Suppose the cost function associated with a product is C(x) = cx + F dollars and the revenue function is R(x) = sx, where c denotes the unit cost of production, s the unit selling price, F the fixed cost incurred by the firm, and x the level of production and sales. Find the break-even quantity and the break-even revenue in terms of the constants c, s, and F.
Break-even quantity and cost:
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To find the break-even quantity, we need to determine the level of production and sales at which the cost equals the revenue.
The cost function is given by C(x) = cx + F, where x represents the level of production and sales.
The revenue function is given by R(x) = sx.
At the break-even point, the cost equals the revenue. So, we have:
Cx + F = sx
To find the break-even quantity, we need to solve this equation for x.
First, let's rearrange the equation to isolate x:
Cx - sx = -F
Factor out x:
x(C - s) = -F
Divide both sides by (C - s):
x = -F / (C - s)
This equation gives us the break-even quantity in terms of the constants c, s, and F.
To find the break-even cost, substitute this value of x back into the cost function:
C(x) = c(-F / (C - s)) + F
Simplify the expression:
C(x) = -Fc / (C - s) + F
Thus, the break-even quantity is -F / (C - s) and the break-even cost is -Fc / (C - s).
Next, let's find the break-even revenue.
Substitute the break-even quantity into the revenue function:
R(x) = s(-F / (C - s))
Simplify the expression:
R(x) = -Fs / (C - s)
Hence, the break-even revenue is -Fs / (C - s).
To find the break-even quantity, we need to determine the point at which the cost equals the revenue. In other words, we need to find the value of x for which C(x) = R(x).
Given the cost function C(x) = cx + F and the revenue function R(x) = sx, we can set them equal to each other:
cx + F = sx
Next, let's isolate x by moving the terms with x to one side of the equation:
cx - sx = -F
Factor out x on the left side:
x(c - s) = -F
Divide both sides by (c - s) to solve for x:
x = -F / (c - s)
This is the break-even quantity. It represents the level of production and sales at which the cost equals the revenue.
To find the break-even cost (or break-even revenue), we can substitute the break-even quantity (x) into either the cost function or the revenue function.
Let's use the cost function: C(x) = cx + F
Break-even cost = C(x) = c * ( -F / (c - s) ) + F
Simplifying this expression will give us the break-even cost in terms of c, s, and F.
Similarly, you can find the break-even revenue by substituting the break-even quantity into the revenue function R(x) = sx.
just did this in my homework...
break even quantity
F/(s-c)
Revenue
s(f/(s-c))