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:7 my (3% #5ng ﬁn 2W 5% Assignment 3 Question 3. Quality—Augmented Hotelling model
T wo firms, A and B producing qualities, 5A and 53 respectively.
Consumer value the goods according to: HA : — 1:10.311 —- pA
11.3 = [1' —~ [(1 ~;1:)]SB —~ In; Consmners buy one unit of A or B but not both (and not neither (711.4 . “ '.', . .' ' ' Tm = I — (5.1,. > 0 so ut1hty'15 111c1ea51ng in quality.
.9 s affect ros‘s valuation vert1cal chfferentiatlon : 5/
‘17 B A => 'I‘SA T
7* (5A — 83) measures vertical differentiation
3A, .38 also affect travelling costs (horizontal diﬁ'erentia— tion): SAT => #173ng (b) Given 5A, 35‘ in order to ﬁnd the demand functions for the two
ﬁnns:
Step 1: Find indifferent consumer : (7' — — 1),; = [1' — [,(1 — SB — pB Hence demand for the two ﬁrms: qA(zanels-B):r;=m+ SB _M
[a a 11 " t(8/1+SB) SA+SB t(5’A+SB) A 7’ 811—3 3 ’11—) QB (1)/111)BES/17¢SB) :1 — :p : 1 _ (__B_) _ B p 13 t(5A+SB) 811+83 t(SA+SB) (c) In order to ﬁnd the price equlibrium.
The ﬁrm proﬁts: A 7‘ (5A - 53) SB m R PB
H. ,; ,s,s 2913;: + ,_
A (p4 M A B) 11 FA ['5 (8/1 + 83) 3A + 83 t(3/l + 83)] 7‘ (8A — .93) 83 + PA — PB ] H3 PEP/1,511,813 :1) 1—1,: = )3 [1- -
( ) B( ) I t(5A+ SB) SA+83 USA +33) Hence ﬁrm A’s problem: maxHA (13/1;va 8/1753) m
FOC:
BHA a: 03? 0
r I A— : '
010A 327A
:> 7' (5A - 6‘3) + 93 _ PA “ PB __ P4 = 0
t(8/1+83) SA+9B («SA-I593) f(54+93)
7' 5 — s 9 2 — J
:>(A B)+ B _29A 13:0
t(sA + 83) 811+ SB USA + 9.8)
ﬁrm B’s problem:
niggxﬂg (PB; m, 8A, 83)
FCC:
0113 A d?
— : 1 — :1: ) — — 0
0223 1 3012B
7‘ (8A — SB) ‘33 I711 "' p13 p3
=> 1 — — + — = 0
“911+ ‘33) 8A + 33 79(3/1 + 313) ’ (911+ ’93)
7* (3A — 33) 3A 12/; — 2123
=> — + + = 0
{(9/1 + '93) 311+th 75(9/1 + SB) Therefore in order to solve for NE: 'I'(.5AmsB)+ 33 _ 2pA~pB :0 } [(SA-I-SB) 5A+SB t(sA~+-HB)
1 _ 7'(5A=-3B) _ 8:2 + 2.3425 _ 0
[(5144-53) 3144-83 [NA-+33)
2’) — ) ') — 2 )
:> [A 13 Z 1 + IA 13
15(3A+SB) t(sA+33) :> 2PA — PB ~ PA + 21713
25(5/14— SB) =>PA +123 = t (3/1 + 83) ) . 1(8/1-I-SB) t(S/1+SB) =1 219A +pA 1 _ 7 (5A — 93) SB
t(9A+sB) “SA-+93) 911+93
2”“ I Hm;— 93) + gsB+ I (9,13— 93)
Z>pA Z t(s*A—r:253) + Mrs/12— SB)
=>p3 =1:(8A+ SB) “ [(9/1 i283) " I (9A3— SB)
2””? : t(29/13+33) + 7' 9,1) We assume that [(3/1 + 283) > 7‘ (5A — SB)
t(28/1+53)>7‘(sB — 8/1) which are satisﬁed if 133 — [ml is not too big. Therefore the equlibrium
prices: t 1.1 2!. ,. 3/ _(.
m: (51+ SB)+'(5A SB) and
3 3
t 25 + s 7' s —- 3‘.
p3: ( A3 19+ (33 a) and therefore pA — p3 : t (8A + 233) + 7' (8A — SB) — “2811 + 85) — 7‘ (SB — SA)
75 (M + 83) 3f: (5/1 + 83)
(27' — 25) (SA — 83) 3“ (M + 513) and
it: 7'(s,1 —83) + 83 __ (27' == (3A — 33)
' t(8A+.S‘B) SA’i‘SB (SA-F83)
_ '1" (5A — 33) — (27' ~ (8A —— 83) .93
— +
3t(s/1+SB) 5/1 + 33
SB (7‘ — 15) (3A — SB) SA + 83 315 (SA + 33)
7' (83 — 5/1)+t(3,1+ 253)
3t (3A + 83) l | (d) Proﬁts are given by: 7' 33 — 8A +t 8/1 + 233
HMS/1:83): (t (5.4 + 2313) + '7’ (8A — 53)) 1 1/ 2 l
5
1 (TBA — 7'35 + tsA + 22583) (7’83 — 719A + tsA + 2633)
9
l )—Ceu : — —_ C 5 ( /l)
t 2t,- 2— —/, 2
: (9/1 + $5) (rs/1 r93) _ C(SA)
925 (SA —|— .33)
_ t(28A-+-83) 7'(SB—SA) 7‘(93- 9A)+'l;(.5A+293)
113(8‘4’83)— < 3 + 3 1 31‘ (9/1 + 83) Assuming that 3A 2 33 = s equilibrium prices are p14 2 p3 2 ts and
proﬁts become:
1 (3t3)2 st
H, 2H s,s:— —Cs=~—Cs
Aes m,> m % m 2 <> Note that in this case the model is identical to the simple Hotelling
model with 7' : ts. The Therefore, fﬂ=5—ce>
()s 2 Hence proﬁts are increasing in quality if é— C’ > 0 and decreasing
in quality if g — C’ < 0. Supoose that — C’ (s) > 0 for some 5. Since
C” > 0, as 3 increases, the effect on profts becomes negative. The
proﬁt function is concave in quality, therefore proﬁts are maximized at a
positive quality level. In this model, the firms offering above minimum
symmetric quality can increase their proﬁts (the higher quality allows them to charge a, higher price). Note: Suppose that the question said that the ﬁrms can choose the
quality they offer, and suppose that the ﬁrms determine the quality in 4 the ﬁrst stage and choose prices in the second stage. In order to ﬁnd
the equlibrium s, we would have to ﬁrst ﬁnd the F00 without imposing
symmetry and then use symmetry in the FOC. The FCC for A: 611A _ 1 (3A + 83) [2 (MA + 2t33)2 — 2 (TS/1 — 7‘83) +t — 7'] — (ts/1 + 27533)2 + ('rsA —- 'r'sB)2 (
(98A 9t (3A + 33)2 - Now assuming that in equilibrium 3,; 2 SB 2 3, the FCC becomes: 1 23 (2 (31:9)2 + 'l; — 7‘) — (3ts)2 _ 1/ t :
9t 482 C (9) 02>
l 93152 (45- — 1) + 21, — 27' ,
Q—F “C (S)—0 :>
[(43 — 1) 15—7"
— _(}I t. 2
4 + 19th ’ (5) 0:5 This can in principle be solved to ﬁnd the equlibrium 3, given cost
function C . (e) Now 11:11:7'+8A — ta: —pA ’l.l,B='I‘+ $3 - t(1 — :13) —pg This is the same model as having different gross valuations for the
two goods. Simply let m = 1' -|— 3,; and r3 : r + $3 in the model we
solved before. The solution is (refer to class notes): TA — 7‘3 8A f 633 :t+ =t—I— ,
p11 3 3 "‘A - TB : {I _ 8A —* SB
3 ’ 3 where 39A 2 p3 = t if quality is symmetric. Therefore, in this model they
cannot charge a higher price if they offer symmetrically higher quality. Hence ﬁrm proﬁts: P3=t- I; 31—55 2 s —s
HA(SA«,SB):§+—( 118;» + A3 B — C(SA) In this case when qualities are symmetric gross proﬁts doe not depend
on qualify. Therefore, proﬁts are decreasing in quality for as long as
consumers have a large enough gross valuation to be in the market. '15 rH
H(s, s) : i — C(s) :> = —C’ < 0 Suppose ﬁrms Choose their quality in the ﬁrst stage. Firm A’s FOC OIL/l H 2 (SA ~— 33) l I M
an — 18-]; + 3 0 (19A) _ 0 (SOC: § — C”(sA) < O Invoking symmetry the equilibrium quality solves — 0(5) 2 0.
T hus, the firms would choose to oﬁ'er more than minimum quality, even
though their profits decrease in equlibrium. This is again a Prisonerls Dilemma Situation for the ﬁrms. #3 (a) Calculate the optimal contract {(qL*, tL"), (qH*, tH*)} under full information. Maximize 11L = tL — 0.5qL2 subject to BLqL — tL = 0. Since tL = 16111, substitute this
in the profit to obtain TIL = 16qL — 0.5qu. Maximize this with respect to qL to get
qL* = 16. Then tL* = $256. Maximize TIH = tH — 0.5qH2 subject to 64H — tH = 0. Since tH = 20qH, substitute this
in the profit to obtain 11H = 20qH — 0.5qH2. Maximize this with respect to qH to
get qH" = 20. Then tH* = $400. ' (b) Calculate the optimal contract {(qL**, tL**), (qH**, tH**)} under incomplete
information. We know that qH** = qH* and that only the IRL and I CH constraints are binding.
So qH'k‘k = 20,
16qL _ ti = 0, _‘ tH = _ ii.
The firm’s profit is given by
1': = 80[tL — 0.5qu] + 20[tH — 0.5qH2]. From the IRU we get tL 2 16th. Substituting it into the [CH and solving, we get
tH = 20(qH — qL) + 16qL. Substituting for tL and tH as well as qH, we obtain 1': = 80[16qL — 0.543] + 20[400 — 4% — 0.5(400)]. ll Maximizing with respect to qL, we get qL** = 15 and tL** = $240. Then get qH**
20 and tL** = $340. 0 Question 64’ (i) Firm A takes q]; as given and chooses qA to solve 3
max RAMA. £113) = (60 — 3 L14 — q") (1.4-
«1.4 Similarly, Firm B takes (1,; as given and chooses (73 to solve A 3
111'“ Malibu LIE) = (60 — 3113 — (Li) (13- ‘11} (ii) A's first—order condition for a maximum is 0 = EMA/0“ = 60—3q,‘ —q3. Similarly,
0 = Una/0113 2 6‘0 — 3qB - (1,1. Hence, the best response functions are downward
sloping and are given by qA 2 Rm”) 2 20 — q?“ and q” = 123014): 20 — (iii) Solving the two best—response functions yields qA = (“3 = 15. Substituting into the demand functions yield 1,7,4 = 1113 = 45/2 = 22.5. Substituting into the profit
functions yields 7” = 7r}; = 675 2 = 337.5. (iv) Solving for £14 and q]; from the above inverse demand functions obtains 6 4 6 4
£14 =24-Ep,4+gp3 and 93:24—EPB‘l'gPA- (v) Firm A takes p3 as given and chooses p,‘ to solve _ 6 4
1113X7TA(PAy1)B) = p14 24 — -p_4 + FPB -
17A 5 o Firm B takes p4 as given and chooses pH to solve 6 4
max WB(P.4,1JB) = p3 (‘24 — — pa + — m) ~
PM 5 5 (vi) A's first—order condition is: 0 = anti/612A 2 4(30 — 31),; +pB)/5. B's first—order
condition is 0 = Bug/tips; = 4(30 — 3113 +pA)/5. Hence, the upward sloping
best—response functions are 1 1
PA = R.4(PB);10+TPB and P8 = RBlPA)=10+§I-’A- 3
(vii) Solving the two price best-response functions yield 7);; = [)3 = 15. Substituting into
the above direct demand functions obtains (1,4 = qB = 18. The resulting profits are
7m 2 rm 2 270. (viii) The price game yields a lower price than the quantity game (15 versus 22.5). There—
fore, consumers buy 6 more units of each brand (36 versus 30). The price game yields
lower profit to each brand—producing firm (270 as opposed to 337.5). This means
that a price game results in a more intense competition relative to a quantity—setting
competition game. 17 ...

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