The following is a cost function for clinic visits in a small inner city clinic:
a.Determine the marginal cost for each level of output.
b.If the price per visit is given to be $25, at what level of visits will the maximum profit position be? What are the profits at this level? What is the quantity supplied?
c.If the price per visit increases to $45, what will be the quantity supplied (assuming maximizing profits)?
To determine the marginal cost for each level of output, you need to find the derivative of the cost function. Let's assume that the cost function for clinic visits is given as follows:
C(q) = 1000 + 200q - 5q^2 + 0.1q^3
where q is the number of visits (output) and C(q) is the cost.
a. To find the marginal cost (MC), you need to take the derivative of the cost function with respect to q:
MC = dC/dq
Taking the derivative of the cost function, we get:
MC = d/dq (1000 + 200q - 5q^2 + 0.1q^3)
= 200 - 10q + 0.3q^2
Now you have the marginal cost equation. To find the marginal cost at each level of output, simply substitute the value of q into the equation.
b. To determine the level of visits that will result in maximum profit, you need the revenue function. Assuming the price per visit is $25, the revenue function (R) can be calculated as:
R(q) = P * q
= 25q
To maximize profit, set the marginal cost equal to the marginal revenue (MR) and solve for q:
MC = MR
200 - 10q + 0.3q^2 = 25
Solving this equation will give you the level of visits that maximizes profit. Once you find the value of q, substitute it into the cost function to find the cost, and then subtract the cost from the revenue to obtain the profit.
c. If the price per visit increases to $45, you would follow the same steps as in part b, but now using the new price value of $45. Set MC equal to MR and solve for q:
200 - 10q + 0.3q^2 = 45
Solving this equation will give you the quantity supplied (level of visits) that maximizes profit at the higher price.