A monopoly has a total cost function
C = 1000 + 108x + 12x^2
for its product, which has demand function
p = 324 − 3x − 2x^2.
Find the consumer's surplus at the point where the monopoly has maximum profit. (Round your answer to the nearest cent.)
Thank You!
there is a nice discussion here. It should enable you to come up with the result you want.
http://en.wikipedia.org/wiki/Economic_surplus
2312
To find the consumer's surplus at the point where the monopoly has maximum profit, we first need to determine the monopolist's maximum profit point.
The monopolist's profit is given by the formula:
Profit = Total Revenue - Total Cost
We can find the total revenue by multiplying the demand function (p) by the quantity sold (x):
Total Revenue (TR) = p * x
Substituting the demand function p = 324 - 3x - 2x^2 into the above equation, we get:
TR = (324 - 3x - 2x^2) * x
Next, we find the monopolist's total cost (TC) by substituting the cost function C = 1000 + 108x + 12x^2 into the equation:
TC = 1000 + 108x + 12x^2
To maximize profit, we need to find the quantity that maximizes the difference between total revenue and total cost. So, we calculate the profit function (π) as follows:
π = TR - TC
Substituting the expressions for TR and TC, we get:
π = (324 - 3x - 2x^2) * x - (1000 + 108x + 12x^2)
Now, to find the quantity that maximizes profit, we take the derivative of the profit function with respect to x and set it equal to zero:
dπ/dx = 0
Differentiating the profit function and simplifying, we obtain:
-4x^2 - 6x + 324 - 108 = 0
-4x^2 - 6x + 216 = 0
Next, we solve this quadratic equation for x. You can use the quadratic formula or factoring methods to find the solutions. In this case, the quadratic factors nicely as:
-2(x + 6)(2x - 6) = 0
Setting each factor equal to zero:
x + 6 = 0 or 2x - 6 = 0
Solving these equations, we find:
x = -6 or x = 3
Since we are considering the monopolist's profit maximization, we discard the negative root and focus on the positive root, which is x = 3.
Now that we know the quantity (x) at the profit-maximizing point, we can find the consumer's surplus using the demand function. The consumer's surplus (CS) is given by the integral of the demand function from 0 to the quantity sold:
CS = ∫[0,x] (324 - 3x - 2x^2) dx
Evaluating this integral, we get:
CS = [324x - 1.5x^2 - (2/3)x^3] evaluated from 0 to 3
Substituting the upper limit (3) and lower limit (0) into the expression, we obtain:
CS = (324*3 - 1.5*3^2 - (2/3)*3^3) - (324*0 - 1.5*0^2 - (2/3)*0^3)
Simplifying this expression, we find:
CS = 972 - 13.5 - 18 - 0 = 940.50
Therefore, the consumer's surplus at the point where the monopoly has maximum profit is approximately $940.50.