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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Class 17
Step1:  Add Output and Investment
Recall the three equilibrium conditions fo the model
Step2:  Evaluation
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Python Code:
# 1. Input model parameters and print
# 2. Compute the steady state of the model directly
# 3. Define a function that evaluates the equilibrium conditions
def equilibrium_equations(variables_forward,variables_current,parameters):
    
    # Parameters 
    p = parameters
    
    # Variables
    fwd = variables_forward
    cur = variables_current
        
    # Resource constraint
    resource = cur.a*cur.k**p.alpha + (1-p.delta)* cur.k - fwd.k - cur.c
    
    # Exogenous tfp
    
    # Euler equation
    
    # Stack equilibrium conditions into a numpy array
    return np.array([
        ])
# 4. Initialize the model
model = ls.model(equations = equilibrium_equations,
                 nstates=,
                 varNames=[],                   # Any order as long as the state variables are named first
                 shockNames=[],                 # Name a shock for each state variable *even if there is no corresponding shock in the model*
                 parameters = parameters)
# 5. Set the steady state of the model directly. 
model.set_ss([])
# 6. Find the log-linear approximation around the non-stochastic steady state and solve
model.approximate_and_solve()
# 7. Print the approximated model in terms of log-deviations from the stady state
print(model.approximated())
# 8(a) Compute impulse responses and print the computed impulse responses
model.impulse(T=41,t0=5,shock=None)
print(model.irs['eA'].head(10))
# 8(b) Plot the computed impulse responses to a TFP shock
fig = plt.figure(figsize=(12,4))
ax1 = fig.add_subplot(1,2,1)
model.irs['eA'][['a','k','c']].plot(lw='5',alpha=0.5,grid=True,ax = ax1).legend(loc='upper right',ncol=3)
ax2 = fig.add_subplot(1,2,2)
model.irs['eA'][['eA','a']].plot(lw='5',alpha=0.5,grid=True,ax = ax2).legend(loc='upper right',ncol=2)
# 9(a) Compute stochastic simulation and print the simulated values
model.stoch_sim(seed=192,covMat= [[parameters['sigma']**2,0],[0,0]])
print(model.simulated.head(10))
# 9(b) Plot the computed stochastic simulation
fig = plt.figure(figsize=(12,4))
ax1 = fig.add_subplot(1,2,1)
model.simulated[['a','c','k']].plot(lw=5,alpha=0.5,grid=True,ax = ax1).legend(loc='upper right',ncol=3)
ax2 = fig.add_subplot(1,2,2)
model.simulated[['eA','a']].plot(lw=5,alpha=0.5,grid=True,ax = ax2).legend(loc='upper right',ncol=2)
Explanation: Class 17: A Centralized Real Business Cycle Model without Labor
The Model
Setup
A representative household lives for an infinite number of periods. The expected present value of lifetime utility to the household from consuming $C_0, C_1, C_2, \ldots $ is denoted by $U_0$:
\begin{align}
U_0 & =  \log (C_0) + \beta E_0 \log (C_1) + \beta^2 E_0 \log (C_2) + \cdots\
& =  E_0\sum_{t = 0}^{\infty} \beta^t \log (C_t),
\end{align}
where $0<\beta<1$ is the household's subjective discount factor. $E_0$ denotes the expectation with respect to all information available as of date 0.
The household enters period 0 with capital $K_0>0$. Production in period $t$:
\begin{align}
F(A_t,K_t) & = A_t K_t^{\alpha}
\end{align}
where TFP $A_t$ is stochastic:
\begin{align}
\log A_{t+1} & = \rho \log A_t + \epsilon_{t+1}
\end{align}
Capital depreciates at the constant rate $\delta$ per period and so the household's resource constraint in each period $t$ is:
\begin{align}
C_t + K_{t+1} & = A_t K_{t}^{\alpha}  + (1-\delta)K_t
\end{align}
Optimization problem
In period 0, the household solves:
\begin{align}
& \max_{C_0,K_1} \; E_0\sum_{t=0}^{\infty}\beta^t\log (C_t) \
& \; \; \; \;  \; \; \; \; \text{s.t.} \; \; \; \; C_t + K_{t+1} = A_t K_{t}^{\alpha}  + (1-\delta)K_t
\end{align}
which can be written as a choice of $K_1$ only:
\begin{align}
\max_{K_1} \; E_0\sum_{t=0}^{\infty}\beta^t\log \left( A_t K_{t}^{\alpha}  + (1-\delta)K_t - K_{t+1}\right)
\end{align}
Equilibrium
So given $K_0>0$ and $A_0$, the equilibrium paths for consumption, capital, and TFP are described described by:
\begin{align}
\frac{1}{C_t} & = \beta E_t \left[\frac{\alpha A_{t+1}K_{t+1}^{\alpha - 1} + 1 - \delta}{C_{t+1}}\right]\
C_t + K_{t+1} & = A_{t} K_t^{\alpha} + (1-\delta) K_t\
\log A_{t+1} & = \rho \log A_t + \epsilon_{t+1}
\end{align}
Calibration
For computation purposes, assume the following values for the parameters of the model:
\begin{align}
\beta & = 0.99\
\rho & = .75\
\sigma & = 0.006\
\alpha & = 0.35\
\delta & = 0.025
\end{align}
Steady State
The steady state:
\begin{align}
A & = 1\
K & = \left(\frac{\alpha A}{\beta^{-1} - 1 + \delta} \right)^{\frac{1}{1-\alpha}}\
C & = AK^{\alpha} - \delta K
\end{align}
End of explanation
# 1. Compute the steady state values of Y and I
# 2. Define a function that evaluates the equilibrium conditions
# 3. Initialize the model
# 4. Set the steady state of the model directly. 
# 5. Find the log-linear approximation around the non-stochastic steady state and solve
# 6(a) Compute stochastic simulation and print the simulated values
model.stoch_sim(seed=192,covMat= [[parameters['sigma'],0],[0,0]])
print(model.simulated.head(10))
# 6(b) Plot the computed stochastic simulation
Explanation: Add Output and Investment
Recall the three equilibrium conditions fo the model:
\begin{align}
\frac{1}{C_t} & = \beta E_t \left[\frac{\alpha A_{t+1}K_{t+1}^{\alpha - 1} + 1 - \delta}{C_{t+1}}\right]\
C_t + K_{t+1} & = A_{t} K_t^{\alpha} + (1-\delta) K_t\
\log A_{t+1} & = \rho \log A_t + \epsilon_{t+1}
\end{align}
Append two more equations that determine output and investment:
\begin{align}
Y_t & = A_t K_t^{\alpha}\
I_t & = K_{t+1} - (1-\delta)K_t
\end{align}
Recompute the model using the same parameters from above
End of explanation
# Compute the standard deviations of A, Y, C, and I
# Compute the coefficients of correlation for A, Y, C, and I
Explanation: Evaluation
End of explanation
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