In evaluating whether a farmer should produce crops or shut down operations, it's essential to analyze the costs and revenues involved. The farmer incurs a fixed cost of $1,000 for renting the field, which is a sunk cost and cannot be recovered regardless of the decision made. Additionally, the farmer must consider the variable cost of $200 for seeds when deciding to produce.

Two scenarios illustrate the farmer's potential revenues: one where the revenue from crop sales is $500 and another where it is only $100. In the first scenario, if the farmer chooses not to produce, the loss is $1,000. However, if the farmer decides to produce, the total revenue of $500 minus the total costs of $1,200 (which includes the $1,000 rent and $200 for seeds) results in a loss of $700. Thus, while both options lead to a loss, producing minimizes the loss.

In the second scenario, with a revenue of only $100, if the farmer does not produce, the loss remains at $1,000. If the farmer produces, the total revenue of $100 minus the total costs of $1,200 results in a loss of $1,100. Here, the farmer would be better off not producing, as it results in a smaller loss of $1,000 compared to a larger loss of $1,100.

The decision to shut down production is influenced by the relationship between total revenue and variable costs. The shutdown condition occurs when the price falls below the average variable cost. The average variable cost is crucial because it represents the costs that vary with production levels, such as the cost of seeds. If the revenue generated from sales does not cover these variable costs, it is not economically viable to continue production.

To formalize this, the shutdown point is defined as the minimum point on the average variable cost curve. If total revenue is less than variable costs, the farmer should shut down operations. This can be expressed mathematically: if the average revenue (which equals price) is less than the average variable cost, the farmer will choose to shut down. Thus, the condition for shutting down can be summarized as:

$$ P < AVC $$

where \( P \) is the price and \( AVC \) is the average variable cost. Understanding these concepts helps farmers make informed decisions about production based on their financial situation and market conditions.