VIEWS: 4 PAGES: 17 POSTED ON: 11/5/2012
Intermediate Microeconomic Theory Technology 1 Inputs In order to produce output, firms must employ inputs (or factors of production) Sometimes divided up into categories: Labor Capital Land 2 The Production Function To produce any given amount of a good a firm can only use certain combinations of inputs. Production Function – a function that characterizes how output depends on how many of each input are used. q = f(x1, x2, …, xn) units of output units of input 1 units of input 2…units of input n 3 Examples of Production Functions What might be candidate production functions for producing the following goods? Apple juice – One ounce of apple juice can be produced from ½ apple. So what is production function for apple juice with respect to Washington Apples and Maine Apples Axe Factory – each axe requires exactly one blade & one handle. Shirts – requires both Labor and Machines (i.e., “Capital”), though not necessarily in fixed proportions. For example, 4 shirts can be produced using either 8 labor hours and 2 machine hour, 2 labor hour and 8 machine hours, or 4 labor hours and 4 machine hours. 4 Examples of Production Functions So what are Production functions analogous to? How are they different? 5 Production Functions vs. Utility Functions Unlike in utility theory, the output that gets produced has cardinal properties, not just ordinal properties. For example, consider the following two production functions: f(x1,x2) = x10.5x20.5 f(x1,x2) = x12x22 6 Isoquants Isoquant – set of all possible input bundles that are sufficient to produce a given amount of output. Isoquant for 10 oz of Apple Juice? 20 oz? Isoquant for 10 Axes? 20 Axes? Isoquant for 4 shirts produced? 10 shirts? So what are Isoquants somewhat analogous to? How do they differ? 7 Isoquants Again, like with demand theory, we are most interested in understanding trade-offs, but now on the production side. What aspect of Isoquants tells us about trade-offs in the production process? 8 Marginal Product of an Input Consider how much output changes due to a small change in one input (holding all other inputs constant), or q f ( x1 x1 , x2 ) f ( x1 , x2 ) x1 x1 Now consider the change in output associated with a “very small” change in the input. Marginal Product (of an input) – the rate-of-change in output associated with increasing one input (holding all other inputs constant), or f ( x1 , x2 ) MP1 ( x1 , x2 ) x1 9 Marginal Product of an Input Suppose you run a factory governed by the production function q = f(L, K) = x1a x2b What will be expression for MP1? What will be expression for MP2? 10 Marginal Product of an Input Example: Suppose you run a factory governed by the production function q = f(L, K) = L0.5K0.5 (q = units of output, L = Labor hrs, K = machine hrs.) What will be expression for Marginal Product of Labor? So what will MPL at {L=4, K= 9}? Discrete Approximation? So what will MPL at {L=9, K= 9}? Discrete Approximation? 11 Substitution between Inputs Marginal Product is interesting on its own. x2 MP also helpful for considering how to Δx1 evaluate trade-offs in the production x2’ process. x2” Δx2 Consider again the following thought exercise: f(x1”,x2’) Suppose firm produces using some input combination (x1’,x2’). f(x1’,x2’) If it used a little bit more x1, how much less of x2 would it have to use x1’ x1” x1 to keep output constant? 12 Technical Rate of Substitution (TRS) Technical Rate of Substitution (TRS): 1. TRS = Slope of Isoquant MP1 ( x1 , x2 ) 2. TRS MP2 ( x1 , x2 ) Also referred to as Marginal Rate of Technical Substitution (MRTS) or Marginal Rate of Transformation (MRT) 13 Technical Rate of Substitution (TRS) So what would be the expression for the TRS for a generalized Cobb-Douglas Production function F(x1,x2) = x1ax2b? So if F(x1,x2) = x10.5x20.5 , what will be TRS at {4,9}? {9,4}? What does this imply about shape of Iso-quant? 14 Substitution between Inputs (cont.) We are often interested in production technologies that exhibit: Diminishing Marginal Product (MP) in each input. Diminishing Technical Rate of Substitution (TRS). Will a Cobb-Douglas production function exhibit diminishing MP in both inputs? How about a diminishing TRS? What is the difference between these in terms of graphically? 15 Diminishing MP machine hrs (K) 9 9.5 9 6.7 cars 6 cars 4 5 9 10 worker hrs (L) 16 Diminishing TRS machine hrs (K) 16 4 4 cars 1 4 worker hrs (L) 17