# Profit maximization problem using linear regression

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!

• What have you done so far? Where are you getting stuck? Nov 29, 2018 at 15:56
• (Also, this question is better suited for Economics StackExchange) Nov 29, 2018 at 15:57
• So far I assumed that I have to set up the profit maximization. So: $y_{it}*P_t - x_{it}*W_t$ and differentiate with respect to $x_{it}$. However I don't see where to include the hint. Nov 29, 2018 at 16:01
• The hint is relevant because the firm makes $P_t$ on one unit of output (not log output) and pays $W_t$ on one unit of labor (not log labor). Nov 29, 2018 at 16:02
• So if we differentiate with repsect to $x$, $\alpha_i$ is equal to 0 since it's a constant. So the labor demand does not depend on $\alpha_i$? I don't get the effect of $Ee{^{\epsilon_{it}}} = \lambda$ in this problem. Nov 29, 2018 at 16:06

The hint tells you to transform to level instead of log ...

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

• That makes sense, thanks a lot for your help! Unfortunately I could not upvote your comment since my reputation is too low. Nov 30, 2018 at 11:27