Question

Can you please help me with this problem

INCLASS ASSIGNMENT A movie theater owner knows that demand for a film varies between daytime (matinee) and evening moviegoers. Suppose the demand curve for daytime viewers is Qd = 100 - 10Pp and demand for evening customers is QE = 140 - 10PE. For all customers, marginal cost is $3 per customer. a. What is the profit-maximizing price if the theater manager charges daytime and evening customers the same price? 1) What is the daytime and evening attendance? 2) What is total profit per day? b. Suppose the manager decides to adopt a third-degree pricing strategy, charg) de and standard prices at night.

Answer 1

1. (a) The combined demand for D and E group would be
Q = QE +QD
or
Q = 140 - 10P + 100 - 10P
or
QA = 240 - 204
. Note that QE demand curve would be above the QD demand curve. The QE and QA demand curve would intersect where
QA = QE
or or or . Hence, the aggregate demand demand curve would be as
VA / 140 - 10P for P> 10 240 - 20P for P < 10
.

(1) The marginal cost is as . The inverse aggregate demand would be as
14 -0.19 for P> 10 12 -0.05Q for P < 10
, and the total revenue would be as
TR = PQ= 140-0.102 for P> 10 120 -0.050- for p < 10
, and the aggregate marginal revenue would be
MRA = (PQ) = {12 -0.10 for P<10 _14 -0.20 for P> 10
, ie
_14 -0.20 for Q< 20 MRA= 12-0.10 for Q > 20
.

The total attendance would be where
MRA = MC
or
2 /14 -0.20 for Q<20 12-0.1Q for Q > 20
or
9 55 for Q<20 90 for Q > 20
, and since Q=55 contradicts with Q<20, would be the total attendance. This means that the MRA=MC at the lower part of the kink in MRA and QA. The price charged would be as which is basically the single price charged. The graph is as below.

P,MC,MR (Q=90,P=7.5) 5– MRA NOO 200 300

(2) The total profit would be $\pi = TR - TC$ or
T= PQ-30
or $\pi = 7.5*90 - 3*90$ or $\pi = 405$ dollars.

(b) In this case, we have two different prices charged for different demands. The inverse demand would be as
Pp = 10 -0.10D
and
Pe = 14-0.10E
for the two groups. The marginal revenue functions would be
11 Rp = (TRp) = (PQ) = (10-0.13%
or
MRp = 10 -0.20
and
A RE = (T RE} = ਰਨ (PQ) = (140-0.19%
or
MRE = 14-0.20
.

(1) The quantity produced for D group would be where
MRp = MC
or
10 -0.20 = 3
or
0.20 = 1
or
QD=35
. The price charged would be hence
Pp = 10 -0.10D
or
Pn= 10 -0.1* 35= 6.5
dollars.

The quantity produced for E group would be where
MRE = MC
or
14 -0.20 = 3
or
0.20 = 11
or
QE = 55
. The price charged would be hence
Pe = 14-0.10E
or
PE = 14-0.1* 55 = 8.5
dollars.

(2) The total attendance would be . This would be the same as in part-a.

Q=QD+QE = 35+55 = 90

The graph is as below.

P,MC,MR 10+ A MRE PEY MRD i PDT71 Hier MD DE IN DRE 200 300

(3) The profit from D group would be
Tip = TRp-TC
or
Лp = Pbер — зер-
or
p = 6.5*35-3* 35
or
Tip = 122.5
dollars.

The profit from E group would be
TE = TRE-TC
or
TE = PEQE - 30
or
E = 8.5* 55-3* 55
or
TE=302.5
dollars.

The aggregate profit in this case would be hence $\pi_A = \pi_D + \pi_E$ or
A = 122.5+ 302.5
or
= 425
dollars. As can be seen, the price discrimination increased the aggregate profit by $20.