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Production
Theory of Production and Optimal Input Combinations
1
Production
• Start of analyzing the supply side of the market
• Focus is on the physical aspects of production
• Related to an engineering perspective of how to combine inputs
• The economic perspective focuses a bit less on the mechanical aspects of
production, and instead thinks about how to optimally produce some good
• Entity producing a good is called a “firm”, and the definition
encompasses everything from a small corner market to a giant
organization like ExxonMobile
• Production is an intermediate step for a firm
• Goal is not to produce a good
• Goal is to make a profit
2
The Production Process
• We will often take an abstract approach
• So will not focus on the physical and mechanical processes that
produce a good – that is the realm of engineering
• Our goal will be to find the production process than minimizes the
cost of producing each unit of the good
• You will see we use a lot of concepts that translate almost seamlessly
from consumer theory
•
•
•
•
“Output” is the analog of “utility”
“Marginal product” works like “marginal utility”
“Cost minimization” is the mirror image of “utility maximization”
Keeping these similarities in mind might help you quickly understand
production
3
Preliminary Concepts I: Inputs
• Production process takes “inputs” which are the factors or things
needed to make a good
• For example, to make gasoline, one necessary input is oil
• To make steel, you need iron
• To grow wheat, you need land, seed, fertilizer, tractors, labor and other things
• Sometimes it is easy to vary the amount of input used, sometimes it is
hard – there is a continuum
• Inputs that are easy to vary are called “variable inputs”
• Inputs that are hard to vary are called “fixed inputs”
• We will generally use generic inputs, often calling them “capital” and
“labor”
4
Preliminary Concepts II: The Production Function
• Inputs can be combined in different ways to get the same input;
inputs can (sometimes or often) be substituted for one another
• To make cars, you can have 2 production plants each running 1 shift of
workers, or 1 production plant running 2 shifts of workers
• This would show an ability to substitute capital and labor
• Other examples?
• Economists represent the (often) complicated process of production
with a simplified expression called a “production function”
• The production function tells how much output will be produced if a
certain amount of each input is used in a set period
• It is a simplification of the comprehensive process
• It is a physical representation in ways equivalent to the utility function
5
The Production Function, continued
• We write the production function Q  f ( x1 , x2 ,...., xn ) where Q is the
output produced, each xi represents a different input, and the
function f(∙) represents the production function, so tells how the
inputs are combined to produce the output
• It is a cardinal relationship (unlike the utility function which was
ordinal)
• That means if Q  f ( x1 , x2 ,...., xn )  12 and Q*  f ( x1*, x2 *,...., xn *)  24 where the
* indicates a different amount of input used compared to when there is no *,
then the input combination of the * inputs produces exactly twice as much as
the input combination of the unstarred
• For example, if Q  f ( K1 , L1 ) and 3Q  f (2 K1 , 2 L1 ) where K1 and L1 are
specific amounts of capital and labor used, doubling the amount of inputs
triples the output
6
Preliminary Concepts III: Time Frame
• Economists talk about the short run and long run
• These are not specific lengths of time. The depend on how long it
takes to vary different inputs
• In the very short run, all inputs are fixed. That means the amount
used cannot be changed. Economists don’t talk about the very short
run much
• In the short run (SR) , at least one input is fixed, and cannot be
changed, while the amount used of other inputs may be varied
• In the long run (LR), all inputs are variable
• Generally, we will be considering labor as a variable input that can
change in the SR and capital as an input that is fixed in the SR but
variable in the LR
7
The Production Function; substitutability
• The production function Q  f ( x1 , x2 ,...., xn ) shows all technologically
efficient input combinations
• That means reducing the amount of any single input will reduce output
• So if Q1  f ( K1 , L1 ) and Q2  f ( K 2 , L2 ) and K 2  K1 or L2  L1 and at least one
inequality is a strict inequality, then Q2  Q1
• The form of f(∙) carries important information about the production
process
• First it tell us how easy it is to substitute inputs
• Linear production functions, Q=aK+bL show perfect substitution
• Fixed proportion production functions, Q=min(aK,bL) where, min stands for
minimum show no substitution works
• Most production processes are somewhere in-between
• Example – packing bird food, use machines or workers
8
The Production Function; marginal products
• Next the production function tell us how output changes when only one
input changes. This is called the marginal product
• The marginal product of labor, often written as MPL, is the change in output as
only labor changes, ie, MPL=∆Q/∆L
• The marginal product of labor, often written as MPL, is the change in output as
only labor changes, ie, MPK=∆Q/∆K
• More precisely
Q f ( K , L)
MPL 

L
L
Q f ( K , L)
MPK 

K
K
It is possible that the MPL
depends on the amount of
capital, and the MPK
depends on the amount
of labor
• Example – packing bird food, again. The MPL depends on how much
capital you have
9
The Production Function; average products
• Another useful measure of production is the average product. Average
products tell how much output is produced per unit of input
• More precisely
Q f ( K , L)
APL  
L
L
Q f ( K , L)
APK  
K
K
Almost certainly the APL
depends on the amount of
capital used, and the APK
depends on the amount
of labor used
• Example – packing bird food, again. The MPL depends on how much
capital you have
10
Marginal Product of Labor and Average Product of Labor
Both graphs hold the amount of capital constant
Notice, it is possible for MP<0, but AP>0 always
unless Q=0 (the firm is not producing)
The average product is the output per unit
of input. Hence, the APL=Q/L. The APL is
at its highest when APL=MPL.
11
The Production Function: Returns to Scale
• Another thing a production function shows is returns to scale (RTS)
• RTS is measured by the proportionate change in output that results
from a proportionate change in the inputs.
• RTS is measured by α and β where Q1=f(K1,L1) and
Q2=αQ1=f(βK1,βL1), α>0, β>1
• If α<β there is decreasing returns to scale. Doubling the inputs used results in
less than double the original output
• If α=β there is constant returns to scale. Doubling the inputs used results in
double the original output
• If α>β there is increasing returns to scale. Doubling the inputs used results in
more than double the original output
12
Example Production Functions
Cobb-Douglas:
Q  aK  L
MPL=a  K  L 1
MPK=a K  1 L
RTS=  
Q  aK  bK 2  cL  dL2  fKL where
usually a, c  0; b, d  0; f can be either
MPL=c  2dL  fK MPK=a  2bK  fL
RTS depends on the amounts of K and L
Linear:
Q K  L
MPL= MPK= RTS=constant
Fixed Proportion: Q  min( K ,  L)
MPL=0 MPK=0
RTS=constant
Quadratic:
What are APL and APK for each production function. Where are they highest?
13
The Isoquant
• Isoquants so different input
combinations that result in the same
output, so imply that the
substitution of inputs is possible
• The shape of the isoquant shows
how easy it is to substitute inputs
• What do isoquants of a linear
production function, Q=aK+bL,
look like?
• What do isoquants of a fixed
proportion production function,
Q=min(ak,bL) look like?
14
More on Isoquants
• The negative of the slope of the isoquant is called the Marginal Rate of
Technical Substitution (MRTSLforK) (like the Marginal Rate of Substitution
for consumers)
MRTS LforK
Q
Q
 K 
and MPK 
. Recall MPL 
 

K
L
 L  Q 0
Q
Then
MPL
L  K . For Q  0 we need L MPL  -K MPK

L
MPK Q
K
so MRTS LforK 
MPL
MPK
15
MRTS and the slope of the isoquant
• Notice for a “normal” shaped
isoquant, as we move from
point 1, to point 2 to point 3
the slope of the isoquant is
getting flatter. That means
MPL/MPK is getting smaller
• Each unit of L added replaces
a smaller and smaller amount
of K. It is getting harder to
substitute L for K
• Called diminishing MRTSLforK
16
Elasticity of Substitution (see Munoz, page 10-12)
• The elasticity of substitution indicates how easy it is to substitute L for K

 L
% K
%MRTS LforK
 L
 K

MRTS LforK
MRTS LforK
• Example, Cobb-Douglas:
Q  aK  L

 L
 K
 1
MPL
a K L
K
MRTS LforK  


 1 
MPK
a K L
L
 K
 K
 MRTS LforK 
so -   MRTS LforK
 L
 L

MRTS LforK
 K L
 K L
MRTS LforK
 L
 K

 K L
   K 
-   
   L 
1
 K 


 L 
• I won’t ask you to calculate σ, but might ask you to interpret it.
17
Isoquants and the relationships between inputs
• As noted, inputs can be complements or substitutes in production
• If inputs are complements, they generally need to be used in a set proportion
• If inputs are substitutes, it is possible to decrease the use of one input by
increasing the use of the other
K
K
Q2
Q2
Q1
Q1
L
Linear production function, σ=
L
Fixed proportion
production function, σ=0
A smooth production
function, 0<σ< but getting
smaller
18
Moving towards efficiency: Optimal Input Use
• The isoquant (and production function) offers a menu of input
combinations that produce the same output
• Once a firm has chosen how much to produce, it must decide how to
produce it
• This is called cost minimization, and follows very closely the idea of
utility maximization we used for consumer theory
19
Isocost curves
• Let’s start with set input prices
• The firm must pay r per unit of capital it uses
• The firm must pay w per unit of labor it uses
• The cost of a combination of inputs is
then C=rK+wL. And isocost curve shows
those different sets of capital and labor
(K,L) that cost the same amount
• We have C=rK+wL. Rearrange this to
K=(C/r)-(w/r)L
• We now have K as a function of L , cost, and
the input prices
• The intercept is (C/r) and the slope is –(w/r)
20
Isocost Curves, continued
• The further the line is from
the origin, the greater C
• In the graph, C”>C’
• Notice, the slopes are the
same, because r and w have
not changed
21
Cost minimization
• For a given output, Q′ in the
graph, the cost of producing it is
minimized with the isocost curve
just tangent to the isoquant for Q′
• Points 1, 2 and 3 all produce Q′, but
C1>C2>C3
• You cannot produce Q′ at a cost of
C4
• Since they are tangent, the slope
of the isoquant equals the slope
of the isocost:
MPL w
MRTS 

MPK r
22
The Expansion Path
• The Expansion path shows the cost
minimizing combination of inputs
as output gets larger
• Top graph shows all inputs are
“normal”. As output increases, the
use of both inputs goes up
• Inputs can be inferior too. In the
graph shown, if quantity gets very
large, capital equipment replaces
labor (example, bird food
packaging)
23
Short run and long run expansion paths
• In the short run, capital is fixed
at K . The only way to increase
output is by adding labor
• The short run expansion path
follows the green arrow
• In the long run, capital is also
variable, so at each output
costs can be minimized
• The expansion path follows the
yellow line
24
Changes in the relative prices of inputs
• If the relative prices of inputs
change, so will the optimal
combination of inputs
• C1 shows the isocost when
capital is relatively expensive
• C2 shows the isocost when labor
is relatively expensive
25
Application: changes in input prices, the minimum wage
• Suppose a firm wants to
produce Q
• If a minimum wage
increases to w2 from w1,
what happens to the mix
of inputs used?
• Which technology costs
more? How do you
know?
capital
C2/r
C1/r
2
1
Q
C2/w2
C1/w1
Labor
26
Application 2: Aluminum or Plastic in Airplanes
• Polymer matrix composites, a type of
plastic, are 7.5 times stronger than
aluminum, and about 40 percent lighter
• Light weight is important when building
planes. In 1990 about 4% of new
commercial aircraft and 10% of new
military aircraft were made of polymer
composites
• Over time, polymer plastics became
relatively cheaper, and now planes are
20% polymer plastics
27
Application: Jet fuel price changes
• Jet fuel prices were very high relative to
other inputs (red line)
• Airlines had planes fly higher, slower and
substituted inputs to lower the weight of
places to save fuel
• With the current oil glut, price of jet fuel
fell relative to cost of other inputs (black
line). They are now flying faster and not
worrying so much about saving weight
• If jet fuel gets expensive again (blue line),
costs will be higher, and airlines will be
inefficient until they can again adjust
other inputs
1
2
28
A lesson from the last example
• Adjusting input mixes in response to short run changes in the prices
of inputs can lead to higher long run costs
• If the airlines worry less about weight because fuel prices are now
cheap, they may use less plastic and more aluminum in planes
• There are other ways to save weight, too. For example, the types of straps
and dividers used in the luggage compartment can amount to significant
weight
• If they order heavier planes, if fuel prices increase, they will be stuck
at higher costs (the blue line in the previous slide) until they can again
adjust capital
• So short run cost minimization may not always be the best decision, if
it is done to temporary changes in the input cost ratio
29
Returns to scale, again
• Remember, RTS is measured by α and β
where Q1=f(K1,L1) and Q2=f(βK1,βL1),
with Q2=αQ1, α>0,β>1.
• If α<β there is decreasing returns to scale (DRS)
• If α=β there is constant returns to scale (CRS)
• If α>β there is increasing returns to scale (IRS)
• Scale changes are measured graphically by
a ray from the origin to isoquants
•
•
•
•
As drawn, K2=2K1, L2=2L1
Q2<2Q1, DRS
Q2=2Q1, CRS
Q2>2Q1, IRS
30
Returns to scale can depend on input ratio
• Ray R1 shows a high ratio K1/L1
• Along R1 the RTS are measured by
the change from Q1 to Q2 (ie, the
move from point 1 to point 2)
• If the firm is at a lower ratio K3/L3
(point 5) the RTS are different
• Point 6 shows doubling the inputs from
K3/L3 to K4/L4
• Since point 6 is below the isoquant for
Q2, RTS must be lower along ray R2 than
along R1
K4
L4
31
RTS and optimal input combinations
• Suppose Q2=2Q1, so R1 shows CRS and
R2 shows DRS
• If the input prices are w and r as shown,
the optimal input combination for
producing Q1 occurs at point 5, a
capital-labor ratio that shows
decreasing returns to scale
• With a production function that has
varying returns to scale depending on
the input ratio, the optimal capitallabor ratio may change with output,
even if the input price ratio is constant
• The least cost method of producing Q2
is at point 7, which is not a scale change
from point 5, as capital and labor
changed in different proportions.
32
The mathematics of cost minimization
• At the cost minimizing point we have MRTS 
MPL w

MPK r
• The problem is to minimize costs (C) subject to output (Q) being held
constant where
• C=wL+rK
• Q*=f(K,L) where Q* is a set level of output
• Of course, this easily generalizes to more than two inputs
• Set up a Lagrangian
min ℒ = 𝑤𝐿 + 𝑟𝐾 − 𝜆[𝑄 ∗ − 𝑓 𝐾, 𝐿 ]
𝐿,𝐾
• λ is again a lagrangian multiplier
• The first part is costs, and the constraint holds output constant at Q*
33
Doing the math
min ℒ = 𝑤𝐿 + 𝑟𝐾 − 𝜆[𝑄∗ − 𝑓 𝐾, 𝐿 ]
𝐿,𝐾
𝜕ℒ
𝜕𝑓 𝐾,𝐿
=𝑤−
𝜕𝐿
𝜕𝐿
𝜕ℒ
𝜕𝑓 𝐾,𝐿
=𝑟−
𝜕𝐾
𝜕𝐾
(1) and (2) ⇒
𝜕ℒ
𝜕𝜆
𝑤
𝑟
= 0 which mean 𝑤 = 𝑀𝑃𝐿 (1)
= 0 which mean 𝑟 = 𝑀𝑃𝐾
=
(2)
𝑀𝑃𝐿
𝑀𝑃𝐾
= 𝑄 ∗ − 𝑓 𝐾, 𝐿 = 0 which means hold output constant
34
𝛼 𝛽
An application with 𝑓 𝐾, 𝐿 = 𝐾 𝐿
min ℒ = 𝑤𝐿 + 𝑟𝐾 − 𝜆[𝑄 ∗ − 𝐾 𝛼 𝐿𝛽 ]
𝐿,𝐾
Put in numbers: Let w=2, r=1, w=2, α=1/2, β=1/2 and Q*=10
𝜕ℒ
= 𝑤 − 𝛽𝐾 𝛼 𝐿𝛽−1 = 0
𝜕𝐿
𝜕ℒ
= 𝑟 − 𝛼𝐾 𝛼−1 𝐿𝛽 = 0
𝜕𝐾
(1) and (2) ⇒
𝑟𝛽
⇒𝐿=
𝐾
𝑤𝛼
𝑤
𝑟
=
𝛽𝐾 𝛼 𝐿𝛽−1
𝛼𝐾 𝛼−1 𝐿𝛽
(1)
The 𝐾 = 10 ∗ 21/2 = 2*10
(2)
=
1
2
So 𝐿 = 𝐾 =
𝛽𝐾
=MRTS
𝛼𝐿
Q*= 10 ∗ 2
2
∗10
2
1/2
10 ∗
2
2
1/2
= 100 ∗
2 2
=10
2
𝜕ℒ
= 𝑄 ∗ − 𝐾 𝛼 𝐿𝛽 = 0
𝜕𝜆
So
𝑄∗
=
𝐾 𝛼 𝐿𝛽
=
𝐾𝛼
𝛽
𝑟𝛽
𝐾
𝑤𝛼
=
𝐾 𝛼+𝛽
𝑟𝛽 𝛽
𝑤𝛼
⇒𝐾=
𝑤𝛼 𝛽
∗
𝑄
𝑟𝛽
1
𝛼+𝛽
35
More on the application
We always require that MRTSLforK=w/r
With the Cobb-Douglas production function this always means
αwL=βrK
If α=β cost is allocated so the same amount is spent on each input. If α≠β
then cost is allocated so the amount spent on each input is proportional
𝑟𝐾
𝛼
to the exponents, ie,
= .
𝑤𝐿
𝛽
𝑤 𝛽𝐾
MRTSLforK= =
𝑟 𝛼𝐿
Notice also that the
so as L increases, the MRTSLforK gets
smaller, as we expect.
In the example, what is the MRTSLforK at the cost minimizing input
combination?
If w increases, what substitution will happen with inputs?
36
Doing the math with more than two inputs
min ℒ = 𝑤𝐿 + 𝑟𝐾 + 𝑞𝑍 − 𝜆[𝑄∗ − 𝑓 𝐾, 𝐿, 𝑍 ]
𝐿,𝐾
𝜕ℒ
𝜕𝑓 𝐾,𝐿
=𝑤−
𝜕𝐿
𝜕𝐿
𝜕ℒ
𝜕𝑓 𝐾,𝐿
=𝑟−
𝜕𝐾
𝜕𝐾
𝜕ℒ
𝜕𝑓 𝐾,𝐿
=𝑞−
𝜕𝑍
𝜕𝑍
= 0 which mean 𝑤 = 𝑀𝑃𝐿 (1)
= 0 which mean 𝑟 = 𝑀𝑃𝐾
(2)
= 0 which mean 𝑞 = 𝑀𝑃𝑍
(3)
(1), (2) and (3) ⇒
𝜕ℒ
𝜕𝜆
𝑀𝑃𝐾
𝑟
=
𝑀𝑃𝐿
𝑤
=
𝑀𝑃𝑍
What does this mean?
𝑞
= 𝑄 ∗ − 𝑓 𝐾, 𝐿, 𝑍 = 0 which means hold output constant
37