Mixed Effects Model: Writing Out Level 1/Level 2 Models Suppose I have an LMER output (using Grunfeld data from plm package) as:
summary(lmer(inv ~ value + capital + (1|firm), data = Grunfeld))

> Random effects:
 Groups   Name        Variance Std.Dev. Corr
 firm     (Intercept) 7367     85.83    
 Residual             2781     52.74    0.831
Number of obs: 200, groups:  firm, 10

Fixed effects:
             Estimate Std. Error t value
(Intercept) -57.86442   29.37776   -1.97
value         0.10979    0.01053   10.43
capital       0.30819    0.01717   17.95

Correlation of Fixed Effects:
        (Intr) value 
value   -0.328       
capital -0.019 -0.368

I know we can write a Level 1/Level 2 model based on this as:
Level 1:
y = alpha_0 + alpha_1 * value + alpha_2 * capital + eps
Level 2:
alpha_0 = gamma_00 + u_0
alpha_1 = gamma_01 + u_1
alpha_2 = gamma_02 + u_2

The level 1 model should have alpha_0=-57.864, alpha_1=0.109, alpha_2=0.308.
Now, I'm actually not sure if this is the correct form of the level 2 model. Im wondering what exactly the level 1/2 models would be here (for example, are the gammas in the level 2 model concrete values, or are they RV's. For example, here, it is not clear what those values are supposed to be). And also what the correlation of the fixed effects means versus the correlation of random effects. I can't seem to find definitive answers online.
 A: Note that you have only included a random intercept for the firm grouping variable. I.e., the model you are fitting is the following: 
$$\left\{
\begin{array}{l}
\texttt{Inv}_{ij} = \beta_0 + \beta_1 \texttt{Value}_{ij} + \beta_2 \texttt{Capital}_{ij} + b_i + \varepsilon_{ij},\\\\
b_i \sim \mathcal N(0, \sigma_b^2), \quad \varepsilon_{ij} \sim \mathcal N(0, \sigma^2),
\end{array}
\right.$$
where $\texttt{Inv}_{ij}$ denotes the $j$-th measurement within the $i$-th firm for you outcome variable, and likewise $\texttt{Value}_{ij}$ and $\texttt{Capital}_{ij}$ denote the $j$-th measurement within the $i$-th firm for the two predictors.
The estimated parameters according to the output are:


*

*$\hat\beta_0 = -57.86442$

*$\hat\beta_1 = 0.10979$

*$\hat\beta_2 = 0.30819$

*$\hat\sigma_b = 85.83$

*$\hat\sigma = 52.74$
You could also rewrite the same model as:
$$\left\{
\begin{array}{l}
\texttt{Inv}_{ij} = \alpha_{0i} + \beta_1 \texttt{Value}_{ij} + \beta_2 \texttt{Capital}_{ij} + \varepsilon_{ij},\\\\
\alpha_{0i} = \beta_0 + b_i,\\\\
b_i \sim \mathcal N(0, \sigma_b^2), \quad \varepsilon_{ij} \sim \mathcal N(0, \sigma^2).
\end{array}
\right.$$
A: It seems there are 10 firms in the dataset.
Level 1 model:
$$Y_{ki}=\alpha_{0k} + \alpha_1X_{1ki} + \alpha_2X_{2ki} + \epsilon_{ki}$$
So the level 2 model can be written as:
$$\alpha_{0k} = \alpha_0 + u_k$$
where $k=,...,10$ is index for firm. $i$ is the i-th measurement. $X_{1ki}$ is value and $X_{2ki}$ capital.
$u_k\sim N(0,\sigma_u^2)$ and $\epsilon \sim N(0,\sigma^2)$.
