Step 1: Understand the shut-down condition.
In a perfectly competitive market, a firm maximizes profit by producing at the point where marginal cost equals marginal revenue. However, if the price falls below average variable cost (AVC), the firm cannot cover its variable costs, and it will choose to shut down in the short run.
Step 2: Explain the shut-down condition.
A firm will shut down if it cannot cover its variable costs, which happens when the price is less than the average variable cost (AVC). At this point, the firm incurs a loss equal to its fixed costs.
Thus, the shut-down condition is \( P<AVC \), meaning the price is less than the average variable cost.
Final Answer:
\[
\boxed{P<AVC}
\]