Pooled likelihood in R, how to code it? - r

Question

I don't really know how to construct the following likelihood in R.
I have panel data for several individuals. Each individual is observed over time and I have several observations for each individual. I assume that the observations for each individual are independent and follow a normal distribution with an individual-specific mean and a variance common to all the individuals. In other words, I want to use all the data for estimating the variance, while using individual specific data to estimate the individual means.
Formally, let $i=(1,2,...,N)$ be the $i-th$ individual and $j=(1,2,...,n_i)$ the $j-th$ observation of individual $i$.
The likelihood function for individual $i$ is
$$
L_i(\mu_i,\sigma)=\prod_{j=1}^{n_i}\frac{1}{\sigma\sqrt{2\pi}}\exp(-\frac{(x_{ij}-\mu_i)^2}{2\sigma^2})
$$
L_i(mu_i, sigma) = PRODUCT_j 1/(sigma*sqrt 2pi) * exp( -(x_ij-mu_i)^2/2sigma^2 )
and the full likelihood is then
$$
L(\mu_1,...\mu_N,\sigma)=\prod_{i=1}^N \prod_{j=1}^{n_i}\frac{1}{\sigma\sqrt{2\pi}}\exp(-\frac{(x_{ij}-\mu_i)^2}{2\sigma^2})
$$
L_i(mu_i,...,mu_N, sigma) = PRODUCT_i L_i(mu_i, sigma)
What's a smart way to code the log of the full likelihood in R in order to pass it to some optimizer?
Example data:
set.seed(2)
A <- rep(1,15)
B <- rep(2,11)
ID <- c(A,B)
dA <- rnorm(15,2,3)
dB <- rnorm(11,1,3)
X <- c(dA,dB)
DATA <- data.frame(ID,X)
DATA