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Management Science
MSCI 2150
Sarrah Vakili

Recall that all factors, particularly capital, are variable in the long-run. There are two ways (plus combinations of the two ways) that Capital can change in the long-run. 1. Change in the size of individual firms through additional or improved equipment and buildings or through reduction or depreciation of equipment and buildings. 2. Entry into or exit from the industry by firms of the same size as existing firms. Economies of Scale: We analyze the effect of changes in the size of individual firm capital through the concept of economies of scale. Definition. Economies of Scale (Increasing Returns to Scale) occur when a % increase in all factor inputs causes a greater % increase in output. E.g. Suppose that 5 units of Labour and 3 units of Capital produce 100 units of output but that 10 units of Labour and 6 units of Capital produce 250 units of output. => 100% increase in factor inputs produces a 150% increase in output. Why does the concept of Economies of Scale not contradict Eventually Diminishing Returns (MP)? Definition: Diseconomies of Scale (Decreasing Returns to Scale) occurs when a % increase in all factors causes a smaller % increase in output. Definition: Constant Returns to Scale occurs when a % increase in all factors causes the same % increase in output. Average Cost decreases with Economies of Scale The key element of economies of scale is a decrease in Average Cost with increases in output. Proof: Recall that Average Cost = ACo = (wL + rK)/q Define ‘a’ as the % change in inputs and ‘b’ as the % change in outputs. - 1 - => AC =1(waL + raK)/bq = a/b(wL + rK)/q = a/b * ACo a < b => Economies of Scale (Increasing Returns to Scale) => Average Cost decreases a = b => Constant Returns to Scale => no change in Average Cost a > b => Diseconomies of Scale (Decreasing Returns to Scale) => Average Cost increases Derivation of Long-run Average Cost from Short-run Average Cost Curves Since each quantity of capital gives specific short-run cost functions, we can derive long- run average cost from the short-run average cost function of each capital. Quantity 20 40 60 80 100 120 ACo (Ko) $20 $14 $10 $15 $22 $30 AC 1K >1o) $30 $20 $12 $8 $10 $14 AC (K >K ) $40 $25 $15 $10 $6 $8 2 2 1 AC 3K >3 )2 $60 $45 $32 $20 $14 $10 Cost/unit Long-run Average Cost AC 2 AC 3 AC1 ACo LRAC Quantity Average Cost falls to minimum and then rises for each Capital. Given that fixed cost is larger for larger capitals, average cost for small output is greater for larger capitals relative to smaller capitals. Average Cost is eventually lower for larger capitals relative to smaller capitals - 2 - if there is economies of scale but minimum average cost will rise if there are diseconomies of scale. Note that falling long-run average cost implies economies of scale and rising long-run average cost implies diseconomies of scale. Long-run Average Cost tends to initially decrease due to increasing returns to scale with increased capital and labour and eventually increase due to decreasing returns to scale with increased capital and labour. There is only one minimum average cost attainable though it may occur with different amounts of capital and labour. The smallest capital that gives this minimum average cost is called Minimum Efficient Scale. Long-run Average Cost (LRAC) Long-run Average Cost shows the lowest average cost for each output in the long-run. This defines the size of capital for lowest average cost for each output. If capital is infinitely divisible, Long-run Average Cost is a smooth curve with each point from the short-run average cost of each capital. Average Cost Increasing Decreasing Returns Returns Minimum Efficient Scale q Profit Maximization in the long-run implies that firms change capital and labour until they find the capital with the Short-run Average Cost function that gives minimum Long-run Average Cost for the desired output. - 3 - The diagram below represents the Long-run Average Cost curve for an industry and three sizes of capital in the industry with their attendant Short-run Average Cost curves. Provided that there is no technological change (i.e., no new minimum efficient scale), examination of the diagram shows that in competitive conditions price will fall from Po to P* as new firms enter the industry due to economic profits at Po for all capitals and above P* for the minimum efficient scale capital. In the long-run, no firm could obtain the average return on capital unless the firm was the size (capital) that permitted Minimum Efficient Scale. AC, P Po MES P* q This derivation shows that ‘perfect competition’ – all firms are the same size with similar cost functions – is an evolutionary result of a competitive market provided that there is no technological change. The assumption of perfect competition means that we do not need to draw the Long-run Average Cost function since we know that its minimum is minimum Short-run Average Cost of the firm. Technological change alters the minimum efficient scale, usually increasing it but sometimes decreasing it drastically. Electricity generation was dominated by $Billion plus plants (Hydro, Nuclear, or Coal) up until the 1970s since these huge plants produced the cheapest electricity. The development of Co-generation Gas Turbines in the 1970s introduced small jet engines that could produce electricity competitively because their small size and clean - 4 - combustion of natural gas meant that they could be located in buildings to take advantage of the heat generated in combustion. The low natural gas prices of the 1980s and 1990s made these co- generators a formidable competitive of the huge plants, sparking the collapse of monopoly pricing in various jurisdictions, especially California. LONG-RUN COMPETITIVE EQUILIBRIUM We will assume perfect competition without technological change in this course so we can ignore the possibility of changes in the size of firm capital. Changes in capital can therefore only occur through the entry into or exit from the industry of perfectly competitive firms. Long-run Equilibrium => No change in Capital despite the possibility of Capital Change => No firms enter or exit the industry => 0 Economic Profit => P = min AC (Price = minimum Average Cost) We express long-run equilibrium graphically through the firm/industry diagram Demand and Supply intersecting at minimum Average Cost as in the following diagram. Long-Run Equilibrium Cost/unit Px MC So AC AC0 P0 Do q0 qx Q0 Qx E.g. Suppose that a perfectly competitive firm has the following cost functions. VC = 2q + 30q FC = 1800 MC = 4q + 30 => AVC = VC/q = 2q + 30 AC = TC/q = 2q + 30 + 1800/q - 5 - 1. What is firm Supply? Firm Supply = MC > min AVC => P = 4q + 30 2. What is industry Supply if there are n firms? Industry Supply => P = 4(Q/n) + 30 3. What is long
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