Types of Cost Curves. Chapter Twenty-One. Types of Cost Curves. Types of Cost Curves. Fixed, Variable & Total Cost Functions
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1 Tpes of Cost Chapter Twent-One Cost A total cost curve is the graph of a firm s total cost function. A variable cost curve is the graph of a firm s variable cost function. An average total cost curve is the graph of a firm s average total cost function. Tpes of Cost An average variable cost curve is the graph of a firm s average variable cost function. An average fixed cost curve is the graph of a firm s average fixed cost function. A marginal cost curve is the graph of a firm s marginal cost function. Tpes of Cost How are these cost curves related to each other? How are a firm s long-run and shortrun cost curves related? Fixed, Variable & Total Cost Functions F is the total cost to a firm of its shortrun fixed inputs. F, the firm s fixed cost, does not var with the firm s output level. c v () is the total cost to a firm of its variable inputs when producing output units. c v () is the firm s variable cost function. c v () depends upon the levels of the fixed inputs. Fixed, Variable & Total Cost Functions c() is the total cost of all inputs, fixed and variable, when producing output units. c() is the firm s total cost function; c( ) = F+ cv ( ). 1
2 c v () F c v () c ( ) = F+ cv ( ) c() c v () F F F Av. Fixed, Av. Variable & Av. Total Cost The firm s total cost function is c( ) = F+ cv ( ). For >, the firm s average total cost function is F c AC v ( ) ( ) = + = AFC ( ) + AVC ( ). Av. Fixed, Av. Variable & Av. Total Cost What does an average fixed cost curve look like? F AFC( ) = AFC() is a rectangular hperbola so its graph looks like... 2
3 AFC() as Av. Fixed, Av. Variable & Av. Total Cost In a short-run with a fixed amount of at least one input, the Law of Diminishing (Marginal) Returns must appl, causing the firm s average variable cost of production to increase eventuall. AFC() AVC() AVC() AFC() Av. Fixed, Av. Variable & Av. Total Cost And ATC() = AFC() + AVC() ATC() = AFC() + AVC() ATC() AVC() AFC() 3
4 Since AFC() as, ATC() AVC() as. AFC() = ATC() - AVC() ATC() ATC() AFC AVC() AFC AVC() AFC() AFC() Since AFC() as, ATC() AVC() as. And since short-run AVC() must eventuall increase, ATC() must eventuall increase in a short-run. ATC() AVC() Marginal Cost Function Marginal cost is the rate-of-change of variable production cost as the output level changes. That is, c MC( ) v ( ) =. AFC() Marginal Cost Function The firm s total cost function is c ( ) = F+ cv ( ) and the fixed cost F does not change with the output level, so c MC v ( ) c ( ) ( ) = =. MC is the slope of both the variable cost and the total cost functions. Marginal and Variable Cost Functions Since MC() is the derivative of c v (), c v () must be the integral of MC(). That is, c MC v ( ) ( ) = cv ( ) = MC( z) dz. 4
5 Marginal and Variable Cost Functions cv ( ) = MC( z) dz MC() How is marginal cost related to average variable cost? Area is the variable cost of making units c Since AVC v ( ) ( ) =, AVC( ) MC( ) 1 cv ( ) =. 2 c Since AVC v ( ) ( ) =, AVC( ) MC( ) 1 cv ( ) =. 2 Therefore, AVC ( ) > = MC > as ( ) = c v ( ). < < c Since AVC v ( ) ( ) =, AVC( ) MC( ) 1 cv ( ) =. 2 Therefore, AVC ( ) > = MC > as ( ) = c v ( ). < < AVC( ) > > c MC( ) v ( ) = as = = AVC( ). < < AVC( ) > > = as MC( ) = AVC( ). < < 5
6 AVC( ) MC( ) < AVC( ) < MC() MC() AVC() AVC() AVC( ) MC( ) > AVC( ) > AVC( ) MC( ) = AVC( ) = MC() MC() AVC() AVC() AVC( ) MC( ) = AVC( ) = The short-run MC curve intersects the short-run AVC curve from MC() below at the AVC curve s minimum. c ( ) Similarl, since ATC( ) =, ATC( ) MC( ) 1 c( ) =. 2 AVC() 6
7 c ( ) Similarl, since ATC( ) =, ATC( ) MC( ) 1 c( ) =. 2 Therefore, ATC( ) > > = as MC( ) = c( ). < < c ( ) Similarl, since ATC( ) =, ATC( ) MC( ) 1 c( ) =. 2 Therefore, ATC( ) > > = as MC( ) = c( ). < < ATC( ) > > c ( ) = as MC( ) = = ATC( ). < < ATC( ) > = < as > MC( ) = ATC( ) < MC() ATC() The short-run MC curve intersects the short-run AVC curve from below at the AVC curve s minimum. And, similarl, the short-run MC curve intersects the short-run ATC curve from below at the ATC curve s minimum. MC() ATC() AVC() A firm has a different short-run total cost curve for each possible shortrun circumstance. Suppose the firm can be in one of just three short-runs; x 2 = x 2 or x 2 = x 2 x 2 < x 2 < x 2. or x 2 = x 2. 7
8 = w 2 x 2 c s (;x 2 ) = w 2 x 2 = w 2 x 2 c s (;x 2 ) c s (;x 2 ) = w 2 x 2 = w 2 x 2 A larger amount of the fixed input increases the firm s fixed cost. c s (;x 2 ) c s (;x 2 ) c = w 2 x 2 s (;x 2 ) = w 2 x 2 A larger amount of the fixed c s (;x 2 ) input increases the firm s fixed cost. Wh does a larger amount of the fixed input reduce the slope of the firm s total cost curve? MP 1 is the marginal phsical productivit of the variable input 1, so one extra unit of input 1 gives MP 1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is MP 1 is the marginal phsical productivit of the variable input 1, so one extra unit of input 1 gives MP 1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 1/ MP 1 units of input 1. 8
9 MP 1 is the marginal phsical productivit of the variable input 1, so one extra unit of input 1 gives MP 1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 1/ MP 1 units of input 1. Each unit of input 1 costs w 1, so the firm s extra cost from producing one extra unit of output is MP 1 is the marginal phsical productivit of the variable input 1, so one extra unit of input 1 gives MP 1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 1/ MP 1 units of input 1. Each unit of input 1 costs w 1, so the firm s extra cost from producing one extra unit of output is w MC = 1. MP1 MC = w 1 MP1 is the slope of the firm s total cost curve. MC = w 1 MP1 is the slope of the firm s total cost curve. If input 2 is a complement to input 1 then MP 1 is higher for higher x 2. Hence, MC is lower for higher x 2. That is, a short-run total cost curve starts higher and has a lower slope if x 2 is larger. = w 2 x 2 = w 2 x 2 = w 2 x 2 c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) The firm has three short-run total cost curves. In the long-run the firm is free to choose amongst these three since it is free to select x 2 equal to an of x 2, x 2, or x 2. How does the firm make this choice? 9
10 For, choose x 2 =? c s (;x 2 ) For, choose x 2 = x 2. c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) For, choose x 2 = x 2. For, choose x 2 =? c s (;x 2 ) c s (;x 2 ) For, choose x 2 = x 2. c s (;x 2 ) For, choose x 2 = x 2. c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) For, choose x 2 = x 2. c s (;x 2 ) For, choose x 2 = x 2. For <, choose x 2 =? c s (;x 2 ) For, choose x 2 = x 2. c s (;x 2 ) For, choose x 2 = x 2. For <, choose x 2 = x 2. c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) 1
11 For, choose x 2 = x 2. c s (;x 2 ) For, choose x 2 = x 2. For <, choose x 2 = x 2. c s (;x 2 ) c s (;x 2 ) c(), the firm s longrun total cost curve. The firm s long-run total cost curve consists of the lowest parts of the short-run total cost curves. The long-run total cost curve is the lower envelope of the short-run total cost curves. If input 2 is available in continuous amounts then there is an infinit of short-run total cost curves but the long-run total cost curve is still the lower envelope of all of the short-run total cost curves. c s (;x 2 ) c s (;x 2 ) c s (;x 2 ) c() Short-Run & Long-Run Average Total Cost For an output level, the long-run total cost curve alwas gives the lowest possible total production cost. Therefore, the long-run av. total cost curve must alwas give the lowest possible av. total production cost. The long-run av. total cost curve must be the lower envelope of all of the firm s short-run av. total cost curves. Short-Run & Long-Run Average Total Cost E.g. suppose again that the firm can be in one of just three short-runs; x 2 = x 2 or x 2 = x 2 (x 2 < x 2 < x 2 ) or x 2 = x 2 then the firm s three short-run average total cost curves are... 11
12 AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) Short-Run & Long-Run Average Total Cost The firm s long-run average total cost curve is the lower envelope of the short-run average total cost curves... AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) The long-run av. total cost AC() curve is the lower envelope of the short-run av. total cost curves. Q: Is the long-run marginal cost curve the lower envelope of the firm s short-run marginal cost curves? Q: Is the long-run marginal cost curve the lower envelope of the firm s short-run marginal cost curves? A: No. The firm s three short-run average total cost curves are... 12
13 AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) MC s (;x 2 ) AC() MC s (;x 2 ) AC() AC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) For an output level >, the longrun marginal cost of production is the marginal cost of production for the short-run chosen b the firm. MC s (;x 2 ) MC(), the long-run marginal cost curve. 13
14 AC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) MC(), the long-run marginal cost curve. For an output level >, the longrun marginal cost is the marginal cost for the short-run chosen b the firm. This is alwas true, no matter how man and which short-run circumstances exist for the firm. For an output level >, the longrun marginal cost is the marginal cost for the short-run chosen b the firm. So for the continuous case, where x 2 can be fixed at an value of zero or more, the relationship between the long-run marginal cost and all of the short-run marginal costs is... SRACs AC() SRMCs AC() SRMCs MC() AC() For each >, the long-run MC equals the MC for the short-run chosen b the firm. 14
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