Concepts from the Theory of the Firm:
Kirschen/Strbac Chapter 2 (Parts 2.3 & 2.5)
Important Acknowledgement:
These notes
Th
t on Kirschen/Strbac
Ki h /St b (Ch
(Chapter
t S
Sections
ti
2
2.3
3
and 2.5, 2004) are based on slides prepared by Daniel
Kirschen (U Manchester) with substantial edits by Leigh
T f t i (Iowa
Tesfatsion
(I
State
St t U).
U)
y 2010
Last Revised: 26 January
© 2006 Daniel Kirschen
1
Short-Run Cost: Accountant’s Perspective
• Some production costs vary with
the level of production and some
d not:
do
t
Variable cost:
Total Cost [$]
ƒ Labor,, fuel,, transport…
p
Fixed cost (amortised):
ƒ Equipment, land, overhead
• S
Some production
d ti costs
t are sunk
k
and some are avoidable
Sunk ((Not Avoidable))
ƒ Loan obligations, special assets with no
resale value,…
Avoidable
0
Quantity
ƒ GenCo “start-up” costs (if GenCo self
commits), labor, fuel, transport, …
© 2006 Daniel Kirschen
2
Two Different Decompositions for Total Cost
‰ Let T denote any given production planning period.
‰ Consider a firm at the beginning of period T making its
production plans for period T.
‰ The period-T total cost of production for this firm can be
decomposed in two different ways:
• Total Cost
= Fixed Cost + Variable Cost
• Total Cost
= Sunk Cost + Avoidable Cost
© 2006 Daniel Kirschen
3
Two Different Decompositions for Total Cost at the
Beginning
g
g of Planning
g Period T … Continued
•
Does NOT vary with changes in the firm’s output
Fixed Cost:
level y over period T
•
Variable Cost: VARIES with changes in y over period T
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
•
Sunk Cost:
Payment obligations that the firm has irrevocably
committed to at the beginning of period T, hence
costs that CANNOT BE RECOVERED.
•
Avoidable Cost: Payments that CAN BE RECOVERED by the firm at
the beginning of period T because the firm has not
committed to making them,
them or because they involve
costs for purchased assets that can be recovered
through asset resale or alternative asset use.
© 2006 Daniel Kirschen
4
EXAMPLE: A Farmer’s Costs for 2010
•
•
A corn farmer takes out a loan on 1/1/10 to finance a used tractor worth
$138,000. As down payment for this loan, he gives the bank $28,000.
g to make a $
$1,000
,
p
payment
y
on this loan at the
He is then also obliged
end of every month for the next 10 years, i.e., 2010 - 2019.
The farmer’s labor and other operating expenses incurred if he produces
corn in amount y during 2010 are given by h(y) = ay + by2 .
•
If the farmer goes out of business during 2010, the bank keeps the down
payment, takes ownership of the tractor, and retires the loan (i.e., the
farmer is not obliged to make any additional loan payments).
•
Total Cost for 2010 calculated on 1/2/10: $40,000 + h(y)
Fixed Cost:
Variable Cost:
$40,000 = $28,000 (down payment) + $12,000 (loan payments)
h(y)
Sunk Cost:
$28,000
Avoidable Cost:
[ $12,000 + h(y) ]
© 2006 Daniel Kirschen
= [ Avoidable Fixed Cost + Variable Cost ]
5
Total cost decomposition for a planner at the
beginning of a particular planning period T:
Source: X. H. Wang & B. Z. Yang, “Fixed and Sunk Costs Revisited,”
Journal of Economic Education
Education, Spring 2001
2001, pp
pp. 178
178-185
185
© 2006 Daniel Kirschen
6
Total Cost Decomposition … Continued
Source: X. H. Wang
g & B. Z. Yang,
g, “Fixed and Sunk Costs Revisited,”
,
Journal of Economic Education, Spring 2001, pp. 178-185
© 2006 Daniel Kirschen
7
Two Different Decompositions for Total Cost…Continued
•
•
In any given time period T:
1) Total
T t l Cost
C t
= Fixed
Fi d C
Costt + V
Variable
i bl C
Costt
2) Total Cost
= Sunk Cost + Avoidable Cost
Second way of decomposing cost is easier to understand & use.
Example: How to classify a “start
start up cost”
cost incurred by a GenCo
when it first starts up a generation unit in some period T?
The start-up cost varies with production level y from y=0 to y > 0 but
not thereafter.
thereafter This is an example of Kirschen
Kirschen’s
s “quasi-fixed
quasi fixed cost”
cost .
Easy answer under 2): It’s a sunk cost in period T if the GenCo is
obliged
g to start up
p the g
generation unit in p
period T. Otherwise it’s an
avoidable cost in period T.
© 2006 Daniel Kirschen
8
Costing of Durable-Benefit Expenditures Requires Care
•
How should producers cost out expenditures whose benefits last over
successive time periods?
ƒ
ƒ
Purchase of physical assets with long-term durability such as equipment, buildings,
and land
St t
Start-up
costs
t (once
(
started
t t d up, a generation
ti unitit can be
b run over successive
i h
hours
without incurring additional start-up costs)
•
Prior to such expenditures, there must be SOME production planning period
T over which full recovery of all avoidable cost is anticipated
anticipated, else the cost
should not be incurred.
•
Once incurred, such expenditures become either sunk (e.g., start-up costs) or
avoidable fixed costs (e
(e.g.,
g resale value of purchased equipment)
equipment).
•
When deciding whether or not to produce in the presence of avoidable fixed
costs, there must be SOME planning period T over which full recovery of all
avoidable costs is anticipated
anticipated, including all avoidable fixed costs
costs, else the firm
should shut down and act to reduce its avoidable costs to zero.
•
BOTTOM LINE: When durable-benefit expenditures are involved, planning
periods
i d mustt b
be suitably
it bl llong tto permit
it proper consideration
id ti off costt recovery.
© 2006 Daniel Kirschen
9
Determining Optimal Production for a Competitive (Price(Price-Taking)
Firm at the Beginning of a “Suitably Long” Planning Period T:
• DEFINE:
Net Earnings = [ Revenue – Avoidable Cost ]
• S
Step 1:
Find production level y* ≥ 0 that maximizes net earnings
for a g
given output
p p
price π and input
p p
prices w1, w2,,…
• Step 2:
If net earnings at y*
y are positive,
positive produce y*
y ;
Otherwise shut down (set y = 0) and take action to reduce
all avoidable cost to zero so that net earnings = 0 .
© 2006 Daniel Kirschen
10
Modern Take on Traditional Economic Distinction:
Long Run versus Short Run
• Some factors of production can be adjusted faster
than others
others.
ƒ Example: Fertilizer vs. planting more trees
• “Long Run”: All costs can be avoided
• “Short
Short Run”:
Run : At least some costs are sunk
• Additional assumption for K/S Chapter 2:
For
simplification
i lifi i and
d consistency
i
with
i h notation
i off Ki
Kirschen/Strbac,
h /S b
we will
ill
assume for now that all fixed costs are sunk costs! This implies
(a) Fixed cost = Sunk cost (i.e., there are NO avoidable fixed costs);
(b) Variable cost = Avoidable cost .
© 2006 Daniel Kirschen
11
Production function: A function giving maximum
possible output for each amount of inputs
y = f (x 1 , x 2 )
• y:
output
• x1 , x2:
inputs (or “factors of production”)
y
y
x2 fixed
x1
x1 fixed
x2
**Law of diminishing marginal product**
© 2006 Daniel Kirschen
12
Input-Output Function:
y = f (x 1 , x 2
)
x 2 fixed
The inverse of this relation (solving for x1 as a function of y,
all else equal) is called the input-output function for x1
x 1 = g ( y ) for x
2
= x2
Example: Minimum amount of fuel x1 required to
produce successively higher amounts of electric
power y,
y given a particular generation plant x2
© 2006 Daniel Kirschen
13
Short-Run Cost Function: CSR(y)
Given fixed input x2, (unit) prices w1, w2 for inputs x1, x2,, and
x1 = g(y) = input-output function for variable input x1, define:
cSR (y)=w1 ⋅x1 +w2 ⋅x 2 =w1 ⋅g(y)+w2 ⋅ x2
c SR ( y )
Variable
Cost VC
= Cv(y) + FC
Fixed
Cost FC
FC
© 2006 Daniel Kirschen
y
14
Short-Run Marginal Cost: SRMC(y)
c SR ( y )
CSR(y) is convex
due to the law
of diminishing
marginal product
y
dc SR ( y ) = SRMC(y)
dy
SRMC(y) = dCSR(y)/dy =
rate of change of CSR(y)
is a non-decreasing
function
y
© 2006 Daniel Kirschen
15
SR Average Cost SRAC(y), SR Average Variable Cost
SRAVC(y), and Average Fixed Cost AFC
CCSR
c( y
) = c v ( y ) + cFCf
= Variable Cost + Fixed Cost
( ith Cv(0) = 0)
(with
CSRc ( y )
c v ( y ) cFC
f
SRAVC(y)
AFC( y )
AC (y)
( y )==
=
+
= AVC
( y ) ++AFC
SRAC
y
CSR(y) [$]
y
[$]
y
SRAC(y) [$/unit]
FC
0
© 2006 Daniel Kirschen
Quantity y
Quantity y
16
Relation between SR Marginal Cost and SR Average Cost
N t dSRAC(
Note:
dSRAC(y)/dy
)/d = [ SRMC(
SRMC(y)) – SRAC(y)
SRAC( ) ] / y
SRMC(y)
SRAC(y)
$/unit
0
© 2006 Daniel Kirschen
Q
Quantity
tit y
17
A Fuller Depiction
p
of SR Cost Relationships:
p Two Examples
p
b
f
© 2006 Daniel Kirschen
18
Optimal periodperiod-T production for a competitive (price(price- taking) firm with
short--run cost function CSR(y) = Cv(y) + FC when
short
all of its fixed cost FC is sunk:
• Define: For given output price π and input prices w1, w2,…
Profit (y) = [ Revenues at y – Total Short-Run Costs at y]
= [ π y – CSR(y) ] = [ π y - Cv(y) - FC ]
NetEarnings
g (y) = [ Revenue at y – Avoidable Cost at y ]
= [ π y - Cv(y) ]
‘ Since “sunk
sunk costs are sunk
sunk,”
the firm’s proper focus should
be on NET EARNINGS, not on profit!
© 2006 Daniel Kirschen
Special K/S Chapter 2
assumption made here that
ALL fixed costs FC are
sunk,, so there are no
avoidable fixed costs!
19
Optimal production…Continued
• For given output price π and given input prices w1, w2,…
NetEarnings(y) = [ Revenue at y – Avoidable Cost at y ]
= [ π y - Cv(y) ]
• Step 1:
Recall Cv(0) = 0 !
Find output level y* that maximizes NetEarnings(y) over all y
≥ 0, where y = 0 corresponds to “shut down”
• Step 2:
If y* > 0 , produce y* and attain NetEarnings(y*) ≥ 0 .
If y** = 0 , shut
h td
down and
d attain
tt i N
NetEarnings(0)
tE i
(0) = 0
© 2006 Daniel Kirschen
20
Additional Details About Step 1:
• CAUTION: The following is a necessary condition for
optimal y* ONLY if y* > 0:
max {π ⋅ y − c SR
( y )}
v
y
d {π ⋅ y − c vSR ( y ) }
dy
=0
dc SR
v ( y ) = SRMC(y)
π=
d
dy
© 2006 Daniel Kirschen
Equality here only holds
if firm perceives that
changes in its output y
have no effect on the
market
k t price
i for
f y
Competitive
producer that perceives
no way to affect market
prices to its advantage
through changes in its
supply offer
21
Additional details about Step 1…Continued
•
SRMC = dCSR(y)/dy = dCv(y)/dy = SR cost of producing 1 more unit
•
If SRMC < π , the next unit costs less than it returns.
•
If SRMC > π , the next unit costs
•
Find y^ where SRMC(y^) just equals π (cutting from below)
•
If NetEarnings(y^) > 0, produce y*=y^. Otherwise, set y* = 0 (shut down).
Costs
SRMC1
SRAC1
more than it returns
returns.
Costs
SRMC2
SRAC2
SRAVC2
π
0
0
y2
Avoidable and sunk cost are both fully covered at optimal output level y*=y1 but only
avoidable cost is fully covered at optimal output level y*=y2
© 2006 Daniel Kirschen
y1
22
In figure, below, the output level y^ where SRMC = π results in
NetEarnings(y^)
g (y ) = [[πy^
y - Cv(y
(y^)]
)] < 0,, meaning
g avoidable cost is not covered.
Firm should select y* = 0 (shut down).
Note:
π * y^ = Revenue (y^) and SRAVC(y^) * y^ = Cv(y^)
SRMC(y)
Costs
SRAC(y)
SRAVC(y)
SRAVC(y^)
π
y*= 0
y
© 2006 Daniel Kirschen
yy^
Quantity y
23
When all fixed costs are sunk, the short-run supply curve
for a competitive firm coincides with the portion of its SRMC
curve lying above its SRAVC curve.
SRMC(y)
Costs
π ($/unit)
0
supply
curve
y = So(π)
SRAC(y)
SRAVC(y)
y#
0
© 2006 Daniel Kirschen
y#
Quantity y
24
Market supply =
Aggregation of
i di id l fi
individual
firm
supply curves
Perfect Competition
• Perfect competition
ƒ The volume handled by
each market participant is
small compared to the
overall market volume
Price
Extra-marginal
ƒ No market participant can
price
influence the market p
by its own unilateral actions
ƒ All market participants act
as price takers
supply
Inframarginal
i l
demand
Quantity
Marginal producer
© 2006 Daniel Kirschen
25
Costs: Economist’s Perspective
• Opportunity
O
t it cost:
t
ƒ What is the next best use of the money that is currently being
spent to produce something ?
ƒ Failure to exploit an opportunity to invest this money in a more
advantageous use represents an “opportunity cost”
• Examples:
ƒ Growing apples or growing kiwis?
ƒ Using money to grow apples or instead depositing this money in
a bank where
here it would
o ld earn interest?
• “Avoidable cost” includes income lost by not investing in
next best opportunity (i.e., opportunity cost)
• Selling “at cost” (net earnings = 0) does not necessarily
mean zero dollar gain. It means producer is just
indifferent between current use and next best use
use.
© 2006 Daniel Kirschen
26
Costs: Economist’s Perspective … Continued
• “Avoidable cost” includes income lost by not investing in
next best opportunity (i.e., opportunity cost)
• Selling “at cost” (net earnings = 0) does not necessarily
mean
ea zero
e o do
dollar
a ga
gain. Itt means
ea s p
producer
oduce is
s just
indifferent between current use and next best use.
• Suppose you have $100 to invest for 1 year
year. If you buy
Apple stock for $100 on Jan 1 and resell it on Dec 31,
you know you will receive a net return of $3. Thus you
should only buy IBM stock on Jan 1 if your revenues
cover your “avoidable cost” = $103, where
$103 = Investment Cost $100 + Opportunity Cost $3
$3.
© 2006 Daniel Kirschen
27
Imperfect Competition
• One or more “strategic player” firms can influence the
market price through their actions
• Strategic player firms
ƒ Can influence the market price through their actions
(price SETTERS rather than price takers)
ƒ Perceive and actively exploit potential demand for their output
ƒ Do not have “supply curves” in previously developed sense
ƒ Typically have a large market share of capacity/output/revenues
• A “competitive fringe” could still exist
ƒ Participants with a small market share
ƒ Take the market price as given
• Examples: Monopoly (single seller) models, Cournot
and Bertrand multi-seller models of competition
© 2006 Daniel Kirschen
28
Single-Price Monopoly Models
• A firm is a monopoly if it is the only supplier of a
product y for which there is no close substitute.
• A monopoly that is limited to charging the same
price π for each unit of its product y is called a
single price monopoly
single-price
monopoly.
• Thi
This iis th
the case ttreated
t d iin Ki
Kirschen/Strbac,
h /St b
Chapter 2, Section 2.5.3.
© 2006 Daniel Kirschen
29
Single-Price Monopoly…Continued
• A single-price monopolist perceives it faces a downward
sloping
l i iinverse d
demand
d curve π(y)
( ) = D-11(y).
( )
• The monopolist exploits this knowledge by setting its
price equal to π(y) = D-1(y) = (maximum willingness of
buyers to pay for y), given it produces y.
• The monopolist sets its price with no worry this price can
be undercut by existing rivals.
• However, new market entrants could be a concern (i.e.,
the market could be “contestable”).
© 2006 Daniel Kirschen
30
Marginal Revenue for a Single-Price Monopolist
•
Total Revenue Function: TR(y) = π(y)•y
•
M
Marginal
i l Revenue
R
Function:
F
i
•
Note: MR(y) < π(y) unless demand is perfectly elastic ( i.e., ε(y) = - ∞ )
© 2006 Daniel Kirschen
31
Profix Maximization for a Single-Price Monopolist
• Basic Observation:
A small increase in output by a monopolist will increase its net
earnings if MR > MC and decrease its net earnings if MR < MC.
•
Single-Price Monopolist’s Short-Run Optimization Rule:
ƒ Increase output if MR > MC ;
ƒ Decrease output if MR < MC ;
ƒ Choose output y* where MR = MC and set price equal to the
consumer’s maximum willingness
g
to pay
y at yy*, i.e., set
π*
= π(y*) = D-1(y*)
.
ƒ If revenue π*y* covers avoidable cost at y*, produce y*. Otherwise,
shut down.
© 2006 Daniel Kirschen
32
Single-Price Monopolist’s Profit Maximization:
$
Net Earnings = Revenue –Avoidable Cost
300
MC(y)
250
Demand curve
MR curve
MC curve
AVC curve
y* = 6
π* = $165
AVC(y*)
200
165
AVC(y)
150
100
50
D-1(y)
MR(y)
0
0
2
© 2006 Daniel Kirschen
4
6
8
10
12
14
16
18
Output y
33
Natural Monopoly
• A production process is said to exhibit economies of scale
if average costs of production decrease with increases in
production.
• A monopoly
l iis said
id tto b
be a natural
t
l monopoly
l if it exhibits
hibit
such extensive economies of scale that the one monopolist
firm can produce at a lower cost per unit (i.e., at lower
average cost) than can two or more firms.
• Key
y Issue for Power Industry:
y What aspects
p
of electric
power production (if any) are properly considered to be
“natural monopolies” and hence potential candidates for
regulation?
© 2006 Daniel Kirschen
34
Cournot Duopoly (Two-Firm) Model
of Quantity
y Competition:
p
market price π
π(Y)
D-1(Y)
Market
Y
quantity
Let Y = y1 + y2, and let π(Y) = market price given Y.
Problem for firm 1:
maxπ ( y 1 + y ) y 1 − cv( y 1 )
e
2
y1
y1 = f 1 ( y )
e
2
( ye2 = expected y2 )
Similar problem for firm 2:
y2 = f2 (y )
( ye1 = expected y1 )
e
1
*
Cournot-Nash Equilibrium:
*
y1 = f 1 ( y 2 )
y 2* = f 2 ( y 1* )
Neither firm has any incentive to deviate from the equilibrium
© 2006 Daniel Kirschen
35
Cournot Oligopoly (Multi-Firm) Model:
Total industry output:
Firm i:
Y = y 1 + L + y n , si = yi / Y
max {y i ⋅ π (Y ) − cv( y i ) }
yi
d
dy i
{y i ⋅ π (Y ) − cv( y i ) }= 0
π (Y ) + y i
dπ (Y )
=
d v( y i )
dc
dy i
dy i
y i Y dπ (Y ) ⎫ dcv( y i )
⎧
π (Y ) ⎨1 +
⎬=
Y dy i π (Y ) ⎭
dy i
⎩
⎧⎪
si
π (Y ) ⎨1−
ε (Y )
⎪⎩
© 2006 Daniel Kirschen
⎫⎪ dc
d (y i )
⎬=
dy i
⎪⎭
This equals 1 for
perfect competition
because dπ/dyi=0
This equals
1 – | 1/ε(Y) | for
a monopolist
(i.e., when n = 1)
36
Cournot Oligopoly Model … Continued
⎧⎪
si
π (Y ) ⎨1−
⎪⎩
ε (Y )
⎫⎪ dc ( y i )
= MC(yi)
⎬=
⎪⎭
dy i
< 1 except for perfect competition case (Y) = -∞
•
Cournot firm i operates at a point where its marginal cost MC(yi) is
less than the market price π(Y)
•
Ability off a Cournot
Abilit
C
t firm
fi i to
t advantageously
d
t
l deviate
d i t ffrom
“competitive” π(Y) = MC(yi) point is a function of:
ƒ Market share
si = y i Y
ƒ Inverse of price elasticity of demand
1/ε (Y) =
dπ(Y) Y
dyi π(Y)
© 2006 Daniel Kirschen
<0
37
Cournot Oligopoly Model…Continued
• We can now complete this model in the same way we did for
the duopoly case (N=2)
• Consider the following equation:
si
⎪⎧
π ( Y ) ⎨1 −
ε (Y )
⎪⎩
⎫⎪ dc ( y i )
⎬=
dy i
⎪⎭
• Let Y be replaced by [ yi + h(ye-ii) ] = Sum of yi plus expected
outputs for all firms but firm i
p
yi choice for each firm i can be expressed
p
as a
• Then optimal
function yi* = f(ye-i) of the expected output of other firms.
• Cournot-Nash equilibrium is said to hold at y* = (y*1,…,y*N)
if th
the expectations
t ti
off each
h firm
fi i are fulfilled
f lfill d att y*.
*
© 2006 Daniel Kirschen
38
Bertrand Duopoly (Two-Firm) Model
of Price Competition
• Two identical firms with constant marginal cost of
production, c, are competing for demand
• Each firm offers a price with the promise to supply
all quantity demanded from it at that price
IF a firm was a monopolist, it would solve:
o
o
Max
[
πD
(π)
–
cD
(π) ] Î π* = c [ε*/(1+ε*)]
Price π
π
(with εε* = ε(π
ε(π*)) < -11)
π*
Market Demand Y = Do(π) in ordinary form
C
Y*
© 2006 Daniel Kirschen
Quantity Y
39
Bertrand Duopoly … Continued
• However, the firm that sets the lowest price
captures
p
the entire market,, so firms keep
p
undercutting each other’s price offers
g
cost of
• But neither firm will bid below its marginal
production, c, because it would sell at a loss
• Thus the only equilibrium is when each firm sells
at the same price, equal to the marginal cost of
production c
• Price=MC:
C Equivalent to competitive equilibrium!
• Note that the division of Y between the firms is
indeterminate when π = c
© 2006 Daniel Kirschen
40