Question

The typical cost function is quadratic. how could that then be estimated through regression? please give an example

Answer 1

The typical TC, AC, and MC curves that are based on a quadratic cost function are shown in Figure. 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.

example of using estimated cost function: Using the output-cost data of an industrial firm, the following total cost function was estimated using the quadratic function: TC = 1016 – 3.36Q + 0.021Q2 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) = (a2/Q) + b2 + c2Q

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. Especially, 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. 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 in Figure shown in the previous unit, which is often postulated in microeconomic theory, the long run average cost curve tends to be L-shaped, as shown in Figure.

Problems in Estimation of Cost Function We confront certain problems while attempting to derive empirical cost function from economic data. Some of these problems are briefly discussed below. Long run average cost curveProduction andCost Analysis181. 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.2. We must also try to obtain data on cost and output during a time period when the output has been produced at a 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. Observingcost-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.3. 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 a time period in which they were incurred.4. 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.5. 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 moneycost.6. Finally, there is a problem of choosing the functional form of the equation or curve that would fit the data best. The usefulness of any cost function for a practical application depends, to a large extent, on the 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 the particular function depends upon the correspondence of the economic properties of the data to the mathematical properties of the alternative hypothesis of the 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.

When we want to estimate a quadratic model, we cannot type in something like this:

model_2 <- lm(salary ~ entry_level_salary + experience^2) >> This will reject an error message Most of these functions do not expect that they have to transform your input variables. As a result, they reject an error message if you try. Furthermore, you do not have a sum at the right-hand side of the equation anymore.

Note: You need to compute experience^² before adding it into your model. Thus, you will run: # First, compute the square values of experience experience_2 <- experience^2 # Then add them into your regression model_2 <- lm(salary ~ entry_level_salary + experience_2) In return, you get a nice quadratic function: