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# Cost Minimization

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### Cost Minimization

1. 1. Cost Minimization-by Anil NayakPaper-presentation-ppt.blogspot.in
2. 2. Cost MinimizationA firm is a cost-minimizer if itproduces any given output level y ≥0 at smallest possible total cost.c(y) denotes the firm’s smallestpossible total cost for producing yunits of output.c(y) is the firm’s total cost function.
3. 3. Cost MinimizationWhen the firm faces given inputprices w = (w1,w2,…,wn) the total costfunction will be written asc(w1,…,wn,y).
4. 4. The Cost-Minimization ProblemConsider a firm using two inputs tomake one output.The production function isy = f(x1,x2).Take the output level y ≥ 0 as given.Given the input prices w1 and w2, thecost of an input bundle (x1,x2) isw1x1 + w2x2.
5. 5. The Cost-Minimization ProblemFor given w1, w2 and y, the firm’scost-minimization problem is tosolve min,x xw x w x1 2 01 1 2 2≥+subject to f x x y( , ) .1 2 =
6. 6. The Cost-Minimization ProblemThe levels x1*(w1,w2,y) and x1*(w1,w2,y) inthe least-costly input bundle are thefirm’s conditional demands for inputs1 and 2.The (smallest possible) total cost forproducing y output units is thereforec w w y w x w w yw x w w y( , , ) ( , , )( , , ).**1 2 1 1 1 22 2 1 2=+
7. 7. Conditional Input DemandsGiven w1, w2 and y, how is the leastcostly input bundle located?And how is the total cost functioncomputed?
8. 8. Iso-cost LinesA curve that contains all of the inputbundles that cost the same amountis an iso-cost curve.E.g., given w1 and w2, the \$100 iso-cost line has the equationw x w x1 1 2 2 100+ = .
9. 9. Iso-cost LinesGenerally, given w1 and w2, theequation of the \$c iso-cost line isi.e.Slope is - w1/w2.xwwxcw21212= − + .w x w x c1 1 2 2+ =
10. 10. Iso-cost Linesc’ ≡ w1x1+w2x2c” ≡ w1x1+w2x2c’ < c”x1x2
11. 11. Iso-cost Linesc’ ≡ w1x1+w2x2c” ≡ w1x1+w2x2c’ < c”x1x2 Slopes = -w1/w2.
12. 12. The y’-Output Unit Isoquantx1x2All input bundles yielding y’ unitsof output. Which is the cheapest?f(x1,x2) ≡ y’
13. 13. The Cost-Minimization Problemx1x2All input bundles yielding y’ unitsof output. Which is the cheapest?f(x1,x2) ≡ y’
14. 14. The Cost-Minimization Problemx1x2All input bundles yielding y’ unitsof output. Which is the cheapest?f(x1,x2) ≡ y’
15. 15. The Cost-Minimization Problemx1x2All input bundles yielding y’ unitsof output. Which is the cheapest?f(x1,x2) ≡ y’
16. 16. The Cost-Minimization Problemx1x2All input bundles yielding y’ unitsof output. Which is the cheapest?f(x1,x2) ≡ y’x1*x2*
17. 17. The Cost-Minimization Problemx1x2f(x1,x2) ≡ y’x1*x2*At an interior cost-min input bundle:(a) f x x y( , )* *1 2 = ′
18. 18. The Cost-Minimization Problemx1x2f(x1,x2) ≡ y’x1*x2*At an interior cost-min input bundle:(a) and(b) slope of isocost = slope ofisoquantf x x y( , )* *1 2 = ′
19. 19. The Cost-Minimization Problemx1x2f(x1,x2) ≡ y’x1*x2*At an interior cost-min input bundle:(a) and(b) slope of isocost = slope ofisoquant; i.e.f x x y( , )* *1 2 = ′− = = −wwTRSMPMPat x x12121 2( , ).* *
20. 20. A Cobb-Douglas Example of CostMinimizationA firm’s Cobb-Douglas productionfunction isInput prices are w1 and w2.What are the firm’s conditional inputdemand functions?y f x x x x= =( , ) ./ /1 2 11 322 3
21. 21. A Cobb-Douglas Example of CostMinimizationAt the input bundle (x1*,x2*) which minimizesthe cost of producing y output units:(a)(b)y x x= ( ) ( )* / * /11 322 3and− = − = −= −−−wwy xy xx xx xxx121212 322 311 321 3211 32 32∂ ∂∂ ∂//( / )( ) ( )( / )( ) ( ).* / * /* / * /**
22. 22. A Cobb-Douglas Example of CostMinimizationy x x= ( ) ( )* / * /11 322 3 wwxx12212=**.(a) (b)
23. 23. A Cobb-Douglas Example of CostMinimizationy x x= ( ) ( )* / * /11 322 3 wwxx12212=**.(a) (b)From (b), xwwx21212* *.=
24. 24. A Cobb-Douglas Example of CostMinimizationy x x= ( ) ( )* / * /11 322 3 wwxx12212=**.(a) (b)From (b), xwwx21212* *.=Now substitute into (a) to gety xwwx=( )* / */11 3 1212 32
25. 25. A Cobb-Douglas Example of CostMinimizationy x x= ( ) ( )* / * /11 322 3 wwxx12212=**.(a) (b)From (b), xwwx21212* *.=Now substitute into (a) to gety xwwxwwx= =( ) .* / */ /*11 3 1212 3122 312 2
26. 26. A Cobb-Douglas Example of CostMinimizationy x x= ( ) ( )* / * /11 322 3 wwxx12212=**.(a) (b)From (b), xwwx21212* *.=Now substitute into (a) to gety xwwxwwx= =( ) .* / */ /*11 3 1212 3122 312 2xwwy1212 32*/=So is the firm’s conditionaldemand for input 1.
27. 27. A Cobb-Douglas Example of CostMinimizationxwwx21212* *= xwwy1212 32*/=is the firm’s conditional demand for input 2.Since andxwwwwywwy212212 3121 3222*/ /= =
28. 28. A Cobb-Douglas Example of CostMinimizationSo the cheapest input bundle yielding youtput units is( )x w w y x w w ywwywwy1 1 2 2 1 2212 3121 322* */ /( , , ), ( , , ), .=
29. 29. x2x1Fixed w1 and w2.Conditional Input Demand Curves′y′′y′′′y
30. 30. x2x1Fixed w1 and w2.Conditional Input Demand Curvesx y1*( )′x y2*( )′′y′′y′′′y′y′yx y2*( )′x y1*( )′x2*x1*yy
31. 31. x2x1Fixed w1 and w2.Conditional Input Demand Curvesx y1*( )′x y1*( )′′x y2*( )′′x y2*( )′′y′′y′′′y′′y′′y′y′yx y2*( )′′x y2*( )′x y1*( )′′x y1*( )′x2*x1*yy
32. 32. x2x1Fixed w1 and w2.Conditional Input Demand Curvesx y1*( )′′′x y2*( )′′′x y1*( )′x y1*( )′′x y2*( )′′x y2*( )′′y′′y′′′y′′′y′′′y′′y′′y′y′yx y2*( )′′′x y2*( )′′x y2*( )′x y1*( )′′′x y1*( )′′x y1*( )′x2*x1*yy
33. 33. x2x1Fixed w1 and w2.Conditional Input Demand Curvesx y1*( )′′′x y2*( )′′′x y1*( )′x y1*( )′′x y2*( )′′x y2*( )′outputexpansionpath′y′′y′′′yx y2*( )′′′x y2*( )′′x y2*( )′x y1*( )′′′x y1*( )′′x y1*( )′′′′y′′′y′′y′′y′y′yx2*x1*yy
34. 34. x2x1Fixed w1 and w2.Conditional Input Demand Curvesx y1*( )′′′x y2*( )′′′x y1*( )′x y1*( )′′x y2*( )′′x y2*( )′outputexpansionpath′y′′y′′′y′′′y′′′y′′y′′y′y′yx y2*( )′′′x y2*( )′′x y2*( )′x y1*( )′′′x y1*( )′′x y1*( )′Cond. demandforinput 2Cond.demandforinput 1x2*x1*yy
35. 35. A Cobb-Douglas Example of CostMinimizationFor the production functionthe cheapest input bundle yielding y outputunits is( )x w w y x w w ywwywwy1 1 2 2 1 2212 3121 322* */ /( , , ), ( , , ), .=3/223/1121 xx)x,x(fy ==
36. 36. A Cobb-Douglas Example of CostMinimizationSo the firm’s total cost function isc w w y w x w w y w x w w y( , , ) ( , , ) ( , , )* *1 2 1 1 1 2 2 2 1 2= +
37. 37. A Cobb-Douglas Example of CostMinimizationSo the firm’s total cost function isc w w y w x w w y w x w w ywwwy wwwy( , , ) ( , , ) ( , , )* */ /1 2 1 1 1 2 2 2 1 21212 32121 322= += +
38. 38. A Cobb-Douglas Example of CostMinimizationSo the firm’s total cost function isc w w y w x w w y w x w w ywwwy wwwyw w y w w y( , , ) ( , , ) ( , , )* */ /// / / / /1 2 1 1 1 2 2 2 1 21212 32121 32 311 322 3 1 311 322 322122= += += +
39. 39. A Cobb-Douglas Example of CostMinimizationc w w y w x w w y w x w w ywwwy wwwyw w y w w yw wy( , , ) ( , , ) ( , , ).* */ /// / / / //1 2 1 1 1 2 2 2 1 21212 32121 32 311 322 3 1 311 322 31 22 1 32212234= += += +=So the firm’s total cost function is
40. 40. A Perfect Complements Example ofCost MinimizationThe firm’s production function isInput prices w1 and w2 are given.What are the firm’s conditionaldemands for inputs 1 and 2?What is the firm’s total costfunction?y x x= min{ , }.4 1 2
41. 41. A Perfect Complements Example ofCost Minimizationx1x2min{4x1,x2} ≡ y’4x1 = x2
42. 42. A Perfect Complements Example ofCost Minimizationx1x24x1 = x2min{4x1,x2} ≡ y’
43. 43. A Perfect Complements Example ofCost Minimizationx1x24x1 = x2min{4x1,x2} ≡ y’Where is the least costlyinput bundle yieldingy’ output units?
44. 44. A Perfect Complements Example ofCost Minimizationx1x2x1*= y/4x2* = y4x1 = x2min{4x1,x2} ≡ y’Where is the least costlyinput bundle yieldingy’ output units?
45. 45. A Perfect Complements Example ofCost Minimizationy x x= min{ , }4 1 2The firm’s production function isand the conditional input demands arex w w yy1 1 24*( , , ) = x w w y y2 1 2*( , , ) .=and
46. 46. A Perfect Complements Example ofCost Minimizationy x x= min{ , }4 1 2The firm’s production function isand the conditional input demands arex w w yy1 1 24*( , , ) = x w w y y2 1 2*( , , ) .=andSo the firm’s total cost function isc w w y w x w w yw x w w y( , , ) ( , , )( , , )**1 2 1 1 1 22 2 1 2=+
47. 47. A Perfect Complements Example ofCost Minimizationy x x= min{ , }4 1 2The firm’s production function isand the conditional input demands arex w w yy1 1 24*( , , ) = x w w y y2 1 2*( , , ) .=andSo the firm’s total cost function isc w w y w x w w yw x w w ywyw yww y( , , ) ( , , )( , , ).**1 2 1 1 1 22 2 1 21 2124 4=+= + = +
48. 48. Average Total Production CostsFor positive output levels y, a firm’saverage total cost of producing yunits isAC w w yc w w yy( , , )( , , ).1 21 2=
49. 49. Returns-to-Scale and Av. Total CostsThe returns-to-scale properties of afirm’s technology determine howaverage production costs change withoutput level.Our firm is presently producing y’output units.How does the firm’s averageproduction cost change if it insteadproduces 2y’ units of output?
50. 50. Constant Returns-to-Scale and AverageTotal CostsIf a firm’s technology exhibitsconstant returns-to-scale thendoubling its output level from y’ to2y’ requires doubling all input levels.
51. 51. Constant Returns-to-Scale and AverageTotal CostsIf a firm’s technology exhibitsconstant returns-to-scale thendoubling its output level from y’ to2y’ requires doubling all input levels.Total production cost doubles.
52. 52. Constant Returns-to-Scale and AverageTotal CostsIf a firm’s technology exhibitsconstant returns-to-scale thendoubling its output level from y’ to2y’ requires doubling all input levels.Total production cost doubles.Average production cost does notchange.
53. 53. Decreasing Returns-to-Scale andAverage Total CostsIf a firm’s technology exhibitsdecreasing returns-to-scale thendoubling its output level from y’ to2y’ requires more than doubling allinput levels.
54. 54. Decreasing Returns-to-Scale andAverage Total CostsIf a firm’s technology exhibitsdecreasing returns-to-scale thendoubling its output level from y’ to2y’ requires more than doubling allinput levels.Total production cost more thandoubles.
55. 55. Decreasing Returns-to-Scale andAverage Total CostsIf a firm’s technology exhibitsdecreasing returns-to-scale thendoubling its output level from y’ to2y’ requires more than doubling allinput levels.Total production cost more thandoubles.Average production cost increases.
56. 56. Increasing Returns-to-Scale andAverage Total CostsIf a firm’s technology exhibitsincreasing returns-to-scale thendoubling its output level from y’ to2y’ requires less than doubling allinput levels.
57. 57. Increasing Returns-to-Scale andAverage Total CostsIf a firm’s technology exhibitsincreasing returns-to-scale thendoubling its output level from y’ to2y’ requires less than doubling allinput levels.Total production cost less thandoubles.
58. 58. Increasing Returns-to-Scale andAverage Total CostsIf a firm’s technology exhibitsincreasing returns-to-scale thendoubling its output level from y’ to2y’ requires less than doubling allinput levels.Total production cost less thandoubles.Average production cost decreases.
59. 59. Returns-to-Scale and Av. Total Costsy\$/output unitconstant r.t.s.decreasing r.t.s.increasing r.t.s.AC(y)
60. 60. Returns-to-Scale and Total CostsWhat does this imply for the shapesof total cost functions?
61. 61. Returns-to-Scale and Total Costsy\$y’ 2y’c(y’)c(2y’) Slope = c(2y’)/2y’= AC(2y’).Slope = c(y’)/y’= AC(y’).Av. cost increases with y if the firm’stechnology exhibits decreasing r.t.s.
62. 62. Returns-to-Scale and Total Costsy\$c(y)y’ 2y’c(y’)c(2y’) Slope = c(2y’)/2y’= AC(2y’).Slope = c(y’)/y’= AC(y’).Av. cost increases with y if the firm’stechnology exhibits decreasing r.t.s.
63. 63. Returns-to-Scale and Total Costsy\$y’ 2y’c(y’)c(2y’)Slope = c(2y’)/2y’= AC(2y’).Slope = c(y’)/y’= AC(y’).Av. cost decreases with y if the firm’stechnology exhibits increasing r.t.s.
64. 64. Returns-to-Scale and Total Costsy\$c(y)y’ 2y’c(y’)c(2y’)Slope = c(2y’)/2y’= AC(2y’).Slope = c(y’)/y’= AC(y’).Av. cost decreases with y if the firm’stechnology exhibits increasing r.t.s.
65. 65. Returns-to-Scale and Total Costsy\$c(y)y’ 2y’c(y’)c(2y’)=2c(y’) Slope = c(2y’)/2y’= 2c(y’)/2y’= c(y’)/y’soAC(y’) = AC(2y’).Av. cost is constant when the firm’stechnology exhibits constant r.t.s.
66. 66. Short-Run & Long-Run Total CostsIn the long-run a firm can vary all ofits input levels.Consider a firm that cannot changeits input 2 level from x2’ units.How does the short-run total cost ofproducing y output units compare tothe long-run total cost of producing yunits of output?
67. 67. Short-Run & Long-Run Total CostsThe long-run cost-minimizationproblem isThe short-run cost-minimizationproblem ismin,x xw x w x1 2 01 1 2 2≥+subject to f x x y( , ) .1 2 =minxw x w x1 01 1 2 2≥+ ′subject to f x x y( , ) .1 2′ =
68. 68. Short-Run & Long-Run Total CostsThe short-run cost-min. problem is thelong-run problem subject to the extraconstraint that x2 = x2’.If the long-run choice for x2 was x2’then the extra constraint x2 = x2’ is notreally a constraint at all and so thelong-run and short-run total costs ofproducing y output units are the same.
69. 69. Short-Run & Long-Run Total CostsThe short-run cost-min. problem istherefore the long-run problem subjectto the extra constraint that x2 = x2”.But, if the long-run choice for x2 ≠ x2”then the extra constraint x2 = x2”prevents the firm in this short-run fromachieving its long-run production cost,causing the short-run total cost toexceed the long-run total cost ofproducing y output units.
70. 70. Short-Run & Long-Run Total Costsx1x2′′′y′′y′yConsider three output levels.
71. 71. Short-Run & Long-Run Total Costsx1x2′′′y′′y′yIn the long-run when the firmis free to choose both x1 andx2, the least-costly inputbundles are ...
72. 72. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Long-runoutputexpansionpath
73. 73. Short-Run & Long-Run Total Costsx1x2′′′y′′y′yLong-runoutputexpansionpath′x1 ′′x1 ′′′x1′x2′′x2′′′x2Long-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2
74. 74. Short-Run & Long-Run Total CostsNow suppose the firm becomessubject to the short-run constraintthat x2 = x2”.
75. 75. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Short-runoutputexpansionpathLong-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2
76. 76. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Short-runoutputexpansionpathLong-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2
77. 77. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Short-runoutputexpansionpathLong-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2Short-run costs are:c y c ys( ) ( )′ > ′
78. 78. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Short-runoutputexpansionpathLong-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2Short-run costs are:c y c yc y c yss( ) ( )( ) ( )′ > ′′′ = ′′
79. 79. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Short-runoutputexpansionpathLong-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2Short-run costs are:c y c yc y c yss( ) ( )( ) ( )′ > ′′′ = ′′
80. 80. Short-Run & Long-Run Total Costsx1x2′′′y′′y′y′x1 ′′x1 ′′′x1′x2′′x2′′′x2Short-runoutputexpansionpathLong-run costs are:c y w x w xc y w x w xc y w x w x( )( )( )′ = ′ + ′′′ = ′′ + ′′′′′′ = ′′′+ ′′′1 1 2 21 1 2 21 1 2 2Short-run costs are:c y c yc y c yc y c ysss( ) ( )( ) ( )( ) ( )′ > ′′′ = ′′′′′ > ′′′
81. 81. Short-Run & Long-Run Total CostsShort-run total cost exceeds long-run total cost except for the outputlevel where the short-run input levelrestriction is the long-run input levelchoice.This says that the long-run total costcurve always has one point incommon with any particular short-run total cost curve.
82. 82. Short-Run & Long-Run Total Costsy\$c(y)′′′y′′y′ycs(y)Fw x=′′2 2A short-run total cost curve always hasone point in common with the long-runtotal cost curve, and is elsewhere higherthan the long-run total cost curve.
83. 83. Thank You