Chapter 21: Cost Curves

Transcription

1 Econ Microeconomic Analysis Chapter : Cost Curves Instructor: Hiroki Watanabe Fall Watanabe Econ Cost Curves / Introduction Cost Curves Time Frame Summary Watanabe Econ Cost Curves / Introduction PMP & CMP Review Cost Curves Time Frame Summary Watanabe Econ Cost Curves /

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 / PMP & CMP Review CMP (Ch.) relates y directly to total cost (). Watanabe Econ Cost Curves / PMP & CMP Review Then we can write Problem. (Profit Maximization Problem in Terms of Output y) Watanabe Econ Cost Curves 6 /

3 Introduction Cost Curves Variable Cost & Fixed Cost Average Cost & Average Variable Cost & Reconstructing Costs Time Frame Summary Watanabe Econ Cost Curves 7 / 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 8 / 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 9 /

4 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 / 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 / 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 /

5 y Chef C Kitchen K Short-run MC(y). + =... + = = =. Watanabe Econ Cost Curves /. Short Run. Total Cost ($)... Watanabe Econ Cost Curves / 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 . Short Run MC(y). ($).. Watanabe Econ Cost Curves 6 / Marginal cost is not always flat. Watanabe Econ Cost Curves 7 / 6 MC(y) Costs [$] 6 8 Cheesecake y [slices] Watanabe Econ Cost Curves 8 /

7 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 9 / AC(y) Costs (dollars) 6 8 Watanabe Econ Cost Curves / Total Cost, Unit Cost (dollars) AC(y) Watanabe Econ Cost Curves /

8 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 / 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 / Average Cost & 6 AC(y) MC(y) Smallest AC Costs (dollars) 6 8 Watanabe Econ Cost Curves /

9 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 / 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 7 /

10 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 8 / Average Variable Cost & AVC(y) MC(y). MinAVC Costs (dollars) Watanabe Econ Cost Curves 9 / 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 /

11 Average Variable Cost & AC(y) AVC(y). MC(y) MinAC MinAVC Costs (dollars). 6 8 Watanabe Econ Cost Curves / 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 long run) If so, how many cheesecakes should I produce? (Need to know marginal cost) More on this in Ch.. Watanabe Econ Cost Curves / 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 /

12 Average Variable Cost &. AC(y) AVC(y) MC(y) Costs (dollars) Watanabe Econ Cost Curves / Reconstructing Costs Definition. (Average Fixed Cost) Average fixed cost is defined by AFC(y) = FC(y). y Watanabe Econ Cost Curves / 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 /

13 Reconstructing Costs 6 VC(6) VC(y) AVC(y) Costs [$] 6 8 Cheesecake y [slices] Watanabe Econ Cost Curves 7 / Reconstructing Costs 6 Costs (dollars) FC(6) FC AFC(y) 6 8 Watanabe Econ Cost Curves 8 / Reconstructing Costs. Costs [$].. FC(6) VC(6) A AVC(y). 6 8 Cheesecake y [slices] Watanabe Econ Cost Curves 9 /

14 Introduction Cost Curves Time Frame Short-Run Total Cost under Different Fixed Inputs Long-Run Total Cost Short-Run Total Cost Long-Run ATC Short-Run ATC Summary Watanabe Econ Cost Curves / Short-Run Total Cost under Different Fixed Inputs What if K is not fixed? Watanabe Econ Cost Curves / 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 /

15 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 / Short-Run Total Cost under Different Fixed Inputs 6 7 Kitchen Size x K 8 Total Cost [$] Cheesecake y [slices] Watanabe Econ Cost Curves / 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 /

16 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 ATC Short-Run ATC 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 7 / Long-Run ATC Short-Run ATC 7 Average Total Cost [$] 6 67 Kitchen Size x K Cheesecake y [slices] Watanabe Econ Cost Curves 8 /

17 Introduction Cost Curves Time Frame Summary Watanabe Econ Cost Curves 9 / Average cost and its relation to marginal cost in short-run and long-run environments. Watanabe Econ Cost Curves / 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 /

18 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 / Appendix : Proof Proof. Proof of Proposition.8 : At the minimum efficient scale y MES, A does not increase nor decrease. That is, da dy = : da d dy = dy y = d dy y + d dy y = MC(y)y y = y MC(y MES ) = ATC(y MES ). Watanabe Econ Cost Curves / AC(y), see average cost AFC(y), see average fixed cost AVC(y), see average variable cost average cost, 9,,, long-run, 7 short-run, 7 average fixed cost,, 6 average variable cost,, 8,, 6 factor price, FC, see fixed cost fixed cost, 9,, 6, increasing returns to scale, marginal cost,,,, 7,, 8,, MC(y), see marginal cost minimum efficient scale,,,, output, profit maximization problem,, 6 shutdown point, 7,,,, see total cost total cost,, 8, long-run, 6 short-run,, variable cost, 9,, 6 VC(y), see variable cost, see factor price y, see output y MES, see minimum efficient scale y SD, see shutdown point