Quản trị kinh doanh - Chapter 6: Inputs and production functions

1Inputs and Production Functions Chapter 6Copyright (c)2014 John Wiley & Sons, Inc.2Chapter Six OverviewMotivationThe Production FunctionMarginal and Average ProductsIsoquantsThe Marginal Rate of Technical SubstitutionTechnical ProgressReturns to ScaleSome Special Functional FormsChapter SixCopyright (c)2014 John Wiley & Sons, Inc.3Chapter SixProduction of Semiconductor Chips “Fabs” cost $1 to $2 billion to construct and are obsolete in 3 to 5 years Must get fab design “right” Choice: Robots or Humans? Up-front investment in robotics vs. better chip yields and lower labor costs? Capital-intensive or labor-intensive production process?Copyright (c)2014 John Wiley & Sons, Inc.4Chapter SixProductive resources, such as labor and capital equipment, that firms use to manufacture goods and services are called inputs or factors of production.The amount of goods and services produces by the firm is the firm’s output.Production transforms a set of inputs into a set of outputsTechnology determines the quantity of output that is feasible to attain for a given set of inputs. Key ConceptsCopyright (c)2014 John Wiley & Sons, Inc.5Chapter SixKey ConceptsThe production function tells us the maximum possible output that can be attained by the firm for any given quantity of inputs.The production set is a set of technically feasible combinations of inputs and outputs.Production Function:Q = outputK = CapitalL = Labor Copyright (c)2014 John Wiley & Sons, Inc.6Q = f(L)LQ ••••CDABProduction SetProduction FunctionChapter SixThe Production Function & Technical EfficiencyCopyright (c)2014 John Wiley & Sons, Inc.7Chapter SixThe Production Function & Technical EfficiencyTechnically efficient: Sets of points in the production function that maximizes output given input (labor)Technically inefficient: Sets of points that produces less output than possible for a given set of input (labor)Copyright (c)2014 John Wiley & Sons, Inc.8Chapter SixThe Production Function & Technical EfficiencyCopyright (c)2014 John Wiley & Sons, Inc.9Chapter SixLabor Requirements FunctionLabor requirements functionExample: for production functionCopyright (c)2014 John Wiley & Sons, Inc.10Chapter SixThe Production & Utility FunctionsCopyright (c)2014 John Wiley & Sons, Inc.11Chapter SixThe Production & Utility FunctionsCopyright (c)2014 John Wiley & Sons, Inc.12Chapter SixThe Production Function & Technical EfficiencyCopyright (c)2014 John Wiley & Sons, Inc.13Chapter SixTotal ProductTotal Product Function: A single-input production function. It shows how total output depends on the level of the inputIncreasing Marginal Returns to Labor: An increase in the quantity of labor increases total output at an increasing rate.Diminishing Marginal Returns to Labor: An increase in the quantity of labor increases total output but at a decreasing rate.Diminishing Total Returns to Labor: An increase in the quantity of labor decreases total output.Copyright (c)2014 John Wiley & Sons, Inc.14Chapter SixTotal ProductCopyright (c)2014 John Wiley & Sons, Inc.15Definition: The marginal product of an input is the change in output that results from a small change in an input holding the levels of all other inputs constant. MPL = Q/L (holding constant all other inputs)MPK = Q/K (holding constant all other inputs)Chapter SixExample: Q = K1/2L1/2MPL = (1/2)L-1/2K1/2MPK = (1/2)K-1/2L1/2The Marginal ProductCopyright (c)2014 John Wiley & Sons, Inc.16Definition: The law of diminishing marginal returns states that marginal products (eventually) decline as the quantity used of a single input increases.Chapter SixDefinition: The average product of an input is equal to the total output that is to be produced divided by the quantity of the input that is used in its production: APL = Q/L APK = Q/KExample:APL = [K1/2L1/2]/L = K1/2L-1/2APK = [K1/2L1/2]/K = L1/2K-1/2The Average Product & Diminishing ReturnsCopyright (c)2014 John Wiley & Sons, Inc.17Chapter SixTotal, Average, and Marginal ProductsLQAPLMPL6305- 1296811181629112419285301505-7Copyright (c)2014 John Wiley & Sons, Inc.18Chapter SixTotal, Average, and Marginal ProductsCopyright (c)2014 John Wiley & Sons, Inc.19TPL maximized where MPL is zero. TPL falls where MPL is negative; TPL rises where MPL is positive.Chapter SixTotal, Average, and Marginal MagnitudesCopyright (c)2014 John Wiley & Sons, Inc.20Chapter SixProduction Functions with 2 InputsMarginal product: Change in total product holding other inputs fixed.Copyright (c)2014 John Wiley & Sons, Inc.21Chapter SixIsoquantsDefinition: An isoquant traces out all the combinations of inputs (labor and capital) that allow that firm to produce the same quantity of outputAndCopyright (c)2014 John Wiley & Sons, Inc.22Chapter SixIsoquantsCopyright (c)2014 John Wiley & Sons, Inc.23IsoquantsLKQ = 10Q = 20All combinations of (L,K) along theisoquant produce 20 units of output.0Slope=K/LChapter SixExample:Copyright (c)2014 John Wiley & Sons, Inc.24Definition: The marginal rate of technical substitution measures the amount of an input, L, the firm would require in exchange for using a little less of another input, K, in order to just be able to produce the same output as before.MRTSL,K = -K/L (for a constant level of output)Marginal products and the MRTS are related: MPL(L) + MPK(K) = 0 => MPL/MPK = -K/L = MRTSL,K Chapter SixMarginal Rate of Technical SubstitutionCopyright (c)2014 John Wiley & Sons, Inc.25 The rate at which the quantity of capital that can be decreased for every unit of increase in the quantity of labor, holding the quantity of output constant, Or The rate at which the quantity of capital that can be increased for every unit of decrease in the quantity of labor, holding the quantity of output constantChapter SixThereforeMarginal Rate of Technical SubstitutionCopyright (c)2014 John Wiley & Sons, Inc.26Chapter SixMarginal Rate of Technical Substitution If both marginal products are positive, the slope of the isoquant is negative. If we have diminishing marginal returns, we also have a diminishing marginal rate of technical substitution - the marginal rate of technical substitution of labor for capital diminishes as the quantity of labor increases, along an isoquant – isoquants are convex to the origin. For many production functions, marginal products eventually become negative. Why don't most graphs of Isoquants include the upwards-sloping portion?Copyright (c)2014 John Wiley & Sons, Inc.27LKQ = 10Q = 200MPK < 0MPL < 0IsoquantsChapter SixExample: The Economic and the Uneconomic Regions of ProductionIsoquantsCopyright (c)2014 John Wiley & Sons, Inc.28Chapter SixMarginal Rate of Technical SubstitutionCopyright (c)2014 John Wiley & Sons, Inc.29Chapter SixElasticity of SubstitutionA measure of how easy is it for a firm to substitute labor for capital.It is the percentage change in the capital-labor ratio for every one percent change in the MRTSL,K along an isoquant.Copyright (c)2014 John Wiley & Sons, Inc.30Definition: The elasticity of substitution, , measures how the capital-labor ratio, K/L, changes relative to the change in the MRTSL,K. Chapter SixElasticity of SubstitutionCopyright (c)2014 John Wiley & Sons, Inc.31Example: Suppose that: MRTSL,KA = 4, KA/LA = 4 MRTSL,KB = 1, KB/LB = 1MRTSL,K = MRTSL,KB - MRTSL,KA = -3  = [(K/L)/MRTSL,K]*[MRTSL,K/(K/L)] = (-3/-3)(4/4) = 1 Chapter SixElasticity of SubstitutionCopyright (c)2014 John Wiley & Sons, Inc.32LK0 = 0 = 1 = 5 = Chapter Six"The shape of the isoquant indicates the degree of substitutability of the inputs"Elasticity of SubstitutionCopyright (c)2014 John Wiley & Sons, Inc.33 How much will output increase when ALL inputs increase by a particular amount?Chapter SixReturns to ScaleCopyright (c)2014 John Wiley & Sons, Inc.34Chapter SixReturns to ScaleLet Φ represent the resulting proportionate increase in output, QLet λ represent the amount by which both inputs, labor and capital, increase.Increasing returns: Decreasing returns:Constant Returns: Copyright (c)2014 John Wiley & Sons, Inc.35 How much will output increase when ALL inputs increase by a particular amount? RTS = [%Q]/[% (all inputs)] If a 1% increase in all inputs results in a greater than 1% increase in output, then the production function exhibits increasing returns to scale. If a 1% increase in all inputs results in exactly a 1% increase in output, then the production function exhibits constant returns to scale. If a 1% increase in all inputs results in a less than 1% increase in output, then the production function exhibits decreasing returns to scale.Chapter SixReturns to ScaleCopyright (c)2014 John Wiley & Sons, Inc.36LKQ = Q0Q = Q10L 2LK2KChapter SixReturns to ScaleCopyright (c)2014 John Wiley & Sons, Inc.37Chapter SixReturns to ScaleCopyright (c)2014 John Wiley & Sons, Inc.38Chapter SixReturns to Scale vs. Marginal Returns• The marginal product of a single factor may diminish while the returns to scale do not• Returns to scale need not be the same at different levels of productionReturns to scale: all inputs are increased simultaneouslyMarginal Returns: Increase in the quantity of a single input holding all others constant.Copyright (c)2014 John Wiley & Sons, Inc.39Chapter SixReturns to Scale vs. Marginal ReturnsProduction function with CRTS but diminishing marginal returns to labor.Copyright (c)2014 John Wiley & Sons, Inc.40Definition: Technological progress (or invention) shifts the production function by allowing the firm to achieve more output from a given combination of inputs (or the same output with fewer inputs).Chapter SixTechnological ProgressCopyright (c)2014 John Wiley & Sons, Inc.41Labor saving technological progress results in a fall in the MRTSL,K along any ray from the originCapital saving technological progress results in a rise in the MRTSL,K along any ray from the origin.Chapter SixTechnological ProgressCopyright (c)2014 John Wiley & Sons, Inc.42Chapter SixNeutral Technological ProgressTechnological progress that decreases the amounts of labor and capital needed to produce a given output. Affects MRTSK,LCopyright (c)2014 John Wiley & Sons, Inc.43Chapter SixLabor Saving Technological ProgressTechnological progress that causes the marginal product of capital to increase relative to the marginal product of laborCopyright (c)2014 John Wiley & Sons, Inc.44Chapter SixCapital Saving Technological ProgressTechnological progress that causes the marginal product of labor to increase relative to the marginal product of capitalCopyright (c)2014 John Wiley & Sons, Inc.