Fixed, Variable & Total Cost Functions
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1 Cost Curves
2 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 vary 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.
3 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 c ( ). v
4 $ c v () F
5 $ c() c( ) F c ( ) v c v () F F
6 Av. Fixed, Av. Variable & Av. Total Cost Curves The firm s total cost function is c( ) F c ( ). For y > 0, the firm s average total cost function is v AC( ) F c v ( ) AFC( ) AVC( ).
7 $/output unit AFC() 0 as AFC() 0
8 Av. Fixed, Av. Variable & Av. Total Cost Curves In a short-run with a fixed amount of at least one input, the Law of Diminishing (Marginal) Returns must apply, causing the firm s average variable cost of production to increase eventually.
9 $/output unit AVC() AFC() 0
10 $/output unit Since AFC() 0 as, ATC() AVC() as ATC() AFC AVC() AFC() 0
11 Marginal Cost Function Marginal cost is the rate-of-change of variable production cost as the output level changes. That is, MC( ) c v ( ).
12 Marginal & Average Cost Functions How is marginal cost related to average variable cost?
13 Marginal & Average Cost Functions Since, ) ( ) ( c AVC v. ) ( 1 ) ( ) ( 2 c MC AVC v Therefore, 0 ) ( AVC ). ( ) ( c MC v as ). ( ) ( ) ( AVC c MC v as 0 ) ( AVC
14 $/output unit MC() AVC()
15 Marginal & Average Cost Functions The short-run MC curve intersects the short-run AVC curve from below at the AVC curve s minimum. And, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC curve s minimum.
16 $/output unit MC() ATC() AVC()
17 Short-Run & Long-Run Total Cost Curves 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.
18 $ F F 0 F = w 2 x 2 F = w 2 x 2 c s (;x 2 ) A larger amount of the fixed C s (;x 2 ) input increases the firm s fixed cost. Why does a larger amount of the fixed input reduce the slope of the firm s total cost curve?
19 Short-Run & Long-Run Total Cost Curves MP 1 is the marginal physical productivity 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 MC w one extra unit of output is 1. MP1
20 Short-Run & Long-Run Total Cost Curves 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.
21 $ F = w 2 x 2 F = w 2 x 2 F = w 2 x 2 c s (;x 2 ) c s (;x 2 ) F F F 0 c s (;x 2 )
22 Short-Run & Long-Run Total Cost Curves 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 eual to any of x 2, x 2, or x 2. How does the firm make this choice?
23 $ For 0, 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 ) F c s (;x 2 ) F F 0
24 $ For 0, choose x c s (;x 2 ) 2 = x 2. For, choose x 2 = x 2. For, choose x 2 = x 2. c s (;x 2 ) F F F 0 c s (;x 2 ) c(), the firm s longrun total cost curve.
25 Short-Run & Long-Run Total Cost Curves 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.
26 Short-Run & Long-Run Total Cost Curves If input 2 is available in continuous amounts then there is an infinity 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.
27 Short-Run & Long-Run Average Total Cost Curves For any output level, the long-run total cost curve always gives the lowest possible total production cost. Therefore, the long-run av. total cost curve must always 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.
28 Short-Run & Long-Run Average Total Cost Curves 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...
29 $/output unit AC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 )
30 $/output unit 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.
31 Short-Run & Long-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.
32 $/output unit AC s (;x 2 ) MC s (;x 2 ) MC s (;x 2 ) AC s (;x 2 ) AC s (;x 2 ) MC s (;x 2 )
33 $/output unit 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()
34 Short-Run & Long-Run Marginal Cost Curves For any output level > 0, the longrun marginal cost is the marginal cost for the short-run chosen by the firm. This is always true, no matter how many and which short-run circumstances exist for the firm.
35 Short-Run & Long-Run Marginal Cost Curves So for the continuous case, where x 2 can be fixed at any value of zero or more, the relationship between the long-run marginal cost and all of the short-run marginal costs is...
36 Short-Run & Long-Run Marginal Cost $/output unit SRACs Curves AC()
37 Short-Run & Long-Run Marginal Cost $/output unit SRMCs Curves AC()
38 Short-Run & Long-Run Marginal Cost $/output unit SRMCs Curves MC() AC() y For each > 0, the long-run MC euals the MC for the short-run chosen by the firm.
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