Download Microeconomics MECN 430 - Spring 2016

final
solutions
a monopolistic bank ………….1
software upgrades ………….2
competition in prices ………….4
price customization ………….5
electricity markets ……..….8
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
final
solutions
the analytics of constrained optimal
decisions
a monopolistic bank
marginal cost/revenue
► Supply function (with Deposits as “quantity” and interest rate on Deposits as “price”):
D = k r D
or equivalently
rD = D/k
► Demand function (with Loans as “quantity” and interest rate on Loans as “price”):
L = A – BrL or equivalently
rL = A/B – L/B
► The supply curve is basically the marginal cost of the bank thus:
MC(D) = D/k
► For a demand function P = a – bQ the marginal revenue is MR = a – bQ, thus
MR(L) = A/B – L/B
monopolist solutions
► The monopolist will choose it’s optimal “quantity” such that MC = MR. In the bank’s case there might be certain restrictions on
how deposits are made available for loans, in particular one restriction is that only fraction f of deposits can be loaned out: L = fD.
► Thus the monopolist will solve:
MR(L) = MR(D)
L = f D
A/B – L/B = D/k
L = f D
A/B – fD/B = D/k
L = f D
D = (kA)/(B + 2fk)
L = f(kA)/(B + 2fk)
Remark: for f = 1 we get the solution for the case in which all deposits are made available for loans.
 2016 Kellogg School of Management
final solutions
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microeconomics
final
solutions
the analytics of constrained optimal
decisions
software upgrades
profit maximization
► The monopolist maximizes its profit by setting
the quantity such that the resulting marginal
revenue equals the corresponding marginal cost:
MR = MC. Remember that for a demand P = a –
b∙Q the marginal revenue is
120
MR = a – 2b∙Q
70
110
monopoly
price
100
90
P(Q)
80
60
►Marginal revenue is:
MR = 100 – 2Q
40
30
► Marginal cost is:
0
QM
= 50
►The market price is obtained from the demand
function for the quantity QM as:
= 100 –
 2016 Kellogg School of Management
QM
MC(Q)
10
► Profit maximization gives:
100 – 2Q = 0 with
MR(Q)
20
MC = 0
PM
monopoly
quantity
(PM ) 50
= 50
0
10
20
30
40
50
(QM )
60
70
80
90
100 110 120
► Graphically we look for the intersection of the marginal
revenue and marginal cost curves – this will give the quantity
QM that maximizes monopolist’s profit. The price charged by the
monopolist is found on the demand curve for the quantity QM.
final solutions
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microeconomics
final
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the analytics of constrained optimal
decisions
software upgrades
profit maximization
► The amount received next year is $40 which is
valued at 0.9∙$40 = $36 this year.
136
140
130
MRnew(Q)
120
► There is an additional revenue of $36 to the MR
calculated above for each additional unit sold thus
the marginal revenue is now
MRnew(Q) = 136 – 2Q
► Graphically only the new marginal revenue line
is relevant; we can drop the initial demand and
initial marginal revenue.
► With a marginal cost of zero the profit
maximization gives:
110
100
90
80
70
60
50
monopoly
quantity
40
30
20
MC(Q)
10
0
0
10
136 – 2Q = 0 with QM = 68
►The market price is obtained from the demand
function for the quantity QM as:
PM = 100 – QM = 32
 2016 Kellogg School of Management
20
30
40
50
60
70
80
90
100 110 120
68
(QM )
► Graphically we look for the intersection of the marginal
revenue and marginal cost curves – this will give the quantity
QM that maximizes monopolist’s profit. The price charged by the
monopolist is found on the demand curve for the quantity QM.
final solutions
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microeconomics
final
solutions
the analytics of constrained optimal
decisions
competition in prices
bertrand solution
► Using the Excel application it is immediate to find:
P1 = P2 = 40
myopic solution
► The reaction functions are now:
firm 1:
firm 2:
P1 = 30 + 0.25·P2
P2 = P1
The solution is P1 = 40 and P2 = 40 (same as in Task 1 but for completely different reason)
► We get the same answer because the reaction function for the second firm is P2 = P1 and so it will produce in equilibrium the same
price for both firms. If we pick a difference reaction function, say a “discount” reaction function, e.g. P2 = P1 – 1, the results in Task 1
and Task 2 will no longer be the same.
 2016 Kellogg School of Management
final solutions
page | 4
microeconomics
final
solutions
the analytics of constrained optimal
decisions
price customization
separate markets and pooled solution
Type 1
Type 2
Pooled
Demand:
Q1 = 5000 – 4P1
Demand:
Q2 = 4000 – 16P2
Demand:
Inverse demand:
P1 = 1250 –Q1/4
Inverse demand:
P2 = 250 – Q2/16
Inverse demand: P = 450 – Q/20
Marginal rev.:
MR1 = 1250 – Q1/2
Marginal rev.:
MR2 = 250 – Q2/8
Marginal rev.:
MR = 50 – Q/10
Marginal cost:
MC1 = 200
Marginal cost:
MC2 = 200
Marginal cost:
MC = 200
Optimal decision: MR1 = MC1
Optimal decision: MR2 = MC2
Q = Q1 + Q2 = 9000 – 20P
Optimal decision: MR = MC
1250 – Q1/2 = 200 → Q1 = 2100
250 – Q2/8 = 200 → Q2 = 400
450 – Q/10 = 200 → Q = 2500
P1 = 1250 – 2100/4 = 725
P2 = 4000 – 400/16 = 225
P = 450 – 2500/20 = 325
 2016 Kellogg School of Management
final solutions
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microeconomics
final
solutions
the analytics of constrained optimal
decisions
price customization
limited supply solution
Type 1
Type 2
► How many units?
1250
We have to find the Q1 for
which MR1 = 250:
MR1
1250 – Q1/2 = 250
Q1 = 2000
250
250
MR2
2500
2000
2000
Since we have 2200 units
available it follows that some
units will also be sent to
Type 2 store.
All these units offer a higher marginal revenue if sent
to Type 1 store than to Type 2 store.
How many units?
For each unit you have to decide to which type of store to send it…
► What’s the decision criteria? Send it to store type that offers the highest MR for that particular unit…
From the diagrams it obvious that you’ll start sending units to Type 1 stores (since you get MR higher than 250 for at least the first
few units)
► But once the marginal revenue from sending units to Type 1 stores hits 250 (or slightly below) you should start sending the units
to Type 2 store…
Again revert to send units to Type 1 store as soon as MR from Type 2 store is below MR from sending to Type 1 store
 2016 Kellogg School of Management
final solutions
page | 6
microeconomics
final
solutions
the analytics of constrained optimal
decisions
price customization
limited supply solution
Type 1
Type 2
► The idea is that as soon as the
monopolist realizes that it will send units
to both markets it must be the case that at
the optimum (the last unit it sends) must
satisfy
1250
MR1
That is
250
230
230
1250 – Q1/2 = 250 – Q2/8
MR2
2500
2040
MR1 = MR2
2000
160
► But there is a constraint on the total
quantity available to distribute between the
two stores:
Q1 + Q2 = 2200
2200
► We get a system of two equations with two unknowns:
1250 – Q1/2 = 250 – Q2/8
Q1 + Q2 = 2200
► Conclusion: Q1 = 2040 units to Type 1 store and Q2 = 160 units to Type 2 store, in total all 2200 units available for distribution.
Notice that the marginal cost plays no role here (sunk cost by now).
► For Type 1 store P1 = 1250 – Q1/4 = 1250 – 2040/4 = 740
► For Type 2 store P2 = 250 – Q2/16 = 250 – 160/16 = 240
 2016 Kellogg School of Management
final solutions
page | 7
microeconomics
final
solutions
the analytics of constrained optimal
decisions
electricity markets
cournot solution
time line
market demand for
electricity is initially
Government commits to supply
electricity QG
with resulting market demand
the two players compete now in a
Cournot model given a demand
P = 120 – Q
● where Q is the total
electricity supplied to the
market by the players
Q = Q1 + Q2 + QG
P = [120 – QG] – (Q1 + Q2)
● this initial commitment of electricity
by the government basically
“reduces” the market size left for the
two players
P = [120 – QG] – (Q1 + Q2)
● the solution to this game follows
the same logic as the solution without
the Government
► For firm 1, residual demand is P1(Q1) = [120 – QG – Q2] – Q1 from which we derive the marginal revenue
MR1(Q1) = [120 – QG – Q2] – 2Q1
► Since marginal cost is zero we get, from MR1 = MC1 the reaction function for firm 1 as Q1 = 60 – 0.5QG – 0.5Q2
► For firm 2, residual demand is P2(Q2) = [120 – QG – Q1] – Q2 from which we derive the marginal revenue
MR2(Q2) = [120 – QG – Q1] – 2Q2
► Since marginal cost is zero we get, from MR2 = MC2 the reaction function for bank 2 as F2 = 60 – 0.5QG – 0.5Q1
 2016 Kellogg School of Management
final solutions
page | 8
microeconomics
final
solutions
the analytics of constrained optimal
decisions
electricity markets
cournot solution
► We have to solve now the system
Q1 = 60 – 0.5QG – 0.5Q2
Q2 = 60 – 0.5QG – 0.5Q1
► The solution is found in the usual way (plug the first equation into the second, solve for Q2, then use this back into the first
equation to find Q1) however the algebra is slightly more cumbersome since we have to carry over the extra term for QG. Anyway:
Q1(QG) = 40 – QG/3
Q2(QG) = 40 – QG/3
► The resulting price is thus
P(QG) = 120 – (Q1 + Q2 + QG) = 120 – (40 – QG/3 + 40 – QG/3 + QG) = 40 – QG/6
► The Government wants a target P* thus solves
with solution
► For P* = 20 we get
and
40 – QG/3 = P*
QG = 120 – 3∙P*
QG = 120 – 3∙20 = 60
Q1 = 40 – 60/3 = 20, F2 = 40 – 60/3 = 20
 2016 Kellogg School of Management
final solutions
page | 9
microeconomics
final
solutions
the analytics of constrained optimal
decisions
electricity markets
cartel solution
time line
market demand for
electricity is initially
Government commits to supply
electricity QG
with resulting market demand
the cartel maximizes its profit given a
demand
P = 120 – Q
● where Q is the total
electricity supplied to the
market by the players
Q = QC + QG
P = [120 – QG] – QC
P = [120 – QG] – QC
● this initial commitment of electricity
by the government basically
“reduces” the market size left for the
cartel
● the solution to this game follows
the same logic as the solution without
the Government
► For the cartel with a demand is P(QC) = [120 – QG] – QC the marginal revenue is
MRC(QC) = [120 – QG] – 2QC
► Since marginal cost is zero we get, from MRC = MCC the profit maximizing quantity for the cartel is
QC = 60 – 0.5QG
 2016 Kellogg School of Management
final solutions
page | 10
microeconomics
final
solutions
the analytics of constrained optimal
decisions
electricity markets
cartel solution
► The resulting price is thus
P(QG) = 120 – (QC + QG) = 120 – (60 – QG/2 + QG) = 60 – QG/2
► The Government wants a target P* thus solves
with solution
► For P* = 20 we get
and
60 – QG/2 = P*
QG = 120 – 2∙P*
QG = 120 – 2∙20 = 80
Q1 = Q2 = QC/2 = (60 – 80/2)/2 = 10
 2016 Kellogg School of Management
final solutions
page | 11