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Lecture13 Technology by IARZh3

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									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

								
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