Suppose you are the manager of a small chemical company operating in a competitive market. Your cost of production can be expressed as C = 100 + Q2, where Q is the level of output and C is total cost.
a. Is this a short-run cost function?
b. What is the marginal cost function?
c. What is the level of total fixed cost?
d. If the price of chemicals is $60, what quantity of chemicals should be produced to maximize profit?
e. What will be the level of profits?
Take a shot, what do you think?
query. By Q2, do you really mean Q^2?
To answer these questions, we will go step-by-step and explain how to approach each one:
a. Is this a short-run cost function?
To determine if this is a short-run cost function, we need to consider the time frame involved. The short run is a period in which at least one factor of production is fixed, while the long run is a period in which all factors of production are variable. In this case, since the cost function does not provide information about any fixed or variable factors, we cannot definitively say if it is a short-run cost function or not.
b. What is the marginal cost function?
The marginal cost (MC) function represents the increase in total cost that results from producing one additional unit of output. To calculate the marginal cost, we need to take the derivative of the cost function with respect to output (Q).
In this case, C = 100 + Q^2, we can find the derivative as:
MC = dC/dQ = d/dQ(100 + Q^2)
MC = 2Q
Therefore, the marginal cost function is MC = 2Q.
c. What is the level of total fixed cost?
To find the level of total fixed cost, we need to identify the part of the cost function that is not dependent on the level of output (Q). In this case, the fixed cost can be identified as the constant component of the cost function, which is $100.
Therefore, the level of total fixed cost is $100.
d. If the price of chemicals is $60, what quantity of chemicals should be produced to maximize profit?
To find the quantity of chemicals that should be produced to maximize profit, we need to equate the marginal cost (MC) to the price (P).
Here, the price of chemicals is given as $60, so we set the MC equal to $60 and solve for Q:
2Q = 60
Q = 30
Therefore, the quantity of chemicals that should be produced to maximize profit is 30 units.
e. What will be the level of profits?
To calculate the level of profits, we need to subtract the total cost (TC) from the total revenue (TR), which is equal to price (P) multiplied by the quantity (Q).
Given that the price of chemicals is $60 and the quantity is 30 units, we can calculate the revenue as:
TR = P * Q = $60 * 30 = $1800
To find the total cost, we substitute the quantity (Q) into the cost function:
C = 100 + Q^2 = 100 + 30^2 = 100 + 900 = $1000
Finally, we calculate the profit by subtracting the total cost from the total revenue:
Profit = TR - TC = $1800 - $1000 = $800
Therefore, the level of profits will be $800.