Chapter 21: Cost Curves

Transcription

1 Econ Microeconomic Analysis Chapter : Cost Curves Instructor: Hiroki Watanabe Spring Watanabe Econ Cost Curves / 6 Introduction Cost Curves Time Frame Summary Watanabe Econ Cost Curves / 6 Introduction PMP & CMP Review Big Guys, Small Guys Cost Curves Time Frame Summary Watanabe Econ Cost Curves / 6

2 PMP & CMP Review Recast: Problem. (Profit Maximization Problem in Terms of Input ) Long Run PMP Problem. in Ch.9: Short Run PMP Problem. in Ch.9: Watanabe Econ Cost Curves / 6 PMP & CMP Review CMP (Ch.) relates y directly to total cost (). Watanabe Econ Cost Curves / 6 PMP & CMP Review Then we can write Problem. (Profit Maximization Problem in Terms of Output y) Watanabe Econ Cost Curves 6 / 6

3 Big Guys, Small Guys Warning. (Big Guys, Small Guys) All the graphs in this chapter fall into either one of the following. Keep track. The big guy features unprocessed, raw economic variables. The small guy represents processed information obtained either by taking the average or differential of the corresponding big guy. Watanabe Econ Cost Curves 7 / 6 Big Guys, Small Guys big guy profit, producer surplus total revenue total cost variable cost fixed cost small guy marginal profit marginal revenue, price marginal cost, wage, average cost marginal cost, wage, average variable cost average fixed cost supply curve Watanabe Econ Cost Curves 8 / 6 Big Guys, Small Guys Conversion for average Big guy divide by quantity small guy. Big guy multiply by quantity small guy. for marginal Big guy take a difference small guy. Big guy take an integral (area size) small guy. Watanabe Econ Cost Curves 9 / 6

4 Big Guys, Small Guys Example. (Big and Small for Fixed Cost) Suppose Jack s fixed cost of production is $. Big guy: fixed cost FC =. Small guy: average fixed cost AFC(y) := FC y = y. Watanabe Econ Cost Curves / 6 Big Guys, Small Guys Fixed Cost (dollars) 6 Watanabe Econ Cost Curves / 6 Big Guys, Small Guys Average Fixed Cost (dollars) 6 6 Watanabe Econ Cost Curves / 6

5 Introduction Cost Curves Variable Cost & Fixed Cost Average Cost & Average Variable Cost & Reconstructing Costs Time Frame Summary Watanabe Econ Cost Curves / 6 Variable Cost & Fixed Cost Consider a short-run Jack first. He can t change the kitchen size K. splits into two parts: = C C + K K = VC(y) + FC. Watanabe Econ Cost Curves / 6 Variable Cost & Fixed Cost Definition. (Two Parts of Total Cost) Total cost consists of Variable cost VC(y) is Jack s expenditure to cover the cost of variable inputs (e.g., labor cost). Fixed cost FC is Jack s outlay on fixed inputs (e.g., capital cost). Note TC(y = ) = VC(y = ) + FC. Watanabe Econ Cost Curves / 6

6 Variable Cost & Fixed Cost Example. (Cost Structure) Jack produces cheesecakes y (output) according to y = ƒ ( C, K ) = C K. Factor price is = ( C, K ) = (, ) and cheesecake sells for $ apiece. If he rents out K = hand mixers to produce y =, or cheesecakes: a y Mixers K Chefs C Fixed cost FC (capital cost) Variable cost VC(y) (labor cost) Total cost a This is a rerun of Example.6 in Ch.9 PMP. Watanabe Econ Cost Curves 6 / 6 A little drawback of recast from to y: marginal cost is a bit hard to come by now. Marginal cost in terms of C was just C Definition. ( in Terms of y) Marginal cost MC(y) denotes the additional cost required to produce one more cheesecake. See Appendix for more details. Watanabe Econ Cost Curves 7 / 6 Example. () Take Example. in Ch. CMP. Jack s production technology is ƒ ( C, K ) = C K, with MRTS( C, K ) = K. Factor price is C = ( C, K ) = (., ). Suppose that K = in the short run. Watanabe Econ Cost Curves 8 / 6

7 y Chef C Kitchen K Short-run MC(y). + =... + = = =. Watanabe Econ Cost Curves 9 / 6. Short Run. Total Cost ($)... Watanabe Econ Cost Curves / 6 Fact. (Slope of Is MC(y)) In general, the slope of marks the marginal cost at y: MC(y) = Δ. Δy Watanabe Econ Cost Curves / 6

8 . Short Run MC(y). ($).. Watanabe Econ Cost Curves / 6 Marginal cost is not always flat. Watanabe Econ Cost Curves / 6 6 MC(y) Costs [$] 6 8 Cheesecake y [slices] Watanabe Econ Cost Curves / 6

9 Recall Definition. in Ch. CMP: Definition.6 (Average Cost) Jack s average cost is a unit production cost defined by: AC(y) :=. y Also, Fact. in Ch. CMP: Fact.7 (Average Cost on the Graph) Average cost at y appears as the slope of a ray connecting (, ) to (y, ). Watanabe Econ Cost Curves / 6 AC(y) Costs (dollars) 6 8 Watanabe Econ Cost Curves 6 / 6 Total Cost, Unit Cost (dollars) AC(y) Watanabe Econ Cost Curves 7 / 6

10 Average Cost & Proposition.8 (Average Cost & ) If average cost has the smallest value AC(y MES ), Superscript MES stands for minimum efficient scale (cf. Ch.). Proof. See Appendix. Watanabe Econ Cost Curves 8 / 6 Average Cost & If MC(y) < AC(y), an additional slice costs less than the current unit cost. Then AC must fall. I.e., AC has not reached its minimum yet. IF MC(y) > AC(y), an additional slice cost more than the current unit cost. Then AC must increase. I.e., y went past the scale where AC is the smallest. If y MES exists, then MC(y MES ) cannot be higher nor lower than AC(y MES ). Watanabe Econ Cost Curves 9 / 6 Average Cost & 6 AC(y) MC(y) Smallest AC Costs (dollars) 6 8 Watanabe Econ Cost Curves / 6

11 Average Variable Cost & Definition.9 (Average Variable Cost) Average variable cost is unit production cost associated with variable component of total cost: AVC(y) = VC(y) FC =. y y Fact. (Average Variable Cost on the Graph) Average cost at y appears as the slope of a ray connecting (, ) to (y, VC(y)). Watanabe Econ Cost Curves / 6 Average Variable Cost & VC(y) AVC(y) TC, VC, AVC (dollars) Watanabe Econ Cost Curves / 6 Average Variable Cost & Proposition. (Average Variable Cost & ) If average variable cost has the smallest value AVC(y SD ), Proof. Superscript SD stands for shutdown point (Cf. Ch.). Similar to Appendix. Watanabe Econ Cost Curves / 6

12 Average Variable Cost & If AVC(y) > MC(y): increasing y costs Jack only MC(y), which is less than the ongoing unit production cost without FC. If AVC(y) < MC(y): increasing y costs Jack MC(y), which is more than the current unit production cost without FC. If AVC(y) = MC(y): Unit production cost (less FC) becomes smallest when it is the same as the cost of producing an additional cheesecake. Watanabe Econ Cost Curves / 6 Average Variable Cost & AVC(y) MC(y). MinAVC Costs (dollars) Watanabe Econ Cost Curves / 6 Average Variable Cost & To summarize AC(y MES ) = MC(y MES ) at the bottom of AC(y). AVC(y SD ) = MC(y SD ) at the bottom of AVC(y). Watanabe Econ Cost Curves 6 / 6

13 Average Variable Cost & AC(y) AVC(y). MC(y) MinAC MinAVC Costs (dollars). 6 8 Watanabe Econ Cost Curves 7 / 6 Average Variable Cost & What s all the fuss about minimum efficient scale and shutdown point business? Jack s decision making is two-fold: Am I making a positive profit? (Need to know y MES in the long run and y SD in the short run) If so, how many cheesecakes should I produce? (Need to know marginal cost) More on this in Ch.. Watanabe Econ Cost Curves 8 / 6 Average Variable Cost & Note minimum efficient scale and shutdown point may not exist. Recall increasing returns to scale technology in Ch. CMP. For more on this, see Ch.. Watanabe Econ Cost Curves 9 / 6

14 Average Variable Cost &. AC(y) AVC(y) MC(y) Costs (dollars) Watanabe Econ Cost Curves / 6 Reconstructing Costs Definition. (Average Fixed Cost) Average fixed cost is defined by AFC(y) = FC y. Watanabe Econ Cost Curves / 6 Reconstructing Costs These become handy when computing costs as an area size on a graph. VC(y) = AVC(y) y. FC = AFC(y) y. Watanabe Econ Cost Curves / 6

15 Reconstructing Costs 6 VC(6) VC(y) AVC(y) Costs [$] 6 8 Cheesecake y [slices] Watanabe Econ Cost Curves / 6 Reconstructing Costs 6 Costs (dollars) FC(6) FC AFC(y) 6 8 Watanabe Econ Cost Curves / 6 Reconstructing Costs. Costs [$].. FC(6) VC(6) A AVC(y). 6 8 Cheesecake y [slices] Watanabe Econ Cost Curves / 6

16 Introduction Cost Curves Time Frame Short-Run Total Cost under Different Fixed Inputs Long-Run Total Cost Short-Run Total Cost Long-Run AC Short-Run AC Summary Watanabe Econ Cost Curves 6 / 6 Short-Run Total Cost under Different Fixed Inputs What if K is not fixed? Watanabe Econ Cost Curves 7 / 6 Short-Run Total Cost under Different Fixed Inputs A firm has a different shot-run total cost curve for each possible short-run circumstances. Recall S = C C + K K. Suppose the firm can be in one of 7 possible environments: K =,, 7. Watanabe Econ Cost Curves 8 / 6

17 Short-Run Total Cost under Different Fixed Inputs If Jack has a small kitchen, his initial investment is small (small K and small FC), but then their kitchen will get crowded early on. small FC, rapid short-run total cost increase, high MC(y). If Jack has a large kitchen, his initial investment is large (large K and large FC), but their kitchen will not get crowded early on. large FC, slow short-run total cost increase, low MC(y). Watanabe Econ Cost Curves 9 / 6 Short-Run Total Cost under Different Fixed Inputs 6 7 Kitchen Size x K 8 Total Cost [$] Cheesecake y [slices] Watanabe Econ Cost Curves / 6 Long-Run Total Cost Short-Run Total Cost Jack has 7 short-run total cost curves. Start with a small kitchen K =. Start with a medium size kitchen K =. Start with a large kitchen K = 7. In the long run, Jack is free to choose among 7 short-run total cost curves. I.e., he can choose any of K. What does his long-run total cost look like? i.e, in the long run, he can simply remove the bar in K. Watanabe Econ Cost Curves / 6

18 Long-Run Total Cost Short-Run Total Cost Jack 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 cost curves. Watanabe Econ Cost Curves / 6 Long-Run AC Short-Run AC For any output y, the long-run total cost curve always gives the lowest possible total production cost. Therefore, the long-run average cost curve must always give the lowest possible average total production cost. The long-run average cost curve must be the lower envelope of all of the Jack s short-run average cost curves. Watanabe Econ Cost Curves / 6 Long-Run AC Short-Run AC 7 6 Average Cost [$] 67 Kitchen Size x K Cheesecake y [slices] Watanabe Econ Cost Curves / 6

19 Introduction Cost Curves Time Frame Summary Watanabe Econ Cost Curves / 6 Average cost and its relation to marginal cost in short-run and long-run environments. Watanabe Econ Cost Curves 6 / 6 Appendix : in Terms of y Recall marginal product is an additional cheesecake produced by hiring one more chef. Then is additional number of chefs to MP C( C, K ) produce one more slice of cheesecake. Since it cost C to hire one chef, Confirm: C MP C ( C, K ) = MC(y). MP C ( C, K ) = Δy MC(y) = Δ Δy Δ C = Δ( C C ) Δy = Δ C Δy Watanabe Econ Cost Curves 7 / 6

20 Appendix : in Terms of y Diminishing marginal product: additional chef contributes less as Kayak s increases the number of chefs. Diminishing marginal product implies increasing marginal cost: MC(y) = C MP C ( C (y), K). () Watanabe Econ Cost Curves 8 / 6 Appendix : Proof Proof. Proof of Proposition.8 : At the minimum efficient scale y MES, AC(y) does not increase nor decrease. That is, dac(y) dy = : dac(y) d dy = dy y = d dy y + d dy y = MC(y)y y = y MC(y MES ) = AC(y MES ). Watanabe Econ Cost Curves 9 / 6 AC(y), see average cost AFC(y), see average fixed cost AVC(y), see average variable cost average cost, 8,, 8, 9, 6 long-run, short-run, average fixed cost, 8,, average variable cost, 8,,, 6, factor price, 6 FC, see fixed cost fixed cost, 8,, 6,, 9 increasing returns to scale, 9 marginal cost, 7, 8,,, 9,, 6, 8 marginal profit, 8 marginal revenue, 8 MC(y), see marginal cost minimum efficient scale, 8, 6, 8, 9 output, 6 profit, 8 profit maximization problem,, 6 shutdown point,, 6, 8, 9, see total cost total cost,, 8, 6, long-run, short-run, 8, total revenue, 8 variable cost, 8,, 6, VC(y), see variable cost, see factor price wage, 8 y, see output y MES, see minimum efficient scale y SD, see shutdown point