Page 30 - Azerbaijan State University of Economics

Salah Salimian, Kiumars Shahbazi, Mahdi M. Beknazar: Locating The Sale Agents In Spoke Model
Through Uniform Distribution Of Consumers

Where, γ is the total profit of the whole sale agents (γ unit for each sale agent). By
solving and simplification of the above equation, the following result will be achieved:

2
1

2 − (2 − )( ) − + s − s B + +
M
p = 2 1 (9)

( − 2 ) − + 2 − +
2

Therefore, the monopolist's profit in this state equals to:

1

2

M

π = (2 − )( ) − 2 + − s + s B − − 10)
2

The comparison of equations 3 and 10 shows that if the monopolist agrees with
launching one sale agent at center instead of several sale agents at margins and the
following equation is established:

2

− ( ) − F − γ
1

2

≥ (2 − )( ) − 2 + − + −
2
−

Solving this inequality in respect to F will yield the following result:

2
2 ( ) + ( − −2 ) − ( −1)
≥ 2 (11)
−1

This result could have main consequences for monopolist who wants to distribute
several sale agents in city.

This theorem confirms the theorem presented by Mathewson & Winter (1986) who
claimed that if the irrecoverable costs related to launching sale agent are high, then
the manufacturer will launch just one sale agent at city center. Moreover, it is shown
that in case of small city, the manufacturer just agrees with launching a sale agent at
city center (Shy, 1995). In this state, concerning the monopolist's profit function, the
maximum optimal number of sale agents equal to:

1

2
− ( ) −
≤ 2
2

2 − 2 ( ) − + +

Third state: Launching a sale agent at city center and several sale agents at city
margins
Now the next question could be proposed in this way that when would the monopolist
agree with launching a sale agent at city center and several sale agents at city margins
rather than just one sale agent at city center? The answer to this question is presented
in theorem 3.
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