Inefficiency in Childcare Production: Evidence from Italian Microdata

Abstract

The paper provides an empirical analysis of production technology in the childcare sector and offers a comparative analysis of inefficiency between public and private day-care centres. Estimates of multi-output production technology and technical inefficiency are obtained in a stochastic frontier model by using cross-section micro-data from a region of northern Italy over the period 2007/8. We find that production exhibits increasing returns to scale and that separability between inputs and outputs is rejected. The average estimate of technical inefficiency is about 10% and public centres are more inefficient than private centres by 4.1% points.

Appendices

Appendix A

The empirical model is specified by assuming the translog functional form for the input distance function D(x, y). Considering two outputs and three inputs, the resulting translog distance function is

$$\begin{aligned} \log D(x,y)= & {} \alpha _0 + \sum _{h=1}^3 \alpha _h \log x_h + \frac{1}{2} \sum _{h=1}^3 \sum _{k=1}^3\alpha _{hk} \log x_h \log x_k + \sum _{m=1}^2 \beta _m \log y_m \nonumber \\&+ \frac{1}{2} \sum _{m=1}^2 \sum _{n=1}^2\beta _{mn} \log y_m \log y_n + \sum _{h=1}^3 \sum _{m=1}^2\gamma _{hm} \log x_h \log y_m, \end{aligned}$$

(12)

where input and output variables are scaled to have unit means. This normalization is used to interpret first-order parameters as elasticities of D(x, y) evaluated at the variables mean.³³

The properties of distance functions require $\alpha _h$ to be positive and $\beta _m$ to be negative. Moreover, the parameters must satisfy the symmetry restrictions, $\alpha _{hk}= \alpha _{kh}$ for $h,k=1,2,3$ and $\beta _{mn}= \beta _{nm}$ for $m,n=1,2$. Homogeneity of degree one in inputs imposes the following requirements on parameters

$$\begin{aligned} \alpha _1 + \alpha _2 + \alpha _3= & {} 1 \end{aligned}$$

(13)

$$\begin{aligned} {} \alpha _{h1} + \alpha _{h2} + \alpha _{h3}= & {} 0 \qquad {\mathrm {for }} \quad h=1,2,3 \end{aligned}$$

(14)

$$\begin{aligned} \gamma _{1m} + \gamma _{2m} + \gamma _{3m}= & {} 0 \qquad {\mathrm {for }} \quad m=1,2 \end{aligned}$$

(15)

As shown by Lovell et al. (1994), homogeneity restrictions are more easily placed on (12) if one normalizes inputs by $x_3$, which gives $x_3^{-1} D(x,y) = D({{\tilde{x}}}, y),$ where ${{\tilde{x}}}=(x_1/x_3, x_2/x_3, 1)$ is the vector of input ratios. Taking the logs, rearranging terms and adding a stochastic, zero mean and symmetric error, v, yields the empirical model specified by (5) and (6). Once the parameters in (6) are estimated, the remaining parameters in (12) are recovered from symmetry and homogeneity restrictions (13), (14) and (15).

In order to estimate the parameters of the model, further assumptions are introduced. The inefficiency terms $u_i$ are treated as random variables with a half-normal distribution $N^+(0,\sigma ^2_u$), where $\sigma ^2_u$ is the variance of the normal distribution before truncation.³⁴ Moreover, the inefficiency terms are supposed to be independently distributed across observations. The random errors $v_i$ are independently and normally distributed with zero mean and variance $\sigma ^2_v$ and are uncorrelated with explanatory variables.

The empirical model (5) does not suffer from an endogeneity problem, as the regressors (the input ratios) are not correlated with the inefficiency error term. Indeed, notice that the first-order conditions of the cost minimization problem can be written in terms of input ratios as

$$\begin{aligned} \frac{p_h}{p_3} {{\tilde{x}}}_i = \frac{\alpha _h + \sum _{k=1}^2 \alpha _{hk} \log {{\tilde{x}}}_k + \sum _{m=1}^2 \gamma _{hm} \log y_m}{\alpha _3 + \sum _{k=1}^2 \alpha _{3k} \log {{\tilde{x}}}_k + \sum _{m=1}^2 \gamma _{3m} \log y_m} \end{aligned}$$

for $h=1,2$, where $p_h$ and $p_3$ are input prices. The system of two equations can be solved for ${{\tilde{x}}}_1$ and ${{\tilde{x}}}_2$ as functions of only exogenous variables, i.e. input prices $p_1$, $p_2$, $p_3$ and outputs $y_1$ and $y_2$. Hence, input ratios are not affected by inefficiency so that the empirical model (5) does not suffer from endogeneity problems.

The parameters of the half-normal model are estimated by using the Maximum Likelihood (ML) method. The assumptions about $u_i$ and $v_i$ are used to derive the distribution of the composed error term, $\epsilon _i$, and thus the log-likelihood for each observation which is³⁵

$$\begin{aligned} L_i = -\log \left( \frac{1}{2}\right) - \left( \frac{1}{2}\right) \log (\sigma _v^2 + \sigma _u^2 ) + \log \phi \left( \frac{\epsilon _i}{\sqrt{\sigma _v^2 + \sigma _u^2 }}\right) + \log \Phi \left( \frac{\mu _{*i}}{\sigma _*}\right) , \end{aligned}$$

where $\phi$ and $\Phi$ are, respectively, the probability density and the probability distribution functions of the standard normal and

$$\begin{aligned} \mu _{*i}= & {} \frac{-\sigma _u^2\epsilon _i}{\sigma _v^2 + \sigma _u^2 } \end{aligned}$$

(16)

$$\begin{aligned} \sigma _{*}= & {} \frac{\sigma _u^2\sigma _v^2}{\sigma _v^2 + \sigma _u^2 }. \end{aligned}$$

(17)

The ML estimates are obtained by numerical optimization of the sum of the log-likelihood of each observation.

The measure of technical inefficiency for each observation i is computed as the expected value of $u_i$ conditional on the composed error $\epsilon _i$, according to the formula provided by Jondrow et al. (1982)

$$\begin{aligned} E(u_i \mid \epsilon _i) = \frac{\sigma _* \, \phi \left( \frac{\mu _{*i}}{\sigma _*}\right) }{ \Phi \left( \frac{\mu _{*i}}{\sigma _*}\right) } + \mu _{*i} \end{aligned}$$

(18)

where $\mu _{*i}$ and $\sigma _*$ are as in (16) and (17). Similarly the observation-specific technical efficiency, computed as in Battese and Coelli (1995), is given by

$$\begin{aligned} E( e ^{-u_i} \mid \epsilon _i) = \frac{\Phi \left( \frac{\mu _{*i}}{\sigma _*} - \sigma _* \right) }{\Phi \left( \frac{\mu _{*i}}{\sigma _*}\right) }\, e^{- \mu _{*i} + \frac{1}{2} \sigma _*^2} \end{aligned}$$

(19)

In the heteroscedastic half-normal model, the examination of the exogenous determinants of inefficiency is conducted through the analysis of the variables $z_i$ affecting the pre-truncated variance of $u_i$ in (10). The marginal effect of each exogenous variable $z_{ik}$ on the expected value of observation-specific inefficiency, is given by the derivative

$$\begin{aligned} \frac{\partial E(u_i)}{\partial z_{ik}} = \delta _k \sigma _{ui} \phi (0) \end{aligned}$$

(20)

Since $\phi (0)>0$, the sign of the marginal effect is the same as the sign of the $\delta _k$ coefficient.

Finally, a local measure of economies of scale can be computed by using the first-order estimated parameters of D(x, y). In fact, as shown in Färe and Primont (1995) (pp. 39–40), the elasticity of scale at (x, y) can be written in terms of elasticities of the distance function as

$$\begin{aligned} \eta (x,y) = - \frac{1}{\varepsilon _{D,y_1} (x,y) + \varepsilon _{D,y_2}(x,y)} \end{aligned}$$

where

$$\begin{aligned} \varepsilon _{D,y_m} (x,y) = \frac{\partial \log D(x,y)}{\partial \log y_m}, \qquad m=1,2 \end{aligned}$$

As easily seen from (12), $\varepsilon _{D,y_m} (x,y)$ evaluated at the variables mean, i.e. at $(x,y) = (1,1,1,1,1)$, is equal to the first-order coefficient of $\log y_m$, so that the local value of the elasticity of scale is given by

$$\begin{aligned} \eta = - \frac{1}{\beta _1 + \beta _2}. \end{aligned}$$

Appendix B

See Tables 5, 6, 7, 8 and 9

Table 5 Sample composition by attribute and by type of provider, private or public

Table 6 Average characteristics of the service by type of provider, public or private

Table 7 OLS—dependent variable nledu

Table 8 HN—dependent variable nledu

Table 9 HNH—dependent variable nledu

See Figs. 3 and 4.

Fig. 3

Distribution of technical efficiency in the HN model

Fig. 4

Distribution of technical inefficiency in the HNH model