a)The subject of calculus is an area of mathematics called the classical optimization model. It has many business and economic applications. It is concerned mainly with optimization i.e. the best combination of variables that will either maximize benefits or minimize losses. In the context of maximizing contribution, calculus is used in determining the level of production that will maximize profit or contribution. This is achieved through the concept of the rate of change in y, the dependent variable, given a change in x, the independent variable. Once a functional relationship between these two variables is established, then through the process of differentiation an optimal output level is determined.
In calculas if y = f(x), then at maximum or minimum the rate of change of „y? with respect to x will be Zero.
For instance, suppose the Revenue “R” and Quantity “Q” are relate.d by the formula:
R = 15q – ¾ q2 Then at optimal production level:
(R 3
= 15 - q = 0.
(q 2
3 15 x 2
15 = q or q = = 10
2 3
So that maximum revenue should be achieved at a production level of 10 units.
The same concept is used for minimization of costs. Suppose costs are related to the units produced as follows:
C = x2 – 10x + 200.
(C
Then = 2x – 10.
(x
(c
At minimum level, = 0, 2x – 10 = 0
(x
2x = 10
X = 5
The cost will be minimized at a production level of 5 units.
(iThe subject of calculus is an area of mathematics called the classical optimization model. It has many business and economic applications. It is concerned mainly with optimization i.e. the best combination of variables that will either maximize benefits or minimize losses. In the context of maximizing contribution, calculus is used in determining the level of production that will maximize profit or contribution. This is achieved through the concept of the rate of change in y, the dependent variable, given a change in x, the independent variable. Once a functional relationship between these two variables is established, then through the process of differentiation an optimal output level is determined.
In calculas if y = f(x), then at maximum or minimum the rate of change of „y? with respect to x will be Zero.
For instance, suppose the Revenue “R” and Quantity “Q” are related by the formula:
R = 15q – ¾ q2 Then at optimal production level:
(R 3
= 15 - q = 0.
(q 2
3 15 x 2
15 = q or q = = 10
2 3
So that maximum revenue should be achieved at a production level of 10 units.
The same concept is used for minimization of costs. Suppose costs are related to the units produced as follows:
C = x2 – 10x + 200.
(C
Then = 2x – 10.
(x
(c
At minimum level, = 0, 2x – 10 = 0
(x
2x = 10
X = 5
The cost will be minimized at a production level of 5 units.
b)i) Profit will be maximized at a sales level in which marginal Revenue = Marginal Cost.
Company A sells goods at Sh. 200 per unit to company B. The unit selling price to B is fixed at Sh. 200/=. Let “q” be the optimal sales unit.
Then the Revenue function for firm A is:
R = q x 200 = 200q.
(R And Marginal Revenue, = 200.
(q
The cost function for firm A is: C = 2q2 + 40q + 80
(R
The marginal cost is then: ( 4q ( 40
(q
(R (c At maximum profit: (
(q (q
i.e. 200 = 4q + 40 160 = 4q 40 = q.
A will maximize its profit at a weekly sale of 40 units.
Revenue function for firm B R = 1000q – 16q2.
At weekly purchase of 40 units:
R = 1000 x 40 – 16 x (40)2 R = 40,000 – 25,600
R = 14,400 per week.
Weekly cost:
C = 2q2 + 80q + 400
= 2 x (40)2 + 80 + 40 + 400
= 3,200 + 3,200 + 400
= 6,800.
Add cost of sales:
40 units @200 = 8,000 Total cost 14,800
Less Revenue 14,400
Weekly loss (400)
(ii) Company B will maximize profit if its Marginal Revenue = Marginal Cost.
Given the TR function R = 1000q – 16q2, the MR function is obtained as follows:
(R
MR = = 1000 – 32q………….. (1)
(q
Given the Total Cost, C = 2q2 + 80q + 400, the MC function is obtained as follows:
(C MC = = 4q + 80
(q
Equating the two functions and solving for q:1000 – 32q = 4q + 80.
920 = 36q 25.6 = q
q = 25 units
A weekly sales of 25 units will maximize the profit of firm B.
(iii) When the two firms merge, and in the absence of any tangible benefits to be brought by the merger, we simply add their total revenue and cost as if they were operating individually. We can then apply calculus to establish profit maximizing output and weekly profit as shown below:
Total Revenue:
From A Ltd = 200q per week
From B Ltd = 1000q – 16q2
Total R
= 1200q – 16q2
(R
(MR =
(q Total cost:
=
1200 – 32q
A Ltd
=
2q2 + 40q + 80
B Ltd
=
2q2 + 80q + 400
Total cost C =
4q2 + 120q + 480
(c
(MC = = 8q + 120
(q
(c (R At maximum (
(q (q
8q + 120 = 1200 – 32q
40q = 1080 q = 27.
Profit maximizing output = 27 units per week.
The weekly profit is established by substituting q = 27 in the profit function
P = TR – TC.
First, Total Revenue = 1200q – 136q2
= 1200 x 27 – 16 x (27)2 = 20,736
And Total Cost = 4q2 + 120q x 480
= 4 x (27)2 x 120 x 27 + 480 = 6,636
(Weekly profit 14,100
) Profit will be maximized at a sales level in which marginal Revenue = Marginal Cost.
Company A sells goods at Sh. 200 per unit to company B. The unit selling price to B is fixed at Sh. 200/=. Let “q” be the optimal sales unit.
Then the Revenue function for firm A is:
R = q x 200 = 200q.
(R And Marginal Revenue, = 200.
(q
The cost function for firm A is: C = 2q2 + 40q + 80
(R
The marginal cost is then: ( 4q ( 40
(q
(R (c At maximum profit: (
(q (q
i.e. 200 = 4q + 40 160 = 4q 40 = q.
A will maximize its profit at a weekly sale of 40 units.
Revenue function for firm B R = 1000q – 16q2.
At weekly purchase of 40 units:
R = 1000 x 40 – 16 x (40)2 R = 40,000 – 25,600
R = 14,400 per week.
Weekly cost:
C = 2q2 + 80q + 400
= 2 x (40)2 + 80 + 40 + 400
= 3,200 + 3,200 + 400
= 6,800.
Add cost of sales:
40 units @200 = 8,000 Total cost 14,800
Less Revenue 14,400
Weekly loss (400)
(ii) Company B will maximize profit if its Marginal Revenue = Marginal Cost.
Given the TR function R = 1000q – 16q2, the MR function is obtained as follows:
(R
MR = = 1000 – 32q………….. (1)
(q
Given the Total Cost, C = 2q2 + 80q + 400, the MC function is obtained as follows:
(C MC = = 4q + 80
(q
Equating the two functions and solving for q:1000 – 32q = 4q + 80.
920 = 36q 25.6 = q
q = 25 units
A weekly sales of 25 units will maximize the profit of firm B.
(iii) When the two firms merge, and in the absence of any tangible benefits to be brought by the merger, we simply add their total revenue and cost as if they were operating individually. We can then apply calculus to establish profit maximizing output and weekly profit as shown below:
Total Revenue:
From A Ltd = 200q per week
From B Ltd = 1000q – 16q2
Total R
= 1200q – 16q2
(R
(MR =
(q Total cost:
=
1200 – 32q
A Ltd
=
2q2 + 40q + 80
B Ltd
=
2q2 + 80q + 400
Total cost C =
4q2 + 120q + 480
(c
(MC = = 8q + 120
(q
(c (R At maximum (
(q (q
8q + 120 = 1200 – 32q
40q = 1080 q = 27.
Profit maximizing output = 27 units per week.
The weekly profit is established by substituting q = 27 in the profit function
P = TR – TC.
First, Total Revenue = 1200q – 136q2
= 1200 x 27 – 16 x (27)2 = 20,736
And Total Cost = 4q2 + 120q x 480
= 4 x (27)2 x 120 x 27 + 480 = 6,636
(Weekly profit 14,100
Moraa orina answered the question on April 2, 2018 at 18:13
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