}{\left( {1-\gamma } \right) \left( {1-l} \right) \left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l}{n}} \right) ^{\rho }} \right) }dl_{t+1} . \end{aligned}$$
(A.3)

Substituting (A.3) into (A.2), we obtain \(\frac{dl_{t+1} }{dl_t }\) as shown by (18).

Derivation of (21) \(\frac{dl}{dn}\)

With \(\sigma =0\), (19) is given as

$$\begin{aligned} l=\left( {1+r} \right) \left( {1-\alpha -\beta } \right) \left( {\frac{\gamma \left( {1-\tau +\frac{n\tau }{1+r}} \right) }{1-\gamma }\left( {\frac{1-l}{n}} \right) ^{1-\rho }+\frac{1}{1+r}} \right) . \end{aligned}$$
(B.1)

From total differentiation of (B.1) with respect to l and n at the steady state, we obtain the following:

$$\begin{aligned} dl= & {} \left( {1+r} \right) \left( {1-\alpha -\beta } \right) \left( \frac{\gamma \left( {1-\rho } \right) \left( {1-\tau +\frac{n\tau }{1+r}} \right) }{1-\gamma }\left( {\frac{1-l}{n}} \right) ^{-\rho }\right. \nonumber \\&\times \left. \left( {-\frac{1}{n}dl-\frac{1-l}{n^{2}}dn} \right) +\frac{\tau \gamma }{\left( {1+r} \right) \left( {1-\gamma } \right) }\left( {\frac{1-l}{n}} \right) ^{1-\rho }dn \right) . \end{aligned}$$
(B.2)

Then we obtain (21).

Derivation of (23) \(\frac{dl}{d\tau }\)

Total differentiation of (B.1) with respect to l and \(\tau \) at the steady state yields the following:

$$\begin{aligned}&\left( {1+\frac{\left( {1-\rho } \right) \gamma \left( {1+r} \right) \left( {1-\alpha -\beta } \right) }{\left( {1-\gamma } \right) n}\left( {\frac{1-l}{n}} \right) ^{-\rho }\left( {1-\tau +\frac{n\tau }{1+r}} \right) } \right) dl\nonumber \\&\quad =\frac{\gamma \left( {1-\alpha -\beta } \right) \left( {n-\left( {1+r} \right) } \right) }{1-\gamma }\left( {\frac{1-l}{n}} \right) ^{1-\rho }d\tau . \end{aligned}$$
(C.1)

Then we obtain (23).

Social Welfare Maximization Problem

The resource constraint in t period is given by the following equation as

$$\begin{aligned} Y_t +D_{t+1} -\left( {1+r} \right) D_t =C_{1t} +C_{2t} +K_{t+1} , \end{aligned}$$
(D.1)

where \(D_t \) denotes aggregate borrowing from foreign countries. \(C_{1t} \) and \(C_{2t}\), respectively, denote the aggregate consumption of younger people and older people. Dividing with \(L_t \) for (D.1) and notifying \(\frac{K_{t+1} }{L_t }=nk\left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l_{t+1} }{n}} \right) ^{\rho }} \right) ^{\frac{1}{\rho }}\) and (25), we obtain the resource constraint per capita shown by (26).

The first order conditions of Lagrange equation (27) are shown by the following equations, as

$$\begin{aligned} \frac{\partial L}{\partial c_{1t} }= & {} 0 \quad \quad \lambda _t =\frac{\alpha }{c_{1t} }, \end{aligned}$$
(D.2)
$$\begin{aligned} \frac{\partial L}{\partial c_{2t} }= & {} 0 \quad \quad \frac{\beta }{c_{2t} }=\frac{\lambda _t }{n},\end{aligned}$$
(D.3)
$$\begin{aligned} \frac{\partial L}{\partial c_{1t+1} }= & {} 0 \quad \quad \lambda _{t+1} =\frac{\alpha \eta }{c_{1t+1} }, \end{aligned}$$
(D.4)
$$\begin{aligned} \frac{\partial L}{\partial d_{1t+1} }= & {} 0 \quad \quad \lambda _t n=\lambda _{t+1} \left( {1+r} \right) , \end{aligned}$$
(D.5)
$$\begin{aligned} \frac{\partial L}{\partial c_{2t+1} }= & {} 0 \quad \quad \frac{\eta \beta }{c_{2t+1} }=\frac{\lambda _{t+1} }{n}, \end{aligned}$$
(D.6)
$$\begin{aligned} \frac{\partial L}{\partial l_{t+1} }= & {} 0 \quad \quad \begin{array}{l}\frac{\eta \left( {1-\alpha -\beta } \right) }{l_{t+1} }=\left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l_{t+1} }{n}} \right) ^{\rho }} \right) ^{\frac{1}{\rho }-1}\left( {\frac{1-l_{t+1} }{n}} \right) ^{\rho -1}\frac{\left( {1-\gamma } \right) \lambda _{t+1} k^{\theta }}{n} \\ -\lambda _t \left( {1-\gamma } \right) k\left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l_{t+1} }{n}} \right) ^{\rho }} \right) ^{\frac{1}{\rho }-1}\left( {\frac{1-l_{t+1} }{n}} \right) ^{\rho -1}.\end{array}\nonumber \\ \end{aligned}$$
(D.7)

Considering (D.2)–(D.7), we obtain the social optimal allocations (28)–(31).

Substituting (5) and (7) into (31) at the steady state, we obtain the following equation as

$$\begin{aligned} 1+\sigma= & {} \frac{\left( {1+r} \right) \left( {1-\gamma } \right) }{\eta w_2 }\left( {\frac{k^{\theta }}{1+r}-k} \right) \nonumber \\&\left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l}{n}} \right) ^{\rho }} \right) ^{\frac{1}{\rho }-1}\left( {\frac{1-l}{n}} \right) ^{\rho -1}, \end{aligned}$$
(D.8)

Considering (9), (11), and (13), we obtain the optimal subsidy rate for elderly employment \(\sigma ^{*}\) given as (32).

The resource constraint (26) at the steady state can be shown by the following equation because of (28) as

$$\begin{aligned} k\left( {\frac{1+r}{\theta }-n} \right) \left( {\gamma +\left( {1-\gamma } \right) \left( {\frac{1-l}{n}} \right) ^{\rho }} \right) ^{\frac{1}{\rho }}=\frac{\alpha +\beta }{\alpha }c_1 . \end{aligned}$$
(D.9)

Considering (5), (7), and (12), we obtain the household allocation of \(\frac{c_1 }{l}\) at the steady state as

$$\begin{aligned} \frac{c_1 }{l}=\frac{\alpha \left( {1+\sigma } \right) \left( {1-\gamma } \right) \left( {1-\theta } \right) k^{\theta }}{\left( {1-\alpha -\beta } \right) \left( {1+r} \right) }\left( {\gamma \left( {\frac{n}{1-l}} \right) ^{\rho }+\left( {1-\gamma } \right) } \right) ^{\frac{1}{\rho }-1}. \end{aligned}$$
(D.10)

Substituting (D.10) and \(\sigma =\sigma ^{*}\) into (D.9), we obtain the socially optimal leisure time \(l^{*}\) to hold (33).

Substituting \(l=l^{*}\), \(\sigma =\sigma ^{*}\), (14) and (15) into (7), we obtain the following equation at the steady state:

$$\begin{aligned} l^{*}= & {} \frac{\left( {1-\alpha -\beta } \right) \left( {1+r} \right) }{1+\sigma ^{*}}\nonumber \\&\left( {\left( {1-\tau -\frac{\sigma ^{*}\left( {1-l^{*}} \right) }{n}+\frac{n\tau }{1+r}} \right) \frac{w_1 }{w_2 }+\frac{1}{1+r}} \right) . \end{aligned}$$
(D.11)

Considering (13) and solving for \(\tau \), the optimal income tax rate for pension benefit \(\tau ^{*}\) is obtainable from (34).

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Miyake, Y., Yasuoka, M. Subsidy Policy and Elderly Labor. Ital Econ J 4, 331–347 (2018). https://doi.org/10.1007/s40797-017-0067-x

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Keywords

  • Aging society
  • Elderly labor
  • Subsidy

JEL Classifications

  • J14
  • J26
  • H20
  • H55