BEHAVIOR IN THE TOBACCO MARKET 141
3. CONCEPTUAL MODEL
Oligopoly behavior is modeled as a Nash equilibrium with con- jectural variations following Dixit 1988a. Domestic burley pur-
chases Q
1
and foreign burley purchases Q
2
are viewed as imperfect substitutes in the production of cigarettes, but are homogenous
within their country of origin. To simplify the analysis, the produc- tion function for manufacturers is assumed to be quadratic with
two inputs, domestic burley Q
1
purchased from the domestic market or government loan stocks and foreign burley Q
2
. Input prices p
1
and p
2
represent burley at the processed level. Manufac- turer’s output prices are normalized to one leading to the profit
function G Eq. 1:
G 5 a
1
Q
1
1 a
2
Q
2
2 0.5b
1
Q
2 1
1 b
2
Q
2 2
1 2kQ
1
Q
2
2 p
1
Q
1
2 p
2
Q
2
1
The first derivative of Equation 1 with respect to Q
1
and Q
2
can be used to derive the respective inverse demand functions for
domestic and foreign burley Eq. 2, 3:
p
1
5 a
1
2 b
1
Q
1
2 kQ
2
2 p
2
5 a
2
1 kQ
1
2 b
2
Q
2
3
where all of the parameters are positive and b
1
b
2
2 k
2
. 0.
Most burley tobacco used by cigarette manufacturers is pur- chased through contracts with dealers. Dealers purchase tobacco
from the market, from government loan stocks, or from foreign markets. The prices that manufacturers pay includes the purchase
price, financing, container costs, transportation costs, and pro- cessing fees. Thus, in most cases, it is the dealers who purchase
and process burley tobacco and deliver it to the manufacturers.
It is assumed that there are n
1
symmetric, domestic dealers that compete in the home market with n
2
symmetric, foreign dealers. For this analysis, a domestic dealer is a firm that buys and sells
burley tobacco produced in the United States, and a foreign dealer is a firm that buys and sells foreign produced burley. On the supply
side, consider the profit functions p
1
and p
2
for a typical dealer of domestic and foreign burley Eq. 4, 5:
p
1
5 p
1
2 c
1
q
1
4 p
2
5 p
2
2 c
2
2 t q
2
5
All domestic and foreign firms are assumed to have a constant marginal cost c
1
and c
2
, respectively which includes input prices
142 O. D. Chambers, M. R. Reed, and W. M. Snell
and all other costs. As usual, profit is defined as price minus cost times quantity. Equation 5 allows an import tariff t on foreign
burley entering the United States. The parameter v
ij
i,j 5 1,2 represents firm i’s conjecture about the quantity response of competing firm j. For example, each
home firm believes that if it increases output q
i
by one unit then domestic output will increase by one unit plus the increase in
output by the other n
1
2 1 domestic firms. Thus Q
1
will increase by 1 1 n
1
2 1v
11
. Similarly, Q
2
will increase by n
2
v
12
. The v
ii
s are continuous variables that reflect the level of competitiveness in
the industry, ranging from v
ii
5 1 for the collusive outcome to
v
ii
5 2 1 for the competitive outcome. Using this approach, the
first order conditions become Eq. 6, 7:
p
1
2 c
1
1 q
1
3
] p
1
] q
1
1 1 n
1
2 1v
11
1 ]
p
1
] q
2
n
2
v
12
4
6 p
2
2 c
2
2 t 1 q
2
3
] p
2
] q
2
1 1 n
2
2 1v
22
1 ]
p
2
] q
1
n
2
v
21
4
7
By aggregating Equations 6 and 7 over all dealers, the first- order conditions are:
p
1
2 c
1
2 Q
1
V
1
5 8
p
2
2 c
2
2 t 2 Q
2
V
2
5 9
where the aggregate conjectural variations parameters are de- fined as Eq. 10, 11:
V
1
5 [b
1
1 1 n
1
2 1v
11
1 kn
2
v
12
]n
1
10 V
2
5 [b
2
1 1 n
2
2 1v
22
1 kn
1
v
21
]n
2
11
Maximizing Equations 4 and 5, and assuming the firm behaves such that the firm’s own quantity decisions affect price, but the
output decisions do not affect other firms the typical Cournot assumptions, the first-order conditions are much simpler Eq.
12, 13:
p
1
2 c
1
1 q
1
] p
1
] q
1
5 12
p
2
2 c
2
2 t 1 q
2
] p
2
] q
2
5 13
Equilibrium prices and quantities are obtained using Equations 2, 3, 8, and 9 Eq. 14, 15:
BEHAVIOR IN THE TOBACCO MARKET 143
a
1
2 b
1
Q
1
2 kQ
2
2 c
1
2 Q
1
V
1
5 14
a
2
2 kQ
1
2 b
2
Q
2
2 c
2
2 t 2 Q
2
V
2
5 15
The explicit solutions is Eq. 16
3
Q
1
Q
2
4
5 1
D
3
b
2
1 V
2
2k 2k
b
1
1 V
1
4 3
a
1
2 c
1
a
2
2 c
2
2t
4
16
where D 5 b
1
1 V
1
b
2
1 V
2
2 k
2
. Solutions for p
1
and p
2
can be found by similar substitutions.
Assuming equal weights are assigned to each group’s welfare, total domestic welfare is given by the sum of manufacturer pro-
ducer surplus G, dealer profits p
1
, rent to domestic producers rQ
1
, and government revenue tQ
2
.
W 5 G 1 p
1
1 rQ
1
1 tQ
2
17
Using Equation 9 and the total differential of Equation 17 yields the optimal tariff
t 5 Q
2
V
2
1 Q
1
V
1
1 r k
b
1
1 V
1
. 18
4. RESULTS
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