I did mean Q

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?


Economics/Math - economyst, Wednesday, November 4, 2009 at 9:15am
Take a shot, what do you think?

query. By Q2, do you really mean Q^2?

I did mean Q^2. Again, my issue is that I don't know how to calculate the output to maximize profits. I appreciate any help. Thank you.

a) could be short run, could be long run. Not enough information to tell.

b) MC is the first derivitive of the Total Cost. So, MC = 2*Q
c) Fixed cost are cost that the firm incurs, regardless of the level of production. In your problem, FC=100.
d) maximize where MC=MR. Here 2*Q=60. Solve for Q. (I get Q=30)

e) profit is total revenue minus total costs. TR=30*60 = 1800. TC=100+(30)^2 = 1000. Ergo, total profit is $800.

To answer these questions, let's go through each question step by step:

a. Is this a short-run cost function?

To determine whether this is a short-run cost function, we need to consider whether there are any fixed costs involved. In this case, we have a term of "100" in the cost equation, which represents the fixed cost component. Therefore, we can conclude that this is a short-run cost function.

b. What is the marginal cost function?

The marginal cost (MC) represents the additional cost incurred for producing one more unit of output. To find the marginal cost function, we need to take the derivative of the cost function with respect to the quantity (Q).

Here, the cost function is C = 100 + Q^2. So, the derivative of this equation with respect to Q is MC = dC/dQ = 2Q.

Therefore, the marginal cost function is MC = 2Q.

c. What is the level of total fixed cost?

The total fixed cost (TFC) represents the cost that does not change with the level of output. In this case, the fixed cost component is given as "100" in the cost equation, so 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 maximize profit, we need to find the quantity where marginal cost (MC) is equal to the price of chemicals.

Here, the marginal cost function is MC = 2Q and the price is $60. So, we set MC equal to the price and solve for Q:

2Q = 60
Q = 30

Therefore, to maximize profit, the quantity of chemicals that should be produced is 30.

e. What will be the level of profits?

To calculate the level of profits, we need to subtract the total cost (C) from the total revenue (R).

The total revenue (R) is given by the price (P) multiplied by the quantity (Q):

R = P * Q = 60 * 30 = $1800

The total cost (C) is given by the cost function C = 100 + Q^2:

C = 100 + (30)^2 = 100 + 900 = $1000

Therefore, the level of profits will be Total Revenue - Total Cost:

Profit = R - C = $1800 - $1000 = $800.