Suppose a multi-monopolist has marginal cost functions:
[tex]
\begin{array}{l}
MC_1=20+2 q_1, \text { and } \
MC_2=10+5 q_2
\end{array}
[/tex]
Where, [tex]$MC_1$[/tex] is the marginal cost of the first plant and [tex]$MC_2$[/tex] is the marginal cost of the second plant. Besides, [tex]$q_1$[/tex] and [tex]$q_2$[/tex] represent output to be produced using the first and second plants, respectively. If the monopolist is minimizing its short run cost by producing 5 units of output in the first plant, then:
A. How many units of output should the monopolist produce in the second plant in order to meet the market demand for its product by minimizing the total cost of production? Show the steps how you have determined [tex]$q_2$[/tex].
B. Prepare a schedule similar to 6.3 for both marginal costs for 1 up to 10 output levels and show how you can determine the total short run output of the monopolist.
Answer (2)
To minimize total cost, the monopolist equates marginal costs: M C 1 = M C 2 .
Calculate M C 1 when q 1 = 5 : M C 1 = 20 + 2 ( 5 ) = 30 .
Set M C 2 = M C 1 and solve for q 2 : 30 = 10 + 5 q 2 , which gives q 2 = 4 .
The monopolist should produce 4 units in the second plant.
Explanation
Problem Setup We are given the marginal cost functions for two plants: M C 1 = 20 + 2 q 1 and M C 2 = 10 + 5 q 2 . We know that the monopolist is producing 5 units of output in the first plant, so q 1 = 5 . Our goal is to find the quantity q 2 that the monopolist should produce in the second plant to minimize the total cost of production, and then to create a schedule of marginal costs for both plants.
Cost Minimization Condition To minimize the total cost of production, the marginal costs of both plants must be equal. That is, M C 1 = M C 2 . First, we need to find the marginal cost of the first plant when q 1 = 5 .
Calculating MC1 Substitute q 1 = 5 into the equation for M C 1 :
M C 1 = 20 + 2 ( 5 ) = 20 + 10 = 30
Equating Marginal Costs Now, set M C 1 equal to M C 2 and solve for q 2 :
30 = 10 + 5 q 2
Isolating q2 Subtract 10 from both sides: 20 = 5 q 2
Solving for q2 Divide both sides by 5: q 2 = 5 20 = 4 So, the monopolist should produce 4 units of output in the second plant to minimize the total cost of production.
Marginal Cost Schedule Now, let's create a schedule of marginal costs for both plants for output levels from 1 to 10.
Output Level
M C 1 = 20 + 2 q 1
M C 2 = 10 + 5 q 2
1
22
15
2
24
20
3
26
25
4
28
30
5
30
35
6
32
40
7
34
45
8
36
50
9
38
55
10
40
60
From this schedule, we can see how the marginal costs change as output levels increase. The monopolist will adjust production in each plant such that the marginal costs are equal. In the short run, with q 1 = 5 , M C 1 = 30 . To minimize costs, the monopolist sets M C 2 = 30 , which we already found corresponds to q 2 = 4 .
Final Answer The monopolist should produce 4 units of output in the second plant to minimize the total cost of production when it produces 5 units in the first plant. The marginal cost schedule shows how the marginal costs increase with output for both plants.
Examples
Consider a company producing widgets in two different factories. Each factory has its own cost structure, represented by marginal cost functions. By equating the marginal costs across factories, the company can determine the optimal production level for each factory to minimize the overall cost of producing a given quantity of widgets. This principle is widely used in operations management to optimize production and resource allocation across multiple facilities or processes.
The monopolist should produce 4 units in the second plant while producing 5 in the first plant to minimize total costs. A marginal cost schedule for both plants shows how marginal costs vary with output from 1 to 10, aiding in optimal production decisions.
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