Chapter 6 - Inputs and production functions. This chapter presents the following content: Motivation, the production function, technical progress, returns to scale, some special functional forms.
Inputs and Production Functions Copyright (c)2014 John Chapter Chapter Six Overview Motivation The Production Function Ø Ø Marginal and Average Products Isoquants The Marginal Rate of Technical Substitution Technical Progress Returns to Scale Some Special Functional Forms Chapter Six Copyright (c)2014 John Ø Production of Semiconductor Chips “Fabs” cost $1 to $2 billion to construct and are obsolete in to years Ø Must get fab design “right” Ø Choice: Robots or Humans? Ø Ø Copyright (c)2014 John Ø Up-front investment in robotics vs better chip yields and lower labor costs? Capital-intensive or laborintensive production process? Chapter Six Key Concepts 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 outputs Technology determines the quantity of output that is feasible to attain for a given set of inputs Chapter Six Copyright (c)2014 John Productive resources, such as labor and capital equipment, that firms use to manufacture goods and services are called inputs or factors of production Key Concepts Production Function: • • • Q = output K = Capital L = Labor Q f ( L, K ) The production set is a set of technically feasible combinations of inputs and outputs Chapter Six Copyright (c)2014 John The production function tells us the maximum possible output that can be attained by the firm for any given quantity of inputs The Production Function & Technical Efficiency Q D C • •A • •B Production Set L Chapter Six Copyright (c)2014 John Production Function Q = f(L) The Production Function & Technical Efficiency • Technically efficient: Sets of points in the production function that maximizes output given input (labor) Q f ( L, K ) Technically inefficient: Sets of points that produces less output than possible for a given set of input (labor) Q f ( L, K ) Chapter Six Copyright (c)2014 John • Copyright (c)2014 John The Production Function & Technical Efficiency Chapter Six Labor Requirements Function Labor requirements function L Example: L Q g (Q) for production function Q Chapter Six L Copyright (c)2014 John • The Production & Utility Functions Production Function Utility Function Output from inputs Preference level from purchases Derived from preferences Copyright (c)2014 John Derived from technologies Cardinal(Defn: given Ordinal amount of inputs yields a unique and specific amount of output) Marginal Product Marginal Utility Chapter Six 10 Elasticity of Substitution Percentage change in capital labor ratio Percentage change in MRTS L , K K % L % MRTS L , K Chapter Six 30 Copyright (c)2014 John Definition: The elasticity of substitution, , measures how the capital-labor ratio, K/L, changes relative to the change in the MRTSL,K Elasticity of Substitution Example: Suppose that: • MRTSL,KA = 4, KA/LA = MRTSL,KB = 1, KB/LB = MRTSL,K = MRTSL,KB - MRTSL,KA = -3 = [ (K/L)/ MRTSL,K]*[MRTSL,K/(K/L)] = (-3/-3) (4/4) = Chapter Six 31 Copyright (c)2014 John • Elasticity of Substitution K =0 =1 =5 = L Chapter Six 32 Copyright (c)2014 John "The shape of the isoquant indicates the degree of substitutability of the inputs…" Returns to Scale How much will output increase when ALL inputs increase by a particular amount? Copyright (c)2014 John • % (quantity of output) Returns to Scale % (quantity of all inputs) Chapter Six 33 Returns to Scale Let λ represent the amount by which both inputs, labor and capital, increase Q f ( L, K ) for • • • Copyright (c)2014 John Let Φ represent the resulting proportionate increase in output, Q Increasing returns: Decreasing returns: Constant Returns: Chapter Six 34 Returns to Scale 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 Six 35 Copyright (c)2014 John • Returns to Scale K Q = Q1 K Q = Q0 L L 2L Chapter Six 36 Copyright (c)2014 John 2K Copyright (c)2014 John Returns to Scale Chapter Six 37 • • Returns to scale: all inputs are increased simultaneously Marginal Returns: Increase in the quantity of a single input holding all others constant • The marginal product of a single factor may diminish while the returns to scale not • Returns to scale need not be the same at different levels of production Chapter Six 38 Copyright (c)2014 John Returns to Scale vs Marginal Returns • Chapter Six Production function with CRTS but diminishing marginal returns to labor 39 Copyright (c)2014 John Returns to Scale vs Marginal Returns Definition: 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 Six 40 Copyright (c)2014 John Technological Progress Technological Progress Capital saving technological progress results in a rise in the MRTSL,K along any ray from the origin Chapter Six 41 Copyright (c)2014 John Labor saving technological progress results in a fall in the MRTSL,K along any ray from the origin Technological progress that decreases the amounts of labor and capital needed to produce a given output Affects MRTSK,L Chapter Six 42 Copyright (c)2014 John Neutral Technological Progress Technological progress that causes the marginal product of capital to increase relative to the marginal product of labor Chapter Six 43 Copyright (c)2014 John Labor Saving Technological Progress Technological progress that causes the marginal product of labor to increase relative to the marginal product of capital Chapter Six 44 Copyright (c)2014 John Capital Saving Technological Progress ... single input increases Chapter Six 16 Copyright (c)2014 John APL = Q/L APK = Q/K Total, Average, and Marginal Products Q APL MPL 30 96 11 162 11 192 150 -7 Chapter Six Copyright (c)2014 John L... include the Chapter upwards-sloping portion? Six 26 Copyright (c)2014 John • Isoquants Isoquants MPK < Example: The Economic and the Uneconomic Regions of Production Q = 20 MPL < Q = 10 L Chapter. .. = 4, KA/LA = MRTSL,KB = 1, KB/LB = MRTSL,K = MRTSL,KB - MRTSL,KA = -3 = [ (K/L)/ MRTSL,K]*[MRTSL,K/(K/L)] = (-3 /-3 ) (4/4) = Chapter Six 31 Copyright (c)2014 John •