# Does the minimum of short-run average cost equal the minimum of long-run average cost in the long run for perfect competition?

I was recently reading this resource (http://www2.econ.iastate.edu/classes/econ101/hallam/Comp_LongRun_HND.pdf) which states that in the long run for perfect competition, price is equal to both the minimum of the long-run average cost and short-run average cost.

I am unsure of what this is implying. The resource states that because short-run average cost equals price, it shows that there is zero economic profit, while the fact that long-run average cost equals price implies that no identical firms will want to enter or exit. Why is this so? Why does the short run average cost equaling price imply that there is zero economic profit (and not long run average cost equaling price), and why does long run average cost equaling price mean that firms will not have the incentive to enter or exit (and not short run average cost equaling price)?

Furthermore, I am unsure of whether the minimum of the short-run and long-run average cost are equal. I read online that each short-run average cost is tangential to the long run average cost at one point. What is exactly meant by by the short-run average cost curves being "tangential" at one point, does this mean that each short run average cost touches the long run average cost at one point (what exactly is meant by tangential)? And why isn't the minimum point on the short run average cost the point on the long run average cost curve, not the tangential point?

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## 2 Answers

So to understand why the long run average cost curve and short run average cost curve have the same minimum in perfect competition, as well as some of the other stuff you ask, you have to understand the different assumptions that underlie the models you're working with.

Let's start with what the short run and long run mean here. In this context, the short run is when a firm is stuck with some fixed costs. It may be able to vary labor or raw materials, making those things variable costs, but some other things (e.g. contracted salaries, rental rate of capital) are fixed costs. It may be hard to expand the size of a factory quickly or find a new place to move production.

So here we see 3 different short run average cost curves. Each of these factories of course want to minimize cost per unit produced (average cost), and then set a price equal or above that (otherwise they'd be losing money!), so you have 3 different quantities that the firms will want to produce. You will notice that some of these firms will have to set a higher price than that firm with the production in the middle, so you can imagine which firm is going to get the customers in perfect competition.

Why are the SAC's shaped like this U shape? It is because generally there are economies of scale up to a certain point before things get too big to handle. A school can more efficiently allocate teachers by bumping up classroom size from 1 to 5 for example. But if you try to give the teacher a 1000 child classroom, you would expect things to be absolute chaos.

So now what happens in the long run? In the long run all these things we would normally consider fixed costs are actually variable costs. We can change the amount of capital we rent or other such things.

So given enough time, firms can change how much they want to produce along this line (LAC). We assume everyone moves along the same line because in perfect competition, we assume firms are identical, so that includes the technology and efficiency the firms have to produce the goods. So firms will naturally want to set production to the minimum of the long run average cost curve, so that they can set those low low prices meant to attract customers.

At least...that's what we would expect naturally, right? The last piece of putting it all together is seeing that the marginal cost equals average cost when average cost is minimized. I give a boring mathematical proof here. I assume you need no explanation for why firms set price equal to marginal cost.

So, price equal to short run average cost means that the firm produces its goods, sells each of them for the price, and breaks even. Thus we say no economic profit. Price equal to long run average cost means that all firms want to expand or contract their production so that they all produce the same amount of goods and basically all look identical. So all firms compete and push price down to the minimum of the long run average cost curve. They neither make nor lose money. So firms are indifferent between operating and not operating. It doesn't really make a difference to them. Thus, firms don't really want to enter or exit.

Edit: To answer the last part, why is the long run cost curve not tangent at the minimum of each short run cost curve, consider the following: For almost all cases, we should expect the minimum of the short run cost to lie above the long run average cost curve. In the short run, you can vary only some inputs, and in the long run you can vary all inputs. So for any given quantity you are trying to produce, it's easier to find a cheaper combination of inputs in the long run than it is in the short run.

This particular question on the tangency has been asked before, with a good top answer here.

• Thank you very much, just one more thing: my textbook states that "The long-run average cost curve is [the] combination of all minimum short-run average total cost values." But when looking at the long run average cost with all the short run average costs, it seems that the minimum point of the short run average cost is not on the long run average cost. Why is this? Commented Mar 18, 2020 at 6:20
• I'll get back to you on that tomorrow. I may need to edit part of my answer actually. Commented Mar 18, 2020 at 6:30
• Thanks very much. Just basically wondering why the long run average cost has the points of tangnecy rather than the minimum point of the short run average cost Commented Mar 18, 2020 at 6:40
• @ChristopherU Tangent just means LRAC touches a particular SRAC at one point. Just use the definition of tangent to figure things out. See my updated edit at the bottom to see why tangency is not at the minimum. Commented Mar 18, 2020 at 16:54
• Don't overthink it. The minimum of the short run average cost curve is the place where you'd want to produce in the short run, but firms to the left of the LRAC minimum are underutilized since they should try to expand production to the optimal point. Commented Mar 19, 2020 at 4:40

Look at it this way :

There are two timeframes in which firms make decisions - Short Run and Long Run. The actual time that these "Runs" include vary from Industry to Industry. For example, for a wheat grower, a full one year can be seen as long-run whereas, for a thermal power plant, 15 years would imply the long run. The answer to your question lies in identifying which decisions are made in which timeframe.

The decision to enter and exit the market requires the firms to make decisions about incurring the fixed cost of production ( which cant be varied in the short run). Hence the decision to enter and exit a market will be taken while looking at the Long Run Cost Curve.

However, zero economic profits simply imply that Total Cost = Total Revenue. This can be implied from both Long Run Cost Curve as well as Short Run Cost Curve. In both the long and short run, firms earn zero economic profits and hence breakeven. However, as mentioned the decision to enter/exit is a long-run decision and hence that implication can only be based on the Long Run Curve.

• What is meant by "The decision to enter and exit the market requires the firms to make decisions about incurring the fixed cost of production"? Won't they incur the fixed costs in the short-run? Commented Mar 18, 2020 at 2:44
• Yes, they do. But in the short run, the firms are already operating in the market. The fixed costs have already been incurred. It is a sunk cost then. It doesn't affect their marginal decision making. Commented Mar 18, 2020 at 7:14