COST FUNCTION AT VARIOUS LEVELS (Broader View)

Meaning The mathematical relationship between the cost of a product and various determinants of costs is referred to as a cost function. The dependent variable in a cost function is unit cost or total cost, while the independent variables are the price of a factor, the size of the output or capacity utilization, or any other relevant phenomenon that influences cost, such as technology, capacity utilization, efficiency, and the time period under consideration.

The cost function is derived from the production function as well as the market supply of inputs. It expresses the relationship between output and costs. Cost functions are derived from the firms’ real cost data and presented through cost curves. The shape of the cost curves depends upon the cost function. There are two types of cost functions: short-run cost functions and long-run cost functions.

Short Run Cost Curve: 

Before understanding the short-run cost curve, let us understand what we mean by “fixed factors” and “variable factors”: 

• Fixed Factors Some factors, such as buildings, capital equipment, or the top management team, cannot be changed quickly. Changing them takes a great amount of time. It takes time to install new machinery. Similarly, constructing a new factory takes time. Such factors, which cannot be readily varied and require a longer period to adjust, are called fixed factors.
• Variable Factors: With changes in the level of output, there are various factors that can be easily adjusted. If a firm needs to increase output, then it can easily hire more workers.. Similarly, if it needs to expand production, it can buy more raw materials. Variable factors are those that can be easily varied in response to a change in output level.

Based on the above distinction between variable and fixed factors, we can differentiate between short-run and long-run periods of time:

short-run period : In the short-run period, output can be increased or decreased only by changing the quantity of variable factors such as labor, raw materials, and so on. The amounts of fixed factors cannot be changed in response to changes in output in the short run. If the firm wishes to boost output in the short term, it can only do so by increasing the variable factors, such as hiring more people and/or purchasing more raw materials. As a result, a short run is a period of time during which only variable factors can be changed while fixed factors’ quantities stay unchanged or unaltered.

A long run period  is a period of time in which the quantities of all factors may be varied. In other words, all factors become variables in the long run.

A completely fixed cost curve: 

Explanation:  Fixed costs are costs that are unaffected by output changes, i.e., they do not change as output changes. These costs are of a fixed amount that a firm incurs in the short run, regardless of the size of the output. Even if the firm closes down  for a brief period of time but continues to operate, these costs must be met by it. Contractual rent, insurance fees, maintenance costs, property taxes, interest on capital employed, managers’ salaries, and watchman’s wages are examples of fixed costs. The above figure illustrates the fixed cost curve.

Completely variable cost curve: 

Explanation: Variable costs, on the other hand, are those that change when output changes. Payments such as wages for casual labor, raw material prices, fuel and power consumed, transportation costs, and so on are included in these costs. If a company closes down for a short period of time, it may not use the variable factors  of production and hence incur no variable costs. The above figure represents a completely variable cost curve drawn under the assumption that variable costs change linearly with changes in output.

Semi Variable Cost Curve:  

Explanation:  There are some costs that are neither entirely variable nor completely fixed in respect to changes in the output. These are known as Semi-variable costs It is well reflected in the above figure. Electricity prices, for example, contain both a fixed charge and a consumption-based charge.

Stair-Step Variable Cost Curve 

Explanation: There are some costs which may increase in a stair-step fashion, i.e., they remain fixed over a certain range of output but suddenly jump to a new higher level when output goes beyond a given limit. E.g. The costs incurred towards the salary of foremen will have a sudden jump if another foreman is appointed when the output crosses a particular limit. This is illustrated in the above figure. 

short-run total cost curve

Explanation: The total cost of a firm is the cost that must be incurred in order to produce a particular quantity of product. The total cost in the short run is made up of two key components: total fixed cost and total variable cost. Symbolically, total cost is equal to total fixed cost plus total variable cost. Total cost, total variable cost, and fixed cost are all represented graphically in the above figure.The total fixed cost curve (TFC) is a horizontal straight line parallel to the X-axis in the diagram above because TFC remains constant across the whole output range. This curve starts from a point on the Y-axis, meaning that fixed costs will be incurred even if the output is zero. The total variable cost curve, on the other hand, rises upward, showing that total variable cost increases as output increases. Because variable costs are zero when the output is zero, the total variable cost curve begins at the origin. It should be noted that the total variable cost initially increases at a decreasing rate and then at an increasing rate with increases in output.

Interpretation: The operation of the law of decreasing  and diminishing returns to the variable inputs causes this pattern of change in the TVC. Due to the law of diminishing returns, as output increases, larger quantities of variable inputs are required to produce the same quantity of output. As a result, at greater levels of output, the variable cost curve becomes steeper. The total cost curve has been obtained by adding vertically the total fixed cost curve and the total variable cost curve.

Average Fixed Cost Curve, Average Fixed Cost Curve, Average Total Cost Curve and Marginal Cost Curve:

Average Fixed Cost (AFC): AFC is obtained by dividing the total fixed cost by the number of units of output produced, i.e.,

Where Q is the number of units produced, Thus, the average fixed cost is the fixed cost per unit of output. For example, if a firm is producing at a total fixed cost of 2,000/- When output is 100 units, the average fixed cost will be 20. And now, if the output increases to 200 units, the average fixed cost will be 10. Since total fixed costs are a constant amount, the average fixed cost will steadily fall as output increases. Therefore, if we draw an average fixed cost curve, it will slope downwards throughout its length but will not touch the X-axis as AFC cannot be zero. which is shown in the above figure.

Average variable cost (AVC): The average variable cost is found by dividing the total variable cost by the number of units produced. I.e.,

where Q is the number of units produced. Thus, the average variable cost is the variable cost per unit of output. Due to the occurrence of increasing returns to variable factors, the average variable cost falls as production increases from zero to normal capacity output. However, due to the law of diminishing returns, average variable costs will rise steeply. If we draw an average variable cost curve, we can see that it first falls, then reaches a minimum, and then rises, as shown in the above figure.

Average total cost (ATC): Average total cost is the sum of average variable cost and average fixed cost. I.e.,

 ATC = AFC + AVC. It is the total cost divided by the number of units produced, i.e., ATC = TC/Q. The average total cost curve behavior depends upon the behavior of the average variable cost curve and the average fixed cost curve. Because both the AVC and AFC curves fall in the beginning, the ATC curve will also fall sharply. When the AVC curve starts to rise but the AFC curve remains sharply falling, the ATC curve continues to fall. This is because the decline in the AFC curve is greater than the rise in the AVC curve at this point, but as output rises further, the rise in AVC more than offsets the decrease in AFC. As a result, the ATC curve begins to decline, hits its minimum, and then begins to rise. As a result, the average total cost curve is like a “U” shaped curve.

A marginal cost is the addition made to the total cost by the production of an additional unit of output. To put it another way, it’s the entire cost of producing t units rather than t-1 units, where t is any given number. For example, if we are producing 5 units at a cost of 200 and then the 6th unit is created at a total cost of 250, the marginal cost is 250 – 200, or 50. If 10 units are produced at a total cost of 320, the marginal cost will be 24 [(320-200)/(10-5)]. It’s worth noting that marginal costs are unaffected by fixed costs. This is due to the fact that fixed costs do not fluctuate with output. Only the variable costs change in response to a change in output level in the short run. As a result, increases in variable costs are the cause of marginal costs. Symbolically marginal cost may be written as:

Where, 𝝙TC = Change in Total Cost

             𝝙Q=     Change in Output

Or,  MCn = TCn – TCn-1

In the beginning, the marginal cost curve falls as output rises. After a certain level of output, it starts rising. Because of the influence of the law of changing proportions, this occurs. The MC curve becomes the minimum corresponding to the point of inflection on the total cost curve. The fact that marginal product rises first, reaches a maximum and then declines ensures that the marginal cost curve of a firm declines first. It reaches its minimum and then rises. In other words, the marginal cost curve of a firm is “U” shaped. As shown in the above figure, 

The behavior of these costs has also been shown in the table given below:

Various Costs Table:

The above table shows that:

• Fixed costs do not vary when output increases up to a given level.  As a result, with every increase in output, the average fixed cost decreases.
• Variable costs increase, but not necessarily in the same proportion as the increase in output. In the example above, the average variable cost continuously decreases until 4 units are produced. After that, it starts to rise.
• The additional cost is divided by the number of additional units produced to arrive at the marginal cost.This also comes down first and then starts increasing.

 The relationship between marginal cost and average cost  is the same as that between any other marginal-average quantity. The following are the points of relationship between the two:

• A marginal  cost is less than the average cost when the average cost falls as  a result of increased output.
• When the average cost rises due to an increase in output, the marginal cost is greater than the average cost.
• When the average cost is the minimum, the marginal cost equals the average cost. To put it another way, the marginal cost curve cuts the average cost curve at its minimum point (i.e., the optimum point).

The above figure confirms the above relationship.

Long run Average cost curve:

In contrast to the short run, where some inputs are fixed and others are variable, the long run is a span of time during which the firm can vary all of its inputs. In other words, while the firm is bound to a specific plant in the short term, it can develop any size or scale of plant in the long run and so migrate from one plant to another; it can acquire a large plant if it wants to boost output and a small plant if it wants to decrease output.The firm plans ahead to create the most appropriate scale of plant to deliver the future level of output since the long run is a planning horizon. It’s important to remember that once a firm has built a certain scale of plant, production will increase in the short run. To put it another way, the firm actually operates in the short run while planning for the long run. When all individual factors are variable, the long-run cost of production is the least possible cost of producing any given level of output.A long run cost curve depicts the functional relationship between output and the long-run cost of production. In order to understand how the long run average cost curve is derived, we consider three short run average cost curves, as shown in the below figure. SACs (short-run average cost curves) are sometimes known as “plant curves.” Given the size of the plant, the firm can operate on any short-run average cost curve in the short term. Suppose there are just three plants that are theoretically feasible. The firm will change the quantity of variable inputs to increase or decrease its output based on the size of the plant. However, in the long run, the firm chooses between the three plant sizes shown by the short-run average curves (SAC1, SAC2, and SAC3). In the long run, the firm will determine the plant size or short-run average cost curve it should operate to produce a given level of output at the lowest overall cost. The figure shows that the firm will operate on SAC1 up to OB’s amount of output, while it might also produce using SAC2. Production on SAC1 results in a lower cost than production on SAC2 up to the OB amount of output. For example, if the level of output OA is produced with SAC1, it will cost AL per unit. However, if it is produced with SAC2, it will cost AH, indicating that AH is more expensive than AL. Similarly, if the firm plans to produce an output that is larger than OB but less than OD, SAC1 will not be economical to produce on SAC1. The firm will have to use SAC2 for this. Similarly, SAC3 will be used for output greater than OD. It is evident that, in the long run, the firm has a choice in the employment of plants, and it will employ the plant that yields the minimum possible unit for producing a given output.

Short Run Average Cost Curves:

Long Run Average Cost Curves:

Assume the firm has the option of varying a plant by arbitrarily small gradations, resulting in an unlimited number of plants corresponding to which there are numerous  average cost curves. The long-run average cost curve will be a smooth curve enveloping all these short run average cost curves. As shown in the above figure, the long-run average cost curve is drawn in such a way that it is tangent to each of the short-run average cost curves. Every point on the long-term average cost curve will be tangent to some short-term average cost curve. If a firm desires to produce an output, it then builds a corresponding plant for it and operates on the corresponding short-run average cost curve. The corresponding point on the LAC curve for Producing OM is G, as indicated in the figure above, and the short-run average cost curve SAC2 is tangent to the long-run AC at this point. As a result, if a firm desires to produce an output of OM, it will construct a plant that corresponds to SAC2 and operate on this curve at point G. Similarly, the company will produce different levels of output by selecting the plant that best meets its needs at the lowest possible cost of production. As seen in the above figure, larger production can be produced at the lowest cost with larger plants, whereas smaller output can be produced at the lowest cost with smaller plants, as seen in the above figure.For example, the firm will only utilize SAC2 to produce OM; if it uses SAC3, the unit cost will be more than SAC2. But, larger outputs of OV can be produced most economically with a larger plant represented by the SAC3. If we produce OV with a smaller plant, the cost per unit will be higher. Similarly, due to its limited capacity, producing a larger output with a smaller plant will result in higher costs.It is to be noted that the LAC curve is not tangent to the minimum points of the SAC curves. When the LAC curve is declining, it is tangent to the falling portions of the short-run cost curves; when the LAC curve is rising, it is tangent to the rising portions of the short-run cost curves. As a result, the firm will construct the relevant plant and operate it at less than full capacity, i.e., at less than its minimum average cost of production, in order to produce output of less than “OQ” at the lowest possible unit cost. On the other hand, for outputs larger than OQ, the firm will construct a plant and operate it beyond its optimum capacity. “OQ” is the “optimal output.” This is due to the fact that “OQ” is formed at the minimum point of LAC and the corresponding SAC, i.e., SAC4. Other plants are either used at less than their full capacity or more than their full capacity. Only SAC4 is being operated at the minimum point.Because a firm plans to produce any output in the long run by selecting a plant on the long run average cost curve corresponding to the given output, the long run average cost curve is commonly referred to as the “planning curve.” The long-run average cost curve aids the firm in determining the plant size that will produce a specific output at the least possible cost.

An explanation of the “U” shape of the long-run average cost curve: As has been seen in the diagram, the LAC curve is a “U” shaped curve. This shape of the LAC curve has nothing to do with the U-shaped SAC, which is due to variable factor ratio because in the long run, all factors are variable.Returns to scale result in a U-shaped LAC. As previously said, returns to scale increase when a firm expands. The returns to scale finally drop after a range of constant returns to scale. The LAC curve follows the same path, first declining and then finally rising. Decreasing returns to scale induce a fall in long-term average cost, while decreasing returns to scale result in a rise in long-term average cost. Internal and external economies of scale result in falling long run average costs and increasing economies of scale, while internal and external diseconomies of scale result in rising long run average costs and diminishing returns to scale.The long-run average cost curve falls initially, with increased output at first, then rises after a certain point, forming a boat shape. The long-run average cost (LAC) curve is also known as the firm’s planning curve since it aids in the selection of an appropriate plant for a decided level of output. Because it envelopes or supports a family of short-run average cost curves from below, the long-run average cost curve is also known as the “Envelope curve.”The above figure, depicting the long-run average cost curve, is arrived at on the basis of traditional economic analysis. It is flattened and ‘U’ shaped. This type of curve can exist only when the state of technology remains constant. However, empirical evidence suggests that modern firms face a “L-shaped” cost curve across a considerable quantity of output. The L-shaped long run cost curve indicates that as output increases as a result of increased plant size (and associated variable factors), per unit cost declines significantly due to economies of scale. Even with a sufficiently large scale of output, the long-run average cost curve does not increase since it continues to enjoy economies of scale.

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