Download Chapter 7 The Theory and Estimation of Production

The Theory and Estimation of
Production
• The Production Function
• Short-Run Analysis of Total,
Average, and Marginal Product
• Long-Run Production Function
• Estimation of Production Functions
• Importance of Production Functions
in Managerial Decision Making
Learning Objectives
• Define production function and explain
difference between short-run and long-run
production function
• Explain “law of diminishing returns”
• Define the Three Stages of Production
and how it relates to the “law of
diminishing returns”
• Describe different forms of production
functions that are used.
• Briefly describe the Cobb-Douglas
function
Importance of chapter
• To provide a framework for managerial decisions
regarding allocation of firms resources
• Show how managers can determine which inputs
and how much of each input to use to produce
output efficiently
• This chapter serves as the foundation for later
chapters, which describe in detail pricing and
output techniques for managers interested in
profit maximization
Production Function
• Technology available for producing output
• Consider a production process that utilizes only
capital and labor to produce output
• K = quantity of capital L= quantity of labor and
Q= Output level produced in the production
process.
• The production function is a relation that defines
the maximum output that can be produced with a
given set of inputs
• Mathematically, the production function
can be expressed as
Q=f( K, L)
• Q: level of output
• K and L: inputs used in the production
process
• Key assumptions
– Some given “state of the art” in the
production technology.
– Whatever input or input combinations
are included in a particular function, the
output resulting from their utilization is
at the maximum level.
Short run vs. long run decisions
• In the short run some factors of production are
fixed and this limits the choice in making input
decisions
e.g. Car manufacturing company: Capital is fixed but
labor and steel can be adjusted making them
variable inputs

The short run production function is essentially a
function of only labor
• In the long run the manager can adjust all
factors of production in the long run all inputs
are variable.
• If it takes a company 3 years to acquire
additional capital machines, then the long run for
that company is 3 years and the short run is less
than 3 years
In summary:
• The short-run production function shows
the maximum quantity of good or service
that can be produced by a set of inputs,
assuming the amount of at least one of the
inputs used remains unchanged.
• The long-run production function shows
the maximum quantity of good or service
that can be produced by a set of inputs,
assuming the firm is free to vary the
amount of all the inputs being used.
Short run functions
• Assume Q = F(K,L) = K.5 L.5
– K is fixed at 16 units.
– Short run production function:
Q = (16).5 L.5 = 4 L.5
– Production when 100 units of labor are
used?
Q = 4 (100).5 = 4(10) = 40 units
Measures of productivity
• Managers must determine the productivity of
inputs used in the production process
• This is useful for evaluating the effectiveness of
the production process and making input decisions
that maximize profit
• 3 most important measures of productivity are
Total product, Average product and Marginal
product
Short-Run Analysis of Total,
Average, and Marginal Product
• Alternative terms in reference to inputs
–
–
–
–
Inputs
Factors
Factors of production
Resources
–
–
–
–
Output
Quantity (Q)
Total product (TP)
Product
• Alternative terms in reference to outputs
Average Product:
• Manager may wish to know, on
average, how much each worker
contributes to the total output of
the firm.
• AP for an input is
• Total product divided by quantity use
of input
• Average Product of Labor
– APL = Q/L.
– Measures the output of an “average” worker.
– Example: Q = F(K,L) = K.5 L.5
• If the inputs are K = 16 and L = 9, then the
average product of labor is APL = [(16)
• 0.5(9)0.5]/9 = 1.33
Marginal Product:
• Is the change in total output
attributable to the last unit of input
• MP for an input is
• Change in Total product divided by
change in quantity use of input
Average Product (AP):
APX 
Q
X
•Marginal product (MP):
MPX 
Q
X
• If MP > AP then AP
is rising.
• If MP < AP then AP
is falling.
• MP=AP when AP is
maximized.
Phases of Marginal Product:
• As the usage of an input increases,
marginal product initially increases
(increasing marginal returns), then
begins to decline (decreasing
marginal returns) and eventually
becomes negative (negative marginal
returns)
Increasing, Diminishing and
Negative Marginal Returns
Q
Increasing
Marginal
Returns
Diminishing
Marginal
Returns
Negative
Marginal
Returns
Q=F(K,L)
MP
AP
L
• Law of Diminishing Returns: As additional units
of a variable input are combined with a fixed
input, at some point the additional output (i.e.,
marginal product) starts to diminish.
– Nothing says when diminishing returns will start to
take effect, only that it will happen at some point.
– All inputs added to the production process are
exactly the same in individual productivity
• The Three Stages of Production in
the Short Run
– Stage I: From zero units of the variable
input to where AP is maximized (where
MP=AP)
– Stage II: From the maximum AP to
where MP=0
– Stage III: From where MP=0 on
• In the short run, rational firms should only
be operating in Stage II.
• Why not Stage III?
– Firm uses more variable inputs to produce less
output
• Why not Stage I?
– Underutilizing fixed capacity
– Can increase output per unit by increasing the
amount of the variable input
• What level of input usage within
Stage II is best for the firm?
• The answer depends upon how many
units of output the firm can sell, the
price of the product, and the
monetary costs of employing the
variable input.
Manager’s role of using the right level
of input:
e.g. restaurant manager must hire the
“correct” number of servers
If product is sold at $3 on the market
and each unit of labor costs $400,
how many units of labor should be
hired to maximize profit?
• First, determine the benefit of hiring an
additional worker. Each worker increases the
firm’s total output by her marginal product.
• This increase can be sold in a market at a price of
$3
• Thus the benefit from each unit of labor is $3 x
MP of worker
• This number is known as the Value marginal
product of labor = VMP
• VMPL = P x MPL.
• It is profitable to hire units of labor so long as
their additional output value exceeds their cost.
• So, employ labor as long as VMP exceeds their
wage (w)
• To maximize profits, a manager should use inputs
at levels which their marginal benefits equal the
marginal cost.
• Specifically for labor,
• VMPL = w
• For capital:
value of marginal product of capital equals the
rental rate: VMPK = r,
Alternative Terminology (using
Marginal Revenue product)
• Total Revenue Product (TRP): market value
of the firm’s output, computed by
multiplying the total product by the
market price.
– TRP = Q · P
• Marginal Revenue Product (MRP): change in
the firm’s TRP resulting from a unit change
in the number of inputs used.
– MRP = = MP · P
TRP
X
• Total Labor Cost (TLC): total cost of using
the variable input, labor, computed by
multiplying the wage rate by the number
of variable inputs employed.
– TLC = w · X
• Marginal Labor Cost (MLC): change in total
labor cost resulting from a unit change in
the number of variable inputs used.
Because the wage rate is assumed to be
constant regardless of the number of
inputs used, MLC is the same as the wage
rate (w).
• Summary of relationship between demand
for output and demand for input
– A profit-maximizing firm operating in perfectly
competitive output and input markets will be
using the optimal amount of an input at the
point at which the monetary value of the
input’s marginal product is equal to the
additional cost of using that input.
– MRP = MLC
• Multiple variable inputs
– Consider the relationship between the
ratio of the marginal product of one
input and its cost to the ratio of the
marginal product of the other input(s)
and their cost.
MP1 MP2 MPk


w1
w2
wk
– Other factors may outweigh this
relationship
• Political/Economic risk factors`
The Long-Run Production
Function
• In the long run, a firm has enough
time to change the amount of all its
inputs.
– Effectively, all inputs are variable.
• The long run production process is
described by the concept of returns
to scale.
• If all inputs into the production
process are doubled, three things can
happen:
– output can more than double
• increasing returns to scale (IRTS)
– output can exactly double
• constant returns to scale (CRTS)
– output can less than double
• decreasing returns to scale (DRTS)
• One way to measure returns to scale is to
use a coefficient of output elasticity:
Percentage change in Q
EQ 
Percentage change in all inputs
• If EQ > 1 then IRTS
• If EQ = 1 then CRTS
• If EQ < 1 then DRTS
• Returns to scale can also be
described using the following
equation
hQ = f(kX, kY)
• If h > k then IRTS
• If h = k then CRTS
• If h < k then DRTS
• Graphically, the returns to scale
concept can be illustrated using the
following graphs.
Q
IRTS
Q
X,Y
DRTS
CRTS
Q
X,Y
X,Y
Estimation of Production
Functions
• Forms of Production Functions
– Cobb-Douglas Production Function: Q = aLbKc
• Both capital and labor inputs must exist for Q to be
a positive number
• Can be increasing, decreasing, or constant returns to
scale
– b + c > 1, IRTS
– b + c = 1, CRTS
– b + c < 1, DRTS
• Permits us to investigate MP for any factor while
holding all others constant
• Elasticities of factors are equal to their exponents
Estimation of Production
Functions
• Forms of Production Functions
– Cobb-Douglas Production Function
• Can be estimated by linear regression analysis
• Can accommodate any number of independent
variables
• Does not require that technology be held constant
• Shortcomings:
– Cannot show MP going through all three stages in one
specification
– Cannot show a firm or industry passing through
increasing, constant, and decreasing returns to scale
– Specification of data to be used in empirical
estimates`
Marginal Rate of Technical
Substitution (MRTS)
• The rate at which two inputs are
substituted while maintaining the
same output level.
MPL
MRTS KL 
MPK
Cost Minimization
• Marginal product per dollar spent
should be equal for all inputs:
MPL MPK
MPL w



w
r
MPK r
• But, this is just
MRTS KL 
w
r