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Lecture13 Technology by IARZh3

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```									Intermediate Microeconomic Theory

Technology

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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

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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.

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

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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)

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Diminishing TRS

machine hrs (K)

16

4

4 cars

1   4            worker hrs (L)

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