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Table of Contents ____________________________________________________________________________________________________ Subject ECONOMICS Paper No and Title 3- Fundamentals of Microeconomic Theory Module No and Title 19 -Derivation of cost functions Module Tag ECO_P3_M19 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Derivation of Cost Functions 3.1 Derivation of Average fixed cost curve from Total fixed cost curve 3.2 Derivation of Average variable cost from Total Variable cost curve 3.3 Derivation of Average Total cost curve from Total cost curve 3.4 Derivation of marginal cost curve from total cost and total variable cost curves 4. Summary ECONOMICS PAPER No. : Fundamentals of Microeconomic Theory MODULE No. : Derivation of cost functions ____________________________________________________________________________________________________ 1. Learning Outcomes After studying this module, you shall be able to • Know how to derive the different cost curves graphically • Learn the interrelationship between thecost concepts • Identify the average, marginal and total cost concepts • Understand the importance of the cost concepts 2. 2. INTRODUCTION DERIVATION OF THE SHORT – RUN AVERAGE AND MARGINAL COST CURVES FROM TOTAL COST CURVES: Whether it is production or cost, there are three categories of concepts – total, average and marginal. In the short run, there are three types of total cost curves: total fixed cost curve, total variable cost curve and total cost curve whereas in the long run, there is no fixed cost, only variable and total cost. The average and marginal cost curves can be geometrically derived from a total cost curve. 3.1DERIVATION OF AVERAGE FIXED COST CURVE FROM THE TOTAL FIXED COST CURVE The Total fixed cost (TFC) curve is parallel to the X-axis since the costs remain the same irrespective of the level of output. In the table below, we find TFC is Rs 240 at all levels of output. The Average Fixed cost (AFC) equals TFC divided by output. Geometrically, at the output level of one unit, AFC is equal to the slope of ray OA and refers to point’a’ on the AFC curve. At output level of two units, AFC is equal to the slope of ray OB and refers to point ‘b’ on the AFC curve .Similarly; we can plot for other levels of output and get the corresponding points on the AFC curve, as shown in Figure 1. The AFC curve is a rectangular hyperbola – it is asymptotic to the axes, which means that as the AFC curve moves further away from the origin along the axis, it gets closer to the axis but never touches it. TABLE 1 ECONOMICS PAPER No. : Fundamentals of Microeconomic Theory MODULE No. : Derivation of cost functions ____________________________________________________________________________________________________ Output TFC TVC TC AFC AVC ATC MC 0 240 0 240 240 - 240 - 1 240 120 360 240 120 360 120 2 240 160 400 120 80 200 40 3 240 180 420 80 60 140 20 4 240 212 452 60 53 113 32 5 240 280 520 48 56 104 68 6 240 420 660 40 70 110 140 7 240 640 880 30.43 90.14 120.57 220 FIGURE 1 3.2 DERIVATION OF AVERAGE VARIABLE COST CURVE FROM TOTAL VARIABLE COST CURVE: The Average Variable cost (AVC) is equal to the TVC divided by output. The AVC is equal to Rs 120 at output level of one unit (TVC is Rs 120). It is represented by the slope of the ray OA and is plotted as point ‘a’ on the AVC curve. At output level of three units, AVC is Rs 60, and is plotted as point ‘c’ on the AVC curve. Similarly, we can plot other points on the AVC curve. The AVC curve is downward sloping till point‘d’ and then slopes upward. The slope of the ray OCfrom the origin to the TVC curve is the lowest at point ‘ C’ on the TVC curve which implies that AVC at the output level of 4 units is the lowestat d( figure no . 2 ) ECONOMICS PAPER No. : Fundamentals of Microeconomic Theory MODULE No. : Derivation of cost functions ____________________________________________________________________________________________________ FIGURE 2 3.3 DERIVATION OF AVERAGE TOTAL COST CURVE FROM TOTAL COST CURVE: When the level of output is 0, there is no variable cost. Therefore, the TFC = TC=AFC. When the output level is one, TC is Rs 360 and ATC is also Rs 360. When the output level is 2, TC is Rs 400 and ATC is 400 ÷ 2 = Rs 200 . The ATC at different level of output is indicated by the slope of the ray from the origin to the TC curve . Thus, OA, OB ,OC and OD are rays whose slope indicates the ATC at the corresponding level of output. These points are plotted to obtain points ‘a’,’b’, ‘c’,’d’ and so on to get the AVC curve. The slope of a ray from the origin falls up to point ‘C’ on TC curve and rises afterwards. The ray from the origin (OC) is tangent to the TC curve at point ‘C’. Thus, ATC curve falls up to point ’C’ ( point ‘c’ is the lowest point on the ATC curve )and then rises , giving it a ‘U’shape ( figure no. 3 ) ECONOMICS PAPER No. : Fundamentals of Microeconomic Theory MODULE No. : Derivation of cost functions ____________________________________________________________________________________________________ FIGURE 3 3.4 DERIVATION OF THE MARGINAL COST CURVE FROM TOTAL COST AND TOTAL VARIABLE COST CURVE: We can derive the marginal cost curve from the total cost curve and the total variable cost curve. Let the TFC and TVC be given as below in the table no: 2, we can find the MC from it. We can draw the TC, TVC and derive the MC. TABLE 2 OUTPUT TC TFC TVC MC 0 120 120 0 - 1 180 120 60 60 2 200 120 80 20 3 210 120 90 10 4 230 120 110 20 5 270 120 150 40 6 360 120 240 90 ECONOMICS PAPER No. : Fundamentals of Microeconomic Theory MODULE No. : Derivation of cost functions ____________________________________________________________________________________________________ FIGURE 4 From the figureno.4, we find that the slopes of the TVC curve and the TC curve are the same at every level of output. Point A and A1 are the points of inflexion respectively on TC and TVC. The MC falls up to 2.5 units of output and refers to point ‘a‘on the MC curve and then rises. The MC is given by the slope of the TVC curve at point B. At 5 units of output, MC is equal to the slope of the TC curve at point C. Thus, point ‘b’ refers to the lowest AVC and point ’c’ refers to the lowest AC.Marginal cost is sometimes known as ‘incremental cost’ – as it is the increase in TC consequent upon a small increase in output. MC = ΔTC/ΔQ or ΔTVC/ΔQ. ΔTC is the change in total cost due to a small change in output ΔTVC is the change in total variable cost due to a small change in output ΔQ is the small change in output For example, the total cost of producing 4 units of output is Rs 1000 and the total cost of producing 5 units of output is Rs 1200, therefore marginal cost of the fifth unit is Rs 200( Rs 1200 – Rs 1000). ECONOMICS PAPER No. : Fundamentals of Microeconomic Theory MODULE No. : Derivation of cost functions ____________________________________________________________________________________________________ Because, fixed cost remains unchanged in the short run, therefore, marginal cost is also the increase in total variable cost due to a small increase in output. COST CURVES IN THE LONG –RUN: Since in the long-run, all factors are variable, therefore there is only variable cost. In the long run, to increase the level of production, all factors have to be increased and this results in expansion of scale. The relationship between Total, Average and marginal cost concepts is the same. In the long run, the relationship between LRMC and LRAC is the same as it exists in the short run .The derivation of TC, ATC and MC can be explained in the same manner as under short run.The Long –run Average cost curve or the LAC curve is the locus of points denoting the least cost of producing different levels of output in the long run. It shows the minimum average costs of producing the corresponding output in the long-run. The LAC curve is thus a planning curve that guides the firm in deciding on the most optimal size of the industrial plant for producing a given level of output. An optimal sized plant is one which enables the production of the output at the minimum costs per unit of output. Given the technology, the firm is free to choose the plant size which entails the least cost. For example, if the firm decides to produce OQ1 level of output, then it will choose the plant size SAC2 and not SAC1.If the demand for the firm’s output increases to OQ3, then the average costs starts increasing along the plant SAC2 and the firm decides to set up a larger plant size SAC3 to minimize its average costs of production in the long-run. The long-run average curve does not touch the short-run average cost curves on their minimum points. Graphically it can touch the minimum points of SACs only under constant returns to scale. In the phase of increasing returns to scale and decreasing cost, the LAC curve touches the SAC curves to the left of the minimum points of the SAC curves and in the phase of diminishing returns it touches the SAC curves to their right. The LAC is therefore, not the locus of lowest points of SAC curves.
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