Chapter 20: Cost Minimization Problem

Transcription

1 Econ 323 Microeconomic Analysis Chapter : Cost Minimization Problem Instructor: Hiroki Watanabe Spring 13 Watanabe Econ 323 CMP 1 / 91 1 Introduction Overview Cost Minimization Problem (CMP) 2 Cost Minimization Problem 3 Comparative Statics 4 Average Cost & Returns to Scale 5 Short-Run & Long-Run Total Cost 6 Summary Watanabe Econ 323 CMP 2 / 91 Overview PMP: select a production plan ( C, K, y) that makes Jack most profitable. Drawback: no sensible solution exists for 1 IRS Jack 2 CRS Jack with positive profit. Watanabe Econ 323 CMP 3 / 91

2 Overview 25. Cheesecakes y (slices) Isoprofit Prod Fn Chef x C Watanabe Econ 323 CMP 4 / 91 Overview Discussion 1.1 (IRS Jack s Decision) If PMP fails an IRS Jack, what does he do instead? Watanabe Econ 323 CMP 5 / 91 Overview Source of problem with IRS: 1 An exponential growth in total revenue 2 Only a proportional increase in total cost IRS Jack exhausts the market demand before he reach the point of tangency. Price-taking assumption Definition 1.7 in Chapter 19 is inconsistent with an IRS Jack. Play Dumb Jack? If we can get away with the price, we ll be back on track. Watanabe Econ 323 CMP 6 / 91

3 Overview How? Watanabe Econ 323 CMP 7 / 91 Overview Same trick we used before: 1 Long-run PMP: everything s on the table. 2 Short-run PMP: added one constraint K = K and saw what best Jack can do with remaining choice variables C, y for each given K. 1 PMP: everything s on the table. 2 CMP: add one constraint y = ȳ and see what best Jack can do with remaining choice variables ( C, K ) for for each given ȳ. Watanabe Econ 323 CMP 8 / 91 Cost Minimization Problem (CMP) Alternative question to PMP: what is the input bundle ( C, K ) that produces output ȳ most cost effectively? Problem 1.2 (Cost Minimization Problem) For given factor price = ( C, K ), and ȳ, Jack (1) Second line in (1) is called an output requirement. Watanabe Econ 323 CMP 9 / 91

4 Cost Minimization Problem (CMP) π = TR TC, and we just fixed TR to be pȳ. The least Jack can do is to minimize his operating and overhead expenses as much as possible. Watanabe Econ 323 CMP / 91 Cost Minimization Problem (CMP) CMP CRS IRS PMP CRS w/ π = DRS Watanabe Econ 323 CMP 11 / 91 Cost Minimization Problem (CMP) Example 1.3 (CMP) Jack was asked to produce ȳ = cheesecakes. His production function is ƒ ( C, K ) = C K. Factor price is = ( C, K ) = (2, 2). 1 How many chefs C and stand mixers K would he need? 2 Among them, which particular combination is most cost effective? Watanabe Econ 323 CMP 12 / 91

5 Cost Minimization Problem (CMP) ȳ C K labor cost capital cost total cost 1 2 Watanabe Econ 323 CMP 13 / 91 Cost Minimization Problem (CMP) 19 Isoquant at y= Chef x C Kitchen Equipment x K Watanabe Econ 323 CMP 14 / 91 1 Introduction 2 Cost Minimization Problem Trinity on Production Side (Recap) Isocost Lines Trinity Meets Trinity 3 Comparative Statics 4 Average Cost & Returns to Scale 5 Short-Run & Long-Run Total Cost 6 Summary Watanabe Econ 323 CMP 15 / 91

6 Trinity on Production Side (Recap) (1) in Problem 1.2 : min C, K TC( C, K ) = C C + C K subject to ƒ ( C, K ) = ȳ. Trinity on the production side: 1 The slope of isoquant. 2 Marginal rate of technical substitution. 3 The ratio between MP C ( ) and MP K ( ). 1 1 You can forget this one. Watanabe Econ 323 CMP 16 / 91 Trinity on Production Side (Recap) What about the cost side? Watanabe Econ 323 CMP 17 / 91 Isocost Lines Definition 2.1 (Isocost Line) A collection of input bundles that incurs Jack the same expenses (c) constructs the isocost line at c. Watanabe Econ 323 CMP 18 / 91

7 Isocost Lines An input bundle ( C, K ) that costs $c satisfies: Similar to the budget line: 1 Budget line represents the consumption bundle that is just affordable with income $m. 2 Isocost line represents the input bundle that is just affordable with cash outley $c. Watanabe Econ 323 CMP 19 / 91 Isocost Lines Take Example 1.3 : Example 2.2 (CMP) Jack was asked to produce ȳ = cheesecakes. His production technology is ƒ ( C, K ) = C K. Factor price is = ( C, K ) = (2, 2). 1 What is his isocost at $ and $25? Watanabe Econ 323 CMP / 91 Isocost Lines total cost c C labor cost capital cost K $ $ 1 $ 2 $. $ $25 $25 1 $25 2 $25. $25 $25 25 Watanabe Econ 323 CMP 21 / 91

8 Isocost Lines 19 Isocost Lines Chef x C Kitchen Equipment x K 3 4 Watanabe Econ 323 CMP 22 / Isocost Lines In general, C C + K K = c Slope of isocost is C K. Watanabe Econ 323 CMP 23 / 91 Isocost Lines Fact 2.3 (Trinity on the Cost Side) The following are equivalent to each other: 1 The slope of isocost line. 2 The relative price of chefs in terms of kitchen equipment: C K. 3 Opportunity cost of hiring one chef in terms of kitchen equipment. Watanabe Econ 323 CMP 24 / 91

9 Trinity Meets Trinity CMP Problem 1.2 : min C, K TC( C, K ) = C C + K K subject to ƒ ( C, K ) = ȳ. Watanabe Econ 323 CMP 25 / 91 Trinity Meets Trinity 19 Isoquant at y= Chef x C Kitchen Equipment x K Watanabe Econ 323 CMP 26 / 91 Trinity Meets Trinity 19 Isocost Lines Chef x C Kitchen Equipment x K 3 4 Watanabe Econ 323 CMP 27 /

10 Trinity Meets Trinity 19 Isocost Lines 18 Isoquant at y= Chef x C Kitchen Equipment x K 3 4 Watanabe Econ 323 CMP 28 / Trinity Meets Trinity Proposition 2.4 (Tangency Condition) At the optimal input bundle = ( C, ), the slope of K the isoquant (MRTS( C, K )) coincides with the slope of the isocost: C = MRTS( C, K ). K 2 2 Or equivalently, C MPC( C, K ) = K MP K ( C, K ). Watanabe Econ 323 CMP 29 / 91 Trinity Meets Trinity And this is how we interpret p.65 in Chapter 19: PMP. Watanabe Econ 323 CMP 3 / 91

11 Trinity Meets Trinity Take Example 1.3 : Example 2.5 (CMP) Jack was asked to produce ȳ = cheesecakes. His production technology is ƒ ( C, K ) = C K, with MRTS( C, K ) = K. Factor price is C = ( C, K ) = (2, 2). 1 Which combination of = ( C, K ) will produce cheesecakes most efficiently? Watanabe Econ 323 CMP 31 / 91 Trinity Meets Trinity Fact 2.6 (Solution to CMP) Solution to CMP Problem 1.2 is found by 1 tangency condition and 2 Output requirement (see (1) in Problem 1.2 ) Watanabe Econ 323 CMP 32 / 91 Trinity Meets Trinity Exercise 2.7 (CMP with Quasi-Linear Production Funcion) Jack was asked to make ȳ = cheesecakes. His production function is ƒ ( C, K ) = C + K, with MRTS( C, K ) = 2 K. Factor price is = ( C, K ) = (, 1). 1 Which combination of = ( C, K ) will produce cheesecakes most cost effectively? 2 How much does he pay for labor and capital cost? Watanabe Econ 323 CMP 33 / 91

12 Trinity Meets Trinity Isocost Isoquant at y= Kitchen Equipment x K Chef x C Watanabe Econ 323 CMP 34 / Introduction 2 Cost Minimization Problem 3 Comparative Statics Comparative Statics Comparative Statics on Factor Price Comparative Statics on Output Level: Scale Effect Recast: from TC( ) to TC(y) 4 Average Cost & Returns to Scale 5 Short-Run & Long-Run Total Cost 6 Summary Watanabe Econ 323 CMP 35 / 91 Comparative Statics Discussion 3.1 (Analysis on Jack s Response to Situational Chanegs) Take a listen to Marketplace Audio Clip. How do or will airlines respond to those changes: 1 jet fuel prices have risen 37% in the last year. 2 (change in) Ticket price, ticket price, ticket price. 3 (introduction of) new jumbo jets from Airbus and Boeing (that) are more fuel efficient. In particular, in light of Problem 1.2, which part of the problem will be updated? Watanabe Econ 323 CMP 36 / 91

13 Comparative Statics 1 Jet fuel prices increases fuel, which alters the shape of isocost. 2 Change in ticket price won t affect anything 3. 3 New jumbo jets alters ƒ ( ), which changes the shape of isoquant. 3 unless Jack hits the shut-down point. Watanabe Econ 323 CMP 37 / 91 Comparative Statics How exactly do we analyze Jack s response to these changes? Comparative statics: 1 endogenous variables: = ( C, K ). 2 exogenous variables: = ( C, K ), ȳ Watanabe Econ 323 CMP 38 / 91 Comparative Statics Watanabe Econ 323 CMP 39 / 91

14 Comparative Statics on Factor Price Back to Example 1.3 : Example 3.2 (Comparative Statics on Factor Price) Jack was asked to produce ȳ = cheesecakes. His production technology is ƒ ( C, K ) = C K, with MRTS( C, K ) = K. Factor price is C = ( C, K ) = (2, 2). 1 Suppose the wage C dropped down to C =.5. How does Jack respond to this change? Watanabe Econ 323 CMP 4 / 91 Comparative Statics on Factor Price Isocost w/ (w, w )=(2,2) C K Isoquant at y= Kitchen Equipment x K Chef x C Watanabe Econ 323 CMP 41 / 91 Comparative Statics on Factor Price Isocost w/ (w C, w K )=(.5,2) Kitchen Equipment x K Isoquant at y= Chef x C Watanabe Econ 323 CMP 42 / 91

15 Comparative Statics on Factor Price O = ( O C, O K ) = (, ), TC( O ) = 4. T = ( T C, T K ) = (, 5), TC( T ) =. Substitution effect: Δ S = T O = 5 Note that Jack saved $. = What should Jack do with that extra cash? He can scale up his operation.. -5 Watanabe Econ 323 CMP 43 / 91 Comp Stat on y: Scale Effect Back to Example 1.3 : Example 3.3 (Comparative Statics on Factor Price) Jack was asked to produce ȳ = cheesecakes. His production technology is ƒ ( C, K ) = C K, with MRTS( C, K ) = K. Factor price is C = ( C, K ) = (2, 2). 1 Suppose the wage C dropped down to C =.5. How does Jack respond to this change? Watanabe Econ 323 CMP 44 / 91 Comp Stat on y: Scale Effect Kitchen Equipment x K Isocost w/ (w C, w K )=(.5,2) Isoquant at y= Isoquant at y= Chef x C Watanabe Econ 323 CMP 45 / 91

16 Comp Stat on y: Scale Effect T = ( T C, T K ) = (, 5), TC( T ) =. F = ( F C, F K ) = (4, ), TC( F ) = 4. Scale effect: Δ = F T = 4 =. 5 5 (Producer theory s answer to income effect). Watanabe Econ 323 CMP 46 / 91 Comp Stat on y: Scale Effect We have just decomposed the total effect due to wage drop in two: 4 Δ = Δ S + Δ. 1 Substitution effect: due to lower wage, Jack lets cheap labor take over what kitchen equipment used to do. 2 Scale effect: due to lower wage, Jack increases the overall production scale (including kitchen equipment). 4 This is a Hicksian decomposition rather than a Slutsky decomposition just fyi. Watanabe Econ 323 CMP 47 / 91 Recast: from TC( ) to TC(y) Findings from Example 3.3 : Target ȳ Chef C Kitchen K Total cost TC( ) = = 4 We can relate y directly to total cost. Watanabe Econ 323 CMP 48 / 91

17 Recast: from TC( ) to TC(y) Watanabe Econ 323 CMP 49 / 91 Recast: from TC( ) to TC(y) We ll bypass = ( C, K ) and connect y directly to TC(y) (We already know what Jack will do for each target level of y. Why bother). From this point on, we ll denote total cost simply by TC(y). 5 But always keep in mind what Jack does behind: He makes the trinity meet the trinity. So, TC(y) is not just a total cost, but the smallest total cost to produce y cheesecakes. 5 rather than TC( (ȳ)). Lengthy. Watanabe Econ 323 CMP 5 / 91 Recast: from TC( ) to TC(y) Target ȳ Chef C Kitchen K Total cost TC(y) = = = = 4 Watanabe Econ 323 CMP 51 / 91

18 Recast: from TC( ) to TC(y). Kitchen Equipment x K Isoquant at y=25 Isoquant at y= Isoquant at y=225 Isoquant at y= Chef x C Watanabe Econ 323 CMP 52 / Recast: from TC( ) to TC(y) 4 TC(y) 35 3 Total Cost ($) Cheesecake y (slices) Watanabe Econ 323 CMP 53 / 91 1 Introduction 2 Cost Minimization Problem 3 Comparative Statics 4 Average Cost & Returns to Scale Average Cost 5 Short-Run & Long-Run Total Cost 6 Summary Watanabe Econ 323 CMP 54 / 91

19 Average Cost Is total cost function TC(y) always concave like the one we ve just seen? Definition 4.1 (Average Cost) Jack s average cost is a unit production cost defined by: AC(y) := TC(y). y Watanabe Econ 323 CMP 55 / 91 Average Cost = (cream cheese, mascarpone): average cost doesn t change with y (cheesecakes) = Boing 737: average cost decreases with y (# of passengers served) = airport: average cost increases with y (# of passengers served) if you try to process all MSP passengers at DLH. Is there any unified way to tell when average cost drops and when it doesn t? Watanabe Econ 323 CMP 56 / 91 Average Cost Answer: returns to scale: 1 Constant returns to scale: ƒ (2 C, 2 K ) = 2ƒ ( C, K ). 2 Increasing returns to scale: ƒ (2 C, 2 K ) > 2ƒ ( C, K ). 3 Decreasing returns to scale: ƒ (2 C, 2 K ) < 2ƒ ( C, K ). Watanabe Econ 323 CMP 57 / 91

20 Average Cost 1 Constant returns to scale: Doubling its output level from y to 2y requires doubling all input levels: ƒ (2 C, 2 K ) = 2ƒ ( C, K ) = 2y. Consequently, total production cost TC also doubles: TC(2y) = C 2 C + K 2 K = 2TC(y). Average cost does not change: AC(2y) = TC(2y) 2y = 2TC(y) 2y = TC(y) y = AC(y). Unit production cost does not change if technology exhibits constant returns to scale. Watanabe Econ 323 CMP 58 / 91 Average Cost 4 TC(y) 3 Total Cost ($) Cheesecake y (slices) Watanabe Econ 323 CMP 59 / 91 Average Cost 4 AC(y) Average Cost ($/slice) Cheesecake y (slices) Watanabe Econ 323 CMP 6 / 91

21 Average Cost 2 Increasing returns to scale: Doubling its output level from y to 2y requires less than doubling all input levels: Let s say ƒ (2 C, 2 K ) > 2ƒ ( C, K ) = 2y. ƒ (1.5 C, 1.5 K ) = 2y. Consequently, total production cost TC grows less than twice as large: TC(2y) = C 1.5 C + K 1.5 K = 1.5TC(y) < 2TC(y). Average cost decreases with y: AC(2y) = TC(2y) 2y = 1.5TC(y) 2y < TC(y) y = AC(y). Unit production cost becomes smaller as y increases if technology exhibits increasing returns to scale. Watanabe Econ 323 CMP 61 / 91 Average Cost 4 Benchmark TC(y) 3 Total Cost ($) Cheesecake y (slices) Watanabe Econ 323 CMP 62 / 91 Average Cost Fact 4.2 (Average Cost on the Graph) Average cost at y appears as the slope of a ray connecting (, ) to (y, TC(y)). Watanabe Econ 323 CMP 63 / 91

22 Average Cost 4 Benchmark AC(y) Average Cost ($/slice) Cheesecake y (slices) Watanabe Econ 323 CMP 64 / 91 Average Cost 3 Decreasing returns to scale: Doubling its output level from y to 2y requires more than doubling all input levels: Lets say ƒ (2 C, 2 K ) < 2ƒ ( C, K ) = 2y. ƒ (3 C, 3 K ) = 2y. Consequently, total production cost TC more than doubles: TC(2y) = C 3 C + K 3 K = 3TC(y) > 2TC(y). Average cost increases with y: AC(2y) = TC(2y) 2y = 3TC(y) 2y > TC(y) y = AC(y). Unit production cost increases as y if technology exhibits diminishing returns to scale. Watanabe Econ 323 CMP 65 / 91 Average Cost 4 Benchmark TC(y) 3 Total Cost ($) Cheesecake y (slices) Watanabe Econ 323 CMP 66 / 91

23 Average Cost 4 Benchmark AC(y) Average Cost ($/slice) Cheesecake y (slices) Watanabe Econ 323 CMP 67 / 91 1 Introduction 2 Cost Minimization Problem 3 Comparative Statics 4 Average Cost & Returns to Scale 5 Short-Run & Long-Run Total Cost 6 Summary Watanabe Econ 323 CMP 68 / 91 So far, K was endogenous and we derived the long-run total cost. How is short-run total related to the long-run total cost? Watanabe Econ 323 CMP 69 / 91

24 Problem 5.1 (Long-Run CMP) min C C + K K s.t. ƒ ( C, K ) = ȳ. C, K Problem 5.2 (Short-Run CMP) min C C C + K K s.t. ƒ ( C, K ) = ȳ. Watanabe Econ 323 CMP 7 / 91 Short-run CMP is a Long-run CMP subject to the extra constraint that K = K. If the long-run choice for K was actually K : no difference. Otherwise: STC is higher than LTC. Watanabe Econ 323 CMP 71 / 91 Back to Example 3.3 : Example 5.3 (Comparative Statics on Factor Price) Jack s production technology is ƒ ( C, K ) = C K, with MRTS( C, K ) = K. Factor price is C = ( C, K ) = (.5, 2). Suppose that K = 5 in the short run. 1 How is short-run total cost compared to its long-run counterpart? Watanabe Econ 323 CMP 72 / 91

25 y Chef C Kitchen K Long-run TC(y) = = = = 4 y Chef C Kitchen K Short-run TC(y) = = = = 5 Watanabe Econ 323 CMP 73 / 91. Kitchen Equipment x K Isoquant at y=25 Isoquant at y= Isoquant at y=225 Isoquant at y= Chef x C Watanabe Econ 323 CMP 74 / Long Run TC(y) Short Run TC(y) Total Cost ($) Cheesecake y (slices) Watanabe Econ 323 CMP 75 / 91

26 Conclude: Short-run TC(y) Long-run TC(y). Watanabe Econ 323 CMP 76 / 91 Discussion 5.4 (Short-Run Cost Minimization) What is the reason why Jack can t realize the smallest total cost possible in the short run? Is he lazy in the short run or? Watanabe Econ 323 CMP 77 / 91 Recall Fact 2.6. Short-run Jack can t meet the first condition. Consequently, Jack s MRTS is larger or smaller than relative price of chefs. He can do better by switching but he can t in the short run. Watanabe Econ 323 CMP 78 / 91

27 Example 5.5 (CRS) Jack s production function is given by ƒ ( C, K ) = C K, with MRTS( C, K ) = K. In the short run, K =. When = ( C, K ) = (1, 1), his total cost will be... C Watanabe Econ 323 CMP 79 / 91 y Chef C Kitchen K Long-run TC(y) = = = = 4 y Chef C Kitchen K Short-run TC(y) = = = = 5 Watanabe Econ 323 CMP 8 / Kitchen Equipment x K Isoquant at y=5 Isoquant at y= Isoquant at y= Isoquant at y= Chef x C Watanabe Econ 323 CMP 81 / 91

28 5 Long Run TC(y) Short Run TC(y) Total Cost ($) Cheesecake y (slices) Watanabe Econ 323 CMP 82 / 91 Exercise 5.6 (Returns to Scale and Average Cost) Compare Example 5.3 to Example Example 5.3 : ƒ ( C, K ) = C K 2 Example 5.5 : ƒ ( C, K ) = C K. Confirm that ➊ has decreasing average cost and ➋ has constant average cost (Recall Fact 4.2 ). Watanabe Econ 323 CMP 83 / 91 Kitchen Equipment x K Isoquant at y=25 Isoquant at y= Isoquant at y=225 Isoquant at y=4 Kitchen Equipment x K Isoquant at y=5 Isoquant at y= Isoquant at y=15 Isoquant at y= Chef x C Chef x C Watanabe Econ 323 CMP 84 / 91

29 5 Long Run TC(y) Short Run TC(y) 5 Long Run TC(y) Short Run TC(y) 4 4 Total Cost ($) Total Cost ($) Cheesecake y (slices) 5 15 Cheesecake y (slices) Watanabe Econ 323 CMP 85 / 91 1 Long Run AC(y) 3 Long Run AC(y) Average Cost ($/slice) Average Cost ($/slice) Cheesecake y (slices) Cheesecake y (slices) Watanabe Econ 323 CMP 86 / 91 1 Introduction 2 Cost Minimization Problem 3 Comparative Statics 4 Average Cost & Returns to Scale 5 Short-Run & Long-Run Total Cost 6 Summary Watanabe Econ 323 CMP 87 / 91

30 UMP and CMP UMP utility function indifference curve MRS budget line CMP Watanabe Econ 323 CMP 88 / 91 When profit maximization problem fails Cost minimization problem Trinity meets trinity Total coast and returns to scale Time frame Watanabe Econ 323 CMP 89 / 91 This page is left intentionally blank. Watanabe Econ 323 CMP 9 / 91

31 average cost, 55, 83 budget line, 19, 88 c, see expenses comparative statics, 38 constant returns to scale, 58 cost minimization problem, 9, 88 long-run, 7 short-run, 7, 77 costant returnst to scale, 3 Δ, see total effect Δ, see scale effect Δ S, see substitution effect endogenous variables, 38 exogenous variables, 38 expenses, 18 factor price, 9 Hicksian decomposition, 47 income effect, 46 increasing returns to scale, 3, 61 indifference curve, 88 input bundle, 9 isocost, 18, 23, 29, 37, 88 isoquant, 16, 29, 88 marginal rate of substitution, 88 marginal rate of technical substitution, 16, 88 MRTS, see marginal rate of technical substitution optimal input bundle, 29 output, 9 output requriment, 32 p, see price π, see profit price, 6 production function, 88 production plan, 3 profit, 3 quasi-linear production function, 33 returns to scale, 57, 83 scale effect, 46, 47 Slutsky decomposition, 47 substitution effect, 43, 47 tangency condition, 32 total cost, 6 total effect, 47 total revenue, 6 trinity, 16 on the cost side, 24 on the production side, 16 utility function, 88 utility maximization problem, 88, see facto price, see input bundle, see optimal input bundle y, see output