Assuming a profit function as p=90-2q and the cost function as c=10+0.5q2.find the profile maximization output and price
A profit function such as that means maximum profit at zero sold. Am I reading that correctly?
Solution
0.5q2+2q-80
To find the profit maximization output and price, we can use the concept of marginal analysis. The profit function is given as p = 90 - 2q, and the cost function is given as c = 10 + 0.5q^2.
The profit equation is calculated by subtracting the cost function from the revenue function. In this case, the revenue function is given by p*q, where q represents the quantity produced.
So, the profit equation is:
Profit (π) = Revenue (p*q) - Cost (c)
Or, π = p*q - c
To maximize profits, we need to differentiate the profit function with respect to q and set it equal to zero. This will give us the value of q that maximizes profit.
Taking the derivative of the profit function with respect to q:
dπ/dq = p - (dc/dq)
Now, let's calculate the derivative of the cost function with respect to q:
dc/dq = d(10 + 0.5q^2)/dq = 0.5*2q = q
Substituting this into the profit equation:
dπ/dq = p - q
We know that dπ/dq = 0 for profit maximization, so:
0 = p - q
Substituting the profit function p = 90 - 2q into the above equation:
0 = 90 - 2q - q
0 = 90 - 3q
Now, solve for q:
3q = 90
q = 90/3
q = 30
So, the profit-maximizing output is q = 30.
To find the price, substitute the value of q into the profit equation:
p = 90 - 2q
p = 90 - 2*30
p = 90 - 60
p = 30
Therefore, the profit-maximizing price is p = 30.