1) Lowering the price by 10%. What is the break-even level of output?
The company produces specialized glass units and is concerned that, as the market leader, they should be able to make higher profits than they currently are. The firm’s costs are as follows; overheads ‘d’ are £800,000 labour costs per unit ‘e’ are £100 and raw material costs per unit ‘f’ are £258. They also spend £40,000 ‘X’ per period on advertising and are capable of producing up to 1,000 units in each period.
The total cost function is expressed as;
TC = d + eQ + fQ + X
The demand function is as follows;
P = a – bQ + c√X
where a = 4,000, b = 3 and c = 3.
2) Doubling the advertising budget. What is the break- even level of the output?
The company produces specialized glass units and is concerned that, as the market leader, they should be able to make higher profits than they currently are. The firm’s costs are as follows; overheads ‘d’ are £800,000 labour costs per unit ‘e’ are £100 and raw material costs per unit ‘f’ are £258. They also spend £40,000 ‘X’ per period on advertising and are capable of producing up to 1,000 units in each period.
The total cost function is expressed as;
TC = d + eQ + fQ + X
The demand function is as follows;
P = a – bQ + c√X
where a = 4,000, b = 3 and c = 3.
1)Lowering the price by 10%. What is the break-even level of output
2) Doubling the advertising budget. What is the break- even level of the output?
Interestingly, you are asking for the break-even level of output rather than the profit-maximizing level. Hummm.
Anyway. This is simply an algebra problem. First collapse the two Q terms in TC. So, TC=800,000+358Q + 40000
Total Revenue is P*Q. So, TR=4000Q-3Q^2 + 600Q
Set TC=TR and solve for Q (subject to the constraint that Q<=1000)
1) Repeat cept multiply all terms in the TR equation by 0.90
2) Repeat cept change advertising by 40K.
To find the break-even level of output, we need to determine the quantity at which the total revenue equals the total cost. Let's start with the first question.
1) Lowering the price by 10%. What is the break-even level of output?
Total Cost (TC) is given by the equation:
TC = d + eQ + fQ + X
Total Revenue (TR) is calculated as:
TR = P * Q
First, we need to determine the price (P) after lowering it by 10%. Since the original price (a) is £4,000, the new price would be 90% of that:
New price (P) = 0.9 * a = 0.9 * £4,000 = £3,600
The demand function (P = a - bQ + c√X) represents the relationship between price (P), quantity (Q), and advertising budget (X). Rearranging the equation to solve for quantity (Q), we have:
Q = (a - P + c√X) / b
Now, substituting the values into the equation:
Q = (4,000 - 3,600 + 3√X) / 3
To find the break-even level of output, we need to find the quantity (Q) at which total revenue (TR) equals total cost (TC). This means that TR = TC:
P * Q = d + eQ + fQ + X
Substituting the values, we get:
(0.9 * 4,000) * Q = 800,000 + 100Q + 258Q + 40,000
Now, we can solve for Q:
(0.9 * 4,000 - 100 - 258) * Q = 800,000 + 40,000
(3,600 - 358) * Q = 840,000
3,242 * Q = 840,000
Q = 840,000 / 3,242
Q ≈ 259
Therefore, the break-even level of output, after lowering the price by 10%, is approximately 259 units.
Now, let's move on to the second question.
2) Doubling the advertising budget. What is the break-even level of output?
To determine the new break-even level of output, we need to double the advertising budget (X). Let's denote the new advertising budget as 2X.
Using the same demand function (Q = (a - P + c√X) / b), we can substitute 2X for X. Thus, the new equation becomes:
Q = (a - P + c√(2X)) / b
Again, we want to find the quantity (Q) at which total revenue (TR) equals total cost (TC):
P * Q = d + eQ + fQ + 2X
Substituting the values, we get:
(4,000) * Q = 800,000 + 100Q + 258Q + 2(40,000)
4,000 * Q = 800,000 + 100Q + 258Q + 80,000
4,000Q = 880,000 + 358Q
3,642Q = 880,000
Q ≈ 241
Therefore, the break-even level of output, after doubling the advertising budget, is approximately 241 units.