Nechyba_2e_Ch_12ABmicroeconomicsslifessss

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Microeconomics:
An Intuitive Approach, 2E
Chapter 12 –
Production with Multiple Inputs
© 2017 Cengage Learning® May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for
use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved
learning management system for classroom use.

Direct 2-Input Profit Maximization
In the 1-input model, profit maximization typically involves tangencies between
isoprofit curves and the production frontier.
In a 2-input model, a production plan
specifies how much x is produced using
labor and capital k. Production plans
therefore lie in 3 dimensions.
Isoprofit lines then become isoprofit
planes that can be viewed as the firm’s
indifference curves.
The 1-input model is best viewed as a
short run model that holds capital fixed
– with the real underlying production
frontier lying in 3 dimensions.
And profit maximization again (typically)
involves tangencies – except now in 3
dimensions. © 2017 Cengage Learning® May not be scanned, copied or
duplicated, or posted to a publicly accessible website, in whole or
in part, except for use as permitted in a license distributed with a
certain product or service or otherwise on a password-protected
website or school-approved learning management system for
classroom use.

Short and Long Run Production Frontiers
Long run production frontiers illustrate the available technology assuming all inputs can
be varied.
Short run production frontiers are derived from long-run frontiers by assuming capital is
fixed at some level and only labor input can be varied.
For different levels of fixed capital, we get
different “slices” of the long run
production frontier …
When graphed as single-input production
frontiers, they then appear as different
frontiers – despite being drawn from the
same long run production frontier.
The marginal product of labor is still the
increase in output from hiring one more
labor hour – and is thus still measured as
a slope on the short run production
frontier. As capital increases, the
marginal product of labor changes.
© 2017 Cengage Learning® May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or
service or otherwise on a password-protected website or school-approved learning management system for classroom use.

2 Input Production and Returns to Scale
A second way of “slicing” a 2-input production frontier is along a ray from the origin.
If the resulting slice is linear, then increasing all inputs by a factor k results in a k-fold
increase in output. Such a production frontier is said to have constant returns to
scale.
Alternatively, the slice may have
diminishing slope, implying that
increasing all inputs by a factor of k
results in less than a k-fold increase in
output. This is called decreasing
returns to scale.
Or, the slice may have increasing
slope, implying that increasing all
inputs by a factor of k results in more
than a k-fold increase in output. This is
called increasing returns to scale.
Go to Math

Returns to Scale versus Marginal Product
Returns to scale are seen on “slices” along rays from the origin – because the concept
describes what happens to output when all inputs rise at the same time.
Marginal product is seen on “slices” that hold
one input fixed – because the concept
describes what happens to output when one
input rises all else staying unchanged.
It is logically possible to have diminishing
marginal product (of all inputs) …
… and increasing returns to scale.
It is also logically possible to have
increasing returns to scale …
… and increasing marginal product.
Is it possible to have decreasing returns to scale and increasing
marginal product of one input? No
Go to Math

Isoquants with 2-Input Production
A third way of “slicing” 2-input production frontiers is “horizontally” at different levels of
output.
The resulting isoquant map looks similar
to a consumer map of indifference
curves.
But instead of showing combinations of
consumption goods that “produce” the
same level of utility, an isoquant illustrates
all input bundles that can produce a given
level of output without wasting any input.
So far, we have looked at two kinds of “vertical” slices of the 2-input production frontier.
These “slices” can then be projected
onto 2 dimensions – with the “height” of
the slice indicated as a label on each
curve.

(Marginal) Technical Rates of Substitution
The blue isoquant represents all technologically efficient production plans that involve
100 units of output.
A, for instance, is the production plan (2, 10, 100) – the plan that combines 2 labor hours
with 10 units of capital to produce 10 output units. It is a technologically efficient plan
because it lies on the production frontier – which means no inputs are wasted.
The slope of the isoquant at A tells us how many units of capital we could substitute for 1
labor hour and maintain production at 100 units of output. This is called the (marginal)
technical rate of substitution (TRS).

MPl
A
 3MPk
A
.
If TRS = –3, it means that at A, labor must be
approximately 3 times as productive per unit as
capital – because we can replace 3 units of capital
with just 1 unit of labor. Thus
And that implies
TRS  
MPl
MPk
Go to Math

Substitutability in Production
Just as we can compare consumer tastes in terms of the degree of substitutability of goods,
so we can compare technologies in terms of the degree of substitutability of inputs.
We will throughout assume that isoquant maps are homothetic – with the TRS constant
along any ray from the origin.
But this allows for any degree of substitutability between labor and capital, with
greater substitutability causing flatter isoquants …
… and less substitutability causing more L-
shaped isoquants.
These rays from the origin are, of course,
the very rays along which we can slice the
production frontier “vertically” to identify
the technology’s returns to scale.

Returns to Scale in the Isoquant Graph
Consider, for instance, the ray that lies on the 45-degree line.
The vertical “slice” (of the production frontier) that lies on
this ray has output x on the vertical axis and both inputs on
the horizontal.
A linear shape of this vertical slice then indicates
constant returns to scale.
But the same isoquant map – with isoquant labels
changed – can give rise to a diminishing slope on the
vertical slice, and thus decreasing returns to scale.
Similarly, other isoquant labels will give us
increasing returns to scale.
Returns to scale are then seen in the
rate at which isoquant labels change
along rays from the origin.
How could you similarly identify
diminishing versus increasing MP?
On horizontal and vertical lines.

Homothetic Production & 2 Types of “Convexity”
For a producer choice set to be convex, any “slice” of the 3-dimensional set must be convex.
Along rays from the origin (like the 45 degree line), a slice can be convex only if the
technology has decreasing (or constant) returns to scale.
This automatically implies that all marginal
products are diminishing – and thus short
run production slices are convex.
Finally, the isoquant map has to have
the convexity property.
To say that a producer choice set is convex
is then to say that is has decreasing
returns to scale and its isoquants have
convex “upper contour sets”.
Increasing returns to scale make the producer
choice set non-convex – even though the
isoquant map may still be convex.
Go to Math

Indifference Maps versus Isoquant Maps
Despite the many technical similarities between consumer
indifference maps and producer isoquants, there are important
differences:
1. These are representations of tastes for consumers and
constraints for producers.
2. The numerical values attached to them are objectively
measurable and thus inherently meaningful for producers
but not for consumers.
3. This is why we take vertical “slices” in the producer case but
NOT in the consumer case.
4. This is also why the concept of marginal utility of a good is
not emphasized in consumer theory – but marginal product
of an input is.

Back to Direct Profit Maximization
The tangency that occurs at the profit maximizing production plan in the 1-input model
has a specific and intuitive economic interpretation. At the tangency,
In a 2-input model, this tangency will
happen in 3 dimensions.
Isoprofit planes can now be thought of
as “sheets” that will touch the 3-
dimensional production frontier at the
profit maximizing production plan.
And this tangency is a tangency “in
every direction” – which means the
short-run profit maximization condition
must still hold (although it
might lie on a different vertical slice as
capital is adjusted in the long run).
pMPl  w and pMPk  r

pMPl  w.

pMPl  w
But the same tangency holds along the
slice with labor held fixed – and so
Go to
Math

• In the 1-input model, we showed how to split profit
maximization into two steps:
– Step 1: Derive AC and MC
– Step 2: Set x where p = MC as long as p ≥ AC
• Step 1 is the cost minimization part of profit
maximization. In the 1-input model, it simply implies
not wasting inputs.
• But in the 2-input model, producing an output level x
at the lowest possible cost involves more than “not
wasting inputs”.
• This is because there are now many ways – many
combinations of capital and labor – to produce the
output level x “without wasting inputs”.
2-Step Profit Maximization using Cost Curves

Cost-Minimizing
For instance, all input bundles on this isoquant will produce
100 units of x “without wasting inputs” – all represent
technologically efficient ways of producing 100 units .
When w =20 and r =10, a budget – or isocost – of $300 is
sufficient to reach production plans on this isoquant.
But a production plan is not economically efficient unless its
inputs are the least-cost way of reaching the output level.
For every output quantity, there is usually one such
economically efficient input bundle where
And when technology is homothetic, the economically
efficient input bundles lie on the same ray from the origin.
For each output level, we can then read off the cost of
production assuming the firm cost-minimizes.
From the (total) cost curve, we can derive the marginal
and average cost curves exactly as in the 1-input case.

TRS  
w
r
Constant
Returns to Scale

Supply Curve
Suppose A is the cost-minimizing input bundle to produce 10 units of
output. With homothetic technologies, this implies all cost-minimizing
input bundles lie on the same ray.
IRS
DRS
From A, we can calculate the cost (A’)and average
cost (A”) of producing 10 units of output (given
input prices w =20 and r =10).
And repeating this for all other output quantities,
we get the (total) cost and average cost curves.
The slope of the (total) cost curve becomes
the marginal cost curve.
And – just as in
the 1-input case,
the supply curve
is the MC curve
that lies above AC.
Go to Math
© 2017 Cengage Learning® May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for
classroom use.

Profit Maximization with Increasing Returns
If the production process had increasing returns to scale throughout, then AC would
always fall.
And the only way AC can fall is for MC to lie below AC.
Since the total cost is the sum of all marginal costs, it is equal to the shaded area.
Suppose AC and MC converge to the dotted horizontal line. Then:
1. Profit is negative for any price at or below p* unless the firm
produces nothing; and
2. Profit is infinite for any price above p* if the firm produces
infinite output.
Corner
Solutions
But – increasing returns to scale throughout is an
assumption incompatible with price-taking
behavior.
We will return to this
in our discussion of
natural monopolies.

A producer that only minimizes costs has to pay no attention to
output prices – only input prices (w,r) matter for determining the
least cost way of producing different levels of output.
A profit-maximizing producer also thinks about output price.
Bringing Cost Minimization and Profit
Maximization Together
implies
does not imply
(unless p=MC)

Multi-Input Production Functions
A production function with inputs labor and capital simply takes the form
and tells us that output can be produced with the input bundle .

x  f (l ,k)

(l ,k)
It gives rise to the producer choice set
that specifies the production plans that are feasible under the
technology represented by the production function f.
Production functions are mathematically similar to utility functions: In fact, we can
think of a utility function as a production function in which consumer goods are inputs
and utility is the output.
Our focus will be on homothetic production processes – and we will return to
functional forms like the Cobb-Douglas and CES functions.

TRS and MP
The (marginal) technical rate of substitution (TRS) is for the production function what
the marginal rate of substitution (MRS) is for the utility function.

MRS  
u x1
u x2

TRS  
f l
f k
.
The marginal product of labor is the change in output from a
marginal change in labor input (all else equal); and the
marginal product of capital is the change in output from a
marginal change in capital (all else equal). Thus

MPl 
f
l
and MPk 
f
k
.
Combining these, we get
TRS  
f l
f k
 
MPl
MPk
.
Recall that . By the same sequence of steps, we then get
Back to
Graphs

Convexity in Production

min f x1
A
, x2
A
 , f x1
B
, x2
B
 
  f x1
A
 (1)x1
B
, x2
A
 (1)x2
B
 

 f x1
A
, x2
A
  (1) f x1
B
, x2
B
  f x1
A
 (1)x1
B
, x2
A
 (1)x2
B
 .
Our notion of convexity (of upper contour sets) from consumer theory simply says that
“averages are better than extremes” or, in terms of producer theory, “average input
bundles result in greater output than extremes”.
A function is defined as quasiconcave if and only if
which is exactly what our previous notion of convexity (of upper contour sets) says.
The set of points underneath a function is convex if the function is concave. A
function is defined to be concave if and only if
implies
does not imply
Convexity of the producer choice set implies convexity of upper contour sets;
BUT convexity of upper contour sets does NOT imply convexity of producer
choice sets.

Convexity of the producer choice set implies convexity of upper contour sets;
BUT convexity of upper contour sets does NOT imply convexity of producer
choice sets.

f (l ,k)  l
1
3
k
1
3
is concave

f (l ,k)  l
2
3
k
2
3
is quasiconcave
(but not concave)
Convexity in Production
Back to
Graphs

Homogeneous Functions and RTS
The concept of returns to scale is most easily presented for homogeneous functions –
where a function f is homogeneous of degree k if and only if

f (tl ,tk)  tk
f (l ,k).
k > 1 implies Increasing Returns to Scale
k = 1 implies Constant Returns to Scale
k < 1 implies Decreasing Returns to Scale
The Cobb-Douglas production function, for instance, is

f (l ,k)  Al 
k
.
+ > 1 implies Increasing Returns to Scale
+ = 1 implies Constant Returns to Scale
+ < 1 implies Decreasing Returns to Scale
Back to
Graphs

The Cobb-Douglas functions offers an easy way to explore the relationship between
marginal product and returns to scale.

f (l ,k)  Al 
k

MPk  Al 
k 1

MPk
k
 ( 1)Al 
k2

 1 implies
MPk
k
 0
 1 implies
MPk
k
 0

f (l ,k)  l
2
3
k
2
3

f (l ,k)  l
4
3
k
4
3
+ > 1 implies
Increasing Returns to Scale
Diminishing MP
Increasing MP
Back to
Graphs
Marginal Product & Returns to Scale

1-Input Model: 2-Input Model:
This can be written as
The first order conditions are:
or simply
Profit Maximization

Profit Maximization: An Example
This is a Cobb-Douglas production function with exponents summing to less than 1. It
therefore has decreasing returns to scale – giving rise to a fully convex producer choice set.
As a result, we know that the first order conditions are necessary and sufficient for us to
identify the profit maximizing production plan.
The constrained profit maximization problem
can be solved using the Lagrange method, or it can be solved by substituting the
constraint into the objective function.
The latter turns the problem into the unconstrained maximization problem

Profit Maximization: An Example
First Order Conditions:

k 
8p
r






5 3
l 2 3
labor demand
capital
demand
output supply
16384
profit function
Back to
Graphs

2-Step Profit Maximizing via Cost Minimizing
Notice that, aside from notation, cost minimizing is identical to the expenditure
minimization portion of the consumer duality picture.
First order conditions:

An Example Continued:
First Order Conditions:
STEP 2: Set
Output
STEP 1: Minimize Cost
Same input
demand and output
supply functions as
under direct profit
maximization

Completing Producer Duality
Hotelling’s
Lemma
Shephard’
s Lemma

12B The Mathematics Behind the
Multiple-Input Model
12B.1 Producer Choice Sets & Production Functions
– The producer choice set is defined as the set of production
plans (x, ℓ, k) that are technologically feasible:
12B.1.1 Marginal Products and TRS
- The mathematical definition of marginal product is the partial
derivative of the production function with respect to the input:
- The slope of an isoquant derived from a production function,
the marginal technical rate of substitution (TRS) is then:
© 2017 Cengage Learning® May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part,
except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or
school-approved learning management system for classroom use.

12B.1 Producer Choice Sets & Production
Functions
12B.1.1 Marginal Products and TRS (cont)
- Given the expressions for marginal product in equation (12.9),
the technical rate of substitution can then also be expressed as
the fraction of the marginal products of the inputs:
- 12B.1.2 “Averages Are Better than Extremes” and
Quasiconcavity
- Assuming convexity of upper contour sets is equivalent to
assuming that the underlying production function is
quasiconcave

Functions
12B.1.2 “Averages Are Better than Extremes” and
Quasiconcavity (cont)
- This does not imply that production (or utility) functions that
have the “averages are better than extremes” feature must be
concave, only that they must be quasiconcave:
- It is easy to see that every concave function is also
quasiconcave:
- The reverse, however, does not hold

Functions
12B.1.2 “Averages Are Better than Extremes” and
Quasiconcavity (cont)

Functions
12B.1.3 Returns to Scale and Concavity
- Recall that all homogenous functions are homothetic, and a
function is homogenous of degree k if and only if:
– 12B.1.4 Returns to Scale and Diminishing Marginal Product

Functions
12B.1.3 Returns to Scale and Concavity (cont)

12B.2 Isoprofit Planes and Profit Maximization
12B.2.1 Isoprofit Curves with Multiple Inputs
– In the two-input case with labor ℓ and capital k, profit π at a
production plan (x, ℓ, k) is then:
– The isoprofit curve P, is the set of production plans that yield
the same amount of profit in a given economic environment (p,
w, r); defined more formally as:
12B.2.2 Profit Maximization with Multiple Inputs
- The movement to the highest possible isoprofit plane on the
three-dimensional production function in Graph 12.1f is:

12B.2.2 Profit Maximization with Multiple Inputs (cont)
- The problem can also be written as an unconstrained
maximization problem by substituting the constraint into the
objective function and writing:
- The 1st order conditions are the partial derivatives of π (with
respect to the two choice variables) set to zero:
- Which can also be written as:
- Or simply:

12B.2.2 Profit Maximization with Multiple Inputs (cont)
- The two equations in (12.23) can then be solved to give the
input demand functions that tell us how much labor and capital
the producer will hire in any economic environment:
- Are the labor and capital demand functions for this producer.
Plugging these into the production function, we can then derive
the output supply function:

12B.2.3 An Example of Profit Maximization
- Suppose that the technology available to me as a producer can
be represented by the function f(ℓ, k) = 20ℓ2/5k2/5; we can then
set up the profit maximization problem:
- Which can also be written:
- The first order conditions are then:
- Which can be written as:

12B.2.3 An Example of Profit Maximization (cont)
- Solving the second of these two equations for k and plugging it
into the first, we get the labor demand function
- And plugging this in for ℓ in the second equation, we get the
capital demand function:
- Finally, we can derive the output supply function by plugging
equations (12.31 and (12.32) into the production function:

12B.3 Cost Minimization on the Way to Profit
Maximization
12B.3.1 Extending Cost Minimization to Multiple Inputs

Maximization
(cont)
– We can express this process as a constrained minimization
problem in which we are attempting to ascertain the minimum
cost necessary to reach each of the isoquants from our
production function:
– The Lagrange function is then given by:
– With first order conditions:

Maximization
(cont)
– Taking the negative terms in the first two equations to the other
side and dividing the two equations by each other, we get:
– Put differently, we can derive the functions:
– If e know the conditional input demand functions, we can derive
the (total) cost function that tells us the minimum cost of
producing any output level for any set of input prices:

Maximization
12B.3.2 An Example Continued
– Using the cost minimization approach, we first define the
problem as in (12.34):
– The Lagrange function is then given by:
– With first order conditions:

Maximization
12B.3.2 An Example Continued
– Taking the negative terms in the first two equations to the other
side and dividing the questions by one another, we get:
– Substituting the latter into the third first-order condition and
solving for ℓ, we get the conditional labor demand function:
– And substituting this back into (12.43), we can solve for the
conditional capital demand function:

Maximization
12B.3.2 An Example Continued (cont)
– The cost function is then simply the sum of the conditional input
demands multiplied by input prices:
– Setting MC equal to price and solving for x, we get:
– When we now plug x(p, w, r), we get:

12B.4 Duality in Producer Theory

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