Transcript Microeconomics MECN 430 Spring 2016

session four
the perfectly competitive market
the monopolist problem ………….1
the pricing problem ………….2
revenue analysis ………….3
revenue, cost and profit analysis ………….5
standard/uniform pricing ………….8
user fee pricing ………….9
bundling menu pricing ………10
pricing intermediation ……….11
consolidation ………17
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
the monopolist problem
competition
► We discussed so far models in which there were (at least a few if not) several producers competing with each other to sell a
non-differentiated good (commodity markets)
► The equilibrium emerged at the intersection of demand and supply:
- demand curve shows the willingness to pay (or how many units are demanded at each price level)
- supply curve shows the willingness to produce (or how many units are offered at each price level)
the monopolist
► We move now to a new market setup characterized by:
- several consumers, therefore the demand curve definition and analysis remains valid
- there is only one producer (monopolist) serving the market
► The challenge is to determine how the monopolist behaves, i.e. what price it will charge, what quantity it will offer, etc. In
other words we have to study the profit maximization problem for a monopolist.
► Conceptually, the equilibrium is defined in a similar way: a pair of price and quantity such that given that price level
consumers are willing to buy the quantity offered and the monopolist maximizes its profit by offering exactly the quantity
demanded.
 2016 Kellogg School of Management
lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
standard pricing
the pricing problem
I am an airline with a monopoly on the Atlanta-Los Angeles route
I have market power: I can set the price! But how high should I go? Or how low should I go?
I currently price tickets at P = $760, and expect a volume Q = 100 passengers per trip, which is less than my plane’s capacity.
Questions to consider:
► If I lower the price can I get more paying passengers?
► How much revenue does an additional passenger add?
► Is the extra revenue worth the extra cost?
If I am to answer these questions what information do I need?
► Demand sensitivity to change in price
► Sensitivity (a lot of/far fewer additional passengers) not enough for second question
► Demand equation, i.e. Q = a – b∙P
► Cost of adding an additional passenger (marginal cost)
 2016 Kellogg School of Management
lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
standard pricing
revenue analysis
Let’s assume that we know the demand equation as Q = 480 – 0.5P and the constant marginal cost for accommodating each
additional passenger as MC = $160.
► At current price P0 = $760 → Q0 = 480 – 0.5∙760 = 100
passengers for a total revenue
P
TR0 = P0Q0 = $760∙100 = $76,000
lost revenue by
lowering the price
960
► I would like to add an extra passenger… what should be the
price for that to happen?
to get Q1 = 101 → 101 = 480 – 0.5P1 → P1 = $758
760
The total revenue is now
758
TR1 = P1Q1 = $758∙101 = $76,558
(0)
(1)
The change in total revenue is
extra revenue by
increasing number of
passengers
TR1 – TR0 = $558
► Should I add this one extra passenger?
Q
marginal revenue = $558 vs. marginal cost = $160
100 101
I get an extra profit of $398 for this extra passenger…
 2016 Kellogg School of Management
lecture 4
480
page | 3
microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
standard pricing
revenue analysis
► Total revenue:
revenue obtained for all units sold; TR = P∙Q
► Marginal revenue:
change in revenue due to the last unit sold; MR = ∆TR/∆Q
► Marginal cost:
cost for the last unit produces (sold); MC = ∆TC/∆Q
P
► The general definition for the marginal revenue is
“the change in total revenue when quantity changes”
a
► For a linear demand function P = a – b∙Q we get a very
nice result:
(Step 1): TR(Q) = P∙Q = (a – b∙Q)∙Q = a∙Q – b∙Q2
demand, P(Q)
(Step 2): MR(Q) = dTR(Q)/dQ = a – 2b∙Q
MR(Q)
► Conclusion: for a linear demand function
P = a – b∙Q
the marginal revenue is also a linear function
Q
MR(Q) = a – 2b∙Q
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a/(2b)
lecture 4
a/b
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
standard pricing
revenue, cost and profit analysis
The demand equation P(Q) is P = 960 – 2Q giving a marginal revenue function MR(Q) = 960 – 4Q.
We saw above that increasing the number of passengers from 100 to 101 gave a marginal revenue of $558 and we decided that
this is profitable to do since … the marginal revenue is greater than the marginal cost…
► Should we continue to increase the number of passengers (by
decreasing the price)? How many more?
P
► As long as MR(Q) > MC(Q) it is profitable to add more
passengers because each will bring a positive net profit…
► If MR(Q) < MC(Q) we added too many passengers… the net
profit for the last added is negative → better off to decrease the
number of passengers
demand, P(Q)
this gives market
price
Pm
this gives market
output
► Profit is maximized for Qm such that
MC(Q)
MR(Qm) = MC(Qm)
► The corresponding monopoly price Pm is found through the
demand function:
Pm = P(Qm)
 2016 Kellogg School of Management
lecture 4
MR(Q)
Q
Qm
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
standard pricing
revenue, cost and profit analysis
► Profit is maximized for Qm such that
MR(Qm) = MC(Qm)
► The corresponding monopoly price Pm is found through the demand function:
Pm = P(Qm)
P
For our initial example:
► the “marginal” functions:
MR(Q) = 960 – 4Q and MC = 160
960
demand, P(Q)
► profit maximization condition:
MR(Q) = MC
this gives market
price
Pm=560
► optimal output:
this gives market
output
960 – 4Q = 160 Qm = 200
► the demand function:
MC(Q)
P(Q) = 960 – 2Q
► optimal price:
MR(Q)
Pm = P(Qm) = 960 – 2Qm = 560
Q
Qm=200 240
 2016 Kellogg School of Management
lecture 4
480
page | 6
microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
standard pricing
revenue, cost and profit analysis
The benchmark is the perfect competition.
► perfect competition equilibrium: demand = supply (left diagram)
- consumer surplus is shown in the diagram as the light gray area (producer surplus = 0, deadweight loss = 0)
► monopolistic competition equilibrium: MR = MC (right diagram)
- consumer surplus the light gray area, producer surplus the dark gray area, deadweight loss the orange area
P
P
demand, P(Q)
demand, P(Q)
Pm
Pc
Pm
(c)
(m)
Pc
MC(Q)
deadweight loss
(c)
(m)
MR(Q)
MC(Q)
MR(Q)
Q
Qm
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Q
Qm
Qc
lecture 4
Qc
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
price discrimination
standard/uniform pricing
► The pricing method we studied so far is called uniform pricing because it assumes that all units are sold to all consumers at the
same price Pm (this is the $0.99 per song price that iTunes charges)
► We saw how the monopolist is determining the optimal price to maximize its profit. The outcome is shown in the diagram (the dark
gray area is the profit for the monopolist)
► Notice that the consumer still gets some surplus, i.e. the price is still below his/her willingness to pay… is there any way for the
monopolist to “capture” this remaining surplus and leave the consumer with no surplus?
P
► Before advancing let’s calculate (demand function P(Q) = 1.20
– 0.03Q, marginal revenue MR(Q) = 1.20 – 0.06Q and marginal
1.20
cost MC = 0.30):
demand, P(Q)
- consumer surplus = 1/2∙(1.20 – 0.75)∙15 = 3.375
- monopolist profit = (0.75 – 0.30)∙15 = 6.750
Pm=0.75
► First suggestion: charge a “user fee” of $3.375 (to use iTunes)
and then charge $0.75 per song
► Second suggestion: offer a “bundling” menu such as
0.30
- any 1 song for a total of $1.185
- any 2 songs for a total of $2.34
…
- any 15 songs for a total of $14.625
(m)
MR(Q)
Q
Qm=15
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MC(Q)
lecture 4
20
40
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
price discrimination
user fee pricing
► Demand function P(Q) = 1.20 – 0.03Q, marginal revenue MR(Q) = 1.20 – 0.06Q and marginal cost MC = 0.30:
- consumer surplus = 1/2∙(1.20 – 0.75)∙15 = 3.375
- monopolist profit = (0.75 – 0.30)∙15 = 6.750
First suggestion: charge a “user fee” of $3.375 (to use iTunes) and then charge $0.75 per song
► What is the buyer doing?
P
If the buyer pays $3.375 it basically gives up the surplus but
he/she still buys the 15 songs (since his demand is not changed,
i.e. he’s willingness to pay is not changed)
What’s the resulting outcome? The monopolist gets the whole
surplus available, making a total profit of
$3.375 + $6.750 = $10.125
1.20
demand, P(Q)
Pm=0.75
► The total profit is now the sum of what was consumer surplus
before and the initial producer surplus (it basically “absorbed” the
consumer surplus)
0.30
(m)
MC(Q)
MR(Q)
Q
Qm=15
 2016 Kellogg School of Management
lecture 4
20
40
page | 9
microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
price discrimination
bundling menu pricing
► Demand function P(Q) = 1.20 – 0.03Q, marginal revenue MR(Q) = 1.20 – 0.06Q and marginal cost MC = 0.30:
- consumer surplus = 1/2∙(1.20 – 0.75)∙15 = 3.375
- monopolist profit = (0.75 – 0.30)∙15 = 6.750
Second suggestion: offer a “bundling” menu… If the monopolist chooses this suggestion how does he/she calculates the
“bundling” prices?
- any 1 song for a total of $1.185
P
- any 2 songs for a total of $2.34
…
- any 15 songs for a total of $14.625
1.20
demand, P(Q)
► Consumer surplus:
1.17
CS(1st) = ½(1.20 – 1.17)1 + 1.17 = 1.185
1.14
CS(2nd) = ½(1.17 – 1.14)1 + 1.14 = 1.155
In total for two songs = 1.185 + 1.155 = 2.34
► …and so on … for a given number of songs just ask a “bundle” price
equal to the area under demand line up to that number of songs (this is
the total amount the consumer is willing to give up for that number of
songs).
For 15 songs you’ll get the total amount consumer is willing to give up for
those songs, namely 14.625.
 2016 Kellogg School of Management
lecture 4
0.78
0.75
Q
1 2
14 15
40
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
pricing intermediation
free market equilibrium
($)
Buyers and seller can transact with each other directly
and to keep algebra simple let’s assume:
10.0
- demand
D(P) = 10 – P
8.0
- supply
S(P) = P
7.0
The equilibrium is very easy to find by setting
demand equal to supply
10 – P = P P* = $5
demand
9.0
supply
6.0
5.0
4.0
3.0
2.0
1.0
Q
0.0
0.0
 2016 Kellogg School of Management
1.0
lecture 4
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
pricing intermediation
intermediaries
But now buyers and seller can transact with each other
only after paying a fee t to an intermediary (gate
keeper).
► The prices paid by the buyer (PB) and supplier (PS)
are different such that the difference between them is
exactly the fee t:
How do we find the equilibrium? Common sense?
► Prices have to satisfy
PB – PS = t
► Quantity demanded given PB and quantity supplied
given PS should be the same
PB – PS = t
► Demand and supply curves do not change but they
depend for the corresponding prices for buyers and
suppliers:
- demand
D(PB) = 10 – PB
- supply
S(PS) = PS
D(PB) = S(PS)
Thus:
PB – PS = t
10 – PB = PS
We get PB = PS + t which plugged in the second equation
gives
10 – (PS + t) = PS → PS = 5 – 0.5t
but
PB = PS + t
 2016 Kellogg School of Management
lecture 4
→ PB = 5 + 0.5t
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
pricing intermediation
intermediaries
($)
With
10.0
PS = 5 – 0.5t or PB = 5 + 0.5t
demand
9.0
the corresponding quantity is
supply
8.0
Q(t) = 5 – 0.5t
PB
Where do we find this graphically?
6.0
The hint comes from the relation
Say t = 4 … start from the maximum difference of 10
(when Q = 0) and move towards the minimum
difference of 0 (when Q = 5) … somewhere in between
10 and 0 the difference has to be 4.
 2016 Kellogg School of Management
P* = 5
competitive price
(t = 0)
4
5.0
PB – PS = t
The vertical difference between prices has to be
exactly t.
7.0
4.0
PS
3.0
2.0
1.0
Q
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Q**
lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
pricing intermediation
the monopolist
($)
10.0
What is the revenue for gate keepers?
demand
9.0
supply
TR(t) = t ∙ Q(t)
8.0
Graphically TR(t) is the area of the rectangle
between the two prices and up to the transacted
quantity.
What happens with the total revenue if t
changes? Two opposite effects:
7.0
6.0
t’
t
5.0
4.0
► a higher t → TR increases
► a lower Q(t) → TR decreases
What is the “best” t, i.e. the value for which the
total revenue is at a maximum?
3.0
2.0
1.0
Q
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Q(t)
Q(t’)
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lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
pricing intermediation
the monopolist
($)
Back to our simple case, the intermediary is a
monopolist that sells its services at a price t and
faces a demand
Q(t) = 5 – 0.5t
10.0
Since the marginal cost of providing the services is
assumed to be zero, MC = 0, the intermediary
maximizes it’s profit by setting
MR = MC
giving
t** = 5 and Q** = 2.5
supply
8.0
PB
therefore its marginal revenue function is
MR(Q) = 10 – 4Q
demand
9.0
7.0
6.0
t**=5
5.0
4.0
3.0
PS
2.0
1.0
Q
0.0
and profit = 12.5
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Q**= 2.5
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lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
pricing intermediation
surplus analysis
($)
10.0
Winners & Losers?
► intermediaries gain is the green rectangle
9.0
► buyers’ surplus is the blue triangle
8.0
► sellers’ surplus is the red triangle
PB
► DWL is the orange triangle
demand
supply
7.0
6.0
t**=5
5.0
4.0
3.0
PS
2.0
1.0
Q
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Q**= 2.5
 2016 Kellogg School of Management
lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
manufacturer
(monopolist)
physical good Q
consolidation
distributor
(monopolist)
payment Pm per unit
physical good Q
final market
payment Pd per unit
marginal cost of producing one
unit is MCm
demand on the final market is
given by P = a – bQ
► Let’s consider a stylized supply –chain model:
● manufacturer unit produces the good at a constant marginal cost MCm
● the manufacturing unit sells the good at a price Pm to a distributor
● the distributor unit sells the good at a price Pd to the final market
● the distributor incurs no additional costs (it only pays Pm per unit of good to the manufacturer)
● demand on the final market is given by P = a – bQ
► How many units of good will be produced and what are the prices Pm and Pd?
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lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
consolidation
distribution
manufacturer
(monopolist)
physical good Q
distributor
(monopolist)
payment Pm per unit
physical good Q
final market
payment Pd per unit
marginal cost of producing one
unit is MCm
demand on the final market is
given by P = a – bQ
► At the final stage of the chain we analyze a standard monopolist model where:
● the demand for the good is P = a – bQ
● the firm (distributor) has a marginal cost of MCd = Pm
● the firm (distributor) has a marginal revenue of MRd = a – 2bQ
► The solution at this stage is given by the condition MRd = MCd
● this gives a – 2bQd = Pm
● we got a relation between the price set by the manufacturer and the quantity the distributor would like to
order from the manufacturer → we derived the demand curve that the manufacturer faces
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lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
consolidation
production
physical good Q
manufacturer
(monopolist)
distributor
(monopolist)
payment Pm per unit
physical good Q
final market
payment Pd per unit
marginal cost of producing one
unit is MCm
demand on the final market is
given by P = a – bQ
► At the initial stage of the chain we analyze a standard monopolist model where:
● the demand for the good is Pm = a – 2bQ
● the firm (manufacturer) has a marginal cost of MCm
● the firm (manufacturer) has a marginal revenue of MRm = a – 4bQ
► The solution at this stage is given by the condition MRm = MCm
● this gives a – 4bQm = MCm
● we get the optimum (profit maximization) for the manufacturer:
 2016 Kellogg School of Management
lecture 4
Qm 
a  MCm
4b
Pm 
a  MCm
2
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microeconomics
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the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
consolidation
final market
manufacturer
(monopolist)
physical good Q
distributor
(monopolist)
payment Pm per unit
physical good Q
final market
payment Pd per unit
marginal cost of producing one
unit is MCm
demand on the final market is
given by P = a – bQ
► We use the results for the manufacturer to determine now the price that the distributor will set for the final market. For this
remember that
● the demand for the good is P = a – bQ
● the good is produced in quantity Qm determined previously
● we get the optimum (profit maximization) for the distributor:
 2016 Kellogg School of Management
lecture 4
Qd 
a  MCm
4b
Pd 
3a  MCm
4
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microeconomics
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the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
Qm 
manufacturer
(monopolist)
Pm 
supply chain profit
a  MCm
4b
consolidation
distributor
(monopolist)
a  MCm
2
Qd 
a  MCm
4b
Pd 
final market
3a  MCm
4
► Profit for manufacturer:
a  MCm
a  MCm (a  MCm )2
 m  (Pm  MCm )Qm  (
 MCm )

2
4b
8b
► Profit for distributor:
3a  MCm a  MCm a  MCm (a  MCm )2
 d  (Pd  Pm )Qd  (

)

4
2
4b
16b
► Total profit for supply chain:
(a  MCm )2 (a  MCm )2 3(a  MCm )2
SC   m   d 


8b
16b
16b
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
consolidation
surplus analysis
double
marginalization
consumer’s “margin”
distributor’s “margin”
Pd – Pm
manufacturer’s “margin”
Pm – MCm
► It is fairly straightforward to calculate the
surpluses
for
consumers,
distributor
and
manufacturer:
consumers surplus
distributor’s surplus
manufacturer’s surplus
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lecture 4
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microeconomics
lecture 4
the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
consolidation
vertical integration
manufacturer
(monopolist)
physical good Q
physical good Q
distributor
(monopolist)
final market
payment Pd per unit
marginal cost of producing one
unit is MCm
demand on the final market is
given by P = a – bQ
► Let’s consider a VERTICAL INTEGRATION of the supply –chain model:
● manufacturer unit produces the good at a constant marginal cost MCm
● the distributor unit sells the good at a price Pd to the final market
● demand on the final market is given by P = a – bQ
► This is a standard monopolist model with solution
Q* 
a  MCm
2b
P* 
a  MCm
2
► The total profit to the firm (manufacturer plus distributor) is
a  MCm
a  MCm (a  MCm )2
 *  (P * MCm )Q *  (
 MCm )

2
2b
4b
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microeconomics
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the monopoly model (I): standard pricing and consolidation
the analytics of constrained optimal
decisions
consolidation
VERTICAL INTEGRATION
SUPPLY CHAIN
► Clearly the consumers are better off under vertical integration.
► Total profit
● for supply chain
● for vertical integration
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3(a  MCm )2
SC   m   d 
16b
2
(a  MCm )
* 
4b
lecture 4
*  SC
total profit higher under
vertical integration
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