Employment incentives and the disaggregated impact on the economy. The Italian case

Abstract

Over the past two decades, the Italian labour market has undergone a number of profound changes. A thorough analysis of these changes shows that there has been a progressive employment polarisation, although with a very peculiar dynamics. While employment did grow in high-skill and low-skill occupations, and it shrank in the medium-skill ones, these changes did not take place simultaneously, as polarisation assumes. Moreover, wage polarisation is hardly observable in the same period. Quite differently, Italy has been characterised by relatively low or even declining returns to education along with progressively decreasing wages in the low-skill segment of the labour market. In this context, we study the potential of an employment incentive policy, for which we imagine two options, one targeting workers in high-skill and the other in low-skill occupations. The objectives of the policy are enhancing aggregate employment and improving working conditions (wages) either in high-skill or low-skill occupations, depending of the option. For the simulation of the two policy options, we employ an integrated model that combines a macro disaggregated and multi-sectoral Computable General Equilibrium (CGE) model with a micro-simulation model. While the CGE model evaluates how the macroeconomic shock reverberates on the labour demand at industry level, the micro-simulation model computes how the changes in macroeconomic variables affect households’ decisions in terms of labour supply and final consumption.

Appendices

Appendix A: Classification of commodities and activities in the SAM

See Tables 13 and 14.

Table 13 Classification of commodities in SAM

Table 14 Classification of activities in SAM

Appendix B: Main aspects of the CGE model specification

In the CGE model, each activity produces homogeneous goods using a nested constant return to scale technology following the nested scheme reported in Fig. 1. In the first nest from the top-left, the price of each commodity derive from the combination of imported and domestically produced commodities as summarised by the following equation:

$$ p_{{TY_{n} }} = \overline{{p_{{TY_{n} }} }} \left[ {\frac{{\overline{{p_{{Y_{n} }} }} \overline{{Y_{n} }} }}{{\overline{{p_{{TY_{n} }} }} \overline{{TY_{n} }} }}\left( {\frac{{p_{{Y_{n} }} }}{{\overline{{p_{{Y_{n} }} }} }}} \right)^{{1 - \sigma_{n}^{M} }} + \frac{{\overline{{p_{{M_{n} }} }} \overline{{M_{n} }} }}{{\overline{{p_{{TY_{n} }} }} \overline{{TY_{n} }} }}\left( {\frac{{p_{{M_{n} }} }}{{\overline{{p_{{M_{n} }} }} }}} \right)^{{1 - \sigma_{n}^{M} }} } \right]^{{\frac{1}{{1 - \sigma_{n}^{M} }}}} $$

$TY_{n}$ is the total output by commodity, $\frac{{\overline{{p_{{Y_{n} }} }} \overline{{Y_{n} }} }}{{\overline{{p_{{TY_{n} }} }} \overline{{TY_{n} }} }}$ is the value share of domestic production on total output, $\frac{{\overline{{p_{{M_{n} }} }} \overline{{M_{n} }} }}{{\overline{{p_{{TY_{n} }} }} \overline{{TY_{n} }} }}$ is the value share of imports on total output, $M_{n}$ is the imports, $\sigma_{n}^{M}$ is the elasticity of substitution by commodity between imports and domestic output, $p_{{TY_{n} }}$ is the price of commodities, $p_{{Y_{n} }}$ is the price of domestic output, $p_{{M_{n} }}$ is the price of imports (fixed in foreign currency).

The price of domestic output by activity can be formalized as:

$$ p_{{Y_{n} }} = \left[ {\frac{{\overline{{p_{{VA_{n} }} }} \overline{{VA_{n} }} }}{{\overline{{p_{{Y_{n} }} }} \overline{{Y_{n} }} }}\left( {\frac{{p_{{VA_{n} }} }}{{\overline{{p_{{VA_{n} }} }} }}} \right)^{1 - \sigma } + \frac{{\overline{{p_{{B_{n} }} }} \overline{{B_{n} }} }}{{\overline{{p_{{Y_{n} }} }} \overline{{Y_{n} }} }}\left( {\frac{{p_{{B_{n} }} }}{{\overline{{p_{{B_{n} }} }} }}} \right)^{1 - \sigma } } \right]^{{\frac{1}{1 - \sigma }}} $$

$VA_{n}$ is the value added generated by activity, $\frac{{\overline{{p_{{VA_{n} }} }} \overline{{VA_{n} }} }}{{\overline{{p_{{Y_{n} }} }} \overline{{Y_{n} }} }}$ is the value share of value added on total output by activity, $\frac{{\overline{{p_{{B_{n} }} }} \overline{{B_{n} }} }}{{\overline{{p_{{Y_{n} }} }} \overline{{Y_{n} }} }}$ is the value share of intermediate consumption on total output by activity, $B_{n}$ is the intermediate consumption, $\sigma = 0$ elasticity of substitution between value added and intermediate consumption, $p_{{B_{n} }}$ is the price of the intermediate consumption aggregate, $p_{{VA_{n} }}$ is the price of value added.

The value added is obtained combining together the costs for primary inputs by activity. Cost functions for each primary factor are presented as the follows:

$$ p_{L} = \overline{{p_{L} }} \left[ {\mathop \sum \limits_{L = 1}^{24} \frac{{\overline{{w_{L} }} \left( {1 + t_{l,L} } \right)\overline{{L_{L} }} }}{{\overline{{p_{L} }} \overline{L} }}\left( {\frac{{w_{L} }}{{\overline{{w_{L} }} \left( {1 + t_{l,L} } \right)}}} \right)^{{1 - \sigma^{L} }} } \right]^{{\frac{1}{{1 - \sigma^{L} }}}} $$

$$ \begin{aligned} p_{VA} &= \overline{{p_{VA} }} \left[ {\frac{{\overline{{p_{L} }} \left( {1 + t_{l,L} } \right)\overline{L} }}{{\overline{{p_{VA} }} \overline{VA} }}\left( {\frac{{p_{L} }}{{\overline{{p_{L} }} \left( {1 + t_{L} } \right)}}} \right)^{{1 - \sigma^{VA} }} + \frac{{\overline{{p_{MI} }} \left( {1 + t_{MI} } \right)\overline{MI} }}{{\overline{{p_{VA} }} \overline{VA} }}\left( {\frac{{p_{MI} }}{{\overline{{p_{MI} }} \left( {1 + t_{MI} } \right)}}} \right)^{{1 - \sigma^{VA} }}}\right. \\&\quad \left.{ + \frac{{\overline{{p_{K} }} \left( {1 + t_{K} } \right)\overline{K} }}{{\overline{{p_{VA} }} \overline{VA} }}\left( {\frac{{p_{K} }}{{\overline{{p_{K} }} \left( {1 + t_{K} } \right)}}} \right)^{{1 - \sigma^{VA} }} } \right]^{{\frac{1}{{1 - \sigma^{VA} }}}} \end{aligned} $$

$L_{L}$ is the labour demand by labour type, $w_{L}$ is the wages before tax by labour type, $p_{MI}$ is the mixed income, $p_{K}$ is the capital compensation, $\sigma^{L}$. is the elasticity of substitution between labour type, elasticity of substitution between labour, capital and mixed income (differentiated by activity), $t_{L}$ is the payroll tax rate by labour type, $t_{MI}$ is the tax rate on mixed income, $t_{K}$ is the capital income tax rate.

From the demand-side specification of the model market demands are the sum of each consumer’s demands. The economic agents maximize their utility function, restricted to the disposable income condition that is represented by net endowments. In the calibrated share form:

$$ U_{h} = \left[ {\sum \theta_{h}^{C} \left( {\frac{{C_{h} }}{{\overline{{C_{h} }} }}} \right)^{{\frac{{1 - \sigma_{{U_{h} }} }}{{\sigma_{{U_{h} }} }}}} + \left( {1 - \theta_{h}^{C} } \right)\left( {\frac{{S_{h} }}{{\overline{{S_{h} }} }}} \right)^{{\frac{{1 - \sigma_{{U_{h} }} }}{{\sigma_{{U_{h} }} }}}} } \right]^{{\frac{{\sigma_{{U_{h} }} }}{{1 - \sigma_{{U_{h} }} }}}} $$

where $\theta_{h}^{C}$ represents the value share of current consumption on income by Institutional Sector. The spending function associated with the utility function of each Institutional Sector is given by:

$$ p_{{U_{h} }} = \overline{{p_{{U_{h} }} }} \left[ {\mathop \sum \limits_{n = 1}^{63} \frac{{\overline{{p_{{TY_{n} }} }} \overline{{C_{n,h} }} }}{{\overline{{p_{{U_{h} }} }} \overline{{U_{h} }} }}\left( {\frac{{p_{{TY_{n} }} }}{{\overline{{p_{{TY_{n} }} }} }}} \right)^{{1 - \sigma_{{U_{h} }} }} + \frac{{\overline{{p_{{S_{h} }} }} \overline{{S_{h} }} }}{{\overline{{p_{{U_{h} }} }} \overline{{p_{{U_{h} }} }} }}\left( {\frac{{p_{{S_{h} }} }}{{\overline{{p_{{S_{h} }} }} }}} \right)^{{1 - \sigma_{{U_{h} }} }} } \right]^{{\frac{1}{{1 - \sigma_{{U_{h} }} }}}} $$

from which getting the demand function for saving by Institutional Sector:

$$ S_{h} = \overline{{S_{h} }} \left( {\frac{{\overline{{p_{S} }} }}{{\overline{{p_{U,h} }} }}\frac{{p_{U,h} }}{{p_{S} }}} \right)^{{\sigma_{{S_{h} }} }} $$

The total consumption of institutional sectors is distributed between different goods Cn according to the CES function:

$$ \frac{{C_{n} }}{{\overline{{C_{n} }} }} = \frac{C}{{\overline{C} }}\left( {\frac{{\overline{{p_{{C_{n} }} }} }}{{\overline{{p_{C} }} }}\frac{{p_{C} }}{{p_{{C_{n} }} }}} \right)^{{\sigma_{C} }} $$

$p_{C}$ is the consumer price index, $\theta_{h}^{C}$ is the benchmark value share of consumption goods, $p_{{C_{n} }}$ is the producer price of good, $t_{{C_{n} }}$ is the consumption tax, $\sigma_{C}$ is the elasticity of substitution in consumption, $C_{n}$ is the consumption of good n, $C$ is the aggregate consumption.

The present model includes an initial “involuntary” unemployment rate and wages that are differentiated by labour type. All workers are supposed to be represented by Unions and a wage per each typology of labour is determined through the negotiation between Firms and Unions. The approach to wage negotiation is modelled as a “right to manage” Nash-bargaining approach in which Union and Firm bargain over wages but the Firm chooses the level of employment to maximize profits by taking the negotiated wage as it is given (Pissarides 1998). Assuming that all workers are members of the Union, we can describe the Labour Union utility function as follow (Pissarides 1998):

$$ U_{LU} = n_{i} \frac{{w_{{L_{i} }}^{1 - \gamma } }}{1 - \gamma } + u_{i} \frac{{b^{1 - \gamma } }}{1 - \gamma }\;{\text{with}}\;i = 24 $$

where ni is the employment rate per each labour type, ui is the unemployment rate, wi is the wage negotiated, b is the unemployment compensation and γ is the parameter that represents the Labour Union risk aversion. We consider that the Union is risk neutral, thus we set γ = 0. We are considering a bilateral monopoly, where the Union chooses the wage and the Firm chooses employment. The bargaining allows determining the wage that can be summarised as (Severini et al. 2019):

$$ w_{{L_{i} }} = \frac{{\varepsilon_{{n,w_{L} }} *u*b}}{{1 + \varepsilon_{{n,w_{L} }} *u}} $$

with εn,wL representing the elasticity of the number of employees to the negotiated wage. The elasticity ε is obtained applying the Shepard’ lemma¹⁸ for which:

$$ \varepsilon_{{n,w_{L} }} = \mathop \sum \limits_{k = 1}^{n} - \sigma_{k} \varGamma_{k} \mathop \prod \limits_{j = 1}^{k - 1} \left( {1 - \varGamma_{j} } \right) $$

where σk is the elasticity of substitution between the input in the Kth production function stage, n is the number of stages in the production function and Γk is the total share of costs attributable to the aggregate not containing labour in the same stage of production function.

Appendix C: The micro-module specification

The first logit regression computes the probability of being employed against the levels of six explanatory variables (gender, region of residence, education level, age, number of household’s components and number of infants in the household). The associated equation is

$$ \Pr \left( {attivabili1 = 1} \right) = \frac{1}{{1 + e^{{ - \beta_{0} - \beta_{1} \cdot gen - \beta_{2} \cdot reg - \beta_{3} \cdot edu - \beta_{4} \cdot age - \beta_{5} \cdot com - \beta_{6} \cdot { \inf }}} }} $$

(1)

where attivabili1 is the dependent variable, which is 0 for workers who cannot increase their hours worked (i.e. full-timers and voluntary part-time/temporary workers) and 1 for workers who can (i.e. involuntary part-time and temporary workers, unemployed and inactive people members of the potential labor force [1]). Furthermore, $gen$ is the gender dummy (1 = MALE, 0 = FEMALE), $reg$ is a discrete variable indicating the region of residence (from 1 to 20), $edu$ is the educational attainment (edu = 1 for LOW; = 2 for MEDIUM; 3 for HIGH), age (from 15 to 75 years old), $com$ is the number of household components (ranging from 0 to 6) and ${ \inf }$ the number of components under three years (ranging from 0 to 5).

These coefficients are used to compute the probability that any single underemployed individual rises his labor supply. Individuals are ordered according to the decreasing probability of improving their working hours, so that new job opportunities will be attributed to workers with higher scores.

gen prob_inv = 1/probability

browse nquest nord classe probabilty attivabili1

*** INCREASE IN EMPLOYMENT

sort classe attivabili1 prob_inv

* increasing in category(classe) and decreasing in probability/increasing in the inverse of the probability

bysort classe attivabili1: gen ordinamento = _n

bysort classe: sum ordinamento

Individuals ordered by the variable ‘ordinamento’ are eligible to increases in employment established according the CGE model results, until the difference between the progressive sum of the activation margin and the CGE margin is null. In this case, the simulation is run only on workers who have some margins to increase their work effort: that is, involuntary part-time and temporary workers and unemployed, and the inactive in the Potential Labor Force (PLF).¹⁹

As for the reduction in employment, a score estimated on the basis of a LOGIT regression (Eq. 2) by using as the dependent variable the variable employed(dummy with 0 for workers without employment and 1 for workers with employment) is applied. The covariates in the LOGIT model are the gender (1 = MALE, 0 = FEMALE), the regions where people lives (‘reg’ from 1 to 20), the educational attainment (edu = 1 for LOW; = 2 for MEDIUM; 3 for HIGH), age (from 15 to 75 years old), the number of components $\#_{component}$ (discrete variable from 0 to 6) and the number of components under three years old $\#_{component < 3}$ (discrete variable from 0 to 5):

$$ PR\left( {employed = 1 |gender,reg, edu, age, \#_{component} ,\#_{components < 3} } \right) = \frac{1}{{1 + { \exp }\left( { - \beta_{0} - \beta_{1} \cdot gender - \beta_{3} \cdot reg - \beta_{4} \cdot edu - \beta_{5} \cdot age - \beta_{6} \cdot \#_{component} - \beta_{7} \cdot \#_{component < 3} } \right)}} . $$

(2)

This probability is used to order workers absorbing the loss in employment in the case of employment. The following exert shows the procedure to be applied in the case of the decrease in employment. We have to order individuals according an increasing probability to be employed (employed1 = 1), so that the loss in job opportunities will be attributed to workers with lower scores. Probability1 is the probability to be employed, whereas prob_inv1 is the inverse of this probability.

*** DECREASE IN EMPLOYMENT

* Decreasing order of labor category (classe) and increasing in probability to be employed

sort classe employed probabilty1

bysort classe employed: gen ordinamento1 = _n if employed ==1

browse nquest nord ordinamento ordinamento1 classe probabilty* attivabili1* employed

gsort - employed -classe + probabilty1

bysort classe: sum ordinamento1

bysort employed: sum probabilty1

Clearly, we could have an integration of the both methods, if there are labor categories with increases and decreases in employment.

The microsimulation model could be used in the final version of the model in a fully integration approach in the following way. Let gross income be:

$$ Y_{h,i} = w_{h,i} \cdot L_{h,i} + \cdots $$

where $w_{h,i}$ is the wage of individual $i$ of household $h$, $L_{i,h}$ is the number of hours worked is labor income, the income of self-employed $Yaut_{h,i}^{0}$, the income from capital $Ycap_{h,i}^{0}$, the pensions $Ypen_{h,i}^{0}$ and the other incomes $Yothers_{h,i}^{0}$.

Hence, disposable income of individual $i$ in household $h$ is

$$ YD_{h,i} = Y_{h,i} \left( {1 - t} \right) + C_{h,i} $$

(2)

where $t$ is the tax rate, which is a function of the income level:

$$ t = t\left( {Y_{h,i} } \right) $$

and $TaxCredit_{h,i}$ is the tax credits.:Changes in wages and employment obtained at the first stage in the CGE simulation can be used to construct the new disposable income:

$$ Y_{h,i}^{D,CGE} = \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{CGE} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} } \right) \cdot $$

$$ \begin{aligned} &\left( {1 - t^{gross} \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{CGE} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} }\right.}\right.\\&\quad \left.{\left.{+ Yothers_{h,i}^{0} - Deduction_{h,i} } \right)} \right) + TaxCredit_{h,i} . \end{aligned} $$

(3)

We can apply the change in the PIT tax rates $t^{gross, PROVISION}$ of the third, fourth and fifth tax bracket needed to increase the net PIT revenues by an amount coherent with the provisions of the cut in employers’ SSCs:

$$ Y_{h,i}^{D,CGE1} = \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{CGE} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} } \right) \cdot $$

$$ \begin{aligned} &\left( {1 - t^{gross, PROVISION} \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{CGE} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0}}\right.}\right.\\&\quad \left.{\left.{+ Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} } \right)} \right) + TaxCredit_{h,i} . \end{aligned} $$

(4)

The household disposable income is computed as:

$$ YD_{h} = \sum\limits_{i} {YD_{h,i} } $$

by household $\mathop \sum \limits_{i = 1}^{imax} Y_{h,i}^{D,CGE1}$ gives us the new consumption patterns by NACE/CPA $C_{h}^{NACE/CPA}$ according to households’ consumption propensity $c_{h}^{0}$ and the share of consumption by NACE/CPA $\frac{{C_{h}^{0,NACE/CPA} }}{{\mathop \sum \nolimits_{NACE/CPA} C_{h}^{0,NACE/CPA} }}$ (under the assumption that shares are unresponsive to changes in disposable income):

$$ C_{h}^{NACE/CPA} = \mathop \sum \limits_{i = 1}^{imax} Y_{h,i}^{D,CGE1} \cdot c_{h}^{0} \cdot \frac{{C_{h}^{0,NACE/CPA} }}{{\mathop \sum \nolimits_{NACE/CPA} C_{h}^{0,NACE/CPA} }}. $$

(5)

The labor supply by individuals is obtained by calculating the disposable income with the wages obtained at the first CGE stage under different assumptions in terms of worked. There are three aspects to stress: (i) for married persons or life partners, decisions are made on the couple basis, whereas it is individual for the other components; (ii) the set of choices of the first earner in a couple (i.e. 33, 35, 37, 39, 41, 43, 45, 48 and 50 weekly worked hours) is more limited than that one of the second earner in the couple and of other components (i.e. 10, 15, 18, 20, 22, 25, 30, 33, 35, 37, 39, 41, 43, 45, 48 and 50 weekly worked hours²⁰) with a matrix of choices of 144 cells for couples and a vector of 9 choices; (iii) the set of hours worked by individuals, as well the matrix of hours worked by the members of the couple, before and after the change in the tax system is established according the principle of the minimization of the percentage loss (in case of an increase in taxation) or of maximization of the percentage gains (in case of decrease in taxation) in terms of disposable income in the simulation scenario compared to the benchmark scenario.For couples, the mechanism can be described as follows. Let gross income be:

$$ YD_{h,i}^{*, CGE, PROVISION} = [(w_{h,i}^{CGE} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} ) \cdot $$

$$ \begin{aligned} & \left( {1 - t^{gross, PROVISION} \cdot \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0}}\right.}\right. \\&\quad \left.{\left.{+ Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} } \right)} \right) + TaxCredit_{h,i} ] \end{aligned} $$

(6)

which calculates the disposable income obtained by applying the CGE level of wages and the new tax scheme needed to give the provision of the manoeuvre under the different values of the hours worked of the both components 1 and 2 of the couple and for individuals;

$$ YD_{h,i}^{*, CGE, FORCE} = [w_{h,i}^{CGE} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} ) \cdot $$

$$ \begin{aligned} &\left( {1 - t^{gross,} \cdot \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0}}\right.}\right. \\&\quad \left.{\left.{+ Yothers_{h,i}^{0} - Deduction_{h,i} } \right)} \right) + TaxCredit_{h,i} ] \end{aligned} $$

(7)

which calculates the hypothetical disposable income obtained by applying the CGE level of wages and the tax scheme in force under the different values of the hours worked of the both components 1 and 2 of the couple and for individuals;

$$ (HW_{h,1}^{*} , HW_{h,2}^{*} ) = argmin\left( {\mathop \sum \limits_{i = 1}^{2} YD_{h,i}^{*, CGE, PROVISION} / \mathop \sum \limits_{i = 1}^{2} YD_{h,i}^{*, CGE, FORCE} \cdot 100 - 100} \right) $$

(8)

which applies the types of disposable income estimated in both the above mentioned equations and calculates the percentage difference between the scenario with provision and that one under the legislation in force cumulatively for the both members of the couple.

As for individuals, the choice mechanism cam be seen as it follows:

$$ HW_{h,i}^{*} = argmin\left( {YD_{h,i}^{*, CGE, PROVISION} /YD_{h,i}^{*, CGE, FORCE} \cdot 100 - 100} \right) $$

(9)

which estimates the percentage difference of hypothetical disposable income for each individual under the hours worked identified in the hour patterns between the scenario with provision and that one under the legislation in force for each individual.

Currently, the microsimulation stage does not interact with the CGE stage with the exception of the change in the labor supply due to the new tax system to assure the provision of the manoeuvre at the CGE level:

$$ YD_{h,i}^{*, 0, PROVISION} = [(w_{h,i}^{0} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} ) \cdot $$

$$ \begin{aligned} &\left( {1 - t^{gross, PROVISION} \cdot \left( {w_{h,i}^{0} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0}}\right.}\right. \\&\quad \left.{\left.{+ Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{h,i} } \right)} \right) + TaxCredit_{h,i} ] \end{aligned} $$

(10)

which calculates the hypothetical disposable income obtained by applying the benchmark level of wages and the new tax scheme needed to give the provision of the manoeuvre under the different values of the hours worked of the both components 1 and 2 of the couple and for individuals;

$$ \begin{aligned} YD_{h,i}^{*, 0, FORCE} &= [w_{h,i}^{0} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} - Deduction_{i} ) \\ &\quad\cdot \left( {1 - t^{gross,} \cdot \left( {w_{h,i}^{CGE} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} }\right.}\right. \\&\quad \left.{\left.{- Deduction_{h,i} } \vphantom{{w_{h,i}^{CGE} \cdot HW_{h,i}^{*} + Yaut_{h,i}^{0} + Ycap_{h,i}^{0} + Ypen_{h,i}^{0} + Yothers_{h,i}^{0} }}\right)} \right) + TaxCredit_{h,i} ] \end{aligned} $$

(11)

which calculates the hypothetical disposable income obtained by applying the benchmark level of wages and the tax scheme in force under the different values of the hours worked of the both components 1 and 2 of the couple and for individuals;

$$ (HW_{h,1}^{*} , HW_{h,2}^{*} ) = argmin\left( {\mathop \sum \limits_{i = 1}^{2} YD_{h,i}^{*, 0, PROVISION} / \mathop \sum \limits_{i = 1}^{2} YD_{h,i}^{*, 0, FORCE} \cdot 100 - 100} \right). $$

(12)

which applies the types of disposable income estimated in both the above mentioned equations and calculates the percentage difference between the scenario with provision and that one under the legislation in force cumulatively for the both members of the couple.

As for individuals, the choice mechanism cam be seen as it follows:

$$ HW_{h,i}^{*} = argmin\left( {YD_{h,i}^{*, 0, PROVISION} /YD_{h,i}^{*, 0, FORCE} \cdot 100 - 100} \right) $$

(13)

which estimates the percentage difference of hypothetical disposable income for each individual under the hours worked identified in the hour patterns between the scenario with provision and that one under the legislation in force for each individual.

We have to stress that the resolution of the maximization problem does follow a continuous approach, but a discrete one a là Van Soest. In particular, we build the whole matrix of the set of hours worked and the type of tax system (without provision and with provision) and choose the combination of hours worked for the both members of the couple and for individuals.

The procedure is illustrated in A.H.O. Van Soest (1995), ‘Structural models of family labor supply’, and in Kornstad and Thoresen (2007), ‘A Discrete Choice Model for Labor Supply and Child Care’.