14. The demand function for a monopolistβs product is π=β500βπ. If the monopolist produces at least 100 units, but no more than 200 units, how many units should be produced to maximize total revenue,π =ππ?
R = pq = qβ(500-q)
dR/dq = (1000-3q)/(2β(500-q))
Looks like dR/dq=0 at q = 1000/3 = 333
But, we are only considering the interval [100,200]. So the maximum R occurs at q=200
Yer good
To maximize total revenue, we need to find the quantity that corresponds to the peak of the total revenue curve. Here are the steps to find the optimal quantity:
Step 1: Write down the demand function:
p = β500 - q
Step 2: Express the price in terms of quantity:
p = 500^0.5 - q^0.5
Step 3: Write down the total revenue function:
R = p * q
Step 4: Substitute the expression for p from the demand function into the total revenue function:
R = (500^0.5 - q^0.5) * q
Step 5: Expanding the function:
R = (500q^0.5 - q^1.5)
Step 6: Take the derivative of the total revenue function with respect to quantity:
dR/dq = 500 * 0.5 * q^(-0.5) - 1.5 * q^0.5
Step 7: Set the derivative equal to zero and solve equation for q:
0 = 250 * q^(-0.5) - 1.5 * q^0.5
Step 8: Multiply through by q^0.5:
0 = 250 - 1.5 * q
Step 9: Add 1.5q to both sides of the equation:
1.5q = 250
Step 10: Divide both sides of the equation by 1.5:
q = 250 / 1.5
Step 11: Calculate the value of q:
q = 166.67
Step 12: Since the monopolist can only produce at least 100 units but no more than 200 units, the quantity to be produced is 166 units.
Therefore, to maximize total revenue, the monopolist should produce 166 units.
To find the number of units that should be produced to maximize total revenue, we need to find the quantity level at which marginal revenue is zero.
Firstly, we need to find the total revenue function. Total revenue (R) is calculated by multiplying the price (p) by the quantity (q): R = pq.
Given the demand function p = β(500 - q), we can substitute this into the total revenue function: R = (β(500 - q))q.
To find the quantity that maximizes total revenue, we differentiate the total revenue function with respect to q and set it equal to zero to find the critical point.
dR/dq = (β(500 - q)) + q * (1/2)*(500 - q)^(-1/2) * (-1)
Simplifying this equation:
0 = (β(500 - q)) - q/(2β(500 - q))
Multiply both sides of the equation by 2β(500 - q) to eliminate the denominator:
0 = β(500 - q) - q
Next, let's square both sides:
0 = 500 - q - q^2
Rearranging the equation:
q^2 + q - 500 = 0
Now, we can solve this quadratic equation to find the values of q where marginal revenue is zero. Using the quadratic formula, we have:
q = (-1 Β± β(1^2 - 4(1)(-500))) / 2(1)
q = (-1 Β± β(1 + 2000)) / 2
q = (-1 Β± β2001) / 2
Since q must be between 100 and 200, we discard the negative value and consider only the positive value:
q = (-1 + β2001) / 2
Evaluating this expression, we get:
q β 27.05
Therefore, to maximize total revenue, the monopolist should produce approximately 27 units.