Download short-run production function

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
• 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.
Labor is hired for a given K until the
additional revenue (Marginal revenue of
labor) equals the marginal cost of labor
(wage)
Determining 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: capital is rented up to the point where
value of marginal product of capital equals the
rental rate: VMPK = r,
• 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
Size
ft)
(sq
Net Revenue (‘000 of $)
20,0000
165 265
347
420
480
520
550
575
15,0000
145 235
300
350
390
420
442
460
10,0000
120 190
255
315
345
365
380
390
50,000
93
135
180
210
235
255
273
288
L(workers)
10
20
30
40
50
60
70
80
If the wage per worker is $3,000
1. How many workers should a 50,000 sq ft store employ?
2. How many workers should a 200,000 sq ft store employ?
• Multiple variable inputs
– Consider the relationship between the
ratio of the marginal product of one
input and its cost to the ratio of the
marginal product of the other input(s)
and their cost.
MP1 MP2 MPk


w1
w2
wk
– Other factors may outweigh this
relationship
• Political/Economic risk factors`
Case 1
Case 2
Case 3
Case 4
Case
5
A
B
A
B
A
B
A
B
A
B
Output of task 1
5
4
5
5
5
5
5
1
5
1
Output of task 2
10
2
10
3
10 5
10
1
10 5
Wage ($)
15
6
15
6
15 6
15
6
15 6
Workers A and B can perform task 1 and/or task 2.
The table shows their output (in units) per hour and
their hourly wage rate (w)
If worker A and B can flexibly work with the same
fixed stock of capital (equivalent to ignoring
capital), who should perform task 1 and/ or task 2.
(perform this exercise for each case)
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
The Demand for Labor
• Labor is hired for a given K until the
additional revenue (Marginal revenue
of labor) equals the marginal cost of
labor (wage)
• Demand for labor (or for any factor
of production) is a derived demand.
• It is derived from the demand for
the final product
• The demand for labor is affected by
1. Changes in output
2. Use of capital
3. Technology
4. Price of substitute inputs
5. Price of complement inputs
6. Price of the final product
The demand for an input will be
more elastic the
1. Higher the sensitivity of the final
product to changes in price
2. Higher the share of the input in the
buyer’s total cost
3. Higher the availability of alternative
inputs
4. The higher the substitutability of
complement inputs
Factors behind the demand for
inputs
Case 1
Price of wood increases  price of furniture
increases  quantity of furniture decreases
considerably if the demand for furniture is
elastic  quantity of wood decreases
considerably.
Case 2
Price of beef increases  price of beef soup
increases  quantity of soup decreases
considerably if beef has a high share in the cost
of beef soup
Case 3
Price of beef increases  demand for substitute
chicken increases  quantity of chicken increases
considerably if supply of chicken is elastic 
quantity of beef decreases considerably
Case 4
Price of tires decrease  demand for complement
input (wheel covers) increases  substitutes for
complement input are readily available  quantity
of tires increases considerably
Supply of Labor
In a normal case a worker supplies
more hours of work as the wage
increases and the supply of labor is
upward sloping
The incentive to work more hours
outweighs the higher demand for
leisure as income rises
Supply of labor
• In a special case a worker supplies
fewer hours of work when the wage
increases and the supply curve is
backward bending
• The higher demand for leisure as
income rises outweighs the incentive
to work more hours
Remuneration (Pay) and
Reservation Wage
Job A: Pay (benefits) – costs = net benefits of A
Job B: Pay (benefits) – costs = net benefits of B
Instead of Job A, a person can choose B, the best
feasible alternative
Gain from Job a is the incremental net benefit of
Job A relative to the net benefit of Job B
Gain of A = (Pay in A – Cost of A) – (Pay in B –
Cost in B)
• Economic gain of A is
(Pay in A – Cost of A) – Opportunity
cost of A
Reservation wage is the minimum a
worker is willing to accept to work in
A. The reservation wage is then
Costs in A + (Pay in B – Costs of B)
1. What is the effect (on employment
and output) of adopting Labor-saving
technology?
2. Do cheaper (lower rental rate)
machines reduce employment?
The Theory and
Estimation of Cost
• The Importance of Cost in Managerial Decisions
• The Definition and Use of Cost in Economic
Analysis
• The Relationship Between Production and Cost
• The Short Run Cost Function
• The Long Run Cost Function
• The Learning Curve
• Economies of Scope
• Economies of Scale: the Short Run Versus the
Long Run
• Supply Chain Management
• Ways Companies Have Cut Costs to Remain
Competitive
The Importance of Cost
in Managerial Decisions
• Ways to contain or cut costs over
the past decade
– Most common: reduce number of people
on the payroll
– Outsourcing components of the business
– Merge, consolidate, then reduce
headcount
Cost Topology
Opportunity costs I(implicit or
economic costs) and explicit cost
(accounting costs)
Total cost (TC) = Fixed Costs (TFC) +
Variable Costs (TVC)
Fixed costs do not change with output.
Variable cost change with output.
This distinction is only valid in the S/R
• Relevant cost: a cost that is affected by a
management decision.
• Incremental cost: additional cost due to
additional units of output
• Sunk cost: does not vary in accordance
with decision alternatives. Are not
recoverable and should not be considered
The Relationship Between
Production and Cost
• Cost function is simply the
production function expressed in
monetary rather than physical units.
• Assume the firm is a “price taker” in
the input market.
The Relationship Between
Production and Cost
• Total Variable Cost (TVC): the cost
associated with the variable input,
determined by multiplying the
number of units by the unit price.
• Marginal Cost (MC): the rate of
TVC W
MC  cost.
change in total variable
Q
MP
• The law of diminishing returns implies
that MC will eventually increase
The Relationship
Between
Production and Cost
• Plotting TP and
TVC illustrates
that they are
mirror images of
each other.
• When TP increases
at an increasing
rate, TVC
increases at a
decreasing rate.
The Short-Run Cost
Function
• Standard variables in the short-run
cost function:
– Quantity (Q): the amount of output that
a firm can produce in the short run.
– Total fixed cost (TFC): the total cost of
using the fixed input, capital (K)
– Total variable cost (TVC): the total cost
of using the variable input, labor (L)
– Total cost (TC): the total cost of using
all the firm’s inputs, L and K.
TC = TFC + TVC
Total and Variable Costs
TC(Q): Minimum total cost $
of producing alternative
levels of output:
C(Q) = VC + FC
VC(Q)
TC(Q) = TVC(Q) + TFC
TVC(Q): Costs that vary
with output.
TFC: Costs that do not vary
with output.
FC
0
Q
Fixed and Sunk Costs
FC: Costs that do not
change as output
changes.
$
C(Q) = VC + FC
VC(Q)
Sunk Cost: A cost that is
forever lost after it has
been paid.
FC
Q
The Short-Run Cost Function
• Standard variables in the short-run cost
function:
– Average fixed cost (AFC): the average per-unit
cost of using the fixed input K.
AFC = TFC/Q
– Average variable cost (AVC): the average perunit cost of using the variable input L.
AVC = TVC/Q
– Average total cost (AC) is the average per-unit
cost of using all the firm’s inputs.
AC = AFC + AVC = TC/Q
– Marginal cost (MC): the change in a firm’s total
cost (or total variable cost) resulting from a
unit change in output.
MC = TC/Q = TVC/Q
Average Total Cost
ATC = AVC + AFC
ATC = C(Q)/Q
Average Variable Cost
AVC = VC(Q)/Q
$
MC
ATC
AVC
Average Fixed Cost
AFC = FC/Q
MR
Marginal Cost
MC = C/Q
AFC
Q
The Short-Run Cost
Function
• Important Observations
– AFC declines steadily over the range of
production.
– When MC = AVC, AVC is at a minimum.
– When MC < AVC, AVC is falling.
– When MC > AVC, AVC is rising.
– The same three rules apply for average
cost (AC) as for AVC.
The Short-Run Cost
Function
• A reduction in the firm’s fixed cost
would cause the average cost line to
shift downward.
• A reduction in the firm’s variable
cost would cause all three cost lines
(AC, AVC, MC) to shift.
The Short-Run Cost Function
• Alternative specifications of the
Total Cost function
– Most commonly: specified as a cubic
relationship between total cost and
output
• As output increases, total cost first
increases at a decreasing rate, then
increases at an increasing rate.
• TC = a + bQ + cQ2 + dQ3
The Short-Run Cost Function
• .
– Quadratic relationship
• As output increases, total cost increases at
an increasing rate.
• TC = a + bQ + cQ2
– Linear relationship
• As output increases, total cost increases at
a constant rate.
TC = a + bQ
Cubic Cost Function
• C(Q) = f + a Q + b Q2 + cQ3
• Marginal Cost?
dC/dQ = a + 2bQ + 3cQ2
An Example
– Total Cost: C(Q) = 10 + Q + Q2
– Variable cost function:
VC(Q) = Q + Q2
– Variable cost of producing 2 units:
VC(2) = 2 + (2)2 = 6
– Fixed costs:
FC = 10
– Marginal cost function:
MC(Q) = 1 + 2Q
– Marginal cost of producing 2 units:
MC(2) = 1 + 2(2) = 5
The Long-Run Cost
Function
• In the long run, all inputs to a firm’s
production function may be changed.
• Because there are no fixed inputs, there
are no fixed costs.
• The firm’s long run marginal cost pertains
to returns to scale.
– First, increasing returns to scale.
– As firms mature, they achieve constant
returns, then ultimately decreasing
returns to scale.
The Long-Run Cost
Function
• When a firm experiences increasing
returns to scale:
– A proportional increase in all inputs
increases output by a greater proportion.
– As output increases by some percentage,
total cost of production increases by some
lesser percentage.
The Long-Run Cost
Function
• Economies of Scale: situation where a
firm’s long-run average cost (LRAC)
declines as output increases.
• Diseconomies of Scale: situation
where a firm’s LRAC increases as
output increases.
• In general, the LRAC curve is ushaped.
Economies of Scale
$
LRAC
Economies
of Scale
Diseconomies
of Scale
Q
Reasons for long-run economies
– Specialization in the use of labor and
capital.
– Prices of inputs may fall as the firm
realizes volume discounts in its
purchasing.
– Use of capital equipment with better
price-performance ratios.
– Larger firms may be able to raise funds
in capital markets at a lower cost than
smaller firms.
– Management efficiencies (fewer people
run more operations)
Reasons for Diseconomies of Scale
• Reasons for Diseconomies of Scale
– Scale of production becomes so large
that it affects the total market demand
for inputs, so input prices rise.
– Transportation costs tend to rise as
production grows.
• Handling expenses, insurance, security, and
inventory costs affect transportation costs.
Why economies are important
• Economies of scale protect existing
firms from entrants by allowing for
low average costs at high output
levels.
• Barriers to entry sustain profits
The Long-Run Cost Function
• In long run, the firm
can choose any level of
capacity.
• Once it commits to a
level of capacity, at
least one of the inputs
must be fixed. This
then becomes a shortrun problem.
• The LRAC curve is an
envelope of SRAC
curves, and outlines
the lowest per-unit
costs the firm will
incur over a range of
output.
Economies of Scope
• Economies of Scope: reduction of a
firm’s unit cost by producing two or
more goods or services jointly rather
than separately.
• Closely related to economies of scale.
Multi-Product Cost
Function
• C(Q1, Q2): Cost of jointly producing
two outputs.
• General function form:
C Q1 , Q2   f  aQ1Q2  bQ  cQ
2
1
2
2
Economies of Scope
• TC(Q1, 0) + TC(0, Q2) > TC(Q1, Q2).
– It is cheaper to produce the two
outputs jointly instead of separately.
• Example:
– It is cheaper for Time-Warner to
produce Internet connections and
Instant Messaging services jointly than
separately.
Cost Complementarity
• The marginal cost of producing good
1 declines as more of good two is
produced:
MC1Q1,Q2) /Q2 < 0.
• Example:
– Cow hides and steaks.
Quadratic Multi-Product Cost
Function
•
•
•
•
•
TC(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
MC1(Q1, Q2) = aQ2 + 2Q1
MC2(Q1, Q2) = aQ1 + 2Q2
Cost complementarity:
a<0
Economies of scope:
f > aQ1Q2
TC(Q1 ,0) + TC(0, Q2 ) = f + (Q1 )2 + f +
(Q2)2
TC(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
f > aQ1Q2: Joint production is cheaper
A Numerical Example:
• TC(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2
• Cost Complementarity?
Yes, since a = -2 < 0
MC1(Q1, Q2) = -2Q2 + 2Q1
• Economies of Scope?
Yes, since 90 > -2Q1Q2
Production Costs, Organization Costs,
and Transaction Costs as Determinants
of Optimum Firm Size
• The Firm starts small and benefits from economies of scale
and scope in production, horizontal integration, and multiplant production as it grows
• The firm continues growing by integrating vertically with
suppliers, downstream producers and intermediate
customers. The firm reduces transaction costs, replacing
higher transaction costs with lower internal organizational
costs
• However as the firm gets larger, increasing internal
organization costs outweigh the economies of scale and
scope in production, the benefits from horizontal
integration and multi-plant production, and the lower
transactions costs
Economies of Scale and Firm Size
Profits do not depend only on economies of scale in
production
–Diseconomies of scale in areas other than
production may outweigh economies of scale in
production
–Smaller producers may benefit from niche
markets
–Smaller companies may be in better position to
“read” the customer and innovate
–As size increases, organizational (agency) costs
rise