BUS 525: Managerial
Economics
Lecture 5
The Production Process and Costs
Overview
I. Production Analysis
–
–
–
–
Total Product, Marginal Product, Average Product
Isoquants
Isocosts
Cost Minimization
II. Cost Analysis
– Total Cost, Variable Cost, Fixed Costs
– Cubic Cost Function
– Cost Relations
III. Multi-Product Cost Functions
Production Analysis
• Production Function
– A function that defines the maximum amount
output that can be produced with a given set
of inputs
• Q = F(K,L)
• The maximum amount of output that can be
produced with K units of capital and L units of
labor.
• Short-Run vs. Long-Run Decisions
• Fixed vs. Variable Factors of Production
1-4
Refer to Table 5.1 Page 158
Total Product
• The maximum level of output that can be
produced with a given amount of input
• Cobb-Douglas Production Function
• Example: 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
Average Productivity Measures
• Average Product of Labor: A measure of the
output produced per unit of input
• APL = Q/L.
• Measures the output of an “average” worker.
• Average Product of Capital
• APK = Q/K.
• Measures the output of an “average” unit of
capital.
Marginal Productivity Measures
• Marginal product : The change in total output
attributable to the last unit of an input
• Marginal Product of Labor: MPL = DQ/DL
– Measures the output produced by the last worker.
– Slope of the short-run production function (with respect to
labor).
• Marginal Product of Capital: MPK = DQ/DK
– Measures the output produced by the last unit of capital.
– When capital is allowed to vary in the long run, MPK is the
slope of the production function (with respect to capital).
Class Exercise
Given the Cobb-Douglas Production
Function
Q = F(K,L) = K.5 L.5, K = 16, L=64
Find out the
(i) Average Product of Labor (APl),
(ii) Average Product of Capital (APk),
(iii) Marginal Product of Capital (MPl)
(iv) Marginal Product of Capital (MPk)
Increasing, Diminishing and
Negative Marginal Returns
Q
Increasing
Marginal
Returns
Diminishing
Marginal
Returns
Negative
Marginal
Returns
Q=F(K,L)
MP
AP
L
Guiding the Production Process
• Producing on the production function
– Aligning incentives to induce maximum
worker effort, tips, profit sharing.
• Employing the right level of inputs
– When labor or capital vary in the short run, to
maximize profit a manager will hire
• labor until the value of marginal product of labor
equals the wage: VMPL = w, where VMPL = P x
MPL.
• capital until the value of marginal product of capital
equals the rental rate: VMPK = r, where VMPK = P
x MPK .
The profit maximizing usage of inputs
∏ = R- C,
Q = F (K,L), C = wL + rK
∏ = P. F(K,L) - wL - rK
π
F(K,L)
 P.
r  0
K
K
π
F(K,L)
 P.
w 0
L
L
F(K,L)
F(K,L)
Since
 MP k and
 MPL
K
L
P .MPK  r and P .MPL  w
V MPK  r and V MPL  w
i.e each input must be used up t o t hepoint
where it s value of margimalproduct equals it s price
Refer to Table 5.2 Page 163
Isoquant
• 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.
Slope of Isoquant
Slo p e o f iso quan t s
Q  F (K, L)
F(K, L)
F(K, L)
dQ 
.d K 
.d L  0
K
L
[Sin ce o ut p ut do n o t ch an gealo n g an iso qua
dQ  0 ]
dK
F(K, L)/L
MPK
o r,


dL
F(K, L)/K
MPL
Marginal Rate of Technical Substitution
(MRTS)
• The rate at which two inputs are substituted while
maintaining the same output level.
•
MPL
MRTSKL 
MP
The production function satisfies the K
law of diminishing
marginal rate of technical substitution: As a producer
uses less of an input, increasingly more of the other
input must be employed to produce the same level of
output
Linear Isoquants
• Capital and labor are
perfect substitutes
K
• Q = aK + bL
• MRTSKL = b/a
• Linear isoquants imply
that inputs are substituted
at a constant rate,
independent of the input
levels employed.
Increasing
Output
Q1
Q2
Q3
L
Leontief Isoquants
• Capital and labor are
K
perfect complements.
• Capital and labor are used
in fixed-proportions.
• Q = min {bK, cL}
• What is the MRTSKL?
Q3
Q2
Q1
Increasing
Output
L
Cobb-Douglas Isoquants
• Inputs are not perfectly
substitutable.
• Diminishing marginal rate of
technical substitution.
– As less of one input is used in
the production process,
increasingly more of the other
input must be employed to
produce the same output
level.
K
Q3
Q2
Q1
Increasing
Output
• Q = KaLb
• MRTSKL = MPL/MPK
L
Isocost
• The combinations of inputs
K
that produce a given level of
C1/r
output at the same cost:
C0/r
wL + rK = C
• Rearranging,
K= (1/r)C - (w/r)L
• For given input prices,
K
isocosts farther from the
C/r
origin are associated with
higher costs.
• Changes in input prices
change the slope of the
isocost line.
New Isocost Line
associated with higher
costs (C0 < C1).
C0
C0/w
C1
C1/w
L
New Isocost Line for
a decrease in the
wage (price of labor:
w0 > w1).
C/w0
C/w1
L
Slope of isocost
wL  rK  C
1
w
 K  (C )  L
r
r
w
Slope  r
Cost Minimization
K
Slope of Isocost
=
Slope of Isoquant
Point of Cost
Minimization
A
C
Q
L
Cost minimization
C  wL  rK,
Q  F(K, L)
M in im izewL  rK subject t oQ  F(K, L)
La g ra n g ia n
H

wL  rK 
[Q- F (K,L)]
H
μF(K,L)
 w
 0........(1 )
L
L
H
μF(K,L)
 r 
 0........(2 )
K
K
H
 Q  F(K,L)  0........
.......
(3)
μ
d ivid in g 1 b y 2
w
F(K,L)/L


r
F(K,L)/K
MP
MP
L
K

MRTS
KL
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
Optimal Input Substitution
•
•
A firm initially produces Q0 by
employing the combination of
inputs represented by point A
at a cost of C0.
Suppose w0 falls to w1.
– The isocost curve rotates
counterclockwise; which
represents the same cost
level prior to the wage
change.
– To produce the same level of
output, Q0, the firm will
produce on a lower isocost
line (C1) at a point B.
– The slope of the new isocost
line represents the lower
wage relative to the rental
rate of capital.
K
A
K0
B
K1
Q0
0 L0
L1 C0/w0
C1/w1
C0/w1 L
Cost Analysis
• Types of Costs
– Fixed costs (FC)
– Variable costs (VC)
– Total costs (TC)
– Sunk costs
• A cost that is forever lost after it has been
incurred. Once paid they are irrelevant to decision
making
Refer to Table 5.3 Page 179
Total and Variable Costs
C(Q): Minimum total cost $
of producing alternative
levels of output:
C(Q) = VC + FC
VC(Q)
C(Q) = VC(Q) + FC
VC(Q): Costs that vary
with output.
FC: 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
Refer to Table 5.4 Page 181
Refer to Table 5.5 Page 182
1-30
Some Definitions
Average Total Cost
ATC = AVC + AFC
ATC = C(Q)/Q
$
MC
ATC
AVC
Average Variable Cost
AVC = VC(Q)/Q
MR
Average Fixed Cost
AFC = FC/Q
Marginal Cost
MC = DC/DQ
AFC
Q
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?
– Memorize:
MC(Q) = a + 2bQ + 3cQ2
– 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
Economies of Scale
$
LRAC
Economies
of Scale
Diseconomies
of Scale
Q
Multi-Product Cost Function
• A function that defines the cost of
producing given levels of two or more
types of outputs assuming all inputs are
used efficiently
• C(Q1, Q2): Cost of jointly producing two
outputs.
• General multiproduct cost function
CQ1, Q2   f  aQ1Q2  bQ12  cQ22
Economies of Scope
• C(Q1, 0) + C(0, Q2) > C(Q1, Q2).
– It is cheaper to produce the two outputs jointly
instead of separately.
• Example:
– It is cheaper for Citycell 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 2 is produced:
DMC1Q1,Q2) /DQ2 < 0.
• Example:
– Bread and biscuits
Quadratic Multi-Product Cost
Function
•
•
•
•
•
C(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
C(Q1 ,0) + C(0, Q2 ) = f + (Q1 )2 + f + (Q2)2
C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
f > aQ1Q2: Joint production is cheaper
A Numerical Example:
• C(Q1, Q2) = 100 – 1/2Q1Q2 + (Q1 )2 +
(Q2 )2
• Cost Complementarity?
Yes, since a = -2 < 0
MC1(Q1, Q2) = -2Q2 + 2Q1
• Economies of Scope?
Yes, since 90 > -2Q1Q2
Conclusion
• To maximize profits (minimize costs)
managers must use inputs such that the
value of marginal of each input reflects price
the firm must pay to employ the input.
• The optimal mix of inputs is achieved when
the MRTSKL = (w/r).
• Cost functions are the foundation for helping
to determine profit-maximizing behavior in
future chapters.
Mid1 Answers
1.
2.
3.
4.
5.
6.
(a) p5; (b) p21; © p56; (d) p13; (e) p90; (f) p79.
(a)p46; (b) p53(price floor, surplus).
(a)p92; (b) -0.793; © -0.91(inelastic demand);
0.156(substitute, inelastic); 0.32(normal good,
inelastic);0.13(positively related, inelastic).
(a) Px = 115 – ¼ Qxd; (b) 12,800; © 16,200; (d)
Consumer surplus decreases.
(a) p=300; (b) p=200; Q=1000; © p=225; Q=1250.
(a) No, demand is inelastic, revenue will fall; (b)
Income must rise by 22.22%; © Sale of shoes will
increase by 2.25%.