Consider the following equation for $Y>0$: $$ (1) \quad \log(Y)=\log(\gamma)+\log(\alpha+\beta X)+\epsilon. $$ Assume that $E(\epsilon| X)=c\neq 0$. What are the consequences of this assumption on the estimation of $(\gamma, \alpha, \beta)$? More precisely, does $c\neq 0$ only cause bias in the estimated intercept $\gamma$ (as in linear least squares), or the fact that this is a nonlinear least square model implies that $c\neq 0$ may cause bias also in estimated $(\alpha, \beta)$?
Update 1: Thanks to the comment below, we can answer the question as follows. Let $\zeta\equiv \epsilon -c$. Hence, (1) can be rewritten as $$ \begin{aligned} \log(Y)&=\log(\gamma)+\log(\alpha+\beta X)+\zeta+c,\\ &=\log(\gamma\exp(c))+\log(\alpha+\beta X)+\zeta. \end{aligned} $$ Therefore, $c$ only causes a bias in the estimation of $\gamma$.
Update 2: To make sure I understand, let me complicate (1) as follows: $$ (2) \quad \log(Y)=\log(\gamma)+\sum_{k=1}^K D_k[\log(\alpha_k+\beta_k X)+\epsilon_k], $$ where $(D_1,\dots, D_K)$ is a vector of dummy variables in which exactly one element takes the value 1, and $E(\epsilon_k| D_1,\dots, D_K, X)=c\neq 0$. As before, I want to investigate which coefficients are biased by $c$. Let $\zeta_k\equiv \epsilon_k -c$. Hence, (2) can be rewritten as $$ \begin{aligned} \log(Y)&=\log(\gamma)+\sum_k D_k [\log(\alpha_k+\beta_k X+\zeta_k+c)],\\ &=\log(\gamma\exp(c\sum_k D_k ))+\sum_k D_k [\log(\alpha_k+\beta_k X) +\zeta_k],\\ &=\log(\gamma\exp(c))+\sum_k D_k [\log(\alpha_k+\beta_k X) +\zeta_k]. \end{aligned} $$ Hence, same answer as above.
Update 3: The comment below also suggests, more generally, that (1) is not identifiable. This is because (with $c=0$): $$ \begin{aligned} &\log(\gamma )+\log(\alpha+\beta X)+\epsilon\\ &=\log(\gamma (\alpha+\beta X))\\ &=\log(\gamma \alpha + \gamma \beta X)+\epsilon. \end{aligned} $$ Hence, we can only identify $\gamma \alpha$ and $\gamma \beta$. Does the same issue hold in (2)? Yes, I think. In fact, $$ \begin{aligned} &\log(\gamma )+\sum_k D_k[\log(\alpha_k+\beta_k X)+\epsilon_k],\\ &=\log(\gamma) + \sum_k \log[(\alpha_k+\beta_k X)^{D_k}] + \sum_k D_k \epsilon_k,\\ &=\log(\gamma \Pi_{k} (\alpha_k+\beta_k X)^{D_k})+ \sum_k D_k \epsilon_k.\\ \end{aligned} $$
