Transcript Document 7416244

The Production Process and Costs
Production Analysis
• Production Function Q = f(K,L)
• Describes available technology and feasible means of
converting inputs into maximum level of output,
assuming efficient utilization of inputs:
• ensure firm operates on production function (incentives for
workers to put max effort)
• use cost minimizing input mix
• Short-Run vs. Long-Run (increases with capital intensity)
• Fixed vs. Variable Inputs
Total Product
• 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
Marginal Product of Labor
• Continuous case: MPL = dQ/dL
• Discrete case:
arc MPL = DQ/DL
• Measures the output produced by the last
worker.
• Slope of the production function
Average Product of Labor
• APL = Q/L
• Measures the output of an “average”
worker.
• Slope of the line from origin onto the
production function
Law of Diminishing Returns (MPs)
Three significant points are:
Max MPL (TP inflects)
Max APL = MPL
MPL = 0 (Max TP)
25
20
Total Product
TP increases at an increasing
rate (MP > 0 and ) until
inflection , continues to
increase at a diminishing rate
(MP > 0 but ) until max and
then decreases (MP < 0).
15
10
5
04
3
0
2
4
6
8
10
12
8
10
12
Input L
2
1
0
A line from the origin is tangent
to Total Product curve at the
maximum average product.
-1 0
2
4
6
-2
-3
-4
Increasing
MP
Diminishing
MP
Negative
MP
Optimal Level of Inputs
Q
Demand
for labor
Marginal Principle:
continue to hire as long as marginal
benefit > marginal cost of the input,
stop when MB = MC.
PL
L*
L
VMPL
Solve the following equation for L*:
MBL = VMPL = MPL*PQ = PL = MCL
MC (Pinput, monetary units):
cost of hiring the last unit of input.
MB (MPinput, physical units):
contribution of the last unit of input
hired to the total product.
MB (VMPinput, monetary units):
value of the output produced by the
last unit of input = MPinput * Poutput .
Downward sloping portion of the
VMP curve is the demand for input.
The Long Run
Production Function:
Q = 10K1/2L1/2
Isoquants and the Production Surface
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 or Marginal Rate of Technical Substitution can be
derived using total differential of Q=f(K,L) set equal to
zero (no change in Q along an isoquant)
Q
0
Q
Q
DK
L   MPL
DK 
DL 

Q
K
L
DL
MPK
K
Cobb-Douglas Production Function
• Q = KaLb
• Inputs are not perfectly
substitutable (slope changes
along the isoquant)
DK
a L
for b  1  a,

DL
1 a K
• Diminishing MRTS: slope
becomes flatter
• Most production processes
have isoquants of this shape
• Output requires both inputs
K
Q3
Q2
Q1
Increasing
Output
-DK1
||
-DK2
DL1 < DL2
L
Linear Production Function
• Q = aK + bL
• Capital and labor are
perfect substitutes
(slope of isoquant is
constant)
y = ax + b
K = Q/a - (b/a)L
• Output can be
produced using only
one input
K
Increasing
Output
Q1
Q2
Q3
L
Leontief Production Function
• Q = min{aK, bL}
• Capital and labor are
perfect complements and
cannot be substituted (no
MRTS <=> no slope)
• Capital and labor are
used in fixed-proportions
• Both inputs needed to
produce output
Q3
K
Q2
Q1
Increasing
Output
Isocost
• The combinations of
inputs that cost the same
amount of money
C = K*PK + L*PL
• For given input prices,
isocosts farther from the
origin are associated with
higher costs.
• Changes in input prices
change the slope (Market
Rate of Substitution) of the
isocost line
K = C/PK - (PL/PK)L
K
New Isocost for
an increase in the
budget (total cost).
C0
K
C1
L
New Isocost for
a decrease in the
wage (labor price).
L
Long Run Cost Minimization
Min cost where isocost
is tangent to isoquant
(slopes are the same)
MRS KL  
PL
MPL

 MRTS KL
PK
MPK
Expressed differently:
MP (benefit) per dollar
spent (cost) must be
equal for all inputs
MPL MPK

PL
PK
K
-PL/PK < -MPL/MPK
MPK/PK< MPL/PL
Point of Cost
Minimization
-PL/PK = -MPL/MPK
MPK/PK= MPL/PL
-PL/PK > -MPL/MPK
MPK/PK> MPL/PL
Q
L
Returns to Scale
• Return (MP): How TP changes when one input increases
• RTS: How TP changes when all inputs increase by the
same multiple λ > 0
• Q = f(K, L)

Increasing
• If f( K, L)  Q  Constant Returns to Scale

Decreasing
• Q = 50K½L½
Q = 100,000 + 500L + 100K
Q = 0.01K3 + 4K2L + L2K + 0.0001L3
Expansion path and Long-Run Total Cost
K*PK + L*PL =
Long-Run Total Cost is the least cost combination of inputs for each
production quantity (derives from the expansion path)
LTC = 10Q-.6Q2+.02Q3
LTC
LAC 
Q
d ( LTC )
LMC 
dQ
DLTC
arc LMC 
DQ
LTC 2  LTC 1

Q 2  Q1
Effect of a Fixed Input on Cost of Production
In the short run
K is fixed at K0.
 Any input L other
than L0 will result in
other than least TC.
 If I1 is required,
input L will be
reduced to point E,
associated with TC
much higher than
optimal at point A.

LTC as a Lower Envelope of STC
• Every point on LTC
represents a least-cost
combination.
• In the short run one or
more inputs are fixed so
that only a single point
on STC is a least-cost
combination of inputs.
• STC curves intersect
cost axis at the value
of the TFC.
STC = TFC + TVC
= 1000+80Q-6Q2+.2Q3
SAC = STC / Q
= TFC/Q + TVC/Q
= AFC + AVC
AFC = 1000/Q
AVC = 80-6Q+.2Q2
SMC = dSTC/dQ
= dTFC/dQ + dTVC/dQ
= dTVC/dQ
= 80-12Q+.6Q2
Productivity of Variable Input and Short-Run Cost
= Q = f(L)
Short-Run Total Cost, Total Variable Cost & Total Fixed Cost
= TFC + TVC
= PL * L
= PK * K
Average Product and Average Variable Cost
Marginal Product and Short-Run Marginal Cost
LAC as a Lower Envelope of SAC
• In the long run all
total costs represent
least-costs.
• All average costs
must be least cost
as well.
• Various short-run
cost curves for
various values of
the fixed input.
• In the short run only
one point represents
least cost.
Economies
of Scale
Diseconomies
of Scale
Economies of scale (minimum SAC of in the
smaller facility greater than SAC in the larger
facility) exist up to the minimum LAC
(downward sloping portion of LAC curve).
Beyond minimum LAC diseconomies of scale.
Long-Run Average Cost and Returns to Scale
Economies
of Scale
Diseconomies
of Scale
Increasing Returns to Scale:
Q1 = f(K = 20, L = 10) = 100
Economies of Scale:
PK = 20, PL = 50
LTC1 = 20*20 + 50*10 = 900
LAC1 = 900 / 100 = 9
Q2 = f(K = 40, L = 20) = 300 > 2Q2
LTC2 = 20*40 + 50*20 = 1,800
LAC2 = 1,800 / 300 = 6 < LAC1
Economies of Scope
and Cost Complementarity
• Cheaper to produce outputs jointly than separately:
C(Q1, Q2) < C(Q1, 0) + C(0, Q2)
• MC of producing good 1 declines as more of good 2 is produced:
MC1 / Q2 < 0
• Example: Joint processing of deposit accounts and loans in banks
Scope: Single financial advisor eliminates duplicate common
factors of production (computers, loan production offices)
Complementarity: Account and credit information developed
for deposits lowers credit check and monitoring cost for loans.
Expansion of deposit base lowers cost of providing loans.
Quadratic Multi-Product Cost
Function
• C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2
MCi(Qi, Qj) = aQj + 2Qi
• Economies of scope (cheaper joint product) if :
f > aQ1Q2
C(Q1, 0) + C(0, Q2 ) = f + (Q1)2 + f + (Q2)2
• Cost complementarity exists if: a < 0
MCi/ Qj = a < 0
A Numerical Example:
• C(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2
MC1(Q1, Q2) = -2Q2 + 2Q1
• Economies of Scope?
Yes, since 90 > -2Q1Q2
• Cost Complementarity?
Yes, since a = -2 < 0
• Implications for Merger?