Chapter Eighteen

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Chapter Eighteen

• Technology

2
Technologies

• A technology is a process by which inputs are
  converted to an output.
• E.g. labor, a computer, a projector, electricity,
  and software are being combined to produce this
  lecture.
• Inputs labor, a computer, a projector,
  electricity, and software
• Output lecture

3
Input Bundles

• xi denotes the amount used of input i i.e. the
  level of input i.
• An input bundle is a vector of the input levels
  (x1, x2, , xn).
• y denotes the output level.

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Production Functions

• The technologys production function states the
  maximum amount of output possible from an input
  bundle.

5
Production Functions
Ex) One input (x), one output (y)
y f(x) is the production function.
Output Level
y
y f(x) is the maximal output level obtainable
from x input units.
x
x
Input Level
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Technology Sets

• A production plan is an input bundle and an
  output level (x1, , xn, y).
• A production plan is feasible if
• The collection of all feasible production plans
  is the technology set.

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Technology Sets
One input (x), one output (y)
Output Level
y
The technologyset
y
x
x
Input Level
8
Technology Sets
One input, one output
Output Level
Technicallyefficient plans
y
The technologyset
Technicallyinefficientplans
y
x
x
Input Level
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Technologies with Multiple Inputs

• What does a production function look like when
  there is more than one input?
• The two input case Input levels are x1 and x2.
  Output level is y.
• Suppose the production function is

10
Technologies with Multiple Inputs

• Suppose the production function is
• In this case, the production function graph is a
  3-dimensional surface.

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Technologies with Multiple Inputs

• The maximal output level possible from the input
  bundle(x1, x2) (1, 8) is
• And the maximal output level possible from
  (x1,x2) (8,8) is

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Technologies with Multiple Inputs
Output, y
x2
(8,8)
(8,1)
x1
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Technologies with Multiple Inputs

• We could then plot every possible combination of
  inputs and draw the entire production function.

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Technologies with Multiple Inputs
y
(8,8)
(8,1)
x1
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Technologies with Multiple Inputs

• Production functions are like utility functions
  but instead of describing how much utility a
  consumer gets from consuming two goods, it
  describes how much output can be produced by two
  different inputs.

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Technologies with Multiple Inputs

• Instead of drawing the entire utility function,
  we instead drew indifference curves.
• For production functions, the analog to an
  indifference curve is called an isoquant.
• An isoquant is a curve that maps out all the
  combinations of inputs that yield the same level
  of output.

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Isoquants with Two Variable Inputs
Output, y
y º 8
y º 4
x2
x1
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Isoquants with Two Variable Inputs

• More isoquants tell us more about the technology.

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Isoquants with Two Variable Inputs
Output, y
y º 8
y º 6
y º 4
x2
y º 2
x1
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Technologies with Multiple Inputs

• The complete collection of isoquants is the
  isoquant map.
• Instead of looking at an isoquant map in 3-D, we
  can instead just look at it in 2-D by projecting
  the isoquants onto x2 and x1 plane.

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Technologies with Multiple Inputs
y
x1
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Technologies with Multiple Inputs
y
x1
23
Technologies with Multiple Inputs
y
x1
24
Technologies with Multiple Inputs
y
x1
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Technologies with Multiple Inputs
y
x1
26
Technologies with Multiple Inputs
y
x1
27
Technologies with Multiple Inputs
y
x1
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Technologies with Multiple Inputs
y
x1
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Technologies with Multiple Inputs
y
x1
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Technologies with Multiple Inputs
y
x1
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Technologies with Multiple Inputs
x2
y
x1
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Technologies with Multiple Inputs
x2
y
x1
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Technologies with Multiple Inputs
x2
y
x1
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Technologies with Multiple Inputs
x2
y
x1
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Technologies with Multiple Inputs
x2
y
x1
36
Technologies with Multiple Inputs
x2
y
x1
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Cobb-Douglas Technologies

• A Cobb-Douglas production function with two
  inputs is of the form

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Cobb-Douglas Technologies
x2
C-D isoquants are strictly convex and never
touch either axis.
x1
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Cobb-Douglas Technologies
x2
Each isoquant describes all the combinations of
the two Inputs that produce the same
output.
x1
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Cobb-Douglas Technologies
x2
Higher isoquants are associated with higher
levels of output.
x1
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Ex 18.1 Cobb-Douglas Technologies

• Plot the isoquants associated with an output
  level of 100 and an output level of 300 when the
  production function is given by

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Fixed-Proportions Technologies

• A fixed-proportions production function with two
  inputs is of the form

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Ex. 18.2 Fixed-Proportions Technologies

• Draw the isoquants associated with output levels
  of 4, 8, and 14 when the production function is
  given by

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Ex. 18.2 Fixed-Proportions Technologies
x2
x1 2x2
minx1,2x2 14
7
minx1,2x2 8
4
2
minx1,2x2 4
4
8
14
x1
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Perfect-Substitutes Technologies

• A perfect-substitutes production function with
  two inputs is of the form

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Example 18.3 Perfect-Substitution Technologies

• Draw the isoquants associated with output levels
  of 18, 36, and 48 when the production function is
  given by

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Ex. 18.3 Perfect-Substitution Technologies
x2
x1 3x2 18
x1 3x2 36
x1 3x2 48
16
12
All are linear and parallel
6
x1
18
36
48
48
Marginal Products

• The marginal product of input i is the
  rate-of-change of the output level as the level
  of input i changes, holding all other input
  levels fixed.
• That is,

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Marginal Products
E.g. if
then the marginal product of input 1 is
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Marginal Products
E.g. if
then the marginal product of input 1 is
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Marginal Products
E.g. if
then the marginal product of input 1 is
and the marginal product of input 2 is
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Marginal Products
E.g. if
then the marginal product of input 1 is
and the marginal product of input 2 is
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Marginal Products
Typically the marginal product of one input
depends upon the amount used of other inputs.
E.g. if
then,
if x2 8,
and if x2 27 then
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Marginal Products

• The marginal product of input i is diminishing if
  it becomes smaller as the level of input i
  increases. That is, if

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Marginal Products
E.g. if
then
and
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Marginal Products
E.g. if
then
and
so
57
Marginal Products
E.g. if
then
and
so
and
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Marginal Products
E.g. if
then
and
so
and
Both marginal products are diminishing.
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Marginal Products

• What is the economic meaning of diminishing
  marginal product?
• Holding the other input constant, as increase the
  amounts of the other input, output may go up, but
  by smaller and smaller amounts.

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Marginal Products

• Why might diminishing marginal product be
  expected?
• Example Consider the technology for making a
  sandwich in a small sandwich shop. There are two
  main inputs, labor and capital (grill, the small
  building).
• What happens to output and the marginal product
  of labor as more workers are hired? Why?

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Returns-to-Scale

• Marginal products describe the change in output
  level as a single input level changes.
• Returns-to-scale describes how the output level
  changes as all input levels change in direct
  proportion (e.g. all input levels doubled, or
  halved).

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Returns-to-Scale

• A production function exhibits Constant Returns
  to Scale (CRS) if for any number k, it is true
  that

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Returns-to-Scale

• A production function exhibits Decreasing Returns
  to Scale (DRS) if for any number k, it is true
  that

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Returns-to-Scale

• A production function exhibits Increasing Returns
  to Scale (IRS) if for any number k, it is true
  that

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Examples of Returns-to-Scale
The perfect-substitutes productionfunction is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The perfect-substitutes productionfunction is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The perfect-substitutes productionfunction is
Expand all input levels proportionatelyby k.
The output level becomes
The perfect-substitutes productionfunction
exhibits constant returns-to-scale.
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Examples of Returns-to-Scale
The perfect-complements productionfunction is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The perfect-complements productionfunction is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The perfect-complements productionfunction is
Expand all input levels proportionatelyby k.
The output level becomes
The perfect-complements productionfunction
exhibits constant returns-to-scale.
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Examples of Returns-to-Scale
The Cobb-Douglas production function is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The Cobb-Douglas production function is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The Cobb-Douglas production function is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The Cobb-Douglas production function is
Expand all input levels proportionatelyby k.
The output level becomes
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Examples of Returns-to-Scale
The Cobb-Douglas production function is
The Cobb-Douglas technologys returns-to-scale
isconstant if a1a2 1increasing if
a1a2 gt 1decreasing if a1a2 lt 1.
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Returns-to-Scale

• Q Can a technology exhibit increasing
  returns-to-scale even though all of its marginal
  products are diminishing?

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Returns-to-Scale

• Q Can a technology exhibit increasing
  returns-to-scale even if all of its marginal
  products are diminishing?
• A Yes.
• E.g.

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Returns-to-Scale
so this technology exhibitsincreasing
returns-to-scale.
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Returns-to-Scale
so this technology exhibitsincreasing
returns-to-scale.
But
diminishes as x1
increases
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Returns-to-Scale
so this technology exhibitsincreasing
returns-to-scale.
But
diminishes as x1
increases and
diminishes as x2
increases.
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Returns-to-Scale

• So a technology can exhibit increasing
  returns-to-scale even if all of its marginal
  products are diminishing. Why?

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Returns-to-Scale

• A marginal product is the rate-of-change of
  output as one input level increases, holding all
  other input levels fixed.
• Marginal product diminishes because the other
  input levels are fixed, so the increasing inputs
  units have each less and less of other inputs
  with which to work.

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Returns-to-Scale

• When all input levels are increased
  proportionately, there need be no diminution of
  marginal products since each input will always
  have the same amount of other inputs with which
  to work.
• Input productivities need not fall and so
  returns-to-scale can be constant or increasing.

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Technical Rate-of-Substitution

• At what rate can a firm substitute one input for
  another without changing its output level?

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Technical Rate-of-Substitution
x2
Isoquant for an output level of 100.
yº100
x1
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Technical Rate-of-Substitution
The slope of an isoquant is the rate at which
input 2 must be given up as input 1s level is
increased so as not to change the output level.
The slope of an isoquant is its technical
rate-of-substitution.
x2
yº100
x1
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Technical Rate-of-Substitution

• How is a technical rate-of-substitution computed?

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Technical Rate-of-Substitution

• How is a technical rate-of-substitution computed?
• The production function is
• A small change (dx1, dx2) in the input bundle
  causes a change to the output level of

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Technical Rate-of-Substitution
But dy 0 along an isoquant since there is no
change to the output level, so the changes dx1
and dx2 to the input levels must satisfy
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Technical Rate-of-Substitution
rearranges to
so
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Technical Rate-of-Substitution
is the rate at which input 2 must be givenup as
input 1 increases so as to keepthe output level
constant. It is the slopeof the isoquant.
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Example 18.4 TRS-Cobb Douglas

• Calculate the TRS for the production function
  yx11/3x22/3
• At the point (4,8)
• At the point (6,12)
• Are these points on the same isoquant?
• Does the production function exhibit increasing,
  constant, or decreasing returns to scale?

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Well-Behaved Technologies

• A well-behaved technology is
• monotonic, and
• convex.

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Well-Behaved Technologies-Monotonicity

• Monotonicity More of any input generates more
  output.
• Monotonicity implies
• Higher isoquants are associated with higher
  levels of output.
• Isoquants are downward sloping.

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Well-Behaved Technologies
higher output
x2
yº200
yº100
yº50
x1
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Well-Behaved Technologies - Convexity

• Convexity If the input bundles x and x both
  provide y units of output then the mixture of
  these two bundles tx (1-t)x provides at
  least y units of output, for any 0 lt t lt 1.

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Well-Behaved Technologies - Convexity
x2
yº100
x1
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Well-Behaved Technologies - Convexity
x2
yº100
x1
99
Well-Behaved Technologies - Convexity
x2
yº120
yº100
x1
100
Well-Behaved Technologies - Convexity
Convexity implies that the TRSdecreases as x1
increases. In other words, isoquants
become flatter as x1 increases.
x2
x1
101
The Long-Run and the Short-Runs

• The long-run is the circumstance in which a firm
  is unrestricted in its choice of all input
  levels. All inputs can be varied in the long-run.
• A short-run is a circumstance in which a firm the
  firm cannot vary the amount of at least one input
  that it is using.

102
The Long-Run and the Short-Runs

• Examples of restrictions that place a firm into a
  short-run
• temporarily being unable to install, or remove,
  machinery (cant vary amount of machinery over a
  certain time period)
• being required by law to meet affirmative action
  quotas (may not be able to vary the racial mix of
  workers)

103
Example 18.5

• Suppose a firm uses two inputs land (x2) and
  labor (x1).
• The long run production function is given by
  yx11/3 x21/3
• In the short run, the amount of land is fixed.
• Land is thus a fixed input in the short-run.
  Labor remains variable.
• Draw the relationship between output and labor
  when land is fixed at 1 acre and when land is
  fixed at 27 acres.