A shoe company will make a new type of running shoe. The fixed cost for the production will be $54,000. The marginal cost will be $25 per pair of shoes. The shoes will sell for $115 for each pair.
a) Write the cost function
b) Write the revenue function
c) Find the number of shoes that must be sold for the company to break even.
d) How many shoes must be sold for the company to profit $180,000.
To answer these questions, we need to understand some basic economic concepts. Let's start by defining the terms:
- Fixed cost (FC): Costs that do not change with the level of output.
- Marginal cost (MC): The additional cost incurred to produce one more unit of output.
- Revenue (R): The income generated from selling a given quantity of goods.
a) To write the cost function, we need to consider the fixed cost and the marginal cost. The total cost (TC) can be calculated as the sum of the fixed cost and the product of the marginal cost per pair and the number of pairs produced (x). Therefore, the cost function (C) is:
C(x) = FC + MC * x
= $54,000 + $25x
b) The revenue function is determined by multiplying the selling price per pair (P) by the number of pairs sold (x). Therefore, the revenue function (R) is:
R(x) = P * x
= $115x
c) To find the number of shoes that must be sold for the company to break even, we need to equate the revenue and cost functions. In other words, we need to find the breakeven point where R(x) = C(x). Substituting the revenue and cost functions into the equation, we get:
$115x = $54,000 + $25x
Now, let's solve for x:
$115x - $25x = $54,000
$90x = $54,000
x = $54,000 / $90
x ≈ 600
Therefore, the company needs to sell approximately 600 pairs of shoes to break even.
d) To determine the number of shoes that must be sold for the company to profit $180,000, we need to find the level of output where the revenue exceeds the total cost by $180,000. In other words, we need to find the point where R(x) - C(x) = $180,000. Substituting the revenue and cost functions into the equation, we have:
$115x - ($54,000 + $25x) = $180,000
Let's solve for x:
$115x - $54,000 - $25x = $180,000
$90x - $54,000 = $180,000
$90x = $234,000
x = $234,000 / $90
x ≈ 2,600
Therefore, the company needs to sell approximately 2,600 pairs of shoes to make a profit of $180,000.