# 6.2: Short Run Cost Functions for Infrastructure

Economists differentiate between short-run and long-run cost functions. In the short run, capital facilities are fixed. That is, an infrastructure manager must deal with the existing facilities. Any major capital project will take a year or more to be implemented to change facilities. In the long run, capital projects may be implemented, so additional capacity and facilities may be added.

Short and long-run are useful distinctions for developing cost functions, but there are many cases in which intermediate run cost functions may be needed when operational changes might be accomplished. For example, a transit manager may be limited to no changes in operations in the short run. However, schedule and route changes may be made without major capital expenditure. Vehicle fleets can be altered with new purchases in a somewhat longer time frame. Over the long run, capital facilities such as garages, rail lines and busways might be changed.

Costs can be divided into fixed costs of providing a facility and variable costs which depend upon usage. Fixed costs would be the cost of infrastructure services even without usage. Examples include:

• Generating plants, transmission lines and distribution lines for power
• Pumps, pipes, and storage for water systems
• Buildings for office infrastructure.

In many infrastructure cases, these fixed costs may be substantial.

Variable costs are incurred to provide infrastructure use. These costs generally increase as the amount of usage increases. For example, more maintenance is needed as the travel volume on a roadway increases. As another example, more building occupants will result in more power use, bathroom use and elevator trips. In most infrastructure systems, there are capacity constraints in which the variable cost increase rapidly as capacity is approached. An example is the roadway congestion shown in Figure 6.1 in which the user cost of travel is quite high. Buildings often have a maximum allowable occupancy, but crowding may be uncomfortable even before this maximum is attained.

Figure 6.2 illustrates the important short fun cost functions of interest for infrastructure management. The top graph in Figure 6.2.1 shows a fixed cost (F) even with no usage. As usage increases, the short-run total cost (SRTC(q)) increases, where q is a measure of usage such as traffic volume. If no capacity constraints or congestion effects exist, then the SRTC might increase as a straight line.

The bottom graph in Figure 6.2 shows three different short run cost curves:

• Short Run Average Total Cost is the total cost divided by usage:
$$\operatorname{SRATC}(q)=\frac{\operatorname{SRTC}(q)}{q}$$. This curve initially declines as fixed costs are spread over more usage. Eventually, capacity constraints and congestion result in higher costs and the SRATC begins to increase. A line drawn from the origin to the SRTC curve has a slope equal to the short run average total cost. The low point of the SRATC curve occurs where such a line has minimum slope and is tangent to the SRTC curve.
• Short Run Average Variable Cost is the total cost less fixed cost divided by usage:

$$S R A V C(q)=\frac{[S R T C(q)-F]}{q}$$
• This curve increases as capacity constraints and congestion result in higher costs. In the absence of such effects, the SRAVC(q) would be a flat, horizontal line.

Short Run Marginal Cost is the derivative of the SRTC with respect to q (or approximately the change in total cost from an additional unit of usage: $$\operatorname{SRMC}(q)=\frac{\delta \operatorname{SRTC}(q)}{\delta q} \approx \frac{[\operatorname{SRTC}(q)-\operatorname{SRTC}(q-1)]}{q}$$. The SRMC begins at a low value and increases as capacity constraints and congestion effects. The SRMC crosses the SRATC curve at its lowest, inflection point. Beyond this point, the marginal cost of additional usage exceeds the average cost.

As noted in the introduction, these various cost curves will differ depending upon the analysis viewpoint adopted. The major changes occur if external and user costs are included or not included. For a roadway system, user costs would include vehicle operating costs, travel time opportunity cost and potential costs from crashes. Vehicle operating costs include taxes that support roadway maintenance and construction in many cases. Travel time opportunity cost will likely vary with the income (or wealth) of the traveler and the opportunities foregone. A passenger in an autonomous, self-driving vehicle might have low travel time opportunity cost since the passenger could be doing activities other than driving. External costs would include air emissions effects, congestion and crash costs. Many of these ‘external’ costs are external to any individual traveler but are borne by other travelers. For example, an additional vehicle may add congestion that is a travel time penalty for the other vehicles on the road.

For telecommunications infrastructure, these cost functions would differ by type of technology used. For broadcast, over-the-air radio and television stations, all costs are fixed and no congestion effects occur so the SRTC function would be a horizontal line.

As many users can listen or watch as are in the area being served. This is an unusual situation and represents a ‘public’ good in the parlance of economics in which users cannot be easily excluded and do not interfere with other users. In contrast, cellular service infrastructure has capacity limits in base units, so greater usage imposes user costs in the form of inferior service.