# COST FUNCTION ESTIMATION

|**COST FUNCTION AND ITS DETERMINANTS**

Cost function expresses the relationship between cost and its determinants such as the size of plant, level of output, input prices, technology, managerial efficiency, etc. In a mathematical form, it can be expressed as,

C = f (S, O, P, T, E…..)

Where, C = cost (it can be unit cost or total cost)

S = plant size

O = output level

P = prices of inputs used in production

T = nature of technology

E = managerial efficiency

**Determinants of Cost Function**

The cost of production depends on many factors and these factors vary from one firm to another firm in the same industry or from one industry to another industry. The main determinants of a cost function are:

a) plant size

b) output level

c) prices of inputs used in production,

d) nature of technology

e) managerial efficiency

We will discuss briefly the influence of each of these factors on cost.

a) **Plant size: **Plant size is an important variable in determining cost. The scale of operations or plant size and the unit cost are inversely related in the sense that as the former increases, unit cost decreases, and vice versa. Such a relationship gives downward slope of cost function depending upon the different sizes of plants taken into account. Such a cost function gives primarily engineering estimates of cost.

b) **Output level: **Output level and total cost are positively related, as the total cost increases with increase in output and total cost decreases with decrease in output. This is because increased production requires increased use of raw materials, labour, etc., and if the increase is substantial, even fixed inputs like plant and equipment, and managerial staff may have to be increased.

c) **Price of inputs: **Changes in input prices also influence cost, depending on the relative usage of the inputs and relative changes in their prices. This is because more money will have to be paid to those inputs whose prices have increased and there will be no simultaneous reduction in the costs from any other source. Therefore, the cost of production varies directly with the prices of production.

d) **Technology: **Technology is a significant factor in determining cost. By definition, improvement in technology increases production leading to increase in productivity and decrease in production cost. Therefore, cost varies inversely with technological progress. Technology is often quantified as capital-output ratio. Improved technology is generally found to have higher capital-output ratio.

e) **Managerial efficiency: **This is another factor influencing the cost of production. More the managerial efficiency less the cost of production. It is difficult to measure managerial efficiency quantitatively. However, a change in cost at two points of time may explain how organisational or managerial changes within the firm have brought about cost efficiency, provided it is possible to exclude the effect of other factors.

**ESTIMATION OF COST FUNCTION**

Several methods exist for the measurement of the actual cost-output relation for a particular firm or a group of firms, but the three broad approaches – accounting, engineering and econometric – are the most important and commonly used.** **

**Accounting Method**

This method is used by the cost accountants. In this method, the cost-output relationship is estimated by classifying the total cost into **fixed, variable** and **semi-variable costs.** These components are then estimated separately. The average variable cost, the semi-variable cost which is fixed over a certain range of output, and fixed costs are determined on the basis of inspection and experience. The total cost, the average cost and the marginal cost for each level of output can then be obtained through a simple arithmetic procedure.

Although, the accounting method appears to be quite simple, it is a bit cumbersome as one has to maintain a detailed breakdown of costs over a period to arrive at good estimates of actual cost-output relationship. One must have experience with a wide range of fluctuations in output rate to come up with accurate estimates.** **

**Engineering Method**

The engineering method of cost estimation is based directly on the physical relationship of inputs to output, and uses the price of inputs to determine costs. This method of estimating real world cost function rests clearly on the knowledge that the shape of any cost function is dependent on: (a) the production function and (b) the price of inputs.

We have seen earlier post discussing the *estimation of production function* that for a given production function and input prices, the optimum input combination for a given output level can be determined. The resultant cost curve can then be formulated by multiplying each input in the least cost combination by its price, to develop the cost function. This method is called **engineering method as the estimates of least cost combinations are provided by engineers.**

The assumption made while using this method is that both the **technology and factor prices are constant. **This method may not always give the correct estimate of costs as the technology and factor prices do change substantially over a period of time. Therefore, this method is more relevant for the short run. Also, this method may be useful if good historical data is difficult to obtain. But this method requires a sound understanding of engineering and a detailed sampling of the different processes under controlled conditions, which may not always be possible.** **

**Econometric Method**

This method is also some times called statistical method and is widely used for estimating cost functions. Under this method, the historical data on cost and output are used **to estimate the cost-output relationship**. The basic technique of regression is used for this purpose. The data could be a time series data of a firm in the industry or of all firms in the industry or a cross-section data for a particular year from various firms in the industry.

Depending on the kind of data used, we can estimate short run or long run cost functions. For instance, if **time series data** of a firm whose output capacity has not changed much during the sample period is used, the cost function will be short run. On the other hand, if **cross-section data** of many firms with varying sizes, or the time series data of the industry as a whole is used, the estimated cost function will be the long run one.

The procedure for estimation of cost function involves **three steps**. **First,** the determinants of cost are identified. **Second,** the functional form of the cost function is specified. **Third,** the functional form is chosen and then the basic technique of regression is applied to estimate the chosen functional form.

**Functional Forms of Cost Function**

The following are the three common functional forms of cost function in terms of total cost function (TC).

a) Linear cost function: **TC = a _{1} + b_{1}Q**

b) Quadratic cost function: **TC = a _{2} + b_{2}Q + c_{2}Q^{2}**

c) Cubic cost function: TC = **a _{3} + b_{3}Q + c_{3}Q^{2} +d_{3}Q^{3}**

Where, a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, c_{2}, c_{3}, d_{3} are constants.

When all the determinants of cost are chosen and the data collection is complete, the alternative functional forms can be estimated by using regression software package on a computer. The most appropriate form of the cost function for decision-making is then chosen on the basis of the principles of economic theory and statistical inference.

Once the constants in the total cost function are estimated using regression technique, the average cost (AC) and marginal cost (MC) functions for chosen forms of cost function will be calculated. The TC, AC and MC cost functions for different functional forms of total cost function and their typical graphical presentation and interpretation are explained below.

The typical TC, AC, and MC curves that are based on a quadratic cost function are shown in Figure 2. These cost functions have the following properties: TC increases at an increasing rate; MC is a linearly increasing function of output; and AC is a U shaped curve.

The typical TC, AC, and MC curves that are based on a cubic cost function are shown in Figure 3. These cost functions have the following properties:

TC first increases at a decreasing rate up to output rate Q1 in the Figure 3 and then increases at an increasing rate; and both AC and MC cost functions are U shaped functions.

The linear total cost function would give a constant marginal cost and a monotonically falling average cost curve. The quadratic function could yield a U-shaped average cost curve but it would imply a monotonically rising marginal cost curve. The cubic cost function is consistent both with a U-shaped average cost curve and a U-shaped marginal cost curve. Thus, **to check the validity of the theoretical cost-output relationship, one should hypothesize a cubic cost function.**

**An example of using estimated cost function:**

Using the output-cost data of a chemical firm, the following total cost function was estimated using quadratic function:

TC = 1016 – 3.36Q + 0.021Q^{2}

a) Determine average and marginal cost functions.

b) Determine the output rate that will minimize average cost and the per unit cost at that rate of output.

c) The firm proposed a new plant to produce nitrogen. The current market price of this fertilizer is Rs 5.50 per unit of output and is expected to remain at that level for the foreseeable future. Should the plant be built?

i) The average cost function is

AC = (TC/Q) = (a_{2}/Q) + b_{2} + c_{2}Q = (1016/Q) – 3.36 + 0.021Q

and the marginal cost function is

To find the cost at this rate of output, substitute 220 for Q in AC equation and solve it.

AC = (1016/Q) – 3.36 + 0.021Q = (1016/220) – 3.36 + (0.021 * 220) =Rs. 5.88 per unit of output.

iii) Because the lowest possible cost is Rs. 5.88 per unit, which is Rs. 0.38

above the market price (Rs. 5.50), the plant should not be constructed.

**Short Run and Long Run Cost Function Estimation**

The same sorts of regression techniques can be used to estimate short run cost functions and long run cost functions. However, it is very difficult to find cases where the scale of a firm has changed but technology and other relevant factors have remained constant. Thus, it is hard to use time series data to estimate long run cost functions. Generally, regression analysis based on cross section data has been used instead. Specially, a sample of firms of various sizes is chosen, and a firm’s TC is regressed on its output, as well as other independent variables, such as regional differences in wage rates or other input prices.

Many studies of long run cost functions that have been carried out found that there are very significant economies of scale at low output levels, but that these economies of scale tend to diminish as output increases, and that the long run average cost function eventually becomes close to horizontal axis at high output levels. Therefore, in contrast to the U-shaped curve as shown earlier, which is often postulated in micro economic theory, the long run average cost curve tends to be L-shaped, as shown in Figure-4.** **

**Problems in Estimation of Cost Function**

We confront certain problems while attempting to derive empirical cost functions from economic data. Some of these problems are briefly discussed below.

- In collecting cost and output data we must be certain that they are properly paired. That is, the cost data applicable to the corresponding data on output.
- We must also try to obtain data on cost and output during a time period when the output has been produced at relatively even rate. If for example, a month is chosen as the relevant time period over which the variables are measured, it would not be desirable to have wide weekly fluctuations in the rate of output. The monthly data in such a case would represent an average output rate that could disguise the true cost-output relationship. Not only should the output rate be uniform, but it also should be a rate to which the firm is fully adjusted. Furthermore, there should be no disruptions in the output due to external factors such as power failures, delays in receiving necessary supplies, etc. To generate the data necessary for a meaningful statistical analysis, the observations must include a wide range of rates of output. Observing cost-output data for the last 24 months, when the rate of output was the same each month, would provide little information concerning the appropriate cost function.
- The cost data is normally collected and recorded by accountants for their own purposes and in a manner that it makes the information less than perfect from the perspective of economic analysis. While collecting historical data on cost, care must be taken to ensure that all explicit as well as implicit costs have been properly taken into account, and that all the costs are properly identified by time period in which they were incurred.
- For situations in which more than one product is being produced with given productive factors, it may not be possible to separate costs according to output in a meaningful way. One simple approach of allocating costs among various products is based on the relative proportion of each product in the total output. However, this may not always accurately reflect the cost appropriate to each output.
- Since prices change over time, any money value cost would therefore relate partly to output changes and partly to price changes. In order to estimate the cost-output relationship, the impact of price change on cost needs to be eliminated by deflating the cost data by price indices. Wages and equipment price indices are readily available and frequently used to ‘deflate’ the money cost.
- Finally, there is a problem of choosing the functional form of equation or curve that would fit the data best. The usefulness of any cost function for practical application depends, to a large extent, on appropriateness of the functional form chosen. There are three functional forms of cost functions, which are popular, viz.,
**linear, quadratic and cubic.**The choice of a particular function depends upon the correspondence of the economic properties of the data to the mathematical properties of the alternative hypotheses of total cost function.

The accounting and engineering methods are more appropriate than the econometric method for estimating the cost function at the firm level, while the econometric method is more suitable for estimating the cost function at the industry or national level. There has been a growing application of the econometric method at the macro level and there are good prospects for its use even at the micro level. However, it must be understood that the three approaches discussed above are not competitive, but are rather complementary to each other. They supplement each other. The choice of a method therefore depends upon the purpose of study, time and expense considerations.** **

**EMPIRICAL ESTIMATES OF COST FUNCTION**

A number of studies using time series and cross-section data have been conducted to estimate short run and long run cost behaviour of various industries. Table-3 lists a number of well-known studies estimating short run average and marginal cost curves. These and many other studies point one conclusion: in the short run a linear total variable cost function with constant marginal cost is the relationship that appears to describe best the actual cost conditions over the “normal” range of production. U-shaped average cost (AC) and marginal cost (MC) curves have been found, but are less prevalent than one might expect.

Table-4 lists a number of well known, long run average cost studies. In some industries, such as light manufacturing (of baking products), economies of size are relatively unimportant and diseconomies set in rather quickly, implying that a small plant has cost advantages over a large plant. In other industries, such as meat packing or the production of household appliances, the long run average cost curve is found to be flat over an extended range of output, there by indicating that a variety of different plant sizes are all more or less equally efficient. In some other industries such as electricity or metal (aluminum and steel) production, substantial economies of size are found, thereby implying that a large plant is most efficient. Rarely are substantial diseconomies of size found in empirical studies, perhaps because of firms recognizing that production beyond a certain range leads to sharply rising costs. Therefore, they avoid such situations if all possible by building additional plants.