In the long run, all of a firm's costs are variable. In this case the exit criterion for a profit-maximizing firm is to shut down if

The exit criterion for a profit-maximizing firm to shut down in the long run is if the price is lower than average total cost.

Explanation:

In the short run, a firm incurs variable costs and fixed costs. This is because only some factors are variable or can be changed in the short run. The short-run is to the short run to change all the variables. So the cost incurred on variable factors is variable cost and that incurred on fixed factors is fixed costs.

In the long run, though, all factors can be varied, so all the costs are variable. So, in the long run, there is no distinction between fixed or variable costs.

In the short run, the firm shuts down if the price is lower than the average variable cost, but in the long run, all costs can be varied so the firm shuts down if the price is lower than the average total cost.