Suppose a firm faces a downward sloping demand curve givven by the equation 1=100-1/3P. The firm's cost function is given by the equation C=30+1/4Q^2. Find the profit maximizing level of output.
always always always, MC=MR.
First rearrange the demand function to be P=f(Q). That is P=300 - 3Q
Now then Total revenue is P*Q. So TR=300Q -3(Q^2)
MR is the first derivitive of TR. So MR=300 - 6Q
MC is the first derivitive of TC. So MC=(1/2)Q
MC=MR - use algebra and solve for Q. Take it from here
To find the profit-maximizing level of output, we need to determine the level of output at which the firm's marginal revenue (MR) equals its marginal cost (MC).
First, let's find the marginal revenue (MR) function. The marginal revenue is the additional revenue earned from selling one more unit of output. In this case, it is the derivative of the demand function.
The demand function is given by:
1 = 100 - (1/3)P
To find P, we need to rearrange the equation:
P = 300 - 3Q
Now, let's find the marginal revenue (MR):
MR = d(TR) / dQ
The total revenue (TR) is given by:
TR = P * Q
Differentiating TR with respect to Q, we get:
MR = d(PQ) / dQ
MR = P + Q * dP / dQ
Substituting the value of P, we have:
MR = (300 - 3Q) + Q * (-3)
MR = 300 - 3Q - 3Q
MR = 300 - 6Q
Next, let's find the marginal cost (MC) function. The marginal cost is the additional cost incurred from producing one more unit of output. In this case, it is the derivative of the cost function.
The cost function is given by:
C = 30 + (1/4)Q^2
To find MC, we need to differentiate C with respect to Q:
MC = dC / dQ
MC = d(30 + (1/4)Q^2) / dQ
MC = (1/4) * 2 * Q
MC = (1/2)Q
Now we have the marginal revenue (MR) and marginal cost (MC) functions.
To determine the profit-maximizing level of output, we set MR equal to MC and solve for Q:
300 - 6Q = (1/2)Q
First, let's combine like terms:
300 = (6Q + Q) / 2
Multiply both sides by 2 to eliminate the denominator:
600 = 6Q + Q
600 = 7Q
Divide both sides by 7 to solve for Q:
Q = 600 / 7
Therefore, the profit-maximizing level of output is Q = 85.71 (rounded to two decimal places).
Now, to find the corresponding price (P), we can substitute this value of Q into the demand function:
P = 300 - 3Q
P = 300 - 3 * 85.71
P = 300 - 257.14
P = 42.86 (rounded to two decimal places).
Hence, the profit-maximizing level of output is approximately 85.71 units, and the price associated with it is approximately $42.86.