Demand function P=50-Q
Average Cost 5Q + 40 +10/Q
Calculate the firm's total cost function
Find the marginal cost function and evaluate it at Q=2 and Q=3
What is the total revenue function
Find the firms's revenue maximising output level
Find the firm's profit function
Take a shot. What do you think the firm's cost and revenue functions are?
I thought maybe you could say the cost is 5Q + 40 + 10 but only on the basis that the average cost is divided by Q
My problem is I don;t understand the relationship between all the functions so no idea how to find the revenue function
I want to know how to solve it
To calculate the firm's total cost function, you need to find the sum of all costs incurred by the firm for a given level of output. The average cost function provided, 5Q + 40 + 10/Q, gives you the average cost per unit of output. To find the total cost function, you need to multiply the average cost by the quantity of output (Q). So, the firm's total cost function would be:
Total Cost (TC) = (5Q + 40 + 10/Q) * Q
To find the marginal cost function, you need to differentiate the total cost function (TC) with respect to quantity (Q). The derivative of TC with respect to Q will give you the marginal cost (MC) function.
MC = d(TC)/d(Q) = 5 - 10/Q^2
To evaluate the marginal cost function at Q=2 and Q=3, substitute the respective values of Q into the MC function:
For Q=2: MC(Q=2) = 5 - 10/(2^2) = 5 - 10/4 = 5 - 2.5 = 2.5
For Q=3: MC(Q=3) = 5 - 10/(3^2) = 5 - 10/9 ≈ 3.11
To find the total revenue function, you need to multiply the price (P) by the quantity (Q) sold. The demand function provided, P=50-Q, gives you the price as a function of quantity. So, the total revenue function (TR) is:
TR = P * Q = (50 - Q) * Q = 50Q - Q^2
To find the firm's revenue-maximizing output level, you need to find the quantity that maximizes the total revenue function (TR). In this case, you can use calculus to find the maximum point by taking the derivative of TR with respect to Q, which will give you the marginal revenue (MR) function. Set MR equal to zero and solve for Q. Alternatively, you can find the vertex of the parabolic TR function. In this case, the maximum occurs when MR = 0, which happens when Q = 25. So, the revenue-maximizing output level for the firm is 25 units.
To find the firm's profit function, you need to subtract the total cost (TC) from the total revenue (TR). Therefore, the profit function (π) can be calculated as:
π = TR - TC = (50Q - Q^2) - (5Q + 40 + 10/Q) * Q = 45Q - Q^2 - 40 - 10/Q
So, the firm's profit function is 45Q - Q^2 - 40 - 10/Q.