Business Economics
Theory of the Firm II
Production and Cost in the Long Run
Two or more variable input factors
Production Function in the Long-Run
Production function: Again, describes the maximum
output that can be obtained with any combination of
inputs, given a specific technology,
but now (at least) two inputs are variable
Thomas & Maurice, Chapter 9
Herbert Stocker
[email protected]
Institute of International Studies
University of Ramkhamhaeng
&
Department of Economics
University of Innsbruck
Long Run Production Function
Input Substitution:
Labor-intensive method: process that uses large
amounts of labor relative to other inputs.
Capital-intensive method: process that uses large
amounts of capital equipment relative to other
inputs.
Q = Q(L, K )
→ Different combinations of inputs can produce the
same output
→ Input substitution: One input can be substituted
for another
Description of Technology
The Production Function again describes the
maximum output that can be obtained with any
combination of inputs. This can be shown as a table
↓
L
↓
0
1
2
3
4
5
6
7
8
9
10
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1.00
1.23
1.39
1.52
1.62
1.71
1.79
1.87
1.93
2.00
2
0
1.62
2.00
2.26
2.46
2.63
2.78
2.91
3.03
3.14
3.24
3
0
2.16
2.66
3.00
3.27
3.50
3.69
3.87
4.03
4.17
4.31
→ K →
4
5
0
0
2.64
3.09
3.25
3.80
3.67
4.29
4.00
4.68
4.28
5.00
4.52
5.28
4.73
5.53
4.92
5.76
5.10
5.96
5.27
6.16
6
0
3.51
4.32
4.87
5.31
5.68
6.00
6.28
6.54
6.78
6.99
7
0
3.90
4.81
5.43
5.92
6.33
6.68
7.00
7.29
7.55
7.79
or more simply as a function Q = Q(K , L)
8
0
4.29
5.28
5.96
6.50
6.95
7.34
7.69
8.00
8.29
8.55
9
0
4.66
5.73
6.47
7.06
7.55
7.97
8.35
8.69
9.00
9.29
10
0
5.01
6.17
6.97
7.60
8.12
8.58
8.99
9.35
9.69
10.0
Isoquants
Cobb-Douglas Production Function
Isoquants
Input substitution can easily be modeled and
illustrated with isoquants.
Isoquants: show all combinations of inputs that
produce a constant level of output.
Product curves
for K
2
6
4
2
x2
4
x1
6
→
10
8
6
4
2
10
8
Productionfunction
8
10
4
Q1Labor
0
2
6
(L)
→
8
10
Isoquants
0
Isoquants and Factor Substitution
K
L
y
Q2
al
(K
)
10
8
y 6
4
2
0
0
Ca
pit
Isoquants (constant output)
correspond to indifference
curves (constant utility) in the
theory of the household.
They are just like contour lines
on a map!
10
8
6
4
2
0
0
→
Output (Q)
U
10
8
6
4
2
0
0
Q0
higher
output
K USA
10
bc
8
6
4
2
x2
4
x1
2
6
K Africa
8
10
0
K
bc
LUSA
LAfrica
L
The same output (e.g. 100 km road within one year)
can be produced with different factor intensities.
Technology
Well-behaved Technologies:
Monotonic: more inputs produce more output.
Convex: sometimes averages produce more than
extremes.
We can’t take monotonic transformations (like
with utility functions) any more!
Marginal Product
Marginal Product: (MP1 )
MP1 is how much extra output you get from
increasing the input of factor 1, holding factor 2
fixed.
∂Q
∂Q
MPL ≡
,
MPK ≡
∂L
∂K
Diminishing marginal product:
more and more of a single input produces more
output, but at a decreasing rate
→ Law of diminishing returns
→ Important property of almost all technologies!
Properties of Technology
1
2
3
Marginal Product
Marginal Rate of Substitution
Returns to Scale
Marginal Rate of Technical Substitution
The Marginal Rate of Technical Substitution
(MRTS) is a measure for how easily input factors
can be substituted, holding output constant.
The MRTS is the slope of an isoquant
MRTS = −
∆K
∆L
The minus sign is added to make MRTS a positive
number, since ∆K /∆L, the slope of an isoquant, is
negative.
it shows how many units of K are necessary to replace
one unit of L when output is kept constant.
Marginal Rate of Technical Substitution
The MRTS can also be expressed as the ratio of
two marginal products:
Property of MRTS
Isoquant becomes flatter, the further we move
along an axis
⇒ MRTS decreases the more intensively a factor is
used in production: As one input is substituted for
another one, the marginal product of this input
diminishes
∆K
MPL
MRTS = −
=
∆L
MPK
Why?
1
2
Diminishing MRTS
Output is kept constant along an isoquant, therefore
∆Q = 0
Total derivative of the production function yields
Why?
∆Q = MPL ∆L + MPK ∆K
3
Notice: Difference between diminishing MP and
diminishing MRTS
Combining 1 and 2 gives
∆Q = 0 = MPL ∆L + MPK ∆K ⇒ −
∆K
MPL
=
∆L
MPK
MRTS
Special Cases
Infinitesimal
changes:
K
Diskrete changes:
K
small
bc
Fixed-Proportions: inputs cannot be substituted;
e.g. Q = min{K , L}
MRTS is zero except at the edge, where it is not
defined.
K
∆K
bc
∆K
bc
∆K
∆L
∆L
∆L
L
Q
10
8
6
4
2
0
0
10
8
6
4
2
L
K
4
L
2
6
8
10
0
Attention: not differentiable!
L
Special Cases
Returns to Scale
Perfect Substitutes: inputs can perfectly be
substituted; e.g. Q = K + L
MRTS is constant!
Returns to Scale show what happens with
output, if all inputs are doubled.
?
K
Q
10
8
6
4
2
0
0
Q [(2L), (2K )] T 2Q
10
Notice difference to Marginal Product (MP): shows
what happens with output if only one factor of
production increases while all other factors are held
fixed.
8
6
4
2
K
4
L
2
6
8
10
0
Attention: Corner solution!
L
Returns to Scale
Constant Returns to Scale: if all inputs are
doubled output doubles too, i.e.
Returns to Scale
Constant Returns to Scale: Q = L0,6 K 0,4
Q [(2L), (2K )] = 2Q
Decreasing Returns to Scale: if all inputs are
doubled output less than doubles, i.e.
Q [(2L), (2K )] < 2Q
For L = 5 and K = 5:
Q(5, 5) = 50,6 50,4 = 5
10
8
Q 6
4
2
0
0
10
8
4
2
Increasing Returns to Scale: if all inputs are
doubled output more than doubles, i.e.
Q [(2L), (2K )] > 2Q
For L = 10 and K = 10:
6
K
4
L
2
6
8
10
0
Isoquants for Q = 1, 2, 3, . . .
Q(10, 10) = 100,6 100,4 = 10
Returns to Scale
Decreasing: Q = L0,4 K 0,2 Increasing: Q = L0,8 K 0,4
Q
10
8
6
4
2
0
0
10
8
6
4
2
K
4
L
Q
10
8
6
4
2
0
0
10
8
6
4
2
K
4
2
6
L
8
10
Q( 5, 5) = 50,4 50,2 ≈ 2, 6
Q(10, 10) = 100,4 100,2 ≈ 4
2
6
0
8
10
Modeling
decision making
of firms:
Optimal Factor Allocation
0
Q( 5, 5) = 50,8 50,4 ≈ 6, 9
Q(10, 10) = 100,8 100,4 ≈ 16
Isoquants for Q = 1, 2, 3, . . .
Theory of the Firm
Profit is the difference between total revenue and
total (opportunity) cost
Total revenue is determined by the output, which
is sold on markets
⇒ Optimal behavior of firms depends on the market
structure where output is sold
A necessary condition for maximum profit is that
cost are minimal
⇒ Minimum cost can be derived without knowing
the market structure!
Theory of the Firm
Therefore, a firm’s decision making process can be
modeled by artificially splitting it up into two steps:
1
Minimize cost for any output, given technological
restrictions (production function)
⇒ Cost function
2
Maximize profits for a given market structure
(market restrictions)
⇒ Supply function
Notice: Maximizing output for a given amount of cost
delivers the same result as cost minimization!
E.g., nonprofit firms (fixed budget)
Different Perspectives of a Firm
Firms’ Choice
Human resources manager:
Ma’m & Pa:
Technological
restrictions
Cost
C = wL + rK
Optimization
(Minimization)
Economist:
Accountant:
P
MC
AC
AVC
Decisions
bc
PA
PB
A
(Factor Demand &
Cost Function)
bc
B
bc
AFC
bc
QB QA
Q
Cost Minimization
Objective: Minimize cost
⇒ Isocost Curves
C = wL + rK
Constraint: Technological Restriction
⇒ Production function
Q = Q(L, K )
Cost
Firms have to pay for their inputs.
Total costs are defined as the sum of cost for all
inputs. For simplification we restrict ourselves to
two inputs only.
Isocostcurve:
C = wL + rK
Solving for K
C
w
K= − L
r
r
each point on this curve is associated with the
same cost!
Isocostcurve
Cost Minimization
Isocostcurve:
Graphical Solution
K
C
r
C = wL + rK
K = Cr − wr L
∆L
− ∆K
∆L =
w
r
∆K
K=
w
C
− L
r
r
Slope:
dK
w
=−
dL
r
Intercept: for L = 0
L
K |L=0
Consider a market with w = 10 Euro per working
hour (wh) and r = 5 Euro per machine hour (mh)
10 Euro
wh
5 Euro
mh
=
dK
dL
C
r
C∗
r
2 mh
wh
In the market, 2 machine hours cost the same as
one working hour
Further, we assume that a firm produces with
∆K
MRTS = −
=3
∆L
In production, 3 machine hours can be replaced by
one working hour without changing output
=
C
=
K
=
wL + rK
w
C
− L
r
r
− wr
What is the cheapest way to produce the given quantity Q0 ?
Q0
⇒ where the intercept is minimal
and production of Q0 still possible!
⇒ That’s where the isocostline
and the isoquant have the same
slope!
bc
L
C
=
r
Optimal Factor Input: Example
w
=
r
K
Optimal Factor Input: Example
We obtain
2 mh 3 mh
∆K
w
=
<
=−
r
wh
wh
∆L
When the firm uses 3 mh less she saves 15 Euro.
Using one additional unit of labor (wh) leaves the
output unchanged, but this wh costs only 10
Euro, therefore the firm could have saved 5 Euro!
In such a situation, the firm would demand less
capital (mh) and more labor (wh)!
Optimal Factor Input
Cost Minimization with Lagrange
This ‘arbitrage’ works until
∆K
w
=−
≡ MRTS
r
∆L
(1)
Analytical solution with the Lagrange method:
Objective:
min C = wL + rK
K ,L
(Technological) Restriction
Interpretation: Factors are substituted until the
‘exchange ratio in production’ (= MRTS) is equal
to the ‘exchange ratio in the market’ (= relative
factor prices)!
⇒ Eq.(1) defines the optimal factor input ratio
Cost Minimization
Q0 = Q(L, K )
Decision variables: L, K
(Market restriction:
perfect competition ⇒ prices are given!)
Cost Minimization: Lagrange
Optimization Cobb-Douglas production function and
isocostcurve
Problem:
x2
10
Isoquants
8
6
min C = wL + rK
4
Production function
2
L,K
Output (Q)
0
10
MRTS =
dK
dL
=
6
A
4
2
←
0
Ca
pi
ta
l(
K
10
8
6
4
)
2
0
)→
Labor (L
x1
s.t.: Q0 = Q(L, K )
8
− wr
Isocost
‘wall’
Lagrange–Function L:
L = wL + rK + λ [Q0 − Q(L, K )]
Cost Minimization: Lagrange
Setting partial derivatives equal to zero:
∂Q !
∂L
= w −λ
=0
∂L
∂L
∂Q !
∂L
= r −λ
=0
∂K
∂K
∂L
!
= Q0 − Q(L, K ) = 0
∂λ
The solutions of these equations
L∗ = L(w , r , Q0 ), K ∗ = K (w , r , Q0 )
define the factor demand functions!.
Cost Minimization
∂Q
∂Q
dL +
dK
∂L
∂K
rewritting
−
dK
=
dL
Dividing the first two equations yields
w
=
r
∂Q
∂L
∂Q
∂K
≡
MPL
MPK
Re-writting gives
MPL
MPK
=
w
r
Efficient production: Marginal products of labor
and capital per price unit are identical!
Cost Minimization
On a isoquante output is constant, therefore the
change in output (dQ) is zero
dQ = 0 =
Cost Minimization: Lagrange
∂Q
∂L
∂Q
∂K
=
(total derivative)
MPL
MPK
Therefore
MPL
dK
w
=
= MRTS
=−
r
MPK
dL
Expansion Path
Short- and Long-Run Expansion Path
Optimality requires:
MRTS = factor price ratio
K
K
Expansionpath
Slope:
bc
bc
bc
bc
dK dL dQ=0
Slope: − wr dC =0
L
→ Curve along which a firm
expands output in the
long-run
bc
K̄
bc
short run
Expansionpath
Q2
bc
Q1
L
→ Indicates how input usage
changes when output or
cost changes
Restructuring Short-Run Costs
Because managers have greatest flexibility to
choose inputs in the long run, costs are lower in
the long run than in the short run for all output
levels except that for which the fixed input is at
its optimal level
Short-run costs can be reduced by adjusting fixed
inputs to their optimal long-run levels when the
opportunity arises
Assume the firm wants to
expand production from
Q1 to Q2 .
long run
Expansionpath
Expansion path: Each
point represents an
efficient (least-cost) input
combination for each
level of output at
constant input prices
In the short run the firm
can only get additional labor to expand production.
In the long run she can
substitute labor for capital
to reduce cost!
Factor Demand
To derive the factor demand functions (the
function that shows how demand for an input
depends on the prices of inputs and on output) we
have to consider input substitution possibilities,
e.g.
L∗ = L(w , r , Q)
Input substitution depends mainly on
Technology, i.e. the shape of the isoquant.
Prices of Inputs.
The following graph shows how this analysis can
be applied to derive the factor demand equation.
Factor Demand
K
4
Factor Demand:
3
C = wL + rK
√
min
bc
2
L,K
bc
1
bc
0
0
1
2
3
4
Q0 = Q(L, K ) =
s.t.:
Q0 = 1
bc
LK
MRTS: (assumption: r = 1)
L
w
K ! w
dK
=
= 4; = 1; = 0, 25; = 0, 11̇
=
dL
L
r
p
√
Lösung: Q0 = LK = L[(w /r )L]
bc
4
2
L∗ = w −0,5 r 0,5 Q0
bc
bc
0
0
The cheapest way to
produce a given Output
−
3
1
Long-Run Cost Functions
1
2
L∗ = w −0,5 r 0,5 Q0
bc
3
4
L
Long-Run Cost Functions: Recap . . .
Cost minimization problem:
min C = wL + rK
K ,L
s.t. Q = Q(L, K )
Geometric solution: slope of isoquant equals slope
of isocost curve w /r = MP1 /MP2
These optimal choices of factors are the
conditional factor demand functions:
L∗ = L(w , r , Q)
K ∗ = K (w , r , Q)
Long-Run Cost Functions
Inserting the conditional factor demand functions
in the equation of the isocostcurve gives the
Long-Run Cost Function (LTC):
C ∗ = wL∗ (w , r , Q) + rK ∗ (w , r , Q) = C (w , r , Q)
The Long-Run Cost Function shows for given
factor prices the cheapest possibility to produce
any given output.
Its shape is exclusively determined by the
technology and input prices!
Long-Run Average and Marginal Cost
(Dis-)Economies of Scale
Long-run average total cost
LAC(Q; w , r ) =
LTC
Q
→ Cost of a typical output unit
Long-run marginal cost
LMC(Q; w , r ) =
∂LTC
∂Q
→ Cost of an additional output unit
Properties of cost functions
Economies of scale: Long-run average total
cost (LAC) decline as output increases
⇒ Happens when returns to scale are increasing
Diseconomies of scale: Long-run average total
cost (LAC) increase as output increases
⇒ Happens when returns to scale are decreasing
Why?
As in the short-run, the LMC-curve intersects the
LAC-curve in it’s minimum
(Dis-)Economies of Scale
Economies of scale and returns to scale
Doubling all inputs also doubles LTC
However: If returns to scale are increasing
(decreasing), output more (less) than doubles
⇒ LAC = LTC/Q decrease (increase)!
Constant returns to scale are associated with
constant economies of scale, i.e., the LAC-curve is
horizontal
Be careful: Economies of scale applies to the cost
function, returns to scale to the production function
LAC-Curve and Economies of Scale
AC
(Approx.)
Constant
returns
to scale
Economies
of Scale
Diseconomies
of Scale
LAC
bc
MES
Q
Minimum Efficient Scale (MES): Point where
economies of scale are exhausted
Occurrence of (Dis-)Economies of Scale
Economies of scale: Expansion of scale usually
allows ...
to exploit advantages from specialization and
division labor,
to reduce the cost of acquiring and establishing
machines, and
to introduce automation devices
Diseconomies of scale
Information asymmetries between management
and employees typically increase with firm size
⇒ Cost (and unit cost) of management increases
Market Entry Barriers
AC
bc
AC
High
market
entry
barriers
bc
bc
bc
MES
MES
1 MES
2
MES
1 MES
2
Q
MES
C (QX ) + C (QY ) > C (QX , QY )
Reason: Inputs can be jointly used to produce goods
Overheads (e.g., R&D, marketing)
Production of one good results in the production
of another one with little or no extra cost (e.g.,
beef and leather)
Q
Rule of thumb: Average cost at half of the MES
indicates technological market entry barriers
⇒ Important for intensity of competition, M&A . . .
Economies to Scope
Economies to Scope: Exist for a multi-product firm
when producing two or more goods jointly is less
expensive than producing them separately:
Low
market
entry
barriers
Example:
Short- medium- and
long run cost curves
Short- vs. long run
Example
in the short run only few factors are variable, e.g.
labor.
in the medium run more factors are variable, e.g.
capital.
in the long run all factors are variable, e.g.
buildings.
Example
Example with Cobb–Douglas Production function:
1
1
1
Q = L4 K 2 G 4
Output: Q;
Inputs: L, K and G .
Short-run: only L is variable, K and G are fixed,
Medumium-run: L and K are variable, G is still fixed,
Long-run: L, K and G are variable.
Short-run Cost Function
Q = L0.25 K 0.5 G 0.25
min wL + r K̄ + mḠ
L
Returns to scale:
0.25
0.5
0.25
0.25
Short-run:
(tL) K̄ Ḡ
=t Q
⇒ decreasing returns to scale in L
Medium-run:
(tL)0.25 (tK )0.5 Ḡ 0.25 = t 0.75 Q
⇒ decreasing returns to scale in L and K
Long-run:
(tL)0.25 (tK )0.5 (tG )0.25 = t 1 Q
⇒ constant returns to scale in L, K and G
Inserting L∗ =
isocostcurve:
s.t.
1
1
1
Q = L 4 K̄ 2 Ḡ 4
4
in the equation of the
1
Q
1
K̄ 2 Ḡ 4
Q4
C k = wL∗ + r K̄ + mḠ = w 2
+ r|K̄ +
{zmḠ}
| K̄
{z Ḡ}
Fixed cost
Variable C.
Short-run MC and AC
AC
MC
20
ACk1
(K = 5)
MCk1
15
Medium-run Cost Curves
In the medium-run the firm can optimize the input of L
and K .
ACk2
(K = 10)
MCk2 MCk3
MCk4
L,K
ACk4
s.t.
(K = 20)
5
0
min wL + rK + mḠ
ACk3
(K = 15)
10
0
2
4
6
8 10 12 14 16 18 Q
Medium-run Cost Curves
Solution: conditional factor demands:
4
2
2 2
1
L∗ = 2− 3 w − 3 r 3 Ḡ − 3 Q 3
1 1 1
4
1
K ∗ = 2 3 w 3 r − 3 Ḡ − 3 Q 3
Inserting these in the isocostequation gives the
following medium-run cost function:
∗
∗
C m∗ = wL
+ rK + mḠ
1
1 2
1
4
3
=
2 3 w 3 r 3 Ḡ − 3 Q 3 + mḠ
2
1
1
1
Q = L 4 K 2 Ḡ 4
this can be solved e.g. by the Langrange method:
h
i
1
1
1
Lm = wL + rK + mḠ + λ Q − L 4 K 2 Ḡ 4
Short- and medium-run AC
AC
MC
20
ACk1
(K = 5)
ACk3
15
(K = 15)
ACk4
10
(K = 20)
medium-run ACm
(optimal L & K ,
G = 5)
5
0
ACk2
(K = 10)
0
2
4
6
8 10 12 14 16 18 Q
Short- and medium-run AC
AC
12
11
10
9
8
7
6
Medium-run AC and MC
ACk2
Different firm sizes
MC
12
ACk1
ACm
1
ACk3
ACk4
(G = 5)
ACm
1
11
(G = 5)
MCm
1
10
Medium-run
average
cost funcion
for G = 5
(scales changed!)
MCm
2
9
MCm
3
ACm
2
(G = 10)
ACm
3
8
(G = 15)
7
0
5
10 15 20 25 30 35 40 45
Q
Long-run Cost Functions
min wL + rK + mG
s.t.
Q=L K G
5
AC MC
12
(G = 5)
ACm
2
(G = 10)
ACm
3
ACl = MCl
long-run averageand marginal cost function
7
C
∗
= wL + rK + mG
3
1 2
1
= 22 w 4 r 4 m4 Q
ACm
0
ACm
1
8
∗
Q
(G = 1)
11
9
1
4
Solution:
l∗
10 15 20 25 30 35 40 45
10
L,K ,G
1
2
0
Medium- and long-run AC and MC
In the long-run all inputs are variable!
Additionally we can suspect, that most firms produce
with approximately constant returns to scale.
1
4
6
∗
6
0
5
(G = 15)
10 15 20 25 30 35 40 45
Q
Long Run Average Cost Curve
Long Run Average Cost Curve
Though the Long Run Average Cost Curve is
much flatter than the Short Run Average Cost
Curve it might still be U-shaped, because
it is an increasing cost industry (decreasing returns to
scale).
factors are scarce and become more expensive with
increasing demand.
firms are differently efficient or own unique factors.
In this case market price will reflect the minimum
average cost of the marginal supplier.
Any questions?
Thanks!
Other Factors Influencing the Long Run
Average Cost Curve:
Learning by doing: reflects drop in unit costs as total
cumulative production increases because workers
become more efficient as they learn their tasks.
Transportation costs: if these increase with large-scale
production, the LRAC has lower optimum scale of
operation than a curve without these costs.
Factor prices might increase with increasing demand.