Section 3 – Theory of the Firm
•Thus far we have focused on the individual
consumer’s decisions:
– Choosing consumption and leisure to:
• Maximize Utility
• Minimize Income
•Section 3 deals with another economic agent,
the producer, and their decisions:
– Choose inputs, production in order to:
• Minimize Costs (to hopefully maximize
profits)
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Section 3 – Theory of the Firm
In this section we will cover:
Chapter 6: Inputs and Production
Functions
Chapter 7: Costs and Cost
Minimization
Chapter 8: Cost Curves
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Chapter 6: Inputs and Production Functions
Consumer Theory
Theory of the Firm
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Chapter 6: Inputs and Production Functions
In this chapter we will cover:
6.1 Inputs and Production
6.2 Marginal Product (similar to marginal utility)
6.3 Average Product
6.4 Isoquants (similar to indifference curves)
6.5 Marginal rate of technical substitution
(MRTS, similar to MRS)
6.6 Special production functions
(similar to special utility functions)
6.7 Technological Progress
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Inputs: Productive resources, such as labor and
capital, that firms use to manufacture goods and
services (also called factors of production)
Output: The amount of goods and services produced
by the firm
Production: transforms inputs into outputs
Technology: determines the quantity of output possible
for a given set of inputs.
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Production function: tells us the
maximum possible output that can be
attained by the firm for any given
quantity of inputs.
Q = f(L,K,M)
Q = f(P,F,L,A)
Computer Chips = f1(L,K,M)
Econ Mark = f2(Intellect, Study, Bribe)
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Production and Utility Functions
•In Consumer Theory, consumption of
GOODS lead to UTILITY:
U=f(kraft dinner, wieners)
•In Production Theory, use of INPUTS
causes PRODUCTION:
Q=f(Labour, Capital, Technology)
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A technically efficient firm is attaining
the maximum possible output from its
inputs (using whatever technology is
appropriate)
A technically inefficient firm is
attaining less than the maximum possible
output from its inputs (using whatever
technology is appropriate)
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production set : all points on or below
the production function
Note: Capital refers to physical capital
(goods that are themselves produced goods) and not
financial capital (the money required to start or
maintain production).
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Q
Example: The Production Function and Technical
Efficiency
Production Function
Q = f(L)
D
C
•
•
•B
Inefficient point
Production Set
L
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Causes of technical inefficiency:
1) Shirking
-Workers don’t work as hard as they can
-Can be due to laziness or a union
strategy
2) Strategic reasons for technical inefficiency
-Poor production may get government
grants
-Low profits may prevent competition
3) Imperfect information on “best practices”
-inferior technology
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Acme medical equipment faces the
production function:
Q=K1/2L1/2
Given labour of 10 and capital of 20, is Acme
producing efficiently by producing 12 units?
What level of production is technically
efficient?
12
Q
=K1/2L1/2
=201/2101/2
=14.14
Acme is not operating efficiently by producing
12 units. Given labour of 10 and capital of
20, Acme should be producing 14.14 units in
order to be technically efficient.
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6.2 Marginal Product
• The production function calculates TOTAL
PRODUCT
• Marginal Product of an input: the change in
output that results from a small change in an
input holding the levels of all other inputs
constant.
MPL = Q/L (holding constant all other inputs)
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MPK = Q/K (holding constant all other inputs)
Marginal Utility and Marginal Product
•In Consumer Theory, marginal utility was
the slope of the total utility curve
•In Production Theory, marginal product is
the slope of the total product curve:
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Q
L
MPL increasing
MPL
MPL becomes negative
MPL decreasing
L
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Law of Diminishing Returns
•Law of diminishing marginal utility: marginal
utility (eventually) declines as the quantity
consumed of a single good increases.
•Law of diminishing marginal returns states
that marginal products (eventually) decline as the
quantity used of a single input increases.
•Generally the first few inputs are highly
productive, but additional units are less
productive (ie: computer programmers working in
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a small room)
Q
Example: Production as workers increase
Each
Additional
worker
Is equally
productive
Each
Additional
worker
Is more
productive
Each
Additional
worker
Is less
productive
Each
Additional
worker
Decreases
Production
Total Product
L
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6.3 Average Product
Average product: total output that is to be
produced divided by the quantity of the input
that is used in its production:
APL = Q/L
APK = Q/K
Example:
Q=K1/2L1/2
APL = [K1/2L1/2]/L = (K/L)1/2
APK = [K1/2L1/2]/K = (L/K)1/2
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Marginal, and Average Product
When Marginal Product is greater than
average product, average product is increasing
-ie: When you get an assignment mark higher
than your average, your average increases
When Marginal Product is less than average
product, average product is decreasing
-ie: When you get an assignment mark lower
than your average, your average decreases
Therefore Average Product is maximized
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when it equals marginal product
Q
APL increasing
APL
MPL
L
APL decreasing
APL maximized
APL
MPL
L
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Isoquant: traces out all the combinations of inputs (labor and
capital) that allow that firm to produce the same quantity of
output.
Example: Q = 4K1/2L1/2
What is the equation of the isoquant for Q = 40?
40 = 4K1/2L1/2
=> 100 = KL
=> K = 100/L
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…and the isoquant for Q = Q*?
Q* = 4K1/2L1/2
Q*2 = 16KL
K = Q*2/16L
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K
Example: Isoquants
All combinations of (L,K) along the
isoquant produce 40 units of output.
Q = 40
Slope=K/L
0
Q = 20
L
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Indifference and Isoquant Curves
•In Consumer Theory, the indifference curve
showed combinations of goods giving the same
utility
•The slope of the indifference curve was the
marginal rate of substitution
•In Production Theory, the isoquant curve shows
combinations of inputs giving the same product
•The slope of the isoquant curve is the marginal
rate of technical substitution:
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6.5 Marginal Rate of Technical
Substitution (MRS)
Marginal rate of technical substitution
(labor for capital): measures the amount of K
the firm the firm could give up in exchange for
an additional L, in order to just be able to
produce the same output as before.
Marginal products and the MRTS are related:
MPL/MPK = -K/L = MRTSL,K
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Marginal Rate of Technical
Substitution (MRS)
Q  MPL L  MPK K
but since Q  0 as one moves along the isoquantcurve,
-MPLLMPK K
MPL  K
MPK L
 K
L
output constant
MP
L  MRTS

output constant
L,K
MPK
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The marginal rate of technical substitution,
MRTSL,K tells us:
The amount capital can be decreased for every
increase in labour, holding output constant
OR
The amount capital must be increased for every
decrease in labour, holding output constant
-as we move down the isoquant, the slope decreases,
decreasing the MRTSL,K
-this is diminishing marginal rate of technical substitution
-as you focus more on one input, the other
input becomes more productive
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MRTS Example
Let Q=4LK
MPL=4K
MPK=4L
Find MRTSL,K
MRTSL,K = MPL/MPK
MRTSL,K =4K/4L
MRTSL,K =K/L
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Isoquants – Regions of Production
•Due to the law of diminishing marginal returns,
increasing one input will eventually decrease total
output (ie: 50 workers in a small room)
•When this occurs, in order to maintain a level of
output (stay on the same isoquant), the other
input will have to increase
•This type of production is not economical, and
results in backward-bending and upward sloping
sections of the isoquant:
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K
Example: The Economic and the
Uneconomic Regions of Production
Isoquants
MPK < 0
Uneconomic region
Q = 20
MPL < 0
Economic region
0
Q = 10
L
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Isoquants and Substitution
•Different industries have different
production functions resulting in different
substitution possibilities:
–ie: In mowing lawns, hard to substitute
away from lawn mowers
•In general, it is easier to substitute away
from an input when it is abundant
–This is shown on the isoquant curve
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K
MRTSL,K is high;
labour is scarce so a
little more labour
frees up a lot of
capital
•
•
MRTSL,K is low;
labour is
abundant so a
little more labour
barely affects the
need for capital
L
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MRTS Example
Let Q=4LK
MPL=4K
MPK=4L
MRTSL,K =K/L
Show diminishing MRTS when Q=16.
When Q=16, (L,K)=(1,4), (2,2), (4,1)
MRTS(1,4)=4/1=4
MRTS(2,2)=2/2=1
MRTS(4,1)=1/4
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K
MRTSL,K =4
4
•
MRTSL,K =1
MRTSL,K =1/4
•
2
•
1
1
2
4
Q=16
L
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K
When input substitution is easy,
isoquants are nearly straight lines
When input substitution is hard when
inputs are scarce, isoquants are more
L-shaped
170
130
100
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100
L
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How much will output increase when ALL inputs increase by
a particular amount?
RTS = [%Q]/[%(all inputs)]
1% increase in inputs => more than 1% increase in
output, increasing returns to scale.
1% increase in inputs => 1% increase in output
constant returns to scale.
1% increase in inputs => a less than 1% increase in
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output, decreasing returns to scale.
Example 1: Q1 = 500L1+400K1
Q1 * = 500(L1)+400(K1)
Q1 *= 500L1+400K1
Q1 *= (500L1+400K1)
Q1 *= Q1
So this production function exhibits
CONSTANT returns to scale. Ie: if inputs
double (=2), outputs double.
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Example 2: Q1 = AL1K1
Q2 = A(L1)(K1)
= + AL1K1
= +Q1
so returns to scale will depend on the
value of +.
+ = 1 … CRS
+ <1 … DRS
+ >1 … IRS
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Why are returns to scale important?
If an industry faces DECREASING returns to scale, small
factories make sense
-It is easier to have small firms in this industry
If an industry faces INCREASING returns to scale, large
factories make sense
-Large firms have an advantage; natural monopolies
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• The marginal product of a single factor may
diminish while the returns to scale do not
• Marginal product deals with a SINGLE input
increasing, while returns to scale deals with
MULTIPLE inputs increasing
• Returns to scale need not be the same at
different levels of production
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1. Linear Production Function:
Q = aL + bK
MRTS constant
Constant returns to scale
Inputs are PERFECT SUBSTITUTES:
-Ie: 10 CD’s are a perfect substitute
for 1 DVD for storing data.
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K
Example: Linear Production Function
Q = Q1
Q = Q0
0
L
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-ie: 2 pieces of bread and 1 piece of cheese make a
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grilled cheese sandwich: Q=min (c, 1/2b)
Cheese
Example: Fixed Proportion Production Function
2
Q=2
Q=1
1
0
2
4
Bread
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3. Cobb-Douglas Production Function:
Q = aLK
 if  +  > 1 then IRTS
 if  +  = 1 then CRTS
 if  +  < 1 then DRTS
 smooth isoquants
 MRTS varies along isoquants
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K
Example: Cobb-Douglas Production Function
Q = Q1
Q = Q0
0
L
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4. Constant Elasticity of Substitution
Production Function:
Q = [aL+bK]1/
Where  = (-1)/
if  = 0, we get Leontief case
if  = , we get linear case
if  = 1, we get the Cobb-Douglas case
General form of other functions
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Definition: Technological progress shifts
isoquants inward by allowing the firm to achieve
more output from a given combination of inputs
(or the same output with fewer inputs).
Neutral technological progress shifts the
isoquant corresponding to a given level of
output inwards, but leaves the MRTSL,K
unchanged along any ray from the origin
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Example: “neutral technological progress”
K
Q = 100 before
Q = 100 after
MRTS remains same
K/L
L
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Labor saving technological progress results
in a fall in the MRTSL,K along any ray from the
origin
Slope decreases along origin ray
Since MRTSL,K=MPL/MPK, MPk increases more
than MPL
ie: better robots, computers, machines
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Example: Labor Saving Technological Progress
K
Q = 100 before
Q = 100 after
MRTS gets smaller
K/L
L
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Capital saving technological progress results
in a rise in the MRTSL,K along any ray from the
origin.
Slope increases along origin ray
Since MRTSL,K=MPL/MPK, MPk increases less
than MPL
ie: higher education, higher skilled workers
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Example: “capital saving technological progress”
K
Q = 100 before
Q = 100 after
MRTS gets larger
K/L
L
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Examples :
Originally : Q  2 L K
Neutral Tech progress : Q  4 L K
Capital - Saving Tech progress : Q  2L K
Labour - Saving Tech progress : Q  2 K L
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Chapter 6 Key Concepts
Inputs and Production
Technical Efficiency
Marginal Product
Law of Diminishing Returns
Average Product
Isoquants
Marginal rate of technical substitution
Returns to Scale
Special production functions
Technological Progress
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