SOS 325 Problem Set #1: First Principles Answer all questions on this sheet of paper and turn this in by the end of class on Thursday, September 13 th. If you cannot turn your assignment in in-class...

Answer to 1:
P= 40-4qA, implies qA = 10-(P/4) ………… (1)
P = 40-2qB, implies qB = 20 – (P/2) ……….(2)
a)
Maximum price that the consumer A is willing to pay for a unit (i.e. qA =1): P = 40-4*1 = 36
Maximum price that the consumer B is willing to pay for a unit (i.e. qB =1):P = 40-2*1= 38
To determine the quantity at which marginal benefit of more water equals zero can be determined
y substituting P (denoting marginal benefit) = 0 in the demand function for each consumer.
So for consumer A, the quantity at which “marginal benefit of more water equals zero” is given as:
qA = 10-(0/4) = 10
Similarly for consumer B, the quantity at which “marginal benefit of more water equals zero” is
given as: qB = 20-(0/2) = 20
)
Elasticity of demand = (dq/dp)*p/q
In case of consumer A, (dqA/dp) = -(1/4) [from (1)]
So elasticity of demand (in case of consumer A) =-(1/4)*(p/qA) = -p/4qA
In case of consumer B, (dqB/dp) = -(1/2) [from (2)]
So elasticity of demand (in case of consumer B) = -(1/2)*(p/qB) = -p/2qB
Table below shows the price elasticity of demand for A and B at price = 30, 20 and 10 [using the
formula we derived above]
P qA= 10-(P/4) qB = 20-(P/2) Elasticity for A= -p/4qA Elasticity of B = -p/2qB
30 2.5 5 -3 -3
20 5 10 -1 -1
10 7.5 15 -0.333333333 -0.333333333

We note that for both the players, the value of elasticity (in absolute terms) falls as we move down
the demand curve (or as price goes down).
c)
Market demand curve is horizontal summation of individual demand curve. So market demand q=
qA+qB= 10-(P/4) +20 – (P/2) = 30-(3/4)P
So demand curve: q = 30-(3/4)P. It can be rewritten as: P = 40-(4/3)q
P q
40 0
30 7.5
20 15
10 22.5
d)
Elasticity of demand = (dq/dp)*p/q
In case of market demand, (dq/dp) = -(3/4)
So Elasticity of demand = -3p/4q
Market elasticity of demand at the prices given in b) has been given in the table below:
P q (market) price elasticity = -3p/4q
30 7.5 -3
20 15 -1
10 22.5 -0.333333333

e)
0
5
10
15
20
25
30
35
0 5 10 15 20 25
Market demand curve
Market demand curve
At price P = $10, market demand or water consumed overall: q = 30-(3/4)*10 = 22.5
When price become P = $15, then the new market demand or water consumed overall: q = 30-
(3/4)*15 = 18.75
P qA= 10-(P/4) qB = 20-(P/2)
10 7.5 15
15 6.25 12.5
Change -1.25 -2.5

As shown in the table above, when price increases from $10 to $15, consumer A reduces its
consumption by 1.25 units whereas consumer B reduces its consumption by 2.5.
f)
Consumer surplus is the area between price line and market demand curve. It has been represented
y the area shaded with red in the given figure below:
Total Consumer surplus = area of triangle shaded with red = (1/2)*(40-15)*18.75 = 234.375
Consumer surplus shared by consumer A = (1/2)*(40-15)*6.25 = 78.125 [area of triangle formed
y demand curve of A]
Consumer surplus shared by consumer B = (1/2)*(40-15)*12.5 = 156.25 [area of triangle formed
y demand curve of B]
Answer to 2:
The demand function for C:...