A simple model with only neutral productivity

If we fixed the intertemporal elasticity of substitution (IES) for consumption and leisure and consider only one type of non-stationary productivity, labor of TFP, we get a model like the one in this section. Later, I’ll allow for different IESs and the presence of a possibly cointegrated investment-specific productivity process, but this simple case delivers the main message and results can be compared with those in \cite{Edge2007}, who use essentially the same model when producing their results ¹. \cite{Hansen_2010}

Preferences

Since productivity has a stochastic trend some variables will not be stationary. Indeed, in equilibrium all variables, excepting leisure –and therefore labor–, will grow at constant expected rate, defining a Balanced Growth Path. In what follows, variables with tilde (i.e. \(\tilde{C}, \tilde{Y}\), etc.) are non-stationary.

A social planner tries to maximize

\[\label{eq:toy_preferences_u} E_0 \sum_{t=0}^{\infty} \beta^t \tilde{u}_t\]

\[\label{eq:toy_preferences_logs} \tilde{u}_t := \nu \ln \tilde{C}_t + (1-\nu) \ln (1- L_t)\]

Let \(A\) denote the level of total factor productivity in this economy. The following, transformed, variables are stationary: \[\label{eq:toy_stationary_variables} C_t := \frac{\tilde{C}_t}{A_{t-1} }, \quad K_t := \frac{\tilde{K}_t}{A_{t-1} }, \quad I_t := \frac{\tilde{I}_t}{A_{t-1} }\]

We can re-state the Planner’s objective using stationary variables: \[\label{eq:toy_preferences_u_stationary} E_0 \sum_{t=0}^{\infty} \beta^t u_t\] \[\label{eq:toy_preferences_logs_stationary} u_t := \nu \ln C_t + (1-\nu) \ln (1- L_t)\]

Production

There is a single good, produced with Cobb-Douglas technology. The way we have written the production function reminds us that in this case the logarithm of TFP is equal to the logarithm of labor productivity times the labor share.

\[\label{eq:Fisher_prod_tilde} {\tilde{Y}}_{t} = A_t^{1-\alpha} {\tilde{K}}^{\alpha}_{t} L^{1-\alpha}_{t}\]

Output is used for consumption and investment \[\label{eq:toy_CIYtilde} {\tilde{C}}_t + {\tilde{I}}_t \leq {\tilde{Y}}_{t}\]

There is no investment-specific productivity yet: \[\label{eq:toy_Kevol_tilde} {\tilde{K}}_{t+1} = I_t + (1-\delta) {\tilde{K}}_{t}\]

implying \[\label{eq:toy_CKYtilde} {\tilde{C}}_t + {\tilde{K}}_{t+1} - (1-\delta){\tilde{K}}_{t} \leq A_t^{1-\alpha} {\tilde{K}}^{\alpha}_{t} L^{1-\alpha}_{t}\]

Stationary versions of \eqref{eq:toy_CIYtilde}, \eqref{eq:toy_Kevol_tilde} and \eqref{eq:toy_CKYtilde} are goven by

Productivity evolution

Evolution of productivity is given by \[\begin{aligned} A_{t+1} = & A_{t}~ e^{a_{t+1}}, \quad A_0 ~\text{given} \end{aligned}\]

\[\begin{aligned} a_{t} & = \mu_{a} + \zeta_{a,t} + \sigma_{a} w_{a,t} , \quad w_{\zeta_a,t+1} \overset{iid}{\sim} N(0,1) \\ \zeta_{a,t} & = \rho_a \zeta_{a,t-1} + \sigma_{\zeta_a} w_{\zeta_a, t} , \quad w_{a,t} \overset{iid}{\sim} N(0,1) \end{aligned}\]

This is the same processes for productivity specified in \cite{Edge2007}, you can obtain their system of equations by defining ²

\[a_t := p_{t}-p_{t-1}, \, \zeta_{a,t-1} := g_{p,t} - \bar{g}, \, \mu_a := \bar{g}, \, w_{a,t} := \frac{\epsilon_t}{\sigma_{\epsilon}}, \, w_{\zeta_a, t} := \frac{v_t}{\sigma_{v}}, \, \sigma_{a} := \sigma_{\epsilon}, \, \sigma_{\zeta_a} := \sigma_{v}\]