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Cost Curves Consider the total cost function !(# ! , # " , %) , derived earlier, that gives the total cost of producing an output of % , in the cheapest way possible, when the two inputs of production cost # ! , and # " . • !(# ! , # " , %) embeds in it the solution to the cost minimization problem (how to produce an output of % , in the cheapest way possible, when the two inputs of production cost # ! , and # " ) that we studied before. Let’s study how !(# ! , # " , %) changes with % , holding # ! and # " fixed. We will suppress the dependence of !(# ! , # " , %) on input costs, and simply write it as !(%) . Suppose that the firm’s production happens in a plant of some fixed size ’ ( , and this plant size can’t be changed at the moment. We will keep this information in the background for now. Suppose the total cost function associated with this plant size is !(%) . Typically, this will have two parts: !(%) = ! # (%) + + • ! # (%) is ,-.(%) , the Total Variable Cost of producing % units. • + is ,+. , the Total Fixed Costs of production, that have to be incurred whether the firm produces anything (% > 0) or not (% = 0) . Averages Dividing both sides of the equation by % , we get a relationship between average (per unit) costs: !(%) % = ! # (%) % + + % e g claim g a withy or clan we g y.minfwi.ws or claws ye twaly Short run weapfressins the dependence of ceylon't for now e g Cly y't or Ccg too 3oy 85 5

1.(%) = 1-.(%) + 1+.(%) Thus, the Average Cost of producing % units is equal to the Average Variable Cost of producing % units plus the Average Fixed Cost of producing % units . • 1-.(%) will eventually increase with % , because, due to the presence of fixed factors of production (plant size, managerial talent, etc.), the productivity of each successive unit of the variable factors of production would be diminishing, and a greater and greater amount of the variable factors of production would be required to produce another unit of % , increasing 1-.(%). • 1+.(%) decreases with % as the fixed costs of production are spread over a larger and larger amount of output. • 1.(%) will thus initially fall, as it is pulled down by the dramatic fall in 1+.(%) at low levels of % • Notice, AFC reduces by 50% when we go from % = 1 to % = 2 , but only by 33% when we go from % = 2 to % = 3 ), but eventually the increases in 1-.(%) will overpower the decreases in 1+.(%) , pulling 1.(%) up.

Averages and Marginals FACT: the marginal pulls the average towards it: if the marginal is less (greater) than the average, the average falls (rises). At a given % , • If 8.(%) < 1.(%) , then 1. $ (%) < 0. • If 8.(%) > 1.(%) , then 1. $ (%) > 0. • At the minimum of 1.(%) , 8.(%) = 1.(%) . Verify this by setting )* !(#) # + )& = 0. Thus, the 8.(%) curve intersects the 1.(%) curve at its minimum. Since 8.(%) is the “marginal” corresponding to the 1-.(%) as well, the MC curve also intersects the AVC curve at its minimum, by a similar argument. ay me A is y May MVC Cy Verifying Y cloggy so I C'lyrcigy Mccy Acey

• It can be shown rigorously that 8.(0) = 1-.(0). See proof in chapter appendix. • Ignore the statement in the chapter that 8.(1) = 1-.(1) , which is not necessarily true while using the calculus definition of Marginal Cost above. Example: !(%) = 100 + 30A − 8A " + A , How to use the AC come to compute TC Consider some output y AC y shaded area i g ACCY TCG j Tc TFC TVC experts 54mm ay ayy y Gsm's technology Mc c'ey 30 169 35 app pics and ACCY 18 30 8g y AVCly 30 8yty2 AFC Cy 100g

Costs in the Long Run and the Short Run Relabel the cost function !(%) in the previous section as ! - D%; ’ ( F. This explicitly acknowledges the dependence of costs on the plant size ’ ( , which is fixed in the short run, which is denoted by the G subscript. Suppose that in the long run, the firm can change ’ ( , which would change its total, average and marginal cost curves. Each ’ comes with its own Short Run Average Cost (SRAC) and Short Run Marginal Cost (SMC) curves. Figure: For each output level % , denote by ’(%) the optimal plant size—meaning, the ’ that leads to the lowest average cost of production. plant size I shotten suppose a firm had Af only 3 possible plant sizes with corresponding SAC curves og.to ETg

Relationship between long run and short run cost curves. Then the Long Run Cost function of the firm is ! . (%) , where: ! . (%) = ! - D%, ’(%)F The firm produces each output level using the optimal plant size, changing its plant size according to ’(%) as it changes its output. The Long Run Average Cost curve then is the lower envelope of the firm’s Short Run Average Cost curves. The LRAC of any given output % , then, is simply the Short Run Average cost of % corresponding to a plant size of ’(%) , the optimal plant size for an output of % . The Long Run Marginal Cost of any given output % is the Short Run Marginal cost of % corresponding to a plant size of ’(%) . This can be rigorously proved (see chapter appendix). instead of 1 FACT gy.RYDEC.ly

The Shape of the LRAC Curve and Returns to Scale If the underlying technology has constant returns to scale, the LRAC curve is a straight line, as average costs don’t change with output level (e.g. doubling output requires a doubling of inputs, which entails a doubling of costs, so cost divided by output is still the same). 1 If the underlying technology has decreasing returns to scale, the LRAC curve slopes up. If the underlying technology has increasing returns to scale, the LRAC curve slopes up. The underlying technology may exhibit increasing returns to scale at low levels of output and decreasing returns to scale at high levels of output. This would generate a U-shaped LRAC curve. 1 Consider H(I ! , I " ) = √ I ! I " . Suppose I " is plant size. To derive the short run total cost curve, fix I " = I̅ " , and pick the cost minimizing I ! for an output target of % . To derive the short run average cost curve, divide by % . For different levels of I̅ " , every SRAC will have the same cost level at its minimum. LAC Rely Y flee taz Efta Nz hyrizontal down SAC SAC SAC SACY L LRAC y Nott LAC and LRAC are the same thing

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