# 2 and Profit

Topics: 2, The Profit, Integer Pages: 17 (379 words) Published: February 21, 2014
﻿Problem 15 Suppose that any firm intending to produce SOMA must build an integer number of plants: 0, 1, 2,.... Building Q plants costs each firm 3.5 × Q dollars. Each plant produces one unit of SOMA. If firm 1 builds Q1 plants and firm 2 builds Q2 plants, the market price p for one unit of SOMA will be 9 − (Q1 + Q2). For example, if firm 1 builds 2 plants and firm 2 builds 4 plants, the market price will be 9 − (2 + 4) = 3 per unit. At this price firm 1 will make a profit of 2 × 3 − 2 × 3.5 = −1 while firm 2 will make a profit of 4×3−4×3.5 = −2. Assume, no firm will build more than 4 plants.

Cost (Q) = 3.5 * Q
Price = 9 -(Q1+Q2)

1. Set up a 4-by-4 table that records the profits of each firm for each possible choice of the number of plants to build. Then, use your table to answer the questions below.

Price

Firm 2

0
1
2
3
4

0
9
8
7
6
5
Firm 1
1
8
7
6
5
4

2
7
6
5
4
3

3
6
5
4
3
2

4
5
4
3
2
1

Revenue
Firm 2

Rev 1
Rev 2
Rev 1
Rev 2
Rev 1
Rev 2
Rev 1
Rev 2
Rev 1
Rev 2

0
0
1
1
2
2
3
3
4
4
Firm 1
0
0
0
0
8
0
14
0
18
0
20

1
8
0
7
7
6
12
5
15
4
16

2
14
0
12
6
10
10
8
12
6
12

3
18
0
15
5
12
8
9
9
6
8

4
20
0
16
4
12
6
8
6
4
4

Cost
Firm 2

Cost 1
Cost 2
Cost 1
Cost 2
Cost 1
Cost 2
Cost 1
Cost 2
Cost 1
Cost 2

0
0
1
1
2
2
3
3
4
4
Firm 1
0
0.0
0.0
0.0
3.5
0.0
7.0
0.0
10.5
0.0
14.0

1
3.5
0.0
3.5
3.5
3.5
7.0
3.5
10.5
3.5
14.0

2
7.0
0.0
7.0
3.5
7.0
7.0
7.0
10.5
7.0
14.0

3
10.5
0.0
10.5
3.5
10.5
7.0
10.5
10.5
10.5
14.0

4
14.0
0.0
14.0
3.5
14.0
7.0
14.0
10.5
14.0
14.0

Profit
Firm 2

Profit 1
Profit 2
Profit 1
Profit 2
Profit 1
Profit 2
Profit 1
Profit 2
Profit 1
Profit 2

0
0
1
1
2
2
3
3
4
4
Firm 1
0
0.0
0.0
0.0
4.5
0.0
7.0
0.0
7.5
0.0
6.0

1
4.5
0.0
3.5
3.5
2.5
5.0
1.5
4.5
0.5
2.0

2
7.0
0.0
5.0
2.5
3.0
3.0
1.0
1.5
(1.0)
(2.0)

3
7.5
0.0
4.5
1.5
1.5
1.0
(1.5)
(1.5)
(4.5)
(6.0)

4
6.0
0.0
2.0
0.5
(2.0)
(1.0)
(6.0)
(4.5)
(10.0)
(10.0)

2. If there was a single firm capable of producing SOMA, how many plants would it install to maximize profit?

Three plants would need to be installed to maximize the profit by either firm.

Profit
Firm 2

Profit 1
Profit 2
Profit 1
Profit 2
Profit 1
Profit 2
Profit 1
Profit 2
Profit 1
Profit 2

0
0
1
1
2
2
3
3
4
4
Firm 1
0
0.0