# 2. UNIT COST AND COST EQUATIONS

2.1 Introduction
2.2 Example of Cost Equations
2.3 Applications of Cost Equations

## 2.1 Introduction

The use of breakeven and minimum-cost-point formulas require the collection of unit costs. Unit costs can be divided into subunits, each of which measures the cost of a certain part of the total. A typical unit cost formula might be

X = a + b + c

where X is the cost per unit volume such as dollars per cubic meter and the subunits a, b, c will deal with distance, volume, area, or weight. Careful selection of the subunits to express the factors controlling costs is the key to success in all cost studies.

## 2.2 Example of Cost Equations

Let us suppose the cost of harvesting from felling to loading on trucks is being studied. If X is the cost per cubic meter of wood loaded on the truck, we could represent the total cost per unit as

X = A + B + Q + L

where A would be the cost per unit of felling, B the cost of bucking, Q the cost of skidding, and L the cost of loading.

To determine the cost per subunit for felling, bucking, skidding, and loading, the factors which determine production and cost must be specified. Functional forms for production in road construction and harvesting are discussed in Sections 4 and 5. Examples for felling and skidding follow.

For felling, tree diameter may be an important explanatory variable. For a given felling method, the time required to fell the tree might be expressed as

T = a + b D2

where T is the time to fell the tree, b is the felling time required per cm of diameter, D is the tree diameter and "a" represents the felling time not explained by tree diameter-such as for walking between trees. The production rate is equal to the tree volume divided by the time per tree. The unit cost of felling is equal to the cost per hour of the felling operation divided by the hourly production or

A = C/P = C/(V/T) = C (a + B D2)/V

where C is the cost per hour for the felling method being used, P is the production per hour, V is the volume per tree, and T is the time per tree. The hourly cost of operation is referred to as the machine rate and is the combined cost of labor and equipment required for production. (Machine rates are discussed in Section 3.)

EXAMPLE:

Determine the felling unit cost for a 60 cm tree if the cost per hour of a man with power saw is \$5.00, the tree volume is 3 cubic meters, and the time to fell the tree is 3 minutes plus 0.005 times the square of the diameter.

T = 3 + .005 (60) (60) = 21 min = .35 hr
P = V/T = 3.0/.35 = 8.57 m3/hr
A = C/P = 5.00/8.57 = \$0.58/m3

In skidding, for example, if logs were being skidded directly to a road (Figure 2.1), then the distance skidded is an important factor and the stump to truck unit cost might be written as

X = A + B + Q + L
X = A + B + F + C(D/2) + L

where the skidding subunit Q has been replaced by symbol F representing fixed costs of skidding such as hooking, unhooking and decking and C(D/2) represents that part of the skidding cost that varies with distance. C is the cost of skidding a unit distance such as one meter and D/2 represents the average skidding distance in similar units. It is important to note that the average skidding cost occurs at the average skidding distance only when the skidding cost, C does not vary with distance. If C varies with distance, as for example, with animal skidding where the animal can become increasingly tired with distance, the average skidding cost does not occur at the average skidding distance and substantial errors in unit cost calculations can occur if the average skidding distance is used.

If logs were being skidded to a series of secondary roads (Figure 2.1) running into a primary road, then the expression C(D/2) would be replaced by the expression C(S/4) and the cost of truck haul on the secondary roads would appear as a separate item. In the expression C(S/4), the symbol S represents the spacing of the secondary roads and the distance S/4 is the average skidding distance if skidding could take place in both directions. Therefore, the expression C(S/4) would define the variable skidding cost in terms of spacing of the secondary roads.

Figure 2.1 Nomenclature for 2-way Skidding to Continuous Landings Among Spur Roads. A formula for the cost of logs on trucks at the primary road under these circumstances would be

X = A + B + F + C(S/4) + L + H(D/2)

where D/2 is the average hauling distance along the secondary road and H is the variable cost of hauling per unit distance.

The formula can be extended still further to include the cost of the secondary road system by defining the road construction cost per meter R, and the volume per square meter, V. Then, the formula becomes

X = A + B + F + C(S/4) + L + H(D/2) + R/(VS)

## 2.3 Applications of Cost Equations

In the preceding equation, we have a situation where as the spacing between skidding roads increases, skidding unit costs increase, while road unit costs decrease. With the total cost equation, we can look at the cost tradeoffs between skidding distance and road spacing. Calculus can be used to derive the formula for road spacing which minimizes costs as follows:

dX/dS = C/4 - R/(VS2) = 0

or

S = (4R/CV).5

An alternative method is to compare total costs for various road spacings. The total cost method has become less laborious with the use of programmable calculators and microcomputers. It provides information on the sensitivity of total unit cost to road spacing without having to evaluate the derivative of the cost function.

EXAMPLE:

Given the following table of unit costs, what is the effect of alternative spur road spacings on the total cost of wood delivered to the main road if 50 m3 per hectare is being cut and the average length of the spur road is 2 km. The cost of spur roads includes landings.

TABLE 2.1 Table of costs by activity for the road spacing example.

 Activity Unit Cost Fell \$/m3 0.50 Buck \$/m3 0.20 Skid \$/m3 2.00 (fixed cost) Skid \$/m3-km 2.50 (variable cost) Load \$/m3 0.80 Transport \$/m3-km 0.15 Roads \$/km 2000

Since only the skidding costs and spur road costs are affected by the road spacing, the total unit cost can be expressed as

X = A + B + F + C(S/4) + L + H(D/2) + R/(VS)
X + 0.50 + 0.20 + 2.00 + C(S/4) + 0.80 + .15 (1) + R/(VS)
X = 3.65 + C(S/4) + R/(VS)

To evaluate different road spacings, we vary the spur road spacing S and calculate the total unit costs (Table 2.2). It is important to use dimensionally consistent units. That is, if the left side of the equation is in \$/m3, the right side of the equation must be in \$/m3. This is most easily done if all volumes, costs and distances are expressed in meters; such as volume cut per m2, skidding cost per m3 per meter, and road cost per meter. For example, the total cost for a spur road spacing of 200 meters is 3.65 + (2.5/1000) (200/4) + (2000/1000)/[(50/10000) (200)] or \$5.78 per m3.

TABLE 2.2 Total unit cost as a function of road spacing.

 Spur Road Spacing, m Total Unit Cost, \$/m3 200 5.78 400 4.90 600 4.69 800 4.65 1000 4.68 1200 4.73 1400 4.81 1600 4.90 1800 5.00 2000 5.10

The road spacing which minimized total cost could be interpolated from the table or calculated from the formula

S = (4R/CV) .5 S = 800 m.

When costs have been collected in a form which permits unit costs to be developed from them, not only is it possible to predict costs, it is also possible to adjust conditions so that minimum cost can be achieved. Too often, recorded costs are only "experience figures". They are usually made available in a form which can be used to predict costs only under conditions that closely conform to those existing where and when the recorded costs were collected. This is not true of unit costs, which can be fitted into the framework of many different harvesting situations and can be made to tell the story of the future as well as that of the past.

A wide range of cost control formulas can be derived. Typical problems include:

1. The economic location of roads and landings. - The calculation of the optimal spacing between spur roads and landings subject to one-way skidding, two-way skidding, skidding on slopes, linear and nonlinear skidding cost functions.

2. The economic service standard for roads. - The comparison of the benefits of lower haul costs and road maintenance costs as a function of increased initial investment. The calculation of the optimal length of swing roads as a function of the tributary volume.

3. The economic selection of equipment for road systems fixed by topography or other factors. - The identification of the breakeven points between alternative skidding methods which have different fixed and variable operating costs.

4. The economic spacing of roads which will be served by two types of skidding machines. - For example, machines used to skid sawtimber and to relog for fuelwood.

5. The economic spacing of roads which will be reused in future time periods.

Another important application of unit costs is in choosing between alternative harvesting systems.

EXAMPLE:

A forest manager is developing an area and is trying to decide between harvesting methods. He has two choices of skidding systems (small or large), two choices of road standards (high or low), and two choices of trucks (small or large). If larger skidding equipment is selected to bring the logs to the landing, he can still choose to buck them into smaller logs on the landing. We assume that bucking on the landing will not affect log quality or yield.

The managers staff has developed the relevant unit costs, which are summarized in Table 2.3 and Table 2.4. What should he do?

TABLE 2.3 Unit costs for options of using small equipment and large equipment.

 Small Equipment\$/m3 Large Equipment\$/m3 Fall, buck 0.70 0.50 Skid 1.70 2.55 Load 1.00 0.80 Transport 1/ 1/ Unload 0.40 0.30 Process - 0.05 2/

1/See Table 2.4 for transport costs as a function of road standard. Wood for large system could be bucked on landing for \$0.15/m3 and loaded on small trucks.

2/Large logs must be bucked at mill.

TABLE 2.4 Unit costs for road and transport options using small and large equipment.

 Small Equipment \$/m3 Large Equipment \$/m3 Road High Standard 1.30 1.30 Low Standard 1.00 1.00 Transport High Standard 3.50 3.00 Low Standard 4.00 3.40

These choices can be viewed as a network (Figure 2.2). You can verify that the least cost path is obtained by using the larger skidding equipment and trucks and constructing the higher standard road. The total unit cost will be \$8.50 per m3. A key point is the ease at which these problems can be analyzed, once the unit costs have been derived. In turn, the derivation of the unit costs is facilitated by having machine rates available (Section 3).

Figure 2.2 Network Diagram for Equipment Choice 