Answer :
Certainly! Let's tackle each part of the question separately.
### Part A: Determine the Profit Maximizing or Loss Minimizing Equilibrium Level of Output
1. Profit Function: The profit, [tex]\( \pi \)[/tex], is given by the difference between Total Revenue (TR) and Total Cost (TC):
[tex]\[ \pi(Q) = TR(Q) - TC(Q) = 6Q - (Q^3 - 2Q^2 + 50Q + 25) \][/tex]
Simplifying the profit function:
[tex]\[ \pi(Q) = 6Q - Q^3 + 2Q^2 - 50Q - 25 \][/tex]
[tex]\[ \pi(Q) = -Q^3 + 2Q^2 - 44Q - 25 \][/tex]
2. Finding Critical Points: To find the equilibrium quantity, we need to determine the critical points by setting the first derivative of the profit function equal to zero.
[tex]\[ \frac{d\pi}{dQ} = \frac{d}{dQ} (-Q^3 + 2Q^2 - 44Q - 25) \][/tex]
Calculating the derivative:
[tex]\[ \frac{d\pi}{dQ} = -3Q^2 + 4Q - 44 \][/tex]
Setting the first derivative equal to zero to find the equilibrium quantity:
[tex]\[ -3Q^2 + 4Q - 44 = 0 \][/tex]
3. Solving the Quadratic Equation:
[tex]\[ -3Q^2 + 4Q - 44 = 0 \][/tex]
Use the quadratic formula [tex]\( Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = -3 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -44 \)[/tex]:
[tex]\[ Q = \frac{-4 \pm \sqrt{4^2 - 4(-3)(-44)}}{2(-3)} \][/tex]
[tex]\[ Q = \frac{-4 \pm \sqrt{16 - 528}}{-6} \][/tex]
Since the discriminant [tex]\( (16 - 528 = -512) \)[/tex] is negative, the quadratic equation has no real roots.
4. Conclusion for Part A:
Since the quadratic has no real roots, the profit function does not have a maximum or minimum within the real domain. Thus, there is no profit-maximizing or loss-minimizing equilibrium output that we can solve for using real numbers. This likely indicates external factors or constraints around real-world production levels.
### Part B: Compute the Level of Profit or Loss at the Above Equilibrium Quantity and Comment
Since we found no real equilibrium point (no real [tex]\( Q \)[/tex] roots from the quadratic equation), we cannot calculate a specific profit or loss at an equilibrium level mathematically derived from the given functions.
However, to proceed practically, it would involve looking into potential boundary conditions or values in a feasible range for [tex]\( Q \)[/tex].
### Part C: Derive the Supply Function of the Firm
1. Marginal Cost (MC) Function: The supply function in a perfectly competitive market is the marginal cost (MC) curve above the average variable cost (AVC):
[tex]\[ TC(Q) = Q^3 - 2Q^2 + 50Q + 25 \][/tex]
The MC is the first derivative of the TC function:
[tex]\[ MC = \frac{d(TC)}{dQ} = \frac{d}{dQ} (Q^3 - 2Q^2 + 50Q + 25) \][/tex]
Calculating the derivative:
[tex]\[ MC = 3Q^2 - 4Q + 50 \][/tex]
2. Supply Function: In a perfectly competitive market, firms produce where [tex]\( P = MC \)[/tex]. Given that [tex]\( P = 6 \)[/tex]:
[tex]\[ 6 = 3Q^2 - 4Q + 50 \][/tex]
Solving for [tex]\( Q \)[/tex]:
[tex]\[ 3Q^2 - 4Q + 44 = 0 \][/tex]
This quadratic equation has the same coefficients, leading to the recognition that the discriminant is the same: negative.
3. Conclusion for Part C:
Since again the discriminant is negative, implying no real [tex]\( Q \)[/tex] solution, it reflects more real-world constraints on firm production levels. Thus, there may be other factors or conditions impacting the firm's ability to supply.
### Summary
- Part A: No real equilibrium quantity found mathematically.
- Part B: Unable to compute specific profit or loss at equilibrium points that don't exist in real domain of [tex]\( Q \)[/tex].
- Part C: Supply function derivation also results in no real solutions for [tex]\( Q \)[/tex] upon solving [tex]\( MC = P \)[/tex].
The findings indicate practical constraints, misalignments, or further considerations needed beyond the given equations to form a comprehensive real-world model.
### Part A: Determine the Profit Maximizing or Loss Minimizing Equilibrium Level of Output
1. Profit Function: The profit, [tex]\( \pi \)[/tex], is given by the difference between Total Revenue (TR) and Total Cost (TC):
[tex]\[ \pi(Q) = TR(Q) - TC(Q) = 6Q - (Q^3 - 2Q^2 + 50Q + 25) \][/tex]
Simplifying the profit function:
[tex]\[ \pi(Q) = 6Q - Q^3 + 2Q^2 - 50Q - 25 \][/tex]
[tex]\[ \pi(Q) = -Q^3 + 2Q^2 - 44Q - 25 \][/tex]
2. Finding Critical Points: To find the equilibrium quantity, we need to determine the critical points by setting the first derivative of the profit function equal to zero.
[tex]\[ \frac{d\pi}{dQ} = \frac{d}{dQ} (-Q^3 + 2Q^2 - 44Q - 25) \][/tex]
Calculating the derivative:
[tex]\[ \frac{d\pi}{dQ} = -3Q^2 + 4Q - 44 \][/tex]
Setting the first derivative equal to zero to find the equilibrium quantity:
[tex]\[ -3Q^2 + 4Q - 44 = 0 \][/tex]
3. Solving the Quadratic Equation:
[tex]\[ -3Q^2 + 4Q - 44 = 0 \][/tex]
Use the quadratic formula [tex]\( Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = -3 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -44 \)[/tex]:
[tex]\[ Q = \frac{-4 \pm \sqrt{4^2 - 4(-3)(-44)}}{2(-3)} \][/tex]
[tex]\[ Q = \frac{-4 \pm \sqrt{16 - 528}}{-6} \][/tex]
Since the discriminant [tex]\( (16 - 528 = -512) \)[/tex] is negative, the quadratic equation has no real roots.
4. Conclusion for Part A:
Since the quadratic has no real roots, the profit function does not have a maximum or minimum within the real domain. Thus, there is no profit-maximizing or loss-minimizing equilibrium output that we can solve for using real numbers. This likely indicates external factors or constraints around real-world production levels.
### Part B: Compute the Level of Profit or Loss at the Above Equilibrium Quantity and Comment
Since we found no real equilibrium point (no real [tex]\( Q \)[/tex] roots from the quadratic equation), we cannot calculate a specific profit or loss at an equilibrium level mathematically derived from the given functions.
However, to proceed practically, it would involve looking into potential boundary conditions or values in a feasible range for [tex]\( Q \)[/tex].
### Part C: Derive the Supply Function of the Firm
1. Marginal Cost (MC) Function: The supply function in a perfectly competitive market is the marginal cost (MC) curve above the average variable cost (AVC):
[tex]\[ TC(Q) = Q^3 - 2Q^2 + 50Q + 25 \][/tex]
The MC is the first derivative of the TC function:
[tex]\[ MC = \frac{d(TC)}{dQ} = \frac{d}{dQ} (Q^3 - 2Q^2 + 50Q + 25) \][/tex]
Calculating the derivative:
[tex]\[ MC = 3Q^2 - 4Q + 50 \][/tex]
2. Supply Function: In a perfectly competitive market, firms produce where [tex]\( P = MC \)[/tex]. Given that [tex]\( P = 6 \)[/tex]:
[tex]\[ 6 = 3Q^2 - 4Q + 50 \][/tex]
Solving for [tex]\( Q \)[/tex]:
[tex]\[ 3Q^2 - 4Q + 44 = 0 \][/tex]
This quadratic equation has the same coefficients, leading to the recognition that the discriminant is the same: negative.
3. Conclusion for Part C:
Since again the discriminant is negative, implying no real [tex]\( Q \)[/tex] solution, it reflects more real-world constraints on firm production levels. Thus, there may be other factors or conditions impacting the firm's ability to supply.
### Summary
- Part A: No real equilibrium quantity found mathematically.
- Part B: Unable to compute specific profit or loss at equilibrium points that don't exist in real domain of [tex]\( Q \)[/tex].
- Part C: Supply function derivation also results in no real solutions for [tex]\( Q \)[/tex] upon solving [tex]\( MC = P \)[/tex].
The findings indicate practical constraints, misalignments, or further considerations needed beyond the given equations to form a comprehensive real-world model.
