In the long-run, a firm must shut down if its average revenue is

In the long-run, a firm must shut down if its average revenue is less than its average variable cost. This is because the firm is unable to cover even its variable costs, which are the costs it incurs when it produces any output at all. Let me explain further:

• A firm's goal is to maximize profit. To do this, a firm must ensure that it can at least cover its variable costs in the short-term. Variable costs include things like labor and raw materials, which are necessary for production.
• If the firm cannot cover the variable costs, it indicates that producing a unit of output adds more cost than revenue. Therefore, producing any more output will increase losses, making it better for the firm to shut down operations.
• In the long-run, all costs become variable, and if the firm's revenue doesn't cover these costs, the firm will not be able to survive.

To put it simply, if average revenue is less than average variable cost, the firm can't sustain itself in the long-run, and shutting down is a rational decision.