Transcript Applied Economics for Business Management

Lecture #6
• Lecture outline
• Review
• Go over Homework Set #5
• Continue production economic theory
Output elasticity:
Output elasticity or the elasticity of production can be
written as:
This is a special
type of
production
function called
the CobbDouglas form.
For this type of
production
function, the
output elasticity
is equal to the
exponent.
• The production
function can be
divided into
three stages.
• The definition of
each stage is
based solely on a
technological
basis without any
reference to
price (either
product price or
factor price).
Boundary between Stages I & II:
Boundary between Stages II & III:
What’s happening in Stage I?
What’s happening in Stage II?
What’s happening in Stage III?
Which stage does the producer
operate in?
Stages I and III are technically
infeasible. For stage III,
of the input is negative. So an
additional unit of input will decrease
output.
The producer will not choose
Stage I because
increases throughout Stage I.
Increasing
implies that
technical efficiency is increasing
throughout Stage I. Consequently, it
is best to produce beyond Stage I.
The rational or feasible stage of
production is Stage II where
diminishing returns occur
(as denoted by a positive but
declining MPP).
First order condition(s):
What is the economic meaning of
this first order condition?
First order condition(s):
First order condition(s):
What is the economic meaning of
this first order condition?
So the critical value for x1 occurs where the value of the
marginal product for this input is equal to its price.
2nd order condition:
So profit maximization is occurring on the downward
sloping portion of the MPP curve or the area of
diminishing returns or diminishing MPP.
The negative second derivative verifies that the critical
value is a rel max.
You are also given that
Find the profit maximizing level of input usage for x1.
 Use 2nd derivative to verify max:
 critical value represents a rel max
What is the level of profit?
Why does this firm produce in the short run?
For this production period, the loss is less than fixed
costs (recall FC = $100).
In the longer term, if the firm anticipates continued
losses, it will decide to shut down.
In this case, we deal with 2 variable factors, a set of fixed
factors, and a single output.
The production function looks like this:
or simply write the equation as
Technically, we are dealing with three dimensional space
called a production surface. However, we can plot this surface
in two dimensions by assuming one of the three variables is
constant.
Most frequently held constant is output and the curve that is
derived is called the isoquant.
The isoquant is a curve which shows combinations of inputs,
which yield a specific and constant level of
output (y).
The isoquant is very similar to the indifference
curve (from demand theory).
Like the indifference curve, the isoquant is downward
sloping which illustrates the substitution of one input
for the other
while producing a specific
amount of output.
We can illustrate this substitution by examining the slope
of the isoquant. Also, we can specifically measure the
degree of substitution by calculating the slope at a specific
point on the isoquant.
The rate of technical substitution (RTS) is a similar
concept to the rate of commodity substitution (RCS) in
demand theory.
or the slope of the production isoquant is
equal to the negative ratio of marginal
products.
or the slope of the production isoquant is equal to the
negative ratio of marginal products.
There are two ways to illustrate the optimal
combination of inputs in production for the
factor-factor case:
(i) maximize output subject to a cost constraint and
(ii) minimize cost subject to an output constraint
How do we illustrate method (i)?
Assume a given production function:
Specify a given level of costs (a constraint):
Assume a given production function:
Specify a given level of costs (a constraint):
Objective function: maximize output subject to a cost constraint
First order conditions:
Solving for λ in the first two equations:
┌slope of cost line
└slope of isoquant or
RTS x1,x2
The cost line or budget line for production is called
the isocost line.
The first order conditions state that the variable
factors are combined in an optimal manner when
the ratio of marginal products is equal to the ratio of
factor prices. This optimal combination is called the
least cost combination of inputs.
The second order condition is a bordered Hessian:
for maximum
So for the case of constrained output maximization (where
the constraint is costs), the optimal occurs where the ratio
of marginal products is equal to the ratio of factor prices.
This occurs where the isoquant is tangent to the isocost
line.
The optimal combination of inputs can also be determined
in the factor-factor case by constrained cost minimization.
For this case the objective function can be written as
follows:
1st order conditions:
The least cost combination occurs where:
┌slope of isoquant
└slope of isocost
The least cost combination occurs where:
┌slope of isoquant
└slope of isocost
The least cost combination occurs at the tangency between
isoquant and isocost.
This is the same conclusion as the case of constrained output
maximization.
2nd order conditions:
for a minimum
Consistent with finding the optimal combination of inputs,
is the question of determining the optimal level of input use.
To answer this question, we assume that the firm is a profit
maximizer and has the following objective function:
1st order conditions:
The first order conditions state that for
profit maximization inputs should be
utilized such that the value of their marginal
product is equal to the factor price.
2nd order conditions:
Assume perfect competition with output and input prices:
Also, assume also the firm wishes to spend $80 in production costs
with fixed cost (FC) = $20.
One way to find optimal combinations of inputs is
constrained output maximization.
Set up the objective function as:
1st order conditions:
From the 1st order conditions,
Substitute into 3rd equation:
Substitute into 3rd equation:
2nd order conditions:
 output is maximized subject to the given cost constraint when
λ can be interpreted as the change in output (y) given a $1
change in C (costs).
So λ can be interpreted as
or the reciprocal of marginal
cost.
The alternative formulation to solve for the optimal
combination of inputs is constrained cost minimization.
For this method, the objective function is written as:
In this example, y0 is assumed to be 8.
1st order conditions:
First 2 derivatives
Solving these equations simultaneously yields
λ = 10 reflects the change in cost given a one
unit increase in output (the constraint). So for this
formulation, λ represents the MC.
Second order conditions:
 critical values will minimize costs
subject to the output constraint