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# Lecture 8 production, optimal inputs (1)

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### Lecture 8 production, optimal inputs (1)

1. 1. PRODUCTION WITH TWO INPUTS 1
2. 2. Production: Two Variable Inputs• Firm can produce output by combining different amounts of labor and capital• In the long run, capital and labor are both variable 2
3. 3. Production: Two Variable Inputs 3
4. 4. Production: Two Variable Inputs• Isoquant – Curve showing all possible combinations of inputs that yield the same output 4
5. 5. Isoquant MapCapital 5 E Ex: 55 units of outputper year can be produced with 3K & 1L (pt. A) 4 OR 1K & 3L (pt. D) 3 A B C 2 q3 = 90 D q2 = 75 1 q1 = 55 1 2 3 4 5 Labor per year 5
6. 6. Production: Two Variable Inputs• Diminishing Returns to Labor with Isoquants• Holding capital at 3 and increasing labor from 0 to 1 to 2 to 3 – Output increases at a decreasing rate (0, 55, 20, 15) illustrating diminishing marginal returns from labor in the short run 6
7. 7. Diminishing Returns to Capital?Capital 5 Increasing labor holdingper year capital constant (A, B, C) 4 OR Increasing capital holding labor constant 3 (E, D, C A B C D 2 q3 = 90 1 E q2 = 75 q1 = 55 1 2 3 4 5 Labor per year 7
8. 8. Production: Two Variable Inputs• Diminishing Returns to Capital with Isoquants• Holding labor constant at 3 increasing capital from 0 to 1 to 2 to 3 – Output increases at a decreasing rate (0, 55, 20, 15) due to diminishing returns from capital in short run 8
9. 9. ISOQUANTS• Why are isoquant curve downward sloping?
10. 10. Marginal Rate of Technical Substitution– Slope of the isoquant shows how one input can be substituted for the other and keep the level of output the same– The negative of the slope is the marginal rate of technical substitution (MRTS) • Amount by which the quantity of one input can be reduced when one extra unit of another input is used, so that output remains constant 10
11. 11. Production: Two Variable Inputs • The marginal rate of technical substitution equals: Change in Capital Input MRTS Change in Labor Input MRTS K (for a fixed level of q ) L 11
12. 12. MRTS and Marginal Products• If we are holding output constant, the net effect of increasing labor and decreasing capital must be zero• Using changes in output from capital and labor we can see (MPL )( L) (MPK )( K) 0 12
13. 13. MRTS and Marginal Products• Rearranging equation, we can see the relationship between MRTS and MPs (MP )( L) (MP )( K) 0 L K (MP )( L) - (MP )( K) L K (MP ) L L MRTS ( MPK ) K 13
14. 14. ISOQUANT• Why is isoquant convex to the origin?
15. 15. Marginal Rate of Technical SubstitutionCapital 5per year Negative Slope measures MRTS; 2 MRTS decreases as move down 4 the indifference curve 1 3 1 1 2 2/3 1 Q3 =90 1/3 Q2 =75 1 1 Q1 =55 1 2 3 4 5 Labor per month 15
16. 16. Production: Two Variable Inputs• As labor increases to replace capital – Labor becomes relatively less productive – Capital becomes relatively more productive – Isoquant becomes flatter 16
17. 17. Law of Diminishing MRTS• Because of Law of Diminishing MP, MRTS is also diminishing.• Hence, isoquant is convex.• Why is MP curve inverted U shaped?Chapter 6 17
18. 18. SPECIAL ISOQUANTS 18
19. 19. Perfect Substitutes1. Perfect substitutes – MRTS is constant at all points on isoquant – Same output can be produced with a lot of capital or a lot of labor or a balanced mix 19
20. 20. Perfect SubstitutesCapital per A Same output can bemonth reached with mostly capital or mostly labor (A or C) or with equal amount of both (B) B C Q1 Q2 Q3 Labor per month 20
21. 21. Perfect Substitutes• Type of transportation• Type of energy source• Type of protein source
22. 22. Perfect Compliments– There is no substitution available between inputs– The output can be made with only a specific proportion of capital and labor– Cannot increase output unless increase both capital and labor in that specific proportion 22
23. 23. Fixed-Proportions Production FunctionCapital per Same output canmonth only be produced with one set of inputs. Q3 C Q2 B K1 Q1 A Labor per month L1 23
24. 24. Perfect Compliments• Ingredients to prepare a recipe• Parts to make a vehicle• In reality there is no perfect substitute / compliments• Ability to substitute one i/p for the other diminishes as one moves along Isoquant
25. 25. MINIMIZING COSTChapter 6 25
26. 26. Cost Minimizing Input Choice• How do we put all this together to select inputs to produce a given output at minimum cost?• Assumptions – Two Inputs: Labor (L) and capital (K) – Price of labor: wage rate (w) – The price of capital 26
27. 27. ISOCOST CURVE• The Isocost Line – A line showing all combinations of L & K that can be purchased for the same cost, C – Total cost of production is sum of firm’s labor cost, wL, and its capital cost, rK: C = wL + rK – For each different level of cost, the equation shows another isocost line 27
28. 28. ISOCOST CURVE• Rewriting C as an equation for a straight line: K = C/r - (w/r)L – Slope of the isocost: • -(w/r) is the ratio of the wage rate to rental cost of capital. • This shows the rate at which capital can be substituted for labor with no change in cost K w L r 28
29. 29. PowerPoint Slides Prepared byRobert F. Brooker, Ph.D. Slide 29
30. 30. OPTIMAL INPUTS• How to minimize cost for a given level of output by combining isocosts with isoquants• We choose the output we wish to produce and then determine how to do that at minimum cost – Isoquant is the quantity we wish to produce – Isocost is the combination of K and L that gives a set cost 30
31. 31. Producing a Given Output at Minimum CostCapital per Q1 is an isoquant for output Q1. year There are three isocost lines, of which 2 are possible choices in which to produce Q1. K2 Isocost C2 shows quantity Q1 can be produced with combination K2,L2 or K3,L3. However, both of these A are higher cost combinations K1 than K1,L1. Q1 K3 C0 C1 C2 Labor per year L2 L1 L3 31
32. 32. Duality Problem• Optimal inputs –K, L to produce output Q1 and minimize cost• Optimal inputs –K,L with cost C1 and maximize output• Both these problems would give the same optimal input combination
33. 33. Input Substitution When an Input Price Change• If the price of labor changes, then the slope of the isocost line changes, -(w/r)• It now takes a new quantity of labor and capital to produce the output• If price of labor increases relative to price of capital, and capital is substituted for labor 34
34. 34. Input Substitution When an Input Price ChangeCapital per If the price of labor year rises, the isocost curve becomes steeper due to the change in the slope -(w/L). The new combination of K and L is used to produce Q1. B Combination B is used in place K2 of combination A. A K1 Q1 C2 C1 L2 L1 Labor per year 35
35. 35. Optimal Inputs• How does the isocost line relate to the firm’s production process? MRTS - K MPL L MPK Slope of isocost line K w L r MPL w when firmminimizes cost MPK rChapter 7 36
36. 36. Optimal Inputs• The minimum cost combination can then be written as: MPL MPK w r – Increase in output for every dollar spent on an input is same for all inputs.Chapter 7 37
37. 37. OPTIMAL INPUTS• If w = \$10, r = \$2, and MPL = MPK, which input would the producer use more of? – Labor because it is cheaper – Increasing labor lowers MPL – Decreasing capital raises MPK – Substitute labor for capital until MPL MPK w rChapter 7 38
38. 38. Cost in the Long Run• Cost minimization with Varying Output Levels – For each level of output, we can find the cost minimizing inputs. – For each level of output, there is an isocost curve showing minimum cost for that output level – A firm’s expansion path shows the minimum cost combinations of labor and capital at each level of output – Slope equals K/ LChapter 7 39
39. 39. Expansion Path Capital per The expansion path illustrates year the least-cost combinations of labor and capital that can be 150 \$3000 used to produce each level of output in the long-run. Expansion Path \$2000 100 C 75 B 50 300 Units A 25 200 Units Labor per year 50 100 150 200 300Chapter 7 40
40. 40. Expansion Path• It shows optimal input combinations to minimize cost to produce different levels of output• It shows the minimum cost to produce different levels of output• It shows the maximum amount of output that can be produced for different levels of expenditure.
41. 41. A Firm’s Long Run Total Cost Curve Cost/ Year Long Run Total Cost F 3000 E 2000 D 1000 Output, Units/yr 100 200 300Chapter 7 42
42. 42. Long Run Versus Short Run Cost Curves• In the short run, some costs are fixed• In the long run, firm can change anything including plant size – Can produce at a lower average cost in long run than in short run – Capital and labor are both flexible• We can show this by holding capital fixed in the short run and flexible in long runChapter 7 44
43. 43. The Inflexibility of Short Run Production Capital E Capital is fixed at K1. per To produce q1, min cost at K1,L1. year If increase output to Q2, min cost C is K1 and L3 in short run. In LR, can Long-Run change Expansion Path A capital and min costs falls to K2 K2 and L2. Short-Run P Expansion Path K1 Q2 Q1 Labor per year L1 L2 B L3 D FChapter 7 45
44. 44. RETURNS TO SCALE 46
45. 45. BIG CITIES• Metropolis twice the size of one, number of gas stations, length of pipelines, infrastructure decreases by 15%• Why? 47
46. 46. Narayan Hridalaya• Provide health care at full priceTo patients from well to do background• These patients subsidize `poor’ patients• Run at a profit of 7.7%• Why is Narayan Hridalaya able to do this? 48
47. 47. Narayan Hridalaya• Number of Beds, 2001: 225• Current No. of Beds across India: 30,000• How does number of beds play a role in profits? 49
48. 48. Returns to Scale• Rate at which output increases as inputs are increased proportionately – Increasing returns to scale – Constant returns to scale – Decreasing returns to scale 50
49. 49. Returns to Scale• Increasing returns to scale: output more than doubles when all inputs are doubled – What happens to the isoquants? 51
50. 50. Increasing Returns to Scale Capital(machine The isoquants hours) A move closer together 4 30 2 20 10 Labor (hours) 5 10 52
51. 51. Returns to Scale• Constant returns to scale: output doubles when all inputs are doubled – Size does not affect productivity – May have a large number of producers – Isoquants are equidistant apart 53
52. 52. Returns to Scale Capital(machine A hours) 6 30 4 Constant Returns: Isoquants are 2 equally spaced 0 2 10 Labor (hours) 5 10 15 54
53. 53. Returns to Scale• Decreasing returns to scale: output less than doubles when all inputs are doubled – Decreasing efficiency with large size – Reduction of entrepreneurial abilities – Isoquants become farther apart 55
54. 54. Returns to Scale Capital(machine A hours) Decreasing Returns: Isoquants get further 4 apart 30 2 20 10 5 10 Labor (hours) 56