Advice on the Efficiency of Stan Specification

Modeling
5

I have tried the following 2 sets of codes.

Code 1

This code models the probability that consumers will buy each of the product. This utilizes normal distribution.

stan.code = "data {
  int<lower=0> N;              
  int<lower=0> K;               // No. of Choice
  matrix[N,K] Mix;                //outcome
  matrix[N,K] Price;
}

transformed data {
  vector[N*K] y;
  y = to_vector(Mix);
}

parameters {
  vector[K-1] alpha_raw;
  vector<lower=0>[K] beta;
  real<lower=0> sigma;
}

transformed parameters {
  vector[K] alpha;
  matrix[N,K] gamma;
  vector[N*K] mu;
  alpha = append_row(0, alpha_raw);
  for (i in 1:N) {
    gamma[i] = (softmax( alpha - beta .* Price[i]' ))';
  }
  mu = to_vector(gamma);
}

model {
  alpha_raw ~ normal(0,1);
  beta ~ lognormal(0,1);
  sigma ~ normal(0,0.5);
  y ~ normal( mu, sigma);   
}
"

Code 2

This code models sales of each product instead of probability. Hence, it utilizes multinomial distribution.

"data {
  int<lower=0> N;             
  int<lower=0> K;          // No. of Choice
  matrix[N,K] Price;
  int Mix[N,K];              //outcome
}

parameters {
  vector[K-1] alpha_raw;
  vector<lower=0>[K] beta;
}

transformed parameters {
  vector[K] alpha;
  matrix[N, K] pred_shares;
  alpha = append_row(0, alpha_raw);
  for (t in 1:N) {
    pred_shares[t,1:K] = (softmax((alpha - beta .* (Price[t,1:K])')))'; 
  }
}

model {
  alpha_raw ~ normal(0,1);
  beta ~ lognormal(0,1);
  
  for (t in 1:N) {
    Mix[t] ~ multinomial(pred_shares[t]');
  }
}
  "

May I ask which method would be more appropriate? Both methods produce different estimates of the coefficients. In addition, code 1 takes a shorter time to run compared to code 2. Any other advice on how I can improve this further? Thank you!