Chapter 5

```Managerial Economics &
Chapter 5
The Production Process and Costs
Production Analysis
• Production Function
Q = F(K,L)
The maximum amount of output that can be produced
with K units of capital and L units of labor. The role of
the manager is to make sure the firm operates on the
production function
• Short-Run (SR) v. Long-Run (LR) Decisions
In SR some variables fixed. The LR is a period long
enough such that all inputs are variable
Total Product
• Linear Production Function
Perfect substitution among inputs (meaning, constant)
• Example: Q = F(K,L) = aK + bL
Suppose: a = 10, b = 5, and K is fixed at 16 units in
SR.
Short run production function:
Q = 10(16)+5 L = 160 + 5L
Production when 100 units of labor are used?
Q = 160+500 = 660 units
Total Product
• Cobb-Douglas Production Function
Imperfect substitution (K and L are substitutable but
not at a constant rate).
• Example: Q = F(K,L) = Ka Lb
K is fixed at 16 units, let a = b = 0.5
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
Total Product
• Leontief Production Function
Substitution impossible. K and L are complements.
• Example: Q = F(K,L) = min(bK, cL)
Let b=c=1. Let K=16 and L=100.
Production when 100 units of labor are used?
16 units
Marginal Product of Labor
• Measures the output produced by the last worker holding
K constant.
• Discrete: MPL = DQ/DL (discrete)
• Continuous: Slope (derivative) of the production function
(continuous)
Q
MPL 
L
• At times may see “increasing marginal returns” due to
specialization, and “diminishing marginal returns.”
Using Calculus to Find MPL
Linear Production Function
Q  aK  bL
Q
MPL 
b
L
Cobb-Douglas Production Function
QK L
a b
a b 1
MPL  bK L
Average Product of Labor
• APL = Q/L
• Measures the output of an “average”
worker.
• Geometrically, APL is measured as the
slope of a ray from the origin to a point on
the TP curve.
Stages of Production
Q
Increasing
Marginal
Returns
Diminishing
Marginal
Returns
Negative
Marginal
Returns
Slope=0,
MP=0
Q=F(K,L)
Slope=.75,
AP=.75
MP
AP
L
Productivity in SR
• Go to SR and LR production handout
What’s more important? AP or MP
• Suppose in SR and K is fixed. How does a firm
determine the optimal level of L to hire? How
many workers are needed to maximize profits?
If manager knows how much an additional worker
contributes to revenues and how much an additional worker
costs, then manager can decide if hiring the worker will
increase the firm’s profits.
• Therefore, MP more important.
Value Marginal Product of Labor
• Represent the additional revenue generated
by employing an additional unit of L.
VMPL = MR  MPL = dR/dQ  MPL.
• Note: In a perfectly competitive market, the
price of the good is the marginal revenue.
• VMPL=P  MPL
• Similarly, for the value of the MPK,
VMPK=P  MPK
Find Profit-Maximizing Level of
Labor
• Q=2(K)1/2(L)1/2.
• The company has already spent \$10,000 on
4 units of K, and due to current economic
conditions, the company doesn’t have the
• W=\$100 per worker. P=\$200/unit.
• How much L? How much Q is produced?
What are profits?
• L=16, Q=16, = 8400. Because \$8400 is
less than FC of \$10,000, the firm is
covering a portion of its FC (\$1600 of FC).
Cost Analysis
• Types of Costs
Fixed costs (FC)
Variable costs (VC)
Marginal costs (MC)
Total costs (TC)
Sunk costs
• Need to understand
costs to find profitmaximizing output
Total and Variable Costs (SR)
C(Q): Minimum total cost \$
of producing alternative
levels of output:
C(Q) = VC + FC
VC(Q)
C(Q) = VC + FC
VC(Q): Costs that vary
with output
FC: Costs that do not vary
with output
FC
Q
Fixed and Sunk Costs
FC: Costs that do not change as output changes
Sunk Cost: A cost that is forever lost after it has
been paid
Fixed and Sunk Costs
•You lease a building for \$5000 a month.
•This is a FC.
•After you pay, it is a sunk cost, unless some of it
is refundable, then only the nonrefundable portion
is sunk. Sunk costs are irrelevant to marginal
decision-making that takes place during the
month.
•If at the end of the month, the firm is losing
money (negative economic profit), then paying
the lease becomes a marginal decision, and
shouldn’t renew
Fixed and Sunk Costs
•Problem from the book:
“A local restaurateur who had been running a profitable business for many
owner the legal right to sell…spirits in her restaurant. The cost of
licenses are issued by the state. While the license is transferable, only
\$65,000 is refundable if the owner chooses not to use the license. After
selling alcoholic beverages for about 1 year, the restaurateur came to the
realization that she was losing dinner customers and that her profitable
restaurant was turning into a noisy, unprofitable bar. Subsequently, she
restaurant magazines across the state offering to sell the license for
\$70,000. After a long wait, she finally received one offer to purchase the
Would you recommend that she accept the \$66,000 offer?
Fixed and Sunk Costs
•Assuming no re-sale value, how much of the
•Should she accept the \$66,000 offer?
More Cost Definitions (SR)
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
Marginal Cost
dC
MC = DC/DQ =
dQ
AFC
Q
Costs in SR
• Go to SR cost handout
Fixed Cost
Q0(ATC-AVC)
\$
= Q0 AFC
= Q0(FC/ Q0)
MC
ATC
AVC
= FC
ATC
AFC
Fixed Cost
AVC
Q0
Q
Variable Cost
\$
Q0AVC
= Q0[VC(Q0)/ Q0]
= VC(Q0)
MC
ATC
AVC
AVC
Variable Cost
Q0
Q
Total Cost
Q0ATC
\$
= Q0[C(Q0)/ Q0]
= C(Q0)
MC
ATC
AVC
ATC
Total Cost
Q0
Q
Cubic Cost Function
• C(Q) = f + a Q + b Q2 + cQ3
• Marginal Cost?
Calculus:
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
Another Example
•
•
•
•
•
•
•
C(Q) = 5000 + 20Q2 + 10Q
Find AFC, AVC, ATC and MC when Q=10.
C(10) = \$7100.
AFC = \$500 per unit
AVC=\$210 per unit
ATC = \$710 per unit
MC(Q) = 40Q + 10, thus MC(10)=\$410
Inputs in the LR
• Goal: Minimize the costs of producing a given
level of output. Finding the optimal level of K
and L in LR.
• An isoquant shows the combinations of inputs (K,
L) that yield the producer the same level of
output.
• The shape of an isoquant reflects the ease with
which a producer can substitute among inputs
while maintaining the same level of output.
• Go to SR and LR production handout
L
Linear Isoquants
• Capital and labor are
perfect substitutes
Constant slope (e.g., -3/4
always)
• Marginal Rate of
Technical Substitution
(MRTS): the rate of
substitution between 2
inputs while maintaining
the same level of output
• MRTS = MPL / MPK
The negative of the slope of the
isoquant.
K
Increasing
Output
Q1
Q2
Q3
L
Leontief Isoquants
• Capital and labor are
perfect complements
• Capital and labor are
used in fixed-proportions
Q3
K
Q2
Q1
Increasing
Output
Cobb-Douglas Isoquants
• Inputs are not
perfectly substitutable
• Diminishing MRTS
K
Q3
Q2
Q1
Increasing
Output
Nonlinear, nonconstant
slope
• Most production
processes have
isoquants of this shape
L
Isocost
• The combinations of inputs that yield the same total cost
to the producer
C  wL  rK
C w
K  L
r r
• The above equation is just in y=mx + b form. The slope
is the ratio of input prices (w/r). The ratio of input
prices is the market rate of substitution.
Isocost
• The combinations of
inputs that cost the
producer the same amount
of money
• For given input prices,
isocosts farther from the
origin are associated with
higher costs.
• Changes in input prices
change the slope of the
isocost line
K
C0
C1
L
K
New Isocost Line for
a decrease in the
wage (price of
labor).
L
Cost Minimization
• The additional output (i.e., marginal
product) per dollar spent should be equal for
all inputs:
MPL MPK

w
r
• Expressed differently
w
MRTS KL 
r
Cost Minimization
K
Slope of Isocost
=
Slope of Isoquant
Point of Cost
Minimization
That is,
MRTS = w/r
Q=100
CLower
CHigher
L
LR versus SR
K
C1 > C0, where C0
represents the costs of
producing Q0 in LR.
C1
C0
C1: Cost of
producing Q0
in SR
KLR
KSR*
Q0
L
LLR
LSR
The Long-run and costminimization
• The important thing to remember is that the
LR offers flexibility—the ability to
substitute away from more expensive inputs
to lower priced inputs (e.g., see the chapter
Winning the War”).
Cost-minimization problem
• A firm’s production process follows a
Cobb-Douglas production process. At
current production levels, an engineer has
determined that the MPL is 60 units per hour
and the MPK is 120 units per hour. The
wage rate is \$8 per hour and the opportunity
cost of capital is \$12 per hour. Is the firm
producing its output at the lowest cost? If
not, what changes should they make in the
LR?
Relationship between
LR & SR Costs
\$
ATC
ATC
ATC
ATC
LRAC
LRAC is
“envelope” of
SR ATC
curves
10
30
42
Output (millions)
LR Costs and Economies of
Scale
\$
LRAC
Economies
of Scale
Diseconomies
of Scale
Output
Economies of Scale
• When AC decline as Q increases