## Working with Economies of Scale

Consider an example in which there are economies of scale—the more product you produce, the lower the average product price. Say, a factory can produce at

Thus, producing one good costs \$4 + \$10/(1 + 1) = \$9/good, and one-hundred goods costs \$4 + \$10/(100 + 1) = \$4.10/good. The company is currently selling 5 goods domestically, each for a price of \$8.00. It earns a

The company is considering a new foreign sales division that would cost \$16 to open, and that could sell another 5 units at \$8. The average price for the company producing 10 units would be \$4+\$10/11= \$4.91/unit. Therefore, 5 units cost \$24.55 to produce. The total cost of \$16 + \$24.55 = \$40.55 exceeds the total profit of 5 ■ \$8 = \$40. If considered by itself, opening a foreign sales division would not be a positive NPV project.

Now compute the total firm profit if the firm were to open the foreign sales division. Ten units would sell for a profit of \$80. Subtracting the opening costs of \$16 and production costs of 10 ■ \$4.91 = \$49.19 would earn a

This is more than the \$11.67 that the firm earned without the foreign sales division. The reason is that the foreign office has an additional marginal benefit: it reduces the average production cost experienced by the domestic office. This cost improvement is a positive externality that must be credited to the project—or the firm will make the bad decision of not opening the foreign division.

All this is easier to see when you translate it into terms of marginal costs and benefits. The Think of these extra marginal cost of each item changes item by item—it is the difference in total costs of each economies of sca|e in terms of marginal item: costs.

 Units Average Total Marginal 1 \$9.00 \$9.00 \$9.000 2 \$7.33 \$14.67 \$5.667 3 \$6.50 \$19.50 \$4.833 4 \$6.00 \$24.00 \$4.500 5 \$5.66 \$28.33 \$4.333
 Units Average Total Marginal 6 \$5.42 \$32.57 \$4.238 7 \$5.25 \$36.75 \$4.179 8 \$5.11 \$40.89 \$4.139 9 \$5.00 \$45.00 \$4.111 10 \$4.90 \$49.09 \$4.091

Going from 5 items to 10 items, production creates extra costs of \$4.333 to \$4.091 for a marginal cost of \$20.76. There would be an additional marginal cost of \$16 to open the foreign office. The total marginal cost would thus be \$36.76. The marginal benefit of 5 extra items would be \$40. Therefore, the foreign sales division gives you a marginal NPV of \$40 - \$36.76 « \$3.242. This is exactly the difference between \$11.67 from Formula 7.4 and \$14.91 from Formula 7.5. Thinking in terms of marginal costs and benefits is just a more convenient way to compare overall project values.

In Figure 7.1, there are three different cost functions that are based on different externalities. Graphical Display of

When there are positive externalities, as there are when there are economies of scale, then Ewnom^ °f Sca|e each item has a positive effect on the next item produced, so the marginal cost is decreasing with the number of units. When there are negative externalities, as there are when there are diseconomies of scale (it may become harder and harder to find the necessary input materials), then each item has a negative effect on the next item produced, so the marginal cost is increasing with the number of units.

Figure 7.1: The Effects of Economies of Scale on Marginal Costs

- - -0- - - -G - - - o - - -0- - - -G - - - o - - -0- - - -G - - - o -

Constant Costs Neither Benefit Nor Cost To Scale

Decreasing Marginal Costs ____Economies of Scale

Number of Units Produced

Here are three examples to show positive externalities, zero externalities, and negative externalities in the production of goods. The decreasing marginal cost function represents economies of scale. In this case, it also looks as if decreasing marginal costs are themselves decreasing—though the production cost is less for each additional product, it only is a little less good-by-good if you produce many of them. The increasing marginal cost function represents diseconomies of scale—the more units you produce, the more you have to pay for each additional unit.

Economies of scale are often responsible for the big corporate success stories of our time.

In my opinion, decreasing marginal costs are responsible for the biggest corporate success stories. For example, Wal-Mart and Dell have managed not only to use their scale to negotiate considerable supplier discounts, but have also created inventory and distribution systems that allow them to spread their fixed costs very efficiently over the large quantities of goods they sell. They have the lowest costs and highest industry inventory turnover rates—two factors that allow them to benefit tremendously from their economies of scale. Similarly, Microsoft enjoys economies of scale—with a large fixed cost and almost zero variable cost, Microsoft can swamp the planet with copies of Windows. No commercial alternative can compete—Microsoft can always drop its price low enough to drive its competitor out of business. The socially optimal number of operating systems software companies is very small and may even be just one—it is what economists call a natural monopoly. If you think of the economy as one big firm, you would not want to incur the same huge fixed software development cost twice. The same applies to utilities: you would not want two types of cable strung to everyone's house, two types of telephone lines, and two types of power lines. But companies with monopolies can also hurt the economy: they will want to charge higher prices to exploit their monopoly powers. Society has therefore often found it advantageous to regulate monopolists. Unfortunately, the regulatory agencies are themselves often "captured" by the companies that they are supposed to regulate, which sometimes can hurt the economy even more than the monopolies themselves. There are no easy and obvious solutions.

Sunk costs are ubiquitous, and the opposite of costs that should enter decision-making.

An example of how capital investments become sunk, and then how production itself becomes sunk.

Time is a good proxy, but not the deciding factor.

### Working with Sunk Costs

Sunk costs are, in a sense, the opposite of marginal costs. A sunk cost is a cost that cannot be altered and that therefore should not enter into your decisions today. It is what it is. Sunk costs are everywhere, if only because with the passage of time, everything is past or irrevocably decided and thus becomes a sunk cost.

For example, consider circuit board production—a very competitive industry. If you have just completed a circuit board factory for \$1 billion, it is a sunk cost. What matters now is not that you spent \$1 billion, but how much the production of each circuit board costs. Having invested \$1 billion is irrelevant. What remains relevant is that the presence of the factory makes the marginal cost of production of circuit boards very cheap. It is only this marginal cost that matters when you decide whether to produce circuit boards or not. If the marginal board production cost is \$100 each, but you can only sell them for \$90 each, then you should not build boards, regardless of how much you spent on the factory. Though tempting, the logic of "we have spent \$1 billion, so we may as well put it to use" is just plain wrong. Now, presume that the market price for boards is \$180, so you go ahead and manufacture 1 million boards at a cost of \$100 each. Alas, your production run has just finished, and the price of boards—contrary to everyone's best expectations—has dropped from \$180 each to \$10 each. At this point, the board production cost is sunk, too. Whether the boards cost you \$100 to manufacture or \$1 to manufacture is irrelevant. The cost of the production run is sunk. If boards now sell at \$10 each, assuming you cannot store them, you should sell them for \$10 each. Virtually all supply costs eventually become sunk costs, and all that matters when you want to sell a completed product is the demand for the product.

One more note—time itself often, but not always, decides on what is sunk or not. Contracts may allow you to undo things that happened in the past (thereby converting an ex-post sunk cost into a cost about which you still can make decisions), or bind you irrevocably to things that will happen in the future.

IMPORTANT: A sunk cost has no cost contribution on the margin. It should therefore be ignored.

### Overhead Allocation and Unused Capacity

"Capacity" is a subject that is closely related. For example, a garage may be currently only used for half its space. Adding the project "another car" that could also park in the garage would reduce this car's depreciation. The garage would then have a positive externality on project "corporate cars." The marginal cost of storing other cars in the garage should be zero.

Allocating already existing overhead to a project valuation is a common example of bad project decision making.

If capacity is unused, should have a zero price.

### Real World Dilemmas

But should you really charge zippo for parking corporate cars if you suspect that the unused capacity will not be unused forever? What if a new division might come along that wants to rent the five currently unused garage spaces in the future? Do you then kick out all current parkers? Or, how should you charge this new division if it wanted to rent six spaces? Should you give it the five remaining unused parking spots for free? Presuming that garages can only be built in increments of ten parking spots each, should you build another ten-car garage, and charge it entirely to this new division that needs only one extra parking spot in the new garage? Should this new division get a refund if other divisions were to want to use the parking space—but, as otherwise unused parking space, should you not use the garage appropriately by not charging for the nine extra spaces that will then be a free resource?

When there are high fixed and low variable costs, then capacity is often either incredibly cheap (or even free) or it is incredibly expensive—at least in the short run. Still, the right way to think of capacity is in terms of the relevant marginal costs and marginal benefits. From an overall corporate perspective, it does not matter how or who you charge—just as long as you get the optimal capacity utilization. To the extent that cost allocation distorts optimal marginal decision-making, it should be avoided. In our example, if optimal capacity utilization requires zero parking cost for the old garage, then so be it. Of course, when it comes to the decision to build an entirely new garage, you simply weigh the cost of building the 10-spot garage against the reduced deterioration for 1 car.

Unfortunately, real life is not always so simple. Return to the example on Page 166 of an Internet connection, which has a positive influence on all divisions. You know that division managers will not want to pay for it if they can enjoy it for free—you cannot rely on them telling you correctly how much they benefit. Would it solve your problem to charge only divisions that are voluntarily signing up for the Internet connection, and to forcibly exclude those that do not sign up? If you do, then you could solve the problem of everyone claiming that they do not need the Internet connection. However, you are then stuck with the problem that you may have a lot of unused network capacity that sits around, has zero marginal cost, and could be handed to the non-requesters at a zero cost. It would not impose a cost on anyone else and create more profit for the firm. Of course, if you do this, or even if you are suspected to do this, then no division would claim that they need the Internet to begin with, so that they will get it for free. In sum, what makes these problems so difficult in the real world is that as the boss, you often do not know the right marginal benefits and marginal costs, and you end up having to "play games" with your division managers to try to make the right decision. Such is real life!

Often you do not have easy, smooth margins.

Fixed costs are often responsible. The old method works, though—use "on the margin."

It becomes much harder if you do not know the right outcome, so you have to play games with your subordinate managers.

Solve Now!

Q 7.1 Why are zero externalities so convenient for a valuation problem?

Q 7.2 The average production cost per good is estimated at \$5 + \$15/(* + 1). The firm can currently sell 10 units at \$20 per unit.

(a) What is the current total profit of the firm?

(b) How much should the firm value the opportunity to sell one extra good to a new vendor?

(c) The new vendor offers to pay \$19 for one unit. However, your other vendors would find out and demand the same price. What is the marginal cost and benefit of signing up this new vendor now?

Q 7.3 A company must decide if it should move division A to a new location. If division A moves, it will be housed in a new building that reduces its operating costs by \$10,000 per year forever. The new building costs \$120,000. Moving division A allows division B to expand within the old factory. This enables B to increase its profitability by \$3,000 per year forever. If the discount rate is 10%, should division A move?

Q 7.4 A firm can purchase a new punch press for \$10,000. The new press will allow the firm to enter the widget industry, and thereby earn \$2,000 in profits per year forever. However, the punch press will displace several screw machines that produce \$1,500 in profits per year. If the interest rate is 10%, should the new punch press be purchased?

Q 7.5 A company rents 40,000 square feet of space and is using 30,000 square feet for its present operations. It wishes to add a new division that will use the remaining 10,000 square feet. If it adds the division, equipment will cost \$210,000 once, and the operations will generate \$50,000 in profits every year. Presently, the office staff costs \$160,000 per year. However, the expansion requires a larger staff, bringing costs up to \$180,000 per year. If the cost of capital r = 10%, should the firm expand?

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