Managerial Economics, SS14
First Homework Solution
(1) Elasticties
First, we take the logarithm of equation (1):
log(Q) = 100 − 3.1 log(P ) + 2.3 log(I) + 0.1 log(A)
This is a so-called log-log equation. When both sides of the equation are log-transformed, coefficients (= partial effects of each variable) can be interpreted directly as elasticities (i.e. the
percentage change of the left-hand-side variable if a right-hand-side variable is increased by one
percent).1
∂ log(Q)
∂ log(Q)
∂ log(Q)
ηI =
ηA =
ηP =
∂ log(P )
∂ log(I)
∂ log(A)
a) The price elasticity of demand (ηP ) is -3.1, which means that a one percent increase in prices
will decrease quantity demanded by 3.1%. The demand for apple pies is therefore elastic.
b) The income elasticity of demand (ηI ) is 2.3, a one percent increase in per capita monthly
income leads to a 2.3% increase in quantity of apple pies demanded. Since ηI > 0, apple pies
are normal goods; and because ηI > 1, apple pies are actually luxury goods (at least in our
example).
c) The advertising elasticity of demand (ηA ) is 0.1, indicating a 0.1% increase in quantity
demanded when advertising expenditures are increased by one percent.
(2) Perfect competition
First we transform the supply and demand curves:
1
Qs
10,000
5
1
Pd = −
Qd
3 15,000
Ps =
Now we set P ∗ = Ps = Pd (and Q∗ = Qs = Qd ) and solve the system of equations:
5
1
1
Q∗ = −
Q∗
10,000
3 15,000
Q∗ = 10,000 =⇒ P ∗ = 1
(3) Regression analysis
a) A one dollar increase in price decreases quantity sold, ceteris paribus, by 4.1 units. A one
dollar increase in per capita income increases quantity sold, ceteris paribus, by 4.2 units. A
one dollar increase in advertising expenditures increases quantity sold, ceteris paribus, by 3.1
units.
x
Recall the calculus result that ∂ log
= x1 , or ∂ log(x) =
∂x
is the well-known formula for the price elasticity of demand.
1
1
∂x
x ,
thus for example
∂ log(Q)
∂ log(P )
=
∂Q/Q
∂P/P
=
∂Q P
∂P Q ,
which
Managerial Economics, SS14
First Homework Solution
a) Judging from the t-values, all coefficients besides βb2 are statistically significantly different
from zero. Income does not seem to have a significant impact on quantity sold.
c) The R2 indicates that our regression model (or more precisely, the variables we included in
our regression model) explain 64% of the total variation in quantity sold.
d) The error term has a mean of zero by assumption (E[i ] = 0), which means that all variables
which affect quantity sold are included in our regression model.2 For each observation i, the
error term may in fact differ from zero, but on aggregate (over all observations) it is assumed
to be zero.
e) First we substitute the values for I and A in equation (4) and isolate P :
Q = −4.1P + 64,577 =⇒ P = 15,750.49 − 0.244Q
The marginal revenue curve is then:
MR =
∆P Q
∆T R
=
= 15,750.49 − 0.488Q
∆Q
∆Q
(4) Monopoly
P
P
60
60
MC
MC
PM
PC
PC
20
D
QC
D
Q
QM QC
Q
P = 60 − 2Q
T R = P Q = 60Q − 2Q2
M R = ∆T R/∆Q = 60 − 4Q
M C = 2Q
2
If you think about it, this assumption is actually unlikely to hold, since our regression model does not account
for other factors such as prices of rival products, which indeed influence the quantity sold. In this case we say that
our model is misspecified and coefficients are biased.
2
Managerial Economics, SS14
First Homework Solution
a) Under perfect competition, the market equilibrium is where P = M C:
60 − 2Q = 2Q
QC = 15 =⇒ P C = 30
b) The consumer and the producer surplus are depicted in the figure above; the blue-shaded
area is the consumer surplus (CS), and the yellow-shaded area is the producer surplus (P S):
CS C = (60 − 30) ∗ 15 ∗
1
= 225
2
P S C = 30 ∗ 15 ∗
1
= 225
2
The total welfare is therefore T W C = CS C + P S C = 450.
c) A monopolist produces where M R = M C:
M R = 60 − 4Q = 2Q
QM = 10 =⇒ P M = 40
The profit is then3
π M = T R − T C = 60Q − 2Q2 − Q2 = 60 ∗ 10 − 2 ∗ 102 − 102
π M = $300
d) The deadweight loss is depicted as the grey-shaded area in the right figure above. First we
calculate the new consumer and the new producer surplus:
CS
M
1
= (60 − 40) ∗ 10 ∗ = 100
2
PS
M
1
= [(40 − 20) ∗ 10] + 20 ∗ 10 ∗
= 300
2
Thus, the total welfare is T W M = CS M + P S M = 400. The deadweight loss is therefore4
DW L = T W C − T W M = 50
e) The price elasticity of demand in the monopoly equilibrium is
ηP M =
∆Q P M
1 40
=
−
∗
= −2
∆P QM
2 10
Note that we have to express demand as a function of P , Q = 30 − 1/2P , in order to obtain
the marginal effect ∆Q/∆P .
3
Note that we have to take the antiderivative of M C in order to obtain the total cost function:
Z
T C(Q) =
4
M C(Q) dQ = Q2
Alternatively, you could calculate the area of the grey triangle directly in order to obtain the deadweight loss:
QC
Z
15
P − M C dQ = 60Q − 2Q2 DW L =
10
QM
3
= 450 − 400 = 50
Managerial Economics, SS14
First Homework Solution
(5) Price discrimination I
p1 = 160 − 8q1
p2 = 80 − 2q2
T R1 = p1 q1 = 160q1 − 8q12
T R2 = p2 q2 = 80q2 − 2q22
M R1 = ∆T R1 /∆q1 = 160 − 16q1
M R2 = ∆T R2 /∆q2 = 80 − 4q2
M C = 5 + Q = 5 + q1 + q2
First, we set M Ri = M C for each market i ∈ {1, 2}:
160 − 16q1 = 5 + q1 + q2
155 − q2 = 17q1
155 − q2
q1 =
17
80 − 4q2 = 5 + q1 + q2
75 − q1 = 5q2
1
q2 = 15 − q1
5
(1)
(2)
Now we solve this system of equations by plugging (2) into (1) – or vice versa:
155
1
1
−
∗ 15 − q1
17
17
5
140
1
84
140
q1 =
+ q1 =⇒ q1 =
17
85
85
17
q1 =
a) q1∗ = 25/3 = 8.33 ≈ 8
€
b) q2∗ = 40/3 = 13.33 ≈ 13 = 15 − 51 q1∗
Š
c) p∗1 = 93.33 (= 160 − 8q1∗ )
p∗2 = 53.33 (= 80 − 2q2∗ )
(6) Price discrimination II
For the imps, we charge a use fee which is equal to marginal cost (M C = 2) and an entry fee
which is equal to their consumer surplus:
entry feeI = CSI = (30 − 2) ∗ 28 ∗
1
= $392
2
Since no profit is generated through the use fee, the total profit per imp equals the entry fee, which
is $392.
Applying simple monopoly pricing to the wizards yields
M RW = 40 − 2QW = 2 = M CW
2QW = 38
QW = 19 =⇒ PW = 21
Hence, our total profit if we would attract only wizards would be πW = (21 − 2) ∗ 19 = $361. Our
tennis facility should therefore cater for imps.
4