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Chapter 6 Production

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Chapter 6 Production

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    1. Chapter 6 Production Day 1

    2. Announcements

    3. Next exam: 9 Nov Last day to drop: 2 Nov Homework question 2 was graded Elon Innovation Challenge A creativity competition for all college students in North Carolina Hosted by Student Entrepreneurial Enterprise Development (SEED) and the Doherty Center for Entrepreneurial Leadership   Want to know more? http://eloninnovation.org/ Interested? email me Announcements

    4. Course Roadmap

    5. Production Production Function Q = F (M, L, K) where M = materials L = labor K = capital Examples Q = 10 L – 0.5 L2 + 24 K – K2 Q = 12 L0.5K0.5M0.5 Definitions – Round 1 Production Ideas

    6. Auto Parts Production Function

    7. Auto Parts Production Function

    8. Auto Parts Production Function

    9. Short run Fixed inputs Long run Variable input Marginal product Production function: Q = F (L, K, M) Mathematical representation: ?Q / ?L = MPL Example: Q = 10 L – 0.5 L2 + 24 K – K2 MPL = 10 – L Example: Q = 12 L0.5K0.5M0.5 MPL = 6 L-0.5K0.5M0.5 = Definitions – Round 2 More Production Ideas

    10. Auto Part Production Function

    11. Total and Marginal Product (10,000 sq ft factory)

    12. All inputs held constant except one Law of diminishing marginal returns Add successive units of the variable input Eventually marginal product will decline Can marginal product be negative? What then? Law of Diminishing Marginal Returns

    13. Free eucalyptus leaves

    14. Definitions Marginal revenue product (MRP) of a single input MRPL, MRPK, MRPM MRPL = (MR) (MPL) Marginal cost of a single input (MCL) Marginal profit per worker (M?L) M?L = MRPL – MCL Definitions – Round 3 Purchasing Inputs

    15. To decide how much labor to use Set M?L = 0 M?L= MRPL – MCL= 0 MRPL = MCL Solve for L Decision Rule

    16. Information given Production function: Q = 60 L – L2 Price of output = $2 MCL = $16 per hour The problem How much labor to hire? How much output to produce? Example (Example 2, p 223)

    17. Step 1: Decision rule MRPL = MCL Step 2: Calculate MRPL Define MRPL= (MPL) (MR) Define MPL = dQ / dL Given: Q = 60 L – L2 MPL = 60 – 2 L Given: MR = $2 MRPL = 2 (60 – 2L) = 4 (30 – L) Solution

    18. Step 3: Find MCL MCL= $16 per hour Step 4: Use decision rule to solve for L MRPL = MCL 4 (30 – L) = 16 30 – L = 4 L = 30 – 4 = 26 Hire 26 labor-hours Solution (continued)

    19. Step 5: Find output level for L = 26 Q = 60 L – L2 Q = 60 (26) – (26)2 Q = 1,560 – 676 Q = 884 Solution (continued)

    20. Step 0: identify information given and state objectives Step 1: state the decision rule Step 2: calculate MRPL Step 3: find MCL Step 4: use decision rule and solve for L Step 5: find Q using output from Step 4 Solution Recap

    21. Next exam: 9 November Last day to drop: 2 November Next homework: available 30 October, due 6 November Exam 2 appeals are due Friday, October 30 Etiquette Dinner is 12 November at the SAC Lessons in manners AND networking 60 – 80% of jobs are found through networking Speak with business executives at your table 4 dinner scholarships available to students who write me the best email on why I should sponsor you for the Etiquette Dinner. Deadline: 30 Oct, 5 pm Announcements

    22. Production and Costs Overview

    23. Production and Costs Overview

    24. In the short run with K as the fixed input and L as the variable input, the firm maximizes profit by operating where MRPL = MCL Recap from Last Time

    25. All inputs are variable Two decisions: Decide on input mix (capital – labor trade offs) Decide on scale of operations Definition: Returns to scale Constant Increasing Decreasing Definition: Output elasticity Long-Run Production Issues

    26. Returns to Scale in Auto Parts

    27. Q = 10 L – 0.5 L2 + 24 K – K2 Double the amount of labor and capital to see whether output doubles K = 1, L = 2 Q = 10 (2) – 0.5 (2) (2) + 24 (1) – (1) (1) Q = 20 – 2 + 24 – 1 = 41 K = 2, L = 4 Q = 10 (4) – 0.5 (4) (4) + 24 (2) – (2) (2) Q = 40 – 8 + 48 – 4 = 76 Decreasing returns to scale Returns to Scale

    28. Total cost with two inputs, L and K TC = PL L + PK K Decision rule Long run Least-cost combination of inputs What if ? What if Least Cost Production

    29. Given Q = 40 L – L2 + 54 K – 1.5 K2 PL = $10 PK = $15 Problem: Find the least-cost combination of L and K Decision rule MPL / PL = MPK / PK Find MPL and MPK MPL = 40 – 2L MPK = 54 – 3 K Example 3 (page 227 - 8)

    30. Use decision rule (40 – 2 L) / 10 = (54 – 3 K) / 15 15 (40 – 2 L) = 10 (54 – 3 K) 30 (20 – L) = 30 (18 – K) 20 – L = 18 – K L = K + 2 Example 3 (continued)

    31. Many combinations of L and K satisfy the condition. Assume: L = 17, K = 15 Find Q and TC Q = 40 L – L2 + 54 K – 1.5 K2 Q = 40 (17) – (17)2 + 54 (15) – 1.5 (15)2 Q = 680 – 289 + 810 – 1.5 (225) Q = 1201 – 337.5 Q = 863.5 TC = PL L + PK K = 10 (17) + 15 (15) TC = $395 Example 3 (continued)

    32. Definition An isoquant is a curve that shows the different combinations of inputs a firm can use to produce a given level of output. Production trade-offs: home construction Earth moving: Bobcat, Caterpillar, or 10 shovels? Electric nail gun or hammer? Preview: The answer depends on the prices of the inputs Isoquants – Production Graphs

    33. Contour Maps: Equal Elevations

    34. One Isoquant

    35. One Isoquant

    36. Given: PK , PL and Total Cost (TC) Find the combinations of L and K consistent with a given level of total cost. TC = PK K + PL L If all TC is spent on capital, the company can buy TC / PK units of capital. If all TC is spent on labor, the company can buy TC / PL units of labor. Slope of the isocost line is negative and equal to – (TC / PK) / (TC / PL) = – (PL / PK) Isocost Line

    37. Isocost Lines

    38. Isocost Lines

    39. Isocost Assumptions PK = 10 PL = 10 Least-cost production occurs where the isocost is tangent to the isoquant. Higher cost production is possible if sub-optimal combinations of L and K are chosen. Least-Cost Production

    40. If input prices change, the slope of the isocost changes and the optimal combination of K and L changes Use more of the relatively cheaper input Least-Cost Production

    41. Idea: operate where the isoquant is tangent to the isocost curve Slope of the isoquant MTRS = MPL / MPK Slope of the isocost PL / PK Decision rule The Idea of Least-Cost Analysis

    42. Sleeping > Talking

    43. Linear Q = a + b L + c K Marginal product is constant Returns to scale depend on relative sizes of b and c and the sign of a. Fixed Proportions One crane, one crane operator One hot dog, one bun One left shoe, one right shoe L-shaped isoquant Marginal product is undefined Constant returns to scale Types of Production Functions

    44. Polynomial functions' properties depend on the equation Q = a L K – b L2 K2 a > 0, b > 0 Marginal products decline Decreasing returns to scale Q = a1 L K + a2 L2K + a3 L K2 – a4 L3 K – a5 L K3 where all coefficients are positive Marginal product increases then decreases Increasing returns to scale at low output levels and decreasing returns at higher output levels Types of Production Functions (continued)

    45. Cobb-Douglas Function Q = c La Kß Diminishing marginal returns to each input Returns to scale depend on a + ß If a + ß > 1, increasing returns to scale If a + ß = 1, constant returns to scale If a + ß < 1, decreasing returns to scale Estimate it in its logarithmic form: Log (Q) = log (c) + a log (L) + ß log (K) Types of Production Functions (continued)

    46. Characteristics of Production Functions

    47. Several data sources Engineering data and experience Speculative for new products or technologies Does not incorporate non-production inputs Time-series data from the company Assumes process remains constant over time Cross-sectional data from many plants or companies Data on multiple plants for a single company is more likely to be available than across competitors Estimating a Production Function

    48. MRPL = MCL Short-run condition – only one variable input Given production function (or MPL), MR, and MCL Problem: Optimal amount of labor to employ? How much output? MPL / PL = MPK / PK = MPM / PM … Long-run condition – all inputs variable Given production function and input prices Problem: How much of each input? How much output? What is total cost? Review: Two Decision Rules for Inputs

    49. Scenario Short run; one variable input with limited supply Two plants (A and B) with different production conditions Production functions for each are given Problem Allocate the variable input between plants Decision Rule: Allocate input until MPA = MPB Special Case: Multiple Plants

    50. Scenario Short run; one variable input in limited supply Two products (G, F) with different profitability Problem Allocate the limited variable input between products Decision Rule Allocate the input until M?G = M?F Special Case: Alternative Products

    51. Short-run: increase use of variable input until MCL = MRPL With multiple variable inputs, allocate the inputs so the marginal product per dollar spent is equal across all inputs. MPL / PL = MPK / PK = MPM / PM With multiple plants for a given output, equalize marginal products across plants. With multiple outputs, equalize marginal profits across outputs. Chapter 6: Decision-Making Principles

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