I'm currently on a university assignment where I'm stuck more or less in the middle. I have to answer the following problem:
Suppose you are interested in estimating the production function for agricultural output (as in the seminal article Mundlak 1961). You have access to data for a large number of farms $i$ for $T \geq 1$ time periods. The production function you want to estimate is:
\begin{equation} y_{it} = x_{it}\beta + \alpha_i + \epsilon_{it} \end{equation}
where $y_{it}$ is log-output, $x_{it}$ is log-labour (a variable input), $\alpha_i$ is log-soil-quality (a fixed input) and $\epsilon_{it}$ is rainfall (a random input). Each farmer knows the price of output $P_t$, the wage rate $W_t$, and the soil quality of his farm $\alpha_i$. However, as the econometrician you only observe ($y_{it}$, $x_{it}$). Assume that $\epsilon_{it}\sim \text{IID Dist}$ and independent of everything else in the model.
Solve the farmer’s profit maximization problem assuming he sells output at a common (across farmers) market price $P_t$ and pays common wages $W_t$. (Hint: It may help to write down the production function in levels instead of logs.) For notational convenience, assume $\mathbb{E}(e{^{\epsilon_{it}}}) = \lambda$. Does the labor demand depend on $\alpha_i$? Explain the economic intuition behind the result.
Would anyone give me some hints on how to tackle this task? Extremely glad for any help!