Foundations of Microeconomics Topic 3.b Cost Analysis PDF

23/09/2024 FOUNDATIONS OF MICROECONOMICS Topic 3: THE PRODUCTION PROCESS AND COSTS Baye and Price, Chapter 5 FÁTIMA BARROS 1 Topic 3.b Cost Analysis 2...

23/09/2024 FOUNDATIONS OF MICROECONOMICS Topic 3: THE PRODUCTION PROCESS AND COSTS Baye and Price, Chapter 5 FÁTIMA BARROS 1 Topic 3.b Cost Analysis 2 1  23/09/2024 Introduction What are the various costs? How are they related to each other and to output? How are costs different in the short run vs. the long run? What are “Economies of Scale”? And “Economies of Scope”? Foundations of Microeconomics Fátima Barros 3 3 Overview II. Cost Analysis – Total Cost, Variable Cost, Fixed Cost – Marginal Cost – Cost Relations III. Multi-Product Cost Functions Foundations of Microeconomics Fátima Barros 4 4 2  23/09/2024 Cost Analysis Types of Costs – Short-Run Fixed costs (FC) Sunk costs Short-run variable costs (VC) Short-run total costs (TC) – Long-Run All costs are variable No fixed costs Foundations of Microeconomics Fátima Barros 5 5 Cost Function A Cost Function 𝑪(𝑸) indicates the minimum cost that a firm must incur to produce a certain level of output. A cost function is the result of a minimization problem that takes into account the production function (technology) and the market prices of inputs Foundations of Microeconomics Fátima Barros 6 6 3  23/09/2024 Total and Variable Costs 𝑪(𝑸): Minimum total cost of producing alternative levels of $ C(Q) output: Example: 𝐶 𝑄 = 100 + 20𝑄 + 15𝑄 2 + 10𝑄 3 0 Q Foundations of Microeconomics Fátima Barros 7 7 Total and Variable Costs 𝐓𝐨𝐭𝐚𝐥 𝐂𝐨𝐬𝐭, 𝑪(𝑸): Minimum total cost of producing $ alternative levels of output. C(Q) = VC + FC VC(Q) 𝐕𝐚𝐫𝐢𝐚𝐛𝐥𝐞 𝐂𝐨𝐬𝐭, 𝑉𝐶 𝑄 : Costs that vary with output. Fixed Cost Fixed Cost, 𝐹𝐶: Costs that do not FC vary with output. Fixed Cost 𝐶(𝑄) = 𝑉𝐶(𝑄) + 𝐹𝐶 0 Q Foundations of Microeconomics Fátima Barros 8 8 4  23/09/2024 Total and Variable Costs 𝑬𝒙𝒂𝒎𝒑𝒍𝒆 $ 𝐶 𝑄 = 100 + 20𝑄 + 15𝑄 2+10𝑄 3 C(Q) = VC + FC 𝐕𝐚𝐫𝐢𝐚𝐛𝐥𝐞 𝐂𝐨𝐬𝐭 VC(Q) 𝑉𝐶 𝑄 = Fixed Cost Fixed Cost FC 𝐹𝐶= 0 Q Foundations of Microeconomics Fátima Barros 9 9 Fixed and Sunk Costs FC: Costs that do not $ change as output changes. Sunk Cost: A cost that is forever lost after it has been paid. Decision makers should ignore sunk costs to FC maximize profit or minimize losses GM Closes Factories- Econ in Real Life - YouTube Q Foundations of Microeconomics Fátima Barros 10 10 5  23/09/2024 Some Definitions $ Average Total Cost (ATC) ATC 𝐴𝑇𝐶 = 𝐴𝑉𝐶 + 𝐴𝐹𝐶 AVC 𝐶(𝑄) 𝐴𝑇𝐶 = 𝑄 Average Variable Cost (AVC) 𝑉𝐶(𝑄) 𝐴𝑉 = 𝑄 Average Fixed Cost (AFC) 𝐹𝐶 𝐴𝐹𝐶 = 𝑄 AFC Q Foundations of Microeconomics Fátima Barros 11 11 Example 𝐶 𝑄 = 100 + 20𝑄 + 15𝑄 2 +10𝑄 3 𝐶 𝑄 𝐴𝑇𝐶 = = 𝑄 𝑉𝐶 𝑄 𝐴𝑉𝐶 = = 𝑄 𝐹𝐶 𝐴𝐹𝐶 = = 𝑄 Foundations of Microeconomics Fátima Barros 12 12 6  23/09/2024 Example 𝐶 𝑄 = 100 + 20𝑄 + 15𝑄 2 +10𝑄 3 𝐶 𝑄 100 𝐴𝑇𝐶 = = + 20 + 15𝑄 + 10𝑄 2 𝑄 𝑄 𝑉𝐶 𝑄 𝐴𝑉𝐶 = = 20 + 15𝑄 + 10𝑄 2 𝑄 𝐹𝐶 100 𝐴𝐹𝐶 = = 𝑄 Q Foundations of Microeconomics Fátima Barros 13 13 Some Definitions $ Marginal Cost: MC The additional cost the firm incurs when produces an additional unit of output ∆𝐶 𝑀𝐶 = ∆𝑄 In a continuous context, the marginal cost is the slope of the cost function 𝑑𝐶 𝑀𝐶 = 𝑑𝑄 Q Foundations of Microeconomics Fátima Barros 14 14 7  23/09/2024 Example 𝐶 𝑄 = 100 + 20𝑄 + 15𝑄2 +10𝑄3 𝑑𝐶 𝑄 𝑀𝐶 𝑄 = = 0 + 20 + 15 × 2 × 𝑄 + 10 × 3 × 𝑄 2 𝑑𝑄 𝑀𝐶 𝑄 = 20 + 30𝑄 + 30𝑄 2 Foundations of Microeconomics Fátima Barros 15 15 Some Definitions Marginal Cost: $ The MC curve intersects the MC ATC AVC ATC and the AVC at its minimum point Q Foundations of Microeconomics Fátima Barros 16 16 8  23/09/2024 Foundations of Microeconomics Fátima Barros 17 17 An Example Total Cost: 𝐶 𝑄 = 200 + 10𝑄 + 8𝑄 2 + 10𝑄3 Variable cost function: 𝑉𝐶 𝑄 = 10𝑄 + 8𝑄2 + 10𝑄3 Variable cost of producing 2 units: 𝑉𝐶(2) = 10(2) + 8(2)2 + 10(2)3 = 132 Fixed costs: 𝐹𝐶 = 200 Marginal cost function: 𝑑𝐶 𝑄 𝑀𝐶 𝑄 = = 10 + 16𝑄 + 30𝑄2 𝑑𝑄 Marginal cost of producing 2 units: 𝑀𝐶(2) = 10 + 16(2) + 30(2)2 = 162 Foundations of Microeconomics Fátima Barros 18 18 9  23/09/2024 Fixed Cost Q0(ATC-AVC) MC $ = Q0 AFC ATC = Q0(FC/ Q0) AVC = FC ATC AFC Fixed Cost AVC Q0 Q Foundations of Microeconomics Fátima Barros 19 19 Variable Cost Q0AVC MC $ ATC = Q0[VC(Q0)/ Q0] AVC = VC(Q0) AVC Variable Cost Q0 Q Foundations of Microeconomics Fátima Barros 20 20 10  23/09/2024 Total Cost Q0ATC MC $ = Q0[C(Q0)/ Q0] ATC = C(Q0) AVC ATC Total Cost Q0 Q Foundations of Microeconomics Fátima Barros 21 21 Cubic Cost Function 𝐶(𝑄) = 𝑓 + 𝑎 𝑄 + 𝑏 𝑄2 + 𝑐𝑄3 Marginal Cost? 𝑀𝐶(𝑄) = 𝑑𝐶/𝑑𝑄 = 𝑎 + 2𝑏𝑄 + 3𝑐𝑄2 Foundations of Microeconomics Fátima Barros 22 22 11  23/09/2024 Costs in the Short Run & Long Run Short run: Some inputs are fixed (e.g., factories, land). The costs of these inputs are FC. Long run: All inputs are variable (e.g., firms can build more factories or sell existing ones). In the long run, ATC at any Q is cost per unit using the most efficient mix of inputs for that Q (e.g., the factory size with the lowest ATC). Foundations of Microeconomics Fátima Barros 23 23 Example 3: Long Run ATC with 3 Factory Sizes Avg Firm can choose from Total three factory sizes: Cost S, M, L. ATCS ATCM ATCL Each size has its own Short Run ATC curve. The firm can change to a different factory size in the long run, but not in the short Q run. Foundations of Microeconomics Fátima Barros 24 24 12  23/09/2024 Example 3: Long Run ATC with 3 Factory Sizes To produce less Avg than QA, firm will Total Cost choose size S ATCS ATCM in the long run. ATCL To produce between QA and QB, firm will 𝐴𝑇𝐶𝐿𝑅 choose size M in the long run. To produce more than QB, firm will Q choose size L QA QB in the long run. Foundations of Microeconomics Fátima Barros 25 25 A Typical Long Run ATC Curve In the real world, factories come in ATC many sizes, each with its own Short Run ATC 𝐴𝑇𝐶𝐿𝑅 curve. So a typical Long Run ATC curve looks like this: Q Foundations of Microeconomics Fátima Barros 26 26 13  23/09/2024 Long-Run Average Costs Average Cost € 𝐴𝐶𝐿𝑅 Economies Diseconomies of Scale of Scale Q* Q Foundations of Microeconomics Fátima Barros 27 27 How ATC changes as the scale of production changes Economies of scale occur when the long-run average cost decreases as the output goes up. Economies of scale occur because increasing production allows greater specialization: workers are more efficient when they are more specialized. – More common when Q is not very high. Diseconomies of scale are due to coordination problems in large organizations. E.g., management becomes stretched, can’t control costs. – More common when Q is very high. Foundations of Microeconomics Fátima Barros 28 28 14  23/09/2024 Multi-Product Cost Function C(Q1, Q2): Cost of jointly producing two outputs. General function form: C (Q1 , Q2 ) = f + aQ1Q2 + bQ12 + cQ22 Foundations of Microeconomics Fátima Barros 29 29 Economies of Scope 𝐶(𝑄1, 0) + 𝐶(0, 𝑄2) > 𝐶(𝑄1, 𝑄2). Economies of Scope occur when it is cheaper to produce the two outputs jointly instead of separately. Example: It is cheaper for a railroad company to offer passenger and freight rail services jointly than separately. Foundations of Microeconomics Fátima Barros 30 30 15  23/09/2024 A Numerical Example: 𝐶(𝑄1, 𝑄2) = 90 − 2𝑄1𝑄2 + (𝑄1 )2 + (𝑄2 )2 Economies of Scope? 𝐶 𝑄1, 0 = 90 + 𝑄1 2 𝐶(0, 𝑄2) = 90 + (𝑄2 )2 𝐶 𝑄1, 0 + 𝐶 0, 𝑄2 = 180 + 𝑄1 2+ (𝑄2 )2 Economies of Scope if 𝐶 𝑄1, 0 + 𝐶 0, 𝑄2 > 𝐶 𝑄1, 𝑄2 Yes, since 90 > − 2𝑄1𝑄2 Foundations of Microeconomics Fátima Barros 31 31 Quadratic Multi-Product Cost Function 𝐶(𝑄1, 𝑄2) = 𝑓 + 𝑎𝑄1𝑄2 + (𝑄1 )2 + (𝑄2 )2 Economies of scope: 𝑓 > 𝑎𝑄1𝑄2 𝐶 𝑄1 , 0 + 𝐶 0, 𝑄2 = 𝑓 + (𝑄1 )2 + 𝑓 + (𝑄2)2 𝐶(𝑄1, 𝑄2) = 𝑓 + 𝒂𝑸𝟏𝑸𝟐 + (𝑄1 )2 + (𝑄2 )2 𝑓 > 𝒂𝑸𝟏𝑸𝟐: Joint production is cheaper Foundations of Microeconomics Fátima Barros 32 32 16  23/09/2024 Conclusion To maximize profits (minimize costs) managers must use inputs such that the value of marginal product of each input reflects price the firm must pay to employ the input. The optimal mix of inputs is achieved when the MRTSKL = (w/r). Cost functions are essential for firm’s profit maximization Foundations of Microeconomics Fátima Barros 33 33 Appendix: Examples 34 17  23/09/2024 Example 1: Farmer Jack’s Costs Farmer Jack must pay €1000 per month for the land, regardless of how much wheat he grows. The market wage for a farm worker is €2000 per month. So Farmer Jack’s costs are related to how much wheat he produces…. Foundations of Microeconomics Fátima Barros 35 35 Example 1: Farmer Jack’s Costs L Q Cost of Cost of Total (no. of (units land labor cost workers) of wheat) 0 0 €1,000 €0 €1,000 1 1000 €1,000 €2,000 €3,000 2 1800 €1,000 €4,000 €5,000 3 2400 €1,000 €6,000 €7,000 4 2800 €1,000 €8,000 €9,000 5 3000 €1,000 €10,000 €11,000 Foundations of Microeconomics Fátima Barros 36 36 18  23/09/2024 Example 1: Farmer Jack’s Total Cost Curve Q (units Total Total Cost of wheat) Cost 12000 10000 0 €1,000 8000 1000 €3,000 6000 1800 €5,000 4000 2400 €7,000 2000 2800 €9,000 0 0 500 1000 1500 2000 2500 3000 3500 3000 €11,000 Foundations of Microeconomics Fátima Barros 37 37 Marginal Cost Marginal Cost (MC) is the increase in Total Cost from producing one more unit: ∆TC 𝑀𝐶 = ∆Q If the total cost function is 𝑇𝐶(𝑄), then 𝑑𝑇𝐶 𝑀𝐶 = 𝑇𝐶’(𝑄 = 𝑑𝑄 Foundations of Microeconomics Fátima Barros 38 38 19  23/09/2024 Example 1: Total and Marginal Cost Q Total Marginal (units Cost Cost (MC) of wheat) 0 €1,000 ∆Q = 1000 ∆TC = €2000 €2.00 1000 €3,000 ∆Q = 800 ∆TC = €2000 €2.50 1800 €5,000 ∆Q = 600 ∆TC = €2000 €3.33 2400 €7,000 ∆Q = 400 ∆TC = €2000 €5.00 2800 €9,000 ∆Q = 200 ∆TC = €2000 €10.00 3000 €11,000 Foundations of Microeconomics Fátima Barros 39 39 Example 1: The Marginal Cost Curve Q (units TC MC MC usually rises of wheat) as Q rises, as in this example. 0 €1,000 12 € €2.00 10 € 1000 €3,000 Marginal Cost (€) €2.50 8€ 1800 €5,000 6€ €3.33 2400 €7,000 4€ €5.00 2€ 2800 €9,000 €10.00 0€ 3000 €11,000 0 1,000 Q 2,000 3,000 Foundations of Microeconomics Fátima Barros 40 40 20  23/09/2024 Why MC is Important Farmer Jack is rational and wants to maximize his profit. To increase profit, should he produce more or less wheat? To find the answer, Farmer Jack needs to “think at the margin.” If the cost of additional wheat (MC) is less than the revenue he would get from selling it, then Jack’s profits rise if he produces more. Foundations of Microeconomics Fátima Barros 41 41 Fixed and Variable Costs Fixed costs (FC) do not vary with the quantity of output produced. – For Farmer Jack, FC = €1000 for his land – Other examples: cost of equipment, loan payments, rent Variable costs (VC) vary with the quantity produced. – For Farmer Jack, VC = wages he pays workers – Other example: cost of materials Total cost (TC) = FC + VC Foundations of Microeconomics Fátima Barros 42 42 21  23/09/2024 Example 2 Our second example is more general, applies to any type of firm producing any good with any types of inputs. Foundations of Microeconomics Fátima Barros 43 43 Example 2: Costs €800 FC Q FC VC TC VC €700 0 100 0 100 TC €600 1 100 70 170 €500 Costs 2 100 120 220 €400 3 100 160 260 €300 4 100 210 310 €200 5 100 280 380 €100 6 100 380 480 €0 7 100 520 620 0 1 2 3 4 5 6 7 Q Foundations of Microeconomics Fátima Barros 44 44 22  23/09/2024 Example 2: Marginal Cost Q TC MC Recall, Marginal Cost (MC) is the change in total cost from 0 €100 producing one more unit: €70 1 170 ∆TC 50 MC = 2 220 ∆Q 40 3 260 Usually, MC rises as Q rises, due to 50 diminishing marginal product. 4 310 70 Sometimes (as here), MC falls before 5 380 rising. 100 6 480 (In other examples, MC may be 140 constant.) 7 620 Foundations of Microeconomics Fátima Barros 45 45 Example 2: Marginal Cost Q TC MC Marginal Cost 0 €100 160 €70 140 1 170 50 120 2 220 100 40 3 260 80 50 60 4 310 70 40 5 380 20 100 6 480 0 0 1 2 3 4 5 6 7 8 140 7 620 Foundations of Microeconomics Fátima Barros 46 46 23  23/09/2024 Example 2: Average Fixed Cost Q FC AFC Average fixed cost (AFC) is fixed cost divided by the quantity of 0 €100 n/a output: 1 100 100 AFC = FC/Q 2 100 50 Notice that AFC falls as Q rises: The 3 100 33.33 firm is spreading its fixed costs over a 4 100 25 larger and larger number of units. 5 100 20 6 100 16.67 7 100 14.29 Foundations of Microeconomics Fátima Barros 47 47 Example 2: Average Fixed Cost Q FC AFC Average Fixed Cost 120 0 €100 n/a 100 1 100 100 2 100 50 80 3 100 33.33 60 4 100 25 40 5 100 20 20 6 100 16.67 0 0 1 2 3 4 5 6 7 8 7 100 14.29 Foundations of Microeconomics Fátima Barros 48 48 24  23/09/2024 Example 2: Average Variable Cost Q VC AVC Average variable cost (AVC) is variable cost divided by the 0 €0 n/a quantity of output: 1 70 €70 AVC = VC/Q 2 120 60 3 160 53.33 As Q rises, AVC may fall initially. In most cases, AVC will eventually rise 4 210 52.50 as output rises. 5 280 56.00 6 380 63.33 7 520 74.29 Foundations of Microeconomics Fátima Barros 49 49 Example 2: Average Variable Cost Q VC AVC 80 Average Variable Cost €0 70 0 n/a 60 1 70 €70 50 2 120 60 40 30 3 160 53.33 20 4 210 52.50 10 0 5 280 56.00 0 1 2 3 4 5 6 7 6 380 63.33 7 520 74.29 Foundations of Microeconomics Fátima Barros 50 50 25  23/09/2024 Example 2: Average Total Cost Q TC ATC AFC AVC Average total cost (ATC) equals total cost 0 100 n/a n/a n/a divided by the quantity of output: 1 170 170 100 70 ATC = TC/Q 2 220 110 50 60 3 260 86.67 33.33 53.33 Also, 4 310 77.50 25 52.50 ATC = AFC + AVC 5 380 76 20 56.00 6 480 80 16.67 63.33 7 620 88.57 14.29 74.29 Foundations of Microeconomics Fátima Barros 51 51 Example 2: Average Total Cost Q TC ATC 180 160 Average Total Cost 0 100 n/a 140 120 1 170 170 100 80 2 220 110 60 40 3 260 86.67 20 0 4 310 77.50 0 1 2 3 4 5 6 7 5 380 76 6 480 80 Usually, as in this example, the 7 620 88.57 ATC curve is U-shaped. Foundations of Microeconomics Fátima Barros 52 52 26  23/09/2024 Example 2: Why ATC is Usually U-Shaped As Q rises: €200 Initially, €175 falling AFC €150 pulls ATC down. €125 Costs Eventually, rising AVC €100 pulls ATC up. €75 Efficient scale: €50 The quantity that €25 minimizes ATC. €0 0 1 2 3 4 5 6 7 Q Foundations of Microeconomics Fátima Barros 53 53 Example 2: ATC and MC When MC < ATC, ATC is €200 ATC falling. €175 MC When MC > ATC, ATC is rising. €150 €125 Costs The MC curve crosses the ATC curve at the ATC €100 curve’s minimum. €75 d C(q)  €50  q  C ' ( q)q − C ( q) C ' ( q) − C ( q) / q = = , dq q2 q €25 d C(q)  If  q = 0, then C' (q) = C(q) , €0 dq q 0 1 2 3 4 5 6 7 Q Foundations of Microeconomics Fátima Barros 54 54 27  23/09/2024 Summary (1) Implicit costs do not involve a cash outlay, yet are just as important as explicit costs to firms’ decisions. Accounting profit is revenue minus explicit costs. Economic profit is revenue minus total (explicit + implicit) costs. The production function shows the relationship between output and inputs. The marginal product of labor is the increase in output from a one-unit increase in labor, holding other inputs constant. The marginal products of other inputs are defined similarly. Foundations of Microeconomics Fátima Barros 55 55 Summary (2) Marginal product usually diminishes as the input increases. Thus, as output rises, the production function becomes flatter and the total cost curve becomes steeper. Variable costs vary with output; fixed costs do not. Marginal cost is the increase in total cost from an extra unit of production. The MC curve is usually upward-sloping. Average variable cost is variable cost divided by output. Average fixed cost is fixed cost divided by output. AFC always falls as output increases. Foundations of Microeconomics Fátima Barros 56 56 28  23/09/2024 Summary (3) Average total cost (sometimes called “cost per unit”) is total cost divided by the quantity of output. The ATC curve is usually U- shaped. The MC curve intersects the ATC curve at minimum average total cost. When MC < ATC, ATC falls as Q rises. When MC > ATC, ATC rises as Q rises. In the long run, all costs are variable. Economies of scale: ATC falls as Q rises. Diseconomies of scale: ATC rises as Q rises. Constant returns to scale: ATC remains constant as Q rises. Foundations of Microeconomics Fátima Barros 57 57 29  15/09/2024 FOUNDATIONS OF MICROECONOMICS Topic 3: THE PRODUCTION PROCESS AND COSTS Baye and Price, Chapter 5 FÁTIMA BARROS 1 Topic 3.a: The Production Process 2 1  15/09/2024 Introduction What is the most efficient way to produce a certain amount of output? What is the optimal combination of inputs, given a certain technology and input prices? Foundations of Microeconomics Fátima Barros 3 3 Overview 1. Production Function 1.1 Short-run versus Long-run 2. Short Run Analysis 2.1 Total Product 2.2 Average Product 2.3 Marginal Product 2.4 Law of Diminishing Returns 2.5 Optimal Use of a Variable Input 3. Long Run Analysis 3.1 Isoquants 3.2 Marginal Rate of Technical Substitution (MRTS) 3.3 MRTS and the Slope of the Isoquant 3.4 Isocosts 3.5 Cost Minimization 3.6 Returns to Scale Foundations of Microeconomics Fátima Barros 4 4 2  15/09/2024 Tesla Assembly Plant How the Tesla Model S is Made | Tesla Motors Part 1 (WIRED) - Bing video Foundations of Microeconomics Fátima Barros 5 5 Cement Plant https://www.youtube.com/watch?v=TdxPxfeEUSQ Foundations of Microeconomics Fátima Barros 6 6 3  15/09/2024 Services Company Foundations of Microeconomics Fátima Barros 7 7 Textil Manufacturer https://www.youtube.com/watch?v=HVXPurvaK5c https://www.youtube.com/watch?v=KE8fhKoRNbU https://www.youtube.com/watch?v=LUE2pw4wGJ8 Foundations of Microeconomics Fátima Barros 8 8 4  15/09/2024 Amazon Warehouse Inside Amazon's Smart Warehouse (youtube.com) Inside Amazon’s robot revolution - YouTube https://www.youtube.com/watch?v=jwu9SX3YPSk Foundations of Microeconomics Fátima Barros 9 9 1. Production Function Raw Materials Production Output Labor Function Physical Capital Technology Foundations of Microeconomics Fátima Barros 10 10 5  15/09/2024 1. Production Function During the production process firm turns inputs (also called factors of production) into outputs. Broad categories of Inputs: Capital, Labor, Materials Production Function: indicates the maximum amount of output 𝑄 that a firm will produce for every specified combination of inputs. Foundations of Microeconomics Fátima Barros 11 11 1. Production Function In Economics we represent the Production Function as: 𝑄 = 𝐹 𝐾, 𝐿 𝑄 is quantity of output produced. 𝐾 is capital input. 𝐿 is labor input. 𝐹 is a functional form relating the inputs to output. Example: 𝑄 = 𝐹(𝐾, 𝐿) = 𝐾 0.5 𝐿0.5 Foundations of Microeconomics Fátima Barros 12 12 6  15/09/2024 1. Production Function Features 1. Production functions describe what is technically feasible when the firm operates efficiently. 2. Every production function refers to a given technology 3. As technology advances, the production function will change to reflect the higher level of output that can be obtained with the same input. 4. Inputs will not be used if they decrease output. 5. Production function refers to a specific time horizon. – Fixed and variable inputs Foundations of Microeconomics Fátima Barros 13 13 1.1 Short run vs Long run Short run: period of time in which some production factors are fixed. – Introduces restrictions in the decision process – Usually capital/equipment is the fixed factor Long run: when all production factors are adjustable, i.e. all inputs are variable Foundations of Microeconomics Fátima Barros 14 14 7  15/09/2024 2. Short-run Analysis 15 Example 1: Farmer Jack’s Production Function Farmer Jack grows wheat. He has 5 acres of land. He can hire as many workers as he wants. Variable input: Labor Fixed input: Land Foundations of Microeconomics Fátima Barros 16 16 8  15/09/2024 Example 1: Farmer Jack’s Production Function L Q 3,000 (no. of (units workers) 2,500 Quantity of output of wheat) 0 0 2,000 1 1000 1,500 2 1800 1,000 3 2400 500 4 2800 0 0 1 2 3 4 5 5 3000 No. of workers Foundations of Microeconomics Fátima Barros 17 17 Example 1: Total & Average Product Average L Q Product of L: (no. of (units 𝑄 𝑨𝑷𝑳 = workers) of wheat) 𝐿 0 0 1 1000 1000 Average Product of Labour (APL) is the product per 2 1800 900 unit of labour. 3 2400 800 4 2800 700 5 3000 600 Foundations of Microeconomics Fátima Barros 18 18 9  15/09/2024 Example 1: Total & Marginal Product Q Marginal L Product of L: (# of (units ∆𝑄 𝑴𝑷𝑳 = workers) of wheat) ∆𝐿 0 0 ∆L = 1 ∆Q = 1000 1000 Marginal Product of 1 1000 Labour (𝑀𝑃𝐿 ) is the ∆L = 1 ∆Q = 800 800 additional output obtained when one 2 1800 additional worker is ∆L = 1 ∆Q = 600 600 employed 3 2400 ∆L = 1 ∆Q = 400 400 4 2800 ∆L = 1 ∆Q = 200 200 5 3000 Foundations of Microeconomics Fátima Barros 19 19 Example 1: 𝑀𝑃𝐿 = slope of prod. function L Q MPL equals the (no. of (units 𝑀𝑃𝐿 3,000 slope of the production workers) of wheat) function. Quantity of output 2,500 Notice that 0 0 MPL diminishes 2,000 1000 as L increases. 1 1000 1,500 This explains why the 800 production function gets 2 1800 1,000 flatter 600 as L increases. 3 2400 500 400 4 2800 0 200 0 1 2 3 4 5 5 3000 No. of workers Foundations of Microeconomics Fátima Barros 20 20 10  15/09/2024 Example 1: 𝑀𝑃𝐿 = slope of prod. function L Q (no. of (units 𝑀𝑃𝐿 workers) of wheat) 𝑀𝑃𝐿 equals the slope of the 0 0 production function. 1000 Notice that 𝑀𝑃𝐿 diminishes 1 1000 as L increases. 800 2 1800 This explains why the 600 production function gets flatter 3 2400 as L increases. 400 4 2800 200 5 3000 Foundations of Microeconomics Fátima Barros 21 21 Why 𝑀𝑃𝐿 Diminishes Question: Farmer Jack’s output rises by a smaller and smaller amount for each additional worker. Why? Answer: As Jack adds workers, the average worker has less land to work with and will be less productive. In general, 𝑴𝑷𝑳 diminishes as the number of workers, L, rises and the level of capital, K, is fixed (equipment, machines, etc.). Foundations of Microeconomics Fátima Barros 22 22 11  15/09/2024 Short Run Analysis In the short run some factors are fixed. Assume: Capital (𝐾) as a fixed factor Labor (𝐿) as a variable factor 𝑸 = 𝒇(𝑳; 𝑲) therefore the choice of 𝐿 determines the production level 𝑸 = 𝒇(𝑳) Example: Consider a production function 𝑄 = 𝐾. 𝐿 If 𝐾 = 16 then 𝑄 = 16. 𝐿 Foundations of Microeconomics Fátima Barros 23 23 2.1 Total Product Total Product (TP): maximum output produced with given amounts of inputs. Example: assume a production function given by: 𝑄 = 𝐹(𝐾, 𝐿) = 𝐾 0.5 𝐿0.5 and K is fixed at 16 units (𝐾 = 16). Short run production function: 0.5 𝐿0.5 𝑄 = 16 = 4𝐿0.5 Total Product when 100 units of labor are used (𝐿 = 100)? 𝑄 = 4 100 0.5 = 4(10) = 40 𝑢𝑛𝑖𝑡𝑠 Foundations of Microeconomics Fátima Barros 24 24 12  15/09/2024 2.2 Average Product of an Input Average Product of an Input: measure of output produced per unit of input. 𝑸 Average Product of Labor: 𝑨𝑷𝑳 = 𝑳 Measures the output of an “average” worker. 𝑸 Average Product of Capital: 𝑨𝑷𝑲 = 𝑲 Measures the output of an “average” unit of capital. Foundations of Microeconomics Fátima Barros 25 25 Average Product of an Input – Example Example: 𝑄 = 𝐹 𝐾, 𝐿 = 𝐾 0.5 𝐿0.5 the average product of labor is 𝑄 𝐾0.5 𝐿0.5 𝐾0.5 0.5 𝐴𝑃𝐿 = = = 0.5 = 𝐾/𝐿 𝐿 𝐿 𝐿 the average product of capital is 𝑄 𝐾0.5 𝐿0.5 𝐴𝑃𝐾 = 𝐾 = = 𝐿0.5 /𝐾 0.5 = 𝐿/𝐾 0.5 𝐾 If the inputs values are K = 16 and L = 25, then the average product of labor is 𝑄.. 𝐴𝑃𝐿 = 𝐿 = [(16) 0 5(25)0 5]/25 = 4/5 the average product of capital is 𝑄.. 𝐴𝑃𝐾 = 𝐾 = [(16)0 5(25)0 5]/16 = 5/4 Foundations of Microeconomics Fátima Barros 26 26 13  15/09/2024 2.3 Marginal Product of an Input Marginal Product on an Input: is the increase in output arising from an additional unit of that input, holding all other inputs constant. Marginal Product of Labor: 𝑴𝑷𝑳 = ∆𝑸/∆𝑳 Measures the output produced by the last worker. Slope of the production function (with respect to labor). Marginal Product of Capital: 𝑴𝑷𝑲 = ∆𝑸/∆𝑲 Measures the output produced by the last unit of capital. When capital is allowed to vary in the short run, MPK is the slope of the production function (with respect to capital). Foundations of Microeconomics Fátima Barros 27 27 Example 2 Foundations of Microeconomics Fátima Barros 28 28 14  15/09/2024 Example 2 (cont.) Foundations of Microeconomics Fátima Barros 29 29 Relationship between Average Product and Marginal Product When the marginal product is greater than the average product, the average product is increasing. – The last unit of labor contributes more to output than the previous units, therefore the average goes up. When the marginal product is lower than the average product, the average product is decreasing. Foundations of Microeconomics Fátima Barros 30 30 15  15/09/2024 2.4 Law of Diminishing Returns The Law of Diminishing Returns states that as the use of an input increases (with other inputs fixed) it reachs a point at which marginal product begins to decline (decreasing marginal returns). To be efficient the firm must operate in the region of decreasing marginal returns, when marginal producto is decreasing. Foundations of Microeconomics Fátima Barros 31 31 Marginal Product of an Input Example: 𝑄 = 𝐹 𝐾, 𝐿 = 𝐾 0.5 𝐿0.5 the marginal product of labor is 0.5 𝜕𝑄 𝐾 𝑀𝑃𝐿 = = 0.5𝐾 0.5 𝐿(0.5−1) = 0.5𝐾 0.5 𝐿−0.5 = 0.5 𝜕𝐿 𝐿 the marginal product of capital is 𝜕𝑄 𝐿 0.5 𝑀𝑃𝐾 = 𝜕𝐾 = 0.5𝐿0.5 𝐾 (0.5−1) = 0.5𝐿0.5 𝐾 −0.5 = 0.5 𝐾 If the inputs values are K = 16 and L = 25, then the marginal product of labor is 𝜕𝑄 𝐾 0.5 4 2 𝑀𝑃𝐿 = = 0.5 = 0.5 = 𝜕𝐿 𝐿 5 5 the marginal product of capital is 𝜕𝑄 𝐿 0.5 5 2.5 𝑀𝑃𝐾 = = 0.5 = 0.5 = 𝜕𝐾 𝐾 4 4 Foundations of Microeconomics Fátima Barros 32 32 16  15/09/2024 Short Run Analysis What is the Optimal Quantity of a Variable Input? Foundations of Microeconomics Fátima Barros 33 33 2.5 Optimal Quantity of a Variable Input Short-run analysis ― Example Example: Suppose that you produce t-shirts. The market price (P) of t-shirts is 5€. If each worker’s wage is 50€, which is the number of workers that maximizes the firm’s profit? L K Q Q/L Q/ L P*Q/ L 0 10 0 Value of 1 10 10 10 10 50 Marginal Product 2 10 30 15 20 100 of Labour 3 10 60 20 30 150 4 10 80 20 20 100 5 10 95 19 15 75 6 10 108 18 13 65 7 10 112 16 4 20 8 10 112 14 0 0 9 10 108 12 -4 -20 10 10 100 10 -8 -40 Foundations of Microeconomics Fátima Barros 34 34 17  15/09/2024 2.5 Optimal Quantity of a Variable Input Short-run analysis Golden Rule 1: The optimal quantity of a variable input is determined at the point where the Value of Marginal Product (VMP) of the input is equal to the Cost of this Input 𝑉𝑀𝑃𝐿 = 𝑃 × 𝑀𝑃𝐿 = 𝑤 Wage VMPL 𝑉𝑀𝑃𝐿 Profit Maximizing Point w 𝑉𝑀𝑃𝐿 = 𝑤 L* L Foundations of Microeconomics Fátima Barros 35 35 2.5 Optimal Quantity of a Variable Input Short-run analysis To be efficient the firm must operate on the decreasing part of marginal costs. 𝑉𝑀𝑃𝐿 = 𝑃 × 𝑀𝑃𝐿 = 𝑤 Wage VMPL 𝑉𝑀𝑃𝐿 Profit Maximizing Point w 𝑉𝑀𝑃𝐿 = 𝑤 L* L Foundations of Microeconomics Fátima Barros 36 36 18  15/09/2024 Efficient use of Inputs in the Short Run When labor or capital vary in the short run, to maximize profit a manager will hire – labor until the value of marginal product of labor equals the wage: 𝑉𝑀𝑃𝐿 = 𝑤, where 𝑉𝑀𝑃𝐿 = 𝑃. 𝑀𝑃𝐿 – capital until the value of marginal product of capital equals the rental rate: 𝑉𝑀𝑃𝐾 = 𝑟, where 𝑉𝑀𝑃𝐾 = 𝑃. 𝑀𝑃𝐾 Foundations of Microeconomics Fátima Barros 37 37 Another Example Consider a production function 𝑸 = 𝑭(𝑳; 𝑲 = 𝟏) = 𝑳𝟎.𝟓 𝝏𝑸 1 The Marginal Product is 𝑴𝑷𝑳 = then 𝑀𝑃𝐿 = 0.5𝐿−0.5 = 0.5 𝝏𝑳 𝐿 Suppose wage 𝒘 = 𝟏 and the market price of the product is 𝑷 = 𝟒 Question: If the firm wants to maximize profit, how many workers should it employ? 1 𝑃 × 𝑀𝑃𝐿 = 𝑤 then 4 × 0.5 = 1 ⇒ 𝑳∗ = 𝟒 𝐿 In general terms, for any value of w 𝟐 1 𝑷 𝑃 × 𝑀𝑃𝐿 = 𝑤 then 𝑃 × 0.5 = w ⇒ 𝑳∗ = 𝟎. 𝟓 𝐿 𝒘 𝟒 If 𝑷 = 𝟒 then 𝑳∗ = is the Demand for labor 𝒘𝟐 Foundations of Microeconomics Fátima Barros 38 38 19  15/09/2024 Another Example PRODUCTION FUNCTION 3.5 3 𝑸 = 𝑭(𝑳; 𝑲 = 𝟏) = 𝑳𝟎.𝟓 2.5 2 1.5 1 𝑀𝑃𝐿 = 0.5𝐿−0.5 = 0.5 1 𝐿 0.5 0 Market Price 𝑷 = 𝟒 1 1. 5 2 2. 5 3 3. 5 4 4. 5 5 5. 5 6 6. 5 7 7. 5 8 8. 5 9 9. 5 10 10. 5 Value of Marginal Product of Labor Wage 𝒘 = 𝟏 2.5 2 1 𝑃.𝑀𝑃𝐿 𝑃 × 𝑀𝑃𝐿 = 𝑤 then 4 × 0.5 =1 1.5 𝐿 𝑤 1 0.5 𝑳∗ = 𝟒 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 Labor Foundations of Microeconomics Fátima Barros 39 39 Profit Maximization Profit=Revenue-Cost 𝝅 = 𝑷 × 𝑸 − 𝒘𝑳 − 𝒓𝑲 ഥ then 𝑄 = 𝐹 𝐿; 𝐾 Since 𝑄 = 𝐹 𝐿, 𝐾 and K is fixed, (𝐾 = 𝐾), ഥ 𝜋 = 𝑃 × 𝑄 − 𝑤𝐿 − 𝑟 𝐾 Firm chooses 𝐿 to maximize profit, 𝜋, therefore 𝝏𝝅 =𝟎 𝝏𝑳 Or 𝜕𝜋 𝜕𝑄 =𝑃× −𝑤 =0 𝜕𝐿 𝜕𝐿 𝜕𝑄 Therefore 𝑃 × 𝜕𝐿 = 𝑤 or 𝑉𝑀𝑃𝐿 = 𝑤 Golden Rule Foundations of Microeconomics Fátima Barros 40 40 20  15/09/2024 3. Long Run Analysis 41 Long Run Analysis In the long rum all inputs are variable We will consider only labor and capital 𝑸 = 𝒇(𝑳, 𝑲) therefore production can be obtained with different combinations of 𝐾 and 𝐿 Foundations of Microeconomics Fátima Barros 42 42 21  15/09/2024 Countour Lines: Kilimanjaro Mountain Foundations of Microeconomics Fátima Barros 43 43 Contour Lines Foundations of Microeconomics Fátima Barros 44 44 22  15/09/2024 Production Function Foundations of Microeconomics Fátima Barros 45 45 3.1 Isoquant Capital (K) 60 50 40 30 20 10 Q=10 0 2 3 4 5 6 7 8 9 10 Labour (L) Foundations of Microeconomics Fátima Barros 46 46 23  15/09/2024 3.1 Isoquant Isoquant: represents the long-run combinations of inputs (𝐾, 𝐿) that yield the producer the same level of output. Isoquants are typically convex downward sloping curves. Example: consider the production function 𝑄 = 𝐿0.5 𝐾 0.5 What is the expression of the isoquant for 𝑄ത = 100? 𝑄 ത 100 2 10.000 𝑄ത = 𝐿0.5 𝐾 0.5 then 𝐾 0.5 = 𝐿0.5 or 𝐾 = = 𝐿0.5 𝐿 Foundations of Microeconomics Fátima Barros 47 47 A Family of Isoquants Isoquants that represent the same production function never intersect Foundations of Microeconomics Fátima Barros 48 48 24  15/09/2024 3.2 Marginal Rate of Technical Substitution Marginal Rate of Technical Substitution (MRTS) MRTS: The rate at which the producer can substitute one factor by the other while maintaining the same output level. MRTS gives the answer to the following question: If we decrease the amount of labor by one unit (∆𝐿 = −1), by how many units do we need to increase the amount of capital (∆𝐾) in order to maintain the level of output constant (∆𝑄 = 0)? In this case we are moving along an isoquant (upwards) Foundations of Microeconomics Fátima Barros 49 49 3.3 MRTS and the Slope of Isoquant MRTS and the Slope of Isoquant Foundations of Microeconomics Fátima Barros 50 50 25  15/09/2024 3.3 MRTS and the Slope of Isoquant The marginal product of any input is the increase in output arising from an additional unit of that input, holding all other inputs constant. Additional output from Increased use of Labor = 𝑀𝑃𝐿 × ∆𝐿 Reduction in output from Decreased Use of Capital = 𝑀𝑃𝐾 × ∆𝐾 The total output changes by ∆𝑄 = 𝑀𝑃𝐿 × ∆𝐿 + 𝑀𝑃𝐾 × ∆𝐾 Since moving along the isoquant ∆𝑄 = 0 then we can write ∆𝐾 𝑀𝑃𝐿 𝑀𝑃𝐿 × ∆𝐿 + 𝑀𝑃𝐾 × ∆𝐾 = 0 𝑜𝑟 − = ∆𝐿 𝑀𝑃𝐾 and this is the slope of the isoquant. 𝜕𝑄 𝜕𝑄 In continuous terms: 𝑑𝑄 = 𝑑𝐿 + 𝑑𝐾 𝜕𝐿 𝜕𝐾 Foundations of Microeconomics Fátima Barros 51 51 3.3 MRTS is the Slope of Isoquant The slope of an isoquant at a given point is the Marginal Rate of Technical Substitution (MRTS) and we can derive it as ∆𝐾 𝑀𝑃𝐿 𝑀𝑅𝑇𝑆 = − = ∆𝐿 𝑀𝑃𝐾 𝜕𝐹 𝐾,𝐿 𝑑𝐾 𝜕𝐿 Or, using derivatives, 𝑀𝑅𝑇𝑆 = − 𝑑𝐿 = 𝜕𝐹 𝐾,𝐿 𝜕𝐾 The MRTS is the ratio of marginal products. Foundations of Microeconomics Fátima Barros 52 52 26  15/09/2024 Different Isoquants: Perfect Substitutes Capital and labor are perfect substitutes K Linear isoquants imply Increasing that inputs are substituted Output at a constant rate, independent of the input levels employed. 𝑄 = 𝑎𝐾 + 𝑏𝐿 𝑄1 𝜕𝑄 𝜕𝑄 𝑎 𝑀𝑃𝐿 = = 𝑏 𝑀𝑃𝐾 = =𝑎 𝜕𝐿 𝜕𝐾 Q1 Q2 Q3 𝑀𝑃 𝑀𝑅𝑇𝑆𝐾𝐿 = 𝑀𝑃 𝐿 = 𝑏/𝑎 L 𝐾 𝑄1 Slope 𝑏 Foundations of Microeconomics Fátima Barros 53 53 Different Isoquants: Perfect Complements Capital and labor are perfect complements. Q3 K Capital and labor are used in Q2 fixed-proportions. Q1 Increasing Since capital and labor are Output consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL). L Foundations of Microeconomics Fátima Barros 54 54 27  15/09/2024 Different Isoquants: Cobb-Douglas Function Inputs are not perfectly substitutable. Diminishing marginal rate of K technical substitution. Q3 Increasing – As less of one input is used Q2 Output in the production process, Q1 increasingly more of the other input must be employed to produce the same output level. 𝑄 = 𝐾𝑎𝐿𝑏 𝑀𝑃𝐿 L 𝑀𝑅𝑇𝑆𝐾𝐿 = 𝑀𝑃𝐾 Foundations of Microeconomics Fátima Barros 55 55 Example Consider a Cobb-Douglas function: 𝑄 = 𝐾 0.3 𝐿0.6 Then 𝜕𝐹 𝐾,𝐿 𝑀𝑃𝐿 = = 0.6𝐿0.6−1 𝐾 0.3 𝜕𝐿 𝜕𝐹 𝐾,𝐿 𝑀𝑃𝐾 = = 0.3𝐾 0.3−1 𝐿0.6 𝜕𝐾 and 𝑀𝑃𝐿 0.6𝐿−0.4 𝐾 0.3 𝐾 𝑀𝑅𝑇𝑆 = = = 2 𝑀𝑃𝐾 0.3𝐿0.6 𝐾 −0.7 𝐿 Foundations of Microeconomics Fátima Barros 56 56 28  15/09/2024 Cobb-Douglas function: General case Consider a Cobb-Douglas function: 𝑄 = 𝐿𝛼 𝐾𝛽 Then 𝜕𝐹 𝐾,𝐿 𝑀𝑃𝐿 = = 𝛼𝐿𝛼−1 𝐾𝛽 𝜕𝐿 𝜕𝐹 𝐾, 𝐿 𝑀𝑃𝐾 = = 𝛽𝐿𝛼 𝐾𝛽−1 𝜕𝐾 And 𝑀𝑃𝐿 𝛼𝐿𝛼−1 𝐾𝛽 𝛼 𝐾 𝑀𝑅𝑇𝑆 = = = 𝑀𝑃𝐾 𝛽𝐿𝛼 𝐾𝛽−1 𝛽 𝐿 Foundations of Microeconomics Fátima Barros 57 57 Long Run Analysis How to produce a given output with the least cost input combination? Foundations of Microeconomics Fátima Barros 58 58 29  15/09/2024 3.4 Isocost Line Isocost Line: the combinations of inputs that have the same cost for the firm: K New Isocost Line associated with higher 𝐶0 = 𝑤𝐿 + 𝑟𝐾 C1/r costs (C0 < C1). C0/r Rearranging, 0 1 𝑤 𝐶0 𝐶1 𝐾= 𝐶0 − 𝐿 L 𝑟 𝑟 C0/w C1/w Slope of Isocost line is For given input prices, isocosts 𝑤 equal to the ratio farther from the origin are 𝑟 the relative prices of associated with higher costs. production factors Foundations of Microeconomics Fátima Barros 59 59 Example of an Isocost Suppose the price of labor is 𝒘 = 𝟏𝟎 and the price of one unit of capital is 𝒓 = 𝟐𝟎: 𝐶 0 = 10𝐿 + 20𝐾 K Then 6 𝐶0 1 𝐾= − 𝐿 5 20 2 C0 C1 with 𝐶 0 = 100 and 𝐶 1 =120 we 10 12 L obtain 2 different isocosts: 100 1 120 1 𝐾= − 𝐿 and 𝐾 = − 𝐿 20 2 20 2 Foundations of Microeconomics Fátima Barros 60 60 30  15/09/2024 Changes in Isocost Line 1 0 𝑤 𝐾= 𝐶 − 𝐿 Slope of Isocost line 𝑟 𝑟 Changes in input prices change the slope of the isocost line. K New Isocost Line If price of labour decreases (w C/r for a decrease in decreases) the slope of the the wage (price of isocost line also decreases. labor: w0 > w1). L C/w0 C/w1 Foundations of Microeconomics Fátima Barros 61 61 3.5 Least Cost Input Combination We can use isoquants and isocost K lines to find the combinations of 𝐶1 /r Q = F ( K , L) inputs that minimize the cost of 𝐶 𝑜 /r producing a given Q. Point A has K and L sufficient to B produce Q (same isoquant), but at a A cost that is high. By substituting L for K along the 0 𝐶 𝑜 /w 𝐶1 /w L isoquant we reach B, where we can produce Q at a lower cost (lower isoquant). Foundations of Microeconomics Fátima Barros 62 62 31  15/09/2024 3.5 Optimal Point Cost Minimization K Slope of Point of Cost 𝐶 𝑜 /r Isocost Minimization = Slope of Isoquant K* Q L* 𝐶 𝑜 /w L Foundations of Microeconomics Fátima Barros 63 63 3.5 Optimal Combination of Inputs Golden Rule 2: The optimal combination of inputs to produce a given level of output 𝑸 ഥ = 𝑭 𝑲, 𝑳 , given the market prices of the factors, 𝒘 and 𝒓, is the one that minimizes the cost of production and is given by: 𝑤 𝑀𝑅𝑇𝑆 = ቐ 𝑟 𝑄ത = 𝐹 𝐾, 𝐿 𝑀𝑃 𝑀𝑃 𝑤 remember: 𝑀𝑅𝑇𝑆 = 𝑀𝑃 𝐿 therefore 𝑀𝑃 𝐿 = 𝐾 𝐾 𝑟 At the optimal point the isoquant is tangent to the lowest isocost and therefore the slope of the isoquant (MRTS) is equal to the slope of the isocost (𝒘/𝒓) Foundations of Microeconomics Fátima Barros 64 64 32  15/09/2024 Cost Minimization Marginal product per dollar spent should be equal for all inputs: MPL MPK MPL w =  =