1. A firm manufactures and markets a product that sells for Birr 20 per unit. Fixed costs associated with activity total Birr 40,000 a month, while variable cost per unit is Birr 10. A maximum of 10,000 units can be produced and sold.

TR (Total Revenue) function:

TR = Quantity * Selling Price
TR = Q * 20

TC (Total Cost) function:
TC = Fixed Costs + (Variable Cost per unit * Quantity)
TC = 40,000 + (10 * Q)

Total Profit function:
Total Profit = TR - TC
Total Profit = Q * 20 - (40,000 + 10 * Q)

Break-even point:
To find the break-even point, we set Total Profit function equal to zero:
Q * 20 - (40,000 + 10 * Q) = 0

Simplifying the equation:
20Q - 40,000 - 10Q = 0
10Q - 40,000 = 0
10Q = 40,000
Q = 40,000 / 10
Q = 4,000 units

The break-even point is 4,000 units.

New TC function with increased FC:
New TC = (Fixed Costs + 10,000) + (10 * Q)

New break-even point with increased FC:
Solve for Q when New TC = TR

New TC function with decreased unit variable costs:
New TC = Fixed Costs + (0.8 * 10 * Q)

New break-even point with decreased unit variable costs:
Solve for Q when New TC = TR

New TR function with increased unit selling price:
New TR = Q * (1.2 * 20)

New break-even point with increased unit selling price:
Solve for Q when New TR = New TC

Relationship between BEP & FC:
As fixed costs increase, the break-even point will increase. Higher fixed costs mean a larger quantity needs to be sold to cover the increased fixed costs and reach the break-even point.

Relationship between P & BEP:
The break-even point is where the total profit is zero. If the total profit is positive, the company is making a profit, indicating that the break-even point has been surpassed. If the total profit is negative, the company is incurring losses, indicating that the break-even point has not been reached.

Relationship between V & BEP:
Variable costs per unit affect the break-even point. Higher variable costs mean a larger quantity needs to be sold to cover the increased variable costs and reach the break-even point.

Effect of 10% increase in selling price and volume:
To calculate the new break-even point, we need to multiply the original break-even point by 1.1 (10% increase in both price and volume):
New BEP = 4,000 * 1.1 = 4,400 units

Maximum profit and output level:
To find the maximum profit, we need to maximize the Total Profit function. Since fixed costs do not change, the maximum profit will be achieved at the maximum output level, which is 10,000 units.
Maximum Profit = 10,000 * 20 - (40,000 + 10 * 10,000)

Maximum unit variable cost for breaking even:
To calculate the maximum unit variable cost for breaking even, we set the Total Profit function equal to zero and solve for the variable cost per unit:
20Q - (40,000 + V * Q) = 0
V * Q = 20Q - 40,000
V = (20Q - 40,000) / Q
V = 20 - (40,000 / Q)

Profit of Birr 100,000:
Q * 20 - (40,000 + 10 * Q) = 100,000
Solve for Q when Total Profit = 100,000

Loss of Birr 10,000:
Q * 20 - (40,000 + 10 * Q) = -10,000
Solve for Q when Total Profit = -10,000