The Italian productivity slowdown in a Real Business Cycle perspective

Abstract

This paper investigates the structural relation between the Italian weak macroeconomic performances and the productivity decline experienced over the last 15 years, estimating a Dynamic Stochastic General Equilibrium model. Modifying Ireland and Schuh’s (Rev Econ Dyn 11:473–492, 2008) two-sector RBC model in order to account for cointegration between consumption and investment, we interpret the unsatisfactory Italian economic dynamics in the light of a permanent negative shock to the component of productivity which is common across the consumption-good and the investment-good sector. In the light of our results, the common view that the Italian productivity problems involve only the Made in Italy sectors is only partially confirmed, since growth in the investment-good sector relies on the counterbalancing properties of its transitory sector-specific productivity component. Moreover, the model indirectly stresses the importance of the intermediate-good productions in the observed productivity decline. The short- and long-run implications of productivity dynamics for consumption, investment and hours worked are also discussed.

Appendix: Equilibrium conditions in the common-trend model

The Social Planner’s problem consists of choosing

$$\begin{aligned} \left\{ C_{t},H_{ct},H_{it},I_{ct},I_{it},K_{ct+1},K_{it+1}\right\} \quad \forall t \end{aligned}$$

so that the social utility function is maximized, subject to the production and capital accumulation constraints. Denote by $\Lambda _{ct}$ and $\Lambda _{it}$ the non-negative Lagrange multipliers on the technology constraints (2)–(3), and by $\Xi _{ct}$ and $\Xi _{it}$ the non-negative Lagrange multipliers on the capital accumulation constraints (4)–(5), then the first-order conditions (FOC) derived from the Lagrangian of the planner’s problem are given by:

$$\begin{aligned}&\frac{1}{C_{t}-\gamma C_{t-1}}-\beta \gamma E_{t}\left( \frac{1}{C_{t+1}-\gamma C_{t}}\right) =\Lambda _{ct} \end{aligned}$$

(21)

$$\begin{aligned}&\frac{1}{A_{t}}=\frac{\left( 1-\theta _{c}\right) \Lambda _{ct}C_{t}}{H_{ct} }-\phi _{hc}\Lambda _{ct}\left( \frac{H_{ct}}{H_{ct-1}}-\eta _{c}\right) \left( \frac{1}{H_{ct-1}}\right) \left[ 1-\frac{\phi _{kc}}{2}\left( \frac{I_{ct}}{K_{ct}}-\kappa _{c}\right) ^{2}\right] \\&\quad \times \,K_{ct}^{\theta c}\left( Z_{ct}H_{ct}\right) ^{1-\theta c}\,+ \, \beta \phi _{hc}E_{t}\left\{ \Lambda _{ct+1}\left( \frac{H_{ct+1}}{H_{ct}} -\eta _{c}\right) \left( \frac{H_{ct+1}}{H_{ct}}\right) \left( \frac{1}{H_{ct}}\right) \right. \\&\quad \left. \times \,\left[ 1-\frac{\phi _{kc}}{2}\left( \frac{I_{ct+1} }{K_{ct+1}}-\kappa _{c}\right) ^{2}\right] K_{ct+1}^{\theta c}\left( Z_{ct+1}H_{ct+1}\right) ^{1-\theta c}\right\} \end{aligned}$$

(22)

$$\begin{aligned}&\frac{1}{A_{t}}=\frac{\left( 1-\theta _{i}\right) \Lambda _{it}C_{t}}{H_{it} }-\phi _{hi}\Lambda _{it}\left( \frac{H_{it}}{H_{it-1}}-\eta _{i}\right) \left( \frac{1}{H_{it-1}}\right) \left[ 1-\frac{\phi _{ki}}{2}\left( \frac{I_{it}}{K_{it}}-\kappa _{i}\right) ^{2}\right] \\&\quad \times \,K_{it}^{\theta i}\left( Z_{it}H_{it}\right) ^{1-\theta i} + \beta \phi _{hi}E_{t}\left\{ \Lambda _{it+1}\left( \frac{H_{it+1}}{H_{it}} -\eta _{i}\right) \left( \frac{H_{it+1}}{H_{it}}\right) \left( \frac{1}{H_{it}}\right) \right. \\&\quad \left. \times \,\left[ 1-\frac{\phi _{ki}}{2}\left( \frac{I_{it+1} }{K_{it+1}}-\kappa _{i}\right) ^{2}\right] K_{it+1}^{\theta i}\left( Z_{it+1}H_{it+1}\right) ^{1-\theta i}\right\} \end{aligned}$$

(23)

$$\begin{aligned}&\Xi _{ct}=\Lambda _{it}+\phi _{kc}\Lambda _{ct}\left[ 1-\frac{\phi _{hc}}{2}\left( \frac{H_{ct}}{H_{ct-1}}-\eta _{c}\right) ^{2}\right] \left( \frac{I_{ct}}{K_{ct}}-\kappa _{c}\right) \left( \frac{1}{K_{ct}}\right) K_{ct}^{\theta c}\left( Z_{ct}H_{ct}\right) ^{1-\theta c} \end{aligned}$$

(24)

$$\begin{aligned}&\Xi _{it}=\Lambda _{it}\left\{ 1+\phi _{ki}\left[ 1-\frac{\phi _{hi}}{2}\left( \frac{H_{it}}{H_{it-1}}-\eta _{i}\right) ^{2}\right] \left( \frac{I_{it} }{K_{it}}-\kappa _{i}\right) \left( \frac{1}{K_{it}}\right) K_{it}^{\theta i}\left( Z_{it}H_{it}\right) ^{1-\theta i}\right\} \end{aligned}$$

(25)

$$\begin{aligned}&\Xi _{ct}=\beta E_{t}\left[ \left( 1-\delta _{c}\right) \Xi _{ct+1}\right] +\beta \theta _{c}E_{t}\left( \frac{\Lambda _{ct+1}C_{t+1}}{K_{ct+1}}\right) + \beta \phi _{kc}E_{t} \\&\quad \times \,\left\{ \Lambda _{ct+1}\left[ 1-\frac{\phi _{hc}}{2}\left( \frac{H_{ct+1}}{H_{ct}}-\eta _{c}\right) ^{2}\right] \left( \frac{I_{ct+1} }{K_{ct+1}}-\kappa _{c}\right) \left( \frac{I_{ct+1}}{K_{ct+1}}\right) \right. \\&\quad \left. \times \,\left( \frac{1}{K_{ct+1}}\right) K_{ct+1}^{\theta c}\left( Z_{ct+1} H_{ct+1}\right) ^{1-\theta c}\right\} \end{aligned}$$

(26)

$$\begin{aligned}&\Xi _{it}=\beta E_{t}\left[ \left( 1-\delta _{i}\right) \Xi _{it+1}\right] +\beta \theta _{i}E_{t}\left( \frac{\Lambda _{it+1}I_{t+1}}{K_{it+1}}\right) + \beta \phi _{ki}E_{t} \\&\quad \times \,\left\{ \Lambda _{it+1}\left[ 1-\frac{\phi _{hi}}{2}\left( \frac{H_{it+1}}{H_{it}}-\eta _{i}\right) ^{2}\right] \left( \frac{I_{it+1} }{K_{it+1}}-\kappa _{i}\right) \left( \frac{I_{it+1}}{K_{it+1}}\right) \right. \\&\quad \left.\times \, \left( \frac{1}{K_{it+1}}\right) K_{it+1}^{\theta i}\left( Z_{it+1} H_{it+1}\right) ^{1-\theta i}\right\} \end{aligned}$$

(27)

$$\begin{aligned}&C_{t}=\left[ 1-\frac{\phi _{hc}}{2}\left( \frac{H_{ct}}{H_{ct-1}}-\eta _{c}\right) ^{2}\right] \left[ 1-\frac{\phi _{kc}}{2}\left( \frac{I_{ct} }{K_{ct}}-\kappa _{c}\right) ^{2}\right] K_{ct}^{\theta c}\left( Z_{ct}H_{ct}\right) ^{1-\theta c} \end{aligned}$$

(28)

$$\begin{aligned}&I_{t}=\left[ 1-\frac{\phi _{hi}}{2}\left( \frac{H_{it}}{H_{it-1}}-\eta _{i}\right) ^{2}\right] \left[ 1-\frac{\phi _{ki}}{2}\left( \frac{I_{it} }{K_{it}}-\kappa _{i}\right) ^{2}\right] K_{it}^{\theta i}\left( Z_{it}H_{it}\right) ^{1-\theta i} \end{aligned}$$

(29)

$$\begin{aligned}&\left( 1-\delta _{c}\right) K_{ct}+I_{ct}=K_{ct+1} \end{aligned}$$

(30)

$$\begin{aligned}&\left( 1-\delta _{i}\right) K_{it}+I_{it}=K_{it+1} \end{aligned}$$

(31)

for $t=0,1,2,\ldots ,$ where

$$\begin{aligned} I_{t}=I_{ct}+I_{it} \end{aligned}$$

(32)

and

$$\begin{aligned} H_{t}=H_{ct}+H_{it} \end{aligned}$$

(33)

The first-order conditions (21)–(31), along with the aggregation constraints (32)–(33) and the driving processes, as specified in (6)–(13) in the main text, are the equilibrium conditions of the model. This is a system of 21 equations in 21 variables—$C_{t}$, $H_{t}$, $H_{ct}$, $H_{it}$, $I_{t}$, $I_{ct}$, $I_{it}$, $K_{ct}$, $K_{it}$, $\Lambda _{ct}$, $\Lambda _{it}$, $\Xi _{ct}$, $\Xi _{it}$, $A_{t}$, $a_{t}^{l}$, $A_{t}^{g}$, $Z_{ct}$, $z_{ct}^{l}$, $Z_{it}$, $z_{it}^{l}$ and $Z_{t}^{g}$. However, the equilibrium deriving from the solution of this system is not stable, since variables grow and some of them also may also be non-stationary. Following Ireland and Schuh (2008), this system can then be written in terms of the stationarized variables, i.e. variables that are constant in the steady state, obtained dividing the original ones by the sources of non-stationarity assigned to them by the model, so that

$$\begin{aligned} c_{t}&= {} C_{t}/(A_{t-1}^{g}Z_{t-1}^{g})\\ h_{t}&= {} H_{t}/A_{t-1}^{g}\\ h_{ct}&= {} H_{ct}/A_{t-1}^{g}\\ h_{it}&= {} H_{it}/A_{t-1}^{g}\\ i_{t}&= {} I_{t}/(A_{t-1}^{g}Z_{t-1}^{g})\\ i_{ct}&= {} I_{ct}/\left( A_{t-1}^{g}Z_{t-1}^{g}\right) \\ i_{it}&= {} I_{it}/\left( A_{t-1}^{g}Z_{i-1}^{g}\right) \\ k_{ct}&= {} K_{ct}/(A_{t-1}^{g}Z_{t-1}^{g})\\ k_{it}&= {} K_{it}/(A_{t-1}^{g}Z_{t-1}^{g})\\ \lambda _{ct}&= {} A_{t-1}^{g}Z_{t-1}^{g}\Lambda _{ct}\\ \lambda _{it}&= {} A_{t-1}^{g}Z_{t-1}^{g}\Lambda _{it}\\ \xi _{ct}&= {} A_{t-1}^{g}Z_{t-1}^{g}\Xi _{ct}\\ \xi _{it}&= {} A_{t-1}^{g}Z_{t-1}^{g}\Xi _{it}\\ a_{t}&= {} A_{t}/A_{t-1}^{g}\\ a_{t}^{g}&= {} A_{t}^{g}/A_{t-1}^{g}\\ z_{ct}&= {} Z_{ct}/Z_{t-1}^{g}\\ z_{it}&= {} Z_{it}/Z_{it-1}^{g}\\ z_{t}^{g}&= {} Z_{t}^{g}/Z_{t-1}^{g} \end{aligned}$$

while $a_{t}^{l}$, $z_{ct}^{l}$, $z_{it}^{l}$ remain the same, since already stable. The new system is obtained substituting the original variables with their stationary counterparts; however, in order to keep track of the three key original variables, $C_{t}$, $H_{t}$ and $I_{t}$, which are our observables, we need to introduce three further variables, $g_{t}^{c}$, $g_{t}^{i}$ and $g_{t}^{h}$, corresponding to their growth rates and defined as

$$\begin{aligned} g_{t}^{c}&= {} C_{t}/C_{t-1}=a_{t-1}^{g}z_{t-1}^{g}\left( c_{t}/c_{t-1}\right) \\ g_{t}^{i}&= {} I_{t}/I_{t-1}=a_{t-1}^{g}z_{t-1}^{g}\left( i_{t}/i_{t-1}\right) \\ g_{t}^{h}&= {} H_{t}/H_{t-1}=a_{t-1}^{g}\left( h_{t}/h_{t-1}\right) \end{aligned}$$

The final system consists of 24 equations in 24 variables and can be log-linearized around the steady state of the lower-case variables and solved using Klein’s (2000) method.