# Profit maximization problem using linear regression (pooled OLS)

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:

$$$$y_{it} = x_{it}\beta + \alpha_i + \epsilon_{it}$$$$

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 $$\epsilon_{it}$$ is $$iid$$ 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 $$Ee{^{\epsilon_{it}}} = \lambda$$. Does the labor demand depend on $$\alpha_i$$? Explain the economic intuition behind the result.

$$y_{it} = x_{it}\beta + \alpha_i + \epsilon_{it}$$ is log of the production function. So production must be

$$Y_{it} = A_iX_{it}^\beta\cdot u_{it}$$

found by taking exponential $$\exp(y_{it}) = \exp(x_{it}\beta + \alpha_i + \epsilon_{it})$$

and defining $$Y_{it} = exp(y_{it})$$ $$X_{it}=\exp(x_{it})$$ and $$A_i = \exp(\alpha_i)$$ and $$u_{it} = \exp(\epsilon_{it})$$.

Profit max problem is then

$$\max_{X_{it}}\Pi_{it} = P_t A_iX_{it}^\beta\cdot u_{it} - W_t X_{it}$$

Given the iid character of $$\epsilon_{it}$$ and the farmers knowledge the expected profit is the same with $$\mathbb E[u_{it}]$$ substituted for $$u_{it}$$:

$$\max_{X_{it}}\mathbb E [\Pi_{it}] = P_t A_iX_{it}^\beta\cdot \mathbb E [u_{it}] - W_t X_{it}$$

The solution for labour demand is $$X_{it}^\star= \left(\frac{P_tA_i \mathbb E [u_{it}] }{W_t}\right)^{\frac{1}{1-\beta}}$$

higher price, soil quality and expected rainfall are all expected to increase marginal revenue of labor and therfore increase labor demand for a given wage.

Here's the question on which I am stuck:

Under what assumption can you recover a consistent estimate for $$\beta$$ by running (pooled) OLS? Based on what you found in (1a) do you think this assumption is violated in this case? (no proof required)

Would anyone give me some hints on how to tackle this task? Extremely glad for any help!