The Shut-Down Price: Price vs. Minimum AVC

The Logic of the Shutdown Decision

At first glance, it might seem obvious that a firm making a loss should cease operations. However, the decision in the short run is more nuanced, and it all hinges on the distinction between fixed and variable costs.

In the short run, a firm has certain fixed costs that it must pay regardless of whether it produces anything or not. Think of a non-cancellable factory lease or insurance payments. These are considered sunk costs in the context of the short-run production decision. Because they cannot be avoided, they are irrelevant to the choice between producing a positive quantity and producing zero.

The costs that do matter for this decision are the variable costs—the costs of labor, raw materials, and energy that are only incurred if the firm produces.

A firm will choose the option that minimizes its losses. Let's compare the two choices:

1. Shut Down (produce $q=0$): The firm's revenue is zero. Its variable costs are zero. It still has to pay its fixed costs ($FC$). The loss is therefore equal to its total fixed costs. ${\pi }_{shutdown}=-FC$.
2. Continue Operating (produce $q>0$): The firm earns total revenue ($TR$) and incurs both variable costs ($VC$) and fixed costs ($FC$). Its profit (which will be a loss in this case) is ${\pi }_{operate}=TR-VC-FC$.

The firm should continue to operate as long as operating is less costly than shutting down:

By adding $FC$ to both sides, we get a simple and powerful rule:

As long as the total revenue from selling its output is enough to cover the total variable costs of production, the firm is better off continuing to operate. Any revenue left over after covering variable costs ($TR-VC$) can be used to pay down a portion of the fixed costs, thus making the total loss smaller than it would be if the firm shut down and had to bear all its fixed costs with no revenue.

Dividing both sides of the inequality by quantity $q$, we can state the rule in terms of price and average cost:

The firm should continue to produce as long as the price per unit is greater than the average variable cost per unit.

The Shutdown Price and Point

The shutdown price is the market price below which the firm will choose to shut down and produce zero output. It is the point of indifference. This occurs precisely where the price is equal to the minimum possible average variable cost.

If the price falls even a fraction below this minimum AVC, the firm cannot cover its variable costs on any unit it produces and is better off shutting down to limit its losses to its fixed costs.

The following graph visualizes this concept.

Calculating the Shutdown Price

Given your background, you'll appreciate that we can find this price with mathematical precision. There are two equivalent methods for finding the quantity that corresponds to the minimum AVC, which we can then use to find the shutdown price itself.

Method 1: Minimizing the AVC Function using Calculus

The shutdown price is the minimum value of the average variable cost function. From calculus, we know that the minimum of a function occurs where its first derivative is equal to zero.

The procedure is:

1. Start with the firm's total cost function, $TC(q)$.
2. Isolate the variable cost function, $VC(q)$, which includes all terms dependent on $q$.
3. Derive the average variable cost function: $AVC(q)=VC(q)/q$.
4. Take the derivative of the AVC function with respect to $q$: $d(AVC)/dq$.
5. Set this derivative equal to zero and solve for the quantity, $q_{min}$.
6. Substitute $q_{min}$ back into the original $AVC(q)$ function. The result is the minimum average variable cost, which is the shutdown price.

The following video provides a clear walkthrough of this calculus-based method.

Method 2: Equating Marginal Cost and Average Variable Cost

A key property of cost curves we've seen before is that the marginal cost ($MC$) curve intersects the average variable cost ($AVC$) curve precisely at the minimum point of the AVC curve. This gives us a convenient algebraic shortcut.

The procedure is:

1. Start with the firm's total cost function, $TC(q)$.
2. Derive the marginal cost function: $MC(q)=d(TC)/dq$.
3. Derive the average variable cost function: $AVC(q)=VC(q)/q$.
4. Set $MC(q)=AVC(q)$ and solve for the quantity, $q_{min}$.
5. Substitute this quantity back into either the $MC(q)$ or $AVC(q)$ function (they will give the same value) to find the shutdown price.

This next video demonstrates this method from start to finish with a comprehensive example.

Both methods will yield the same result. The first is a direct application of optimization from calculus, while the second leverages a known geometric property of the cost curves.

A Note on Non-Sunk Costs

Our analysis has assumed that all fixed costs are sunk. In reality, some fixed costs might be avoidable if the firm shuts down (e.g., heating or lighting for the factory). These are called non-sunk fixed costs ($NSFC$). In such cases, the firm's avoidable costs are its variable costs plus its non-sunk fixed costs. The shutdown rule becomes slightly more general: a firm should operate as long as price covers the average non-sunk cost ($ANSC$), where $ANSC=(VC+NSFC)/q$. The shutdown price would then be the minimum of the ANSC curve.

For most exam questions, you can assume all fixed costs are sunk unless specified otherwise, making $\min (AVC)$ the relevant threshold.

Conclusion

In this lesson, we established the crucial short-run shutdown rule for a competitive firm. We saw that operating at a loss can be the loss-minimizing strategy, provided the firm can cover its variable costs. This leads to a clear, calculable threshold for this decision.

Key Takeaways:

• Shutdown Rule: A firm should shut down in the short run if the market price falls below its average variable cost ($P<AVC$). If $P\ge AVC$, it should continue to produce.
• Sunk Costs: Fixed costs are sunk in the short run and are irrelevant to the shutdown decision.
• Shutdown Price: The price at which the firm is indifferent between operating and shutting down. It is equal to the minimum of the average variable cost curve: $P_{shutdown}=\min (AVC)$.
• Calculation: The shutdown price can be found by either minimizing the $AVC(q)$ function using calculus or by solving for the price where marginal cost equals average variable cost ($MC=AVC$).
• Short-Run Supply: The firm's short-run supply curve is its marginal cost curve above the shutdown price. Below this price, its supply is zero.

So far, our analysis of perfect competition has been confined to the short run. However, the long run introduces a new dynamic: firms are free to enter or exit the market. In our next lesson, we will explore how this freedom of entry and exit affects market supply, price, and ultimately drives economic profits to zero, leading to the long-run equilibrium.