In a monopolistically competitive market a firm demand is given by the portion q=1500-50p
Tc=1500+3q+0.0025q^2
What is the maximum profit?
To find the maximum profit, we need to determine the quantity and price at which the profit is maximized.
Step 1: Determine the marginal revenue (MR) function.
In a monopolistically competitive market, the firm's marginal revenue is twice as steep as the demand curve. Therefore, the marginal revenue function is given by:
MR = dTR/dq = 1500 - 100p
Step 2: Set MR equal to the marginal cost (MC) to find the profit-maximizing quantity.
Since TC = 1500 + 3q + 0.0025q^2, MC is the derivative of TC with respect to q:
MC = dTC/dq = 3 + 0.005q
Setting MR equal to MC:
1500 - 100p = 3 + 0.005q
Step 3: Substitute the demand function into the equation.
We can substitute q = 1500 - 50p into the equation to obtain an equation with only the variable p:
1500 - 100p = 3 + 0.005(1500 - 50p)
Step 4: Solve for p.
1500 - 100p = 3 + 7.5 - 0.25p
-100p + 0.25p = 3 + 7.5 - 1500
Combining like terms:
-99.75p = -1489.5
Dividing both sides by -99.75:
p = 14.95
Step 5: Substitute the value of p back into the demand function to find q.
q = 1500 - 50(14.95)
q = 1500 - 747.5
q = 752.5
Step 6: Calculate the profit.
Profit = TR - TC
Profit = (p * q) - (1500 + 3q + 0.0025q^2)
Profit = (14.95 * 752.5) - (1500 + 3(752.5) + 0.0025(752.5)^2)
Profit = 11248.375 - (1500 + 2257.5 + 1.12765625 * 564506.25)
Profit = 11248.375 - (1500 + 2257.5 + 637460.6875)
Profit = 11248.375 - 641218.6875
Profit = -629970.3125
Step 7: Interpret the maximum profit.
The calculated profit is negative, which means that the firm is experiencing a loss rather than a profit in this monopolistically competitive market.
To find the maximum profit in a monopolistically competitive market, we need to first determine the optimal level of output for the firm. This can be achieved by maximizing the firm's profit function, which is given by:
Profit = Total Revenue - Total Cost
The profit-maximizing level of output occurs where the marginal revenue equals the marginal cost. In other words, the firm should produce at a level where the additional revenue from selling one more unit is equal to the additional cost of producing that unit.
In this case, the firm's total revenue (TR) can be calculated by multiplying the quantity (q) by the price (p), using the demand function q = 1500 - 50p:
TR = p * q
= p * (1500 - 50p)
= 1500p - 50p^2
The marginal revenue (MR) is the derivative of total revenue with respect to quantity, which can be obtained by differentiating the total revenue equation:
MR = d(TR)/dq
= 1500 - 100p
The total cost (TC) is given by the equation TC = 1500 + 3q + 0.0025q^2.
The marginal cost (MC) is the derivative of total cost with respect to quantity, which can be obtained by differentiating the total cost equation:
MC = d(TC)/dq
= 3 + 0.005q
To find the profit-maximizing level of output, we set MR equal to MC:
1500 - 100p = 3 + 0.005q
Substituting the demand function (q = 1500 - 50p) into the equation, we get:
1500 - 100p = 3 + 0.005(1500 - 50p)
Simplifying the equation, we get:
1500 - 100p = 3 + 7.5 - 0.25p
Combining like terms, we get:
-100p + 0.25p = 3 + 7.5 - 1500
-99.75p = -1489.5
Dividing both sides by -99.75, we get:
p = (-1489.5) / (-99.75)
≈ 14.94
Now that we have the price, we can substitute it back into the demand function to find the corresponding quantity:
q = 1500 - 50p
= 1500 - 50(14.94)
≈ 750.3
Therefore, the profit-maximizing level of output is approximately q = 750.3, and the corresponding price is approximately p = 14.94.
To calculate the maximum profit, we substitute the values of quantity and price into the profit function:
Profit = Total Revenue - Total Cost
= (p*q) - TC
= (14.94 * 750.3) - (1500 + 3*750.3 + 0.0025*(750.3)^2)
Evaluating this expression will give us the maximum profit in the monopolistically competitive market.