The Decentralization of Social Assistance and the Rise of Disability Insurance Enrolment

Abstract

Fiscal decentralization, the decentralization of government expenditures to local governments, may enhance public sector efficiency. Vertical externalities, i.e. spillovers between local and central government, could however undo part of this advantage. In this paper we estimate spillovers from Social Assistance (SA), administered by municipalities, towards central government’s Social Security Disability Insurance (SSDI) scheme in The Netherlands. The latter scheme saw rapidly rising enrolment rates after financial responsibility for Social Assistance was transferred from central to local government. We find that the correlation between local SSDI enrolment and the local stock of SA benefits recipients has increased significantly in the years after decentralization. We show that an increased caseload shifting from Social Assistance to Disability Insurance is the only plausible explanation for this change. Our analysis shows that, following the decentralization of Social Assistance, at least one third of the SSDI inflow was diverted from SA. This caseload shifting increased more rapidly in municipalities experiencing deficits on their SA budgets than in municipalities running a surplus.

Appendix: Consistent Estimation of the Model Parameters and Standard Errors

The substitution model can be thought of as consisting of a structural form, Eqs. (1)–(3), and a reduced form, Eqs. (4)–(6). Consistent estimates of the structural form, which include the substitution parameters $\lambda$ and $\mu$, are obtained in two steps. In the first step, the reduced-form parameters $\theta =\{\beta ,\gamma ,\Sigma \}$ of Eqs. (4)–(6) are estimated using a Maximum Likelihood estimator, where $\Sigma$ represents the elements of the covariance matrix of the error terms.⁷ This gives reduced-form estimates ${\hat{\theta }}$, and the Hessian of the negative log-likelihood at the maximum gives the (inverse) covariance matrix ${\hat{\varOmega }}^{-1}$ corresponding to ${\hat{\theta }}$. In the second step, the structural parameters $\theta _0=\{\lambda ,\mu ;\beta _0,\gamma _0,\Sigma _0\}$ of Eqs. (1)–(3) are estimated using a Minimum Distance estimator.⁸

For the second step we first express the reduced-form parameters in terms of the structural-form parameters. Writing out the covariance matrix of the error terms in the reduced-form model [Eqs. (4)–(6)] as:

$$\begin{aligned} \Sigma = \left( \begin{array}{lll} \sigma ^2_{\mathrm {SA}_\mathrm {in}}&{}\rho _{\mathrm {SA}_\mathrm {in}\mathrm {SA}_\mathrm {out}}\sigma _{\mathrm {SA}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {out}}&{}\rho _{\mathrm {SA}_\mathrm {in}\mathrm {SSDI}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {in}}\sigma _{\mathrm {SSDI}_\mathrm {in}}\\ \rho _{\mathrm {SA}_\mathrm {in}\mathrm {SA}_\mathrm {out}}\sigma _{\mathrm {SA}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {out}}&{}\sigma ^2_{\mathrm {SA}_\mathrm {out}}&{}\rho _{\mathrm {SA}_\mathrm {out}\mathrm {SSDI}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {out}}\sigma _{\mathrm {SSDI}_\mathrm {in}}\\ \rho _{\mathrm {SA}_\mathrm {in}\mathrm {SSDI}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {in}}\sigma _{\mathrm {SSDI}_\mathrm {in}}&{}\rho _{\mathrm {SA}_\mathrm {out}\mathrm {SSDI}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {out}}\sigma _{\mathrm {SSDI}_\mathrm {in}}&{}\sigma ^2_{\mathrm {SSDI}_\mathrm {in}}\\ \end{array} \right) \nonumber \\ \end{aligned}$$

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and allowing for a similarly general covariance matrix $\Sigma _0$ of the structural-form model [Eqs. (1)–(3)], we get the following set of equations (recall that $\beta$ is the regression coefficient of the instrumental variable, and $\lambda$ and $\mu$ are the substitution parameters):

$$\begin{aligned} \beta _{\mathrm {SA}_\mathrm {in}}= & {} (1-\lambda )\beta _{\mathrm {sa}_\mathrm {in}}\nonumber \\ \gamma _{\mathrm {SA}_\mathrm {in}}= & {} (1-\lambda )\gamma _{\mathrm {sa}_\mathrm {in}}\nonumber \\ \beta _{\mathrm {SA}_\mathrm {out}}= & {} (1+\mu )\beta _{\mathrm {sa}_\mathrm {out}}\nonumber \\ \gamma _{\mathrm {SA}_\mathrm {out}}= & {} (1+\mu )\gamma _{\mathrm {sa}_\mathrm {out}}\nonumber \\ \beta _{\mathrm {SSDI}_\mathrm {in}}= & {} \lambda \beta _{\mathrm {sa}_\mathrm {in}}+ \mu \beta _{\mathrm {sa}_\mathrm {out}}\nonumber \\ \gamma _{\mathrm {SSDI}_\mathrm {in}}= & {} \gamma _{\mathrm {ssdi}_\mathrm {in}}+ \lambda \gamma _{\mathrm {sa}_\mathrm {in}}+ \mu \gamma _{\mathrm {sa}_\mathrm {out}}\\ \sigma ^2_{\mathrm {SA}_\mathrm {in}}= & {} (1-\lambda )^2\sigma ^2_{\mathrm {sa}_\mathrm {in}}\nonumber \\ \sigma ^2_{\mathrm {SA}_\mathrm {out}}= & {} (1+\mu )\sigma ^2_{\mathrm {sa}_\mathrm {out}}\nonumber \\ \sigma ^2_{\mathrm {SSDI}_\mathrm {in}}= & {} \sigma ^2_{\mathrm {ssdi}_\mathrm {in}}+ \lambda ^2\sigma ^2_{\mathrm {sa}_\mathrm {in}}+ \mu ^2\sigma ^2_{\mathrm {sa}_\mathrm {out}}+ 2\mu \rho _{\mathrm {sa}_\mathrm {out}\mathrm {ssdi}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {out}}\sigma _{\mathrm {ssdi}_\mathrm {in}}\nonumber \\&\quad + 2\lambda \rho _{\mathrm {sa}_\mathrm {in}\mathrm {ssdi}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {in}}\sigma _{\mathrm {ssdi}_\mathrm {in}}+ 2\lambda \mu \rho _{\mathrm {sa}_\mathrm {in}\mathrm {sa}_\mathrm {out}}\sigma _{\mathrm {sa}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {out}}\nonumber \\ \rho _{\mathrm {SA}_\mathrm {in}\mathrm {SA}_\mathrm {out}}\sigma _{\mathrm {SA}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {out}}= & {} (1-\lambda )(1+\mu )\rho _{\mathrm {sa}_\mathrm {in}\mathrm {sa}_\mathrm {out}}\sigma _{\mathrm {sa}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {out}}\nonumber \\ \rho _{\mathrm {SA}_\mathrm {in}\mathrm {SSDI}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {in}}\sigma _{\mathrm {SSDI}_\mathrm {in}}= & {} (1-\lambda )\rho _{\mathrm {sa}_\mathrm {in}\mathrm {ssdi}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {in}}\sigma _{\mathrm {ssdi}_\mathrm {in}}+ \lambda (1-\lambda )\sigma ^2_{\mathrm {sa}_\mathrm {in}}\nonumber \\&\quad + \mu (1-\lambda )\rho _{\mathrm {sa}_\mathrm {in}\mathrm {sa}_\mathrm {out}}\sigma _{\mathrm {sa}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {out}}\nonumber \\ \rho _{\mathrm {SA}_\mathrm {out}\mathrm {SSDI}_\mathrm {in}}\sigma _{\mathrm {SA}_\mathrm {out}}\sigma _{\mathrm {SSDI}_\mathrm {in}}= & {} (1+\mu )\rho _{\mathrm {sa}_\mathrm {out}\mathrm {ssdi}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {out}}\sigma _{\mathrm {ssdi}_\mathrm {in}}+ \mu (1+\mu )\sigma ^2_{\mathrm {sa}_\mathrm {out}}\nonumber \\&\quad + \lambda (1+\mu )\rho _{\mathrm {sa}_\mathrm {in}\mathrm {sa}_\mathrm {out}}\sigma _{\mathrm {sa}_\mathrm {in}}\sigma _{\mathrm {sa}_\mathrm {out}}\nonumber \end{aligned}$$

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These equations form a set of restrictions on the parameters of the reduced-form model in terms of the parameters of the structural model; they follow directly from Eqs. (1)–(3). Denote this set of restrictions by $\theta =g(\theta _0)$. A consistent estimate of the structural-form parameters ${\hat{\theta }}_0$ then obtains from minimizing the distance between the first-step estimate ${\hat{\theta }}$ and $g(\theta _0)$, defined as:

$$\begin{aligned} \varDelta s^2(\theta _0) = \left[ {\hat{\theta }}-g(\theta _0)\right] {\hat{\varOmega }}^{-1} \left[ {\hat{\theta }}-g(\theta _0)\right] ' \end{aligned}$$

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where the metric tensor ${\hat{\varOmega }}^{-1}$ equals the inverse of the covariance matrix of ${\hat{\theta }}$ (see Chamberlain 1984). The corresponding covariance matrix of the structural-form parameters ${\hat{\theta }}_0$ is

$$\begin{aligned} {\hat{\varOmega }}_0 = \left[ {\hat{\varGamma }}'{\hat{\varOmega }}^{-1}{\hat{\varGamma }}\right] ^{-1}, \end{aligned}$$

(13)

where

$$\begin{aligned} {\hat{\varGamma }} = \left[ \frac{\partial g(\theta _0)}{\partial \theta _0} \right] _{\theta ={\hat{\theta }}}. \end{aligned}$$

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