Transcript Slide 1

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
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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
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Examples of Production Functions

What might be candidate production functions for the
following?



Vodka Distillary – can be made from either potatoes or corn.
Lawn mowing service – requires both Labor and “Capital”,
though not necessarily in fixed proportions.
So what are Production functions analogous to? How are they
different?
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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
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Isoquant – set of all possible input bundles that are sufficient
to produce a given amount of output.
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Isoquants for Vodka?
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Isoquants for acres of lawn mowed?
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Isoquants for Axes (i.e. each axe requires one blade and one
handle)?
So what are Isoquants somewhat analogous to? How do they
differ?
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Isoquants

Again, like with demand theory, we are most interested in
understanding trade-offs.

What aspect of Isoquants tells us about trade-offs in the
production process?
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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

Example:

Suppose you run a car wash business governed by the production
function
q = f(L, K) = L0.5K0.5
 (q = cars washed, L = Labor hrs, K = machine hrs.)
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What will Isoquants look like?
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What will be the Marginal Product of Labor at the input bundle
{L=4, K= 9}?
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What will be the Marginal Product of Labor at the input bundle
{L=9, K= 9}?
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Substitution between Inputs

Marginal Product is interesting on its own
x2

MP also helpful for considering how to
evaluate trade-offs in the production
process.
 Consider again the following thought
exercise:
 Suppose firm produces using some
input combination (x1’,x2’).

If it used a little bit more x1, how
much less of x2 would it have to use
to keep output constant?
x2’
x2”
Δx1
Δx2
f(x1”,x2’)
f(x1’,x2’)
x1’ x1”
x1
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Technical Rate of Substitution (TRS)

Technical Rate of Substitution (TRS):
1.
2.
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TRS = Slope of Isoquant
MP1 ( x1 , x2 )
TRS  
MP2 ( x1 , x2 )
Also referred to as Marginal Rate of Technical Substitution
(MRTS) or Marginal Rate of Transformation (MRT)
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Substitution between Inputs (cont.)

We are often interested in production technologies that exhibit
Diminishing 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?
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Diminishing TRS
machine hrs (K)
16
4
4 cars
1
4
worker hrs (L)
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Economies of Scale
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
What do we mean by the term “economies-of-scale.”
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Increasing Returns-to-Scale
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Decreasing Returns-to-Scale
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Constant Returns-to-Scale
What kind of assumption would these be with respect to a
generic production function q = f(x1,x2)?
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Returns-to-scale graphically
machine hrs (K)
32
16
? cars
4
4 cars
1
4
8
worker hrs (L)
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