# Reinforcement Learning in a New Keynesian Model +
## Appendix C Balanced Growth Steady State  
In recursive form, the zero-growth zero-inflation (\(\Pi=1\)) steady state can be written as  
\[R = \frac{1}{\beta}\] \[\Lambda = \beta\] \[MC=\frac{P^{W}}{P} = 1-\frac{1}{\zeta}\] \[\frac{C}{Y} = 1-g_{y}\] \[H = \frac{\alpha\Delta^{\pi}MC}{\kappa(1-g_{y})}\] \[Y^{W} = (AH)^{a}\] \[Y = \frac{Y^{W}}{\Delta^{a}}\] \[W = \alpha\frac{P^{W}}{P}\frac{Y^{W}}{H}\] \[J = \frac{YMC\,U_{C}}{(1-\frac{1}{\zeta})(1-\xi\beta\Pi^{\zeta})}\] \[JJ = \frac{YU_{C}}{(1-\xi\beta\Pi^{\zeta-1})}\] Hence, with \(\Pi=1,J\) \[= JJ\] \[\Delta = 1\] \[\Gamma = Y-\alpha MCY^{W}\]  
For a particular steady state, the inflation rate \(\Pi>1\), and the NK features of the steady state become  
\[\frac{J}{JJ} = \left(\frac{1-\xi\Pi^{\zeta-1}}{1-\xi}\right)^{\frac{1}{1-\zeta}}\] \[MC=\frac{P^{W}}{P} = \left(1-\frac{1}{\zeta}\right)\frac{J(1-\beta\xi\Pi^{\zeta})}{JJ (1-\beta\xi\Pi^{\zeta-1})}\] \[\Delta = \frac{(1-\xi)^{a}\left(\frac{J}{J}\right)^{-\zeta}}{1-\xi\Pi^{ \zeta}}\]  
then, \(P^{W}Y^{W}/PY=MC\Delta\).  
We can now easily set up the model with a balanced exogenous-growth steady state. Now the process for \(A_{t}\) is replaced with  
\[A_{t} = \bar{A}_{t}A_{t}^{c}\] \[\bar{A}_{t} = (1+g)\bar{A}_{t-1}\exp(\epsilon_{A,t})\] \[\log A_{t}^{c}-\log A^{c} = \rho_{A}(\log A_{t-1}^{c}-\log A^{c})+\epsilon_{A,t}\]  
where \(A_{t}\) is a labour-augmenting technical progress parameter which we decompose into a cyclical component, \(A_{t}^{c}\), modelled as a temporary AR(1) process and a stochastic trend, whose log is a random walk with drift, \(\bar{A}_{t}\). Thus, the balanced growth deterministic steady state path is driven by labour-augmenting technical change growing at a net rate \(g\). If we put \(g=\epsilon_{trend,t}=0\) and \(\bar{A}_{t}=1\), we arrive at our previous formulation with \(A_{t}^{c}=A_{t}\).  
Now stationarise the variables by defining cyclical and stationary components  
\[(Y_{t}^{W})^{c} \equiv \frac{Y_{t}^{W}}{\bar{A}_{t}}=A_{t}^{c}H_{t}^{\kappa}\] \[C_{t}^{c} \equiv \frac{C_{t}}{\bar{A}_{t}}\] \[W_{t}^{c} \equiv \frac{W_{t}}{\bar{A}_{t}}\] \[U_{t}^{c} \equiv \log C_{t}^{c}-\kappa\frac{H_{t}^{1+\phi}}{1+\phi}\] \[U_{C,t}^{c} \equiv \frac{1}{\overline{C}_{t}^{c}}\] \[\Lambda_{t,t+1} = \beta\frac{U_{C,t+1}}{U_{C,t}}=\beta_{g,t+1}\frac{U_{C,t+1}^{c}}{ U_{C,t}^{c}}\]  
for all non-stationary variables where  
\[g_{t} \equiv \frac{(\bar{A}_{t}-\bar{A}_{t-1})}{\bar{A}_{t}}=(1+g)\exp(\epsilon _{A,t})-1\] \[\beta_{g,t} \equiv \beta(1+g_{t})\]  
is the stochastic steady state growth rate; then, the stationarised Euler equation and the Calvo pricing become  
\[E_{t}[\Lambda_{t,t+1}R_{t+1}]=E_{t}\bigg{[}\beta_{g,t+1}\frac{U_{C,t+1}^{c}}{ U_{C,t}^{c}}R_{t+1}\bigg{]}=1\]  
and  
\[\widehat{f}_{t}^{c}-\xi E_{t}[\Pi_{t+1}^{\xi-1}\widehat{f}_{t+1}^ {c}\Lambda_{t,t+1}] = Y_{t}^{c}\] \[\widehat{f}_{t}^{c}-\xi E_{t}[\Pi_{t+1}^{\xi}\widehat{f}_{t+1}^{c }\Lambda_{t,t+1}] = Y_{t}^{c}MC_{t}MS_{t}\]  
or equivalently  
\[\widehat{f}_{t}^{c}-\xi E_{t}[\Pi_{t+1}^{\xi-1}\widehat{f}_{t+1}^ {\xi}\beta_{g,t+1}] = Y_{t}^{c}U_{t}^{c}\] \[\widehat{f}_{t}^{c}-\xi E_{t}[\Pi_{t+1}^{\xi}\widehat{f}_{t+1}^{c }\beta_{g,t+1}] = Y_{t}^{c}U_{t}^{c}MC_{t}MS_{t}\]  
The steady state for the rest of the system is the same as the zero-growth one except for the following relationships:  
\[R = \frac{1}{\beta_{g}}=\frac{R_{n}}{\Pi}\]  
where \(R\) and \(R_{n}\) are the real and nominal steady state interest rates, and \(\Pi\) is inflation.