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**A. Determine the profit-maximizing or loss-minimizing equilibrium level of output:**
The profit-maximizing (or loss-minimizing) output occurs where Marginal Cost (MC) equals Marginal Revenue (MR). In perfect competition, MR equals the price (P). Set MC equal to P to find the equilibrium quantity.
\[MR = MC\]
The given Total Revenue function is \(TR = 6Q\), and the Total Cost function is \(TC = Q^3 - 2Q^2 + 50Q + 150\). First, find Marginal Revenue (\(MR\)) by taking the derivative of the Total Revenue with respect to quantity (\(Q\)).
\[MR = \frac{d(TR)}{dQ}\]
\[MR = 6\]
Now, find Marginal Cost (\(MC\)) by taking the derivative of the Total Cost with respect to quantity (\(Q\)).
\[MC = \frac{d(TC)}{dQ}\]
\[MC = 3Q^2 - 4Q + 50\]
Set \(MR\) equal to \(MC\) and solve for \(Q\):
\[6 = 3Q^2 - 4Q + 50\]
This is a quadratic equation that can be solved to find the equilibrium quantity.
**B. Compute the level of profit or loss at the above equilibrium quantity and comment on the decision of the firm:**
Once you find the equilibrium quantity (\(Q\)), substitute it back into the Total Revenue and Total Cost functions to calculate Total Revenue (\(TR\)), Total Cost (\(TC\)), and profit (\(\pi\)):
\[TR = 6Q\]
\[TC = Q^3 - 2Q^2 + 50Q + 150\]
\(\pi = TR - TC\)
**C. Mathematically and graphically derive the supply function of the firm:**
The supply function in a perfectly competitive market is determined by the portion of the MC curve above the Average Variable Cost (AVC) curve. Mathematically, it is given by:
\[Q_s = \frac{MC}{P} \text{ if } MC > AVC\]
Graphically, the supply curve is the portion of the MC curve that lies above the AVC curve.
Note: Solving the equations and graphing them would require specific numerical values for the constants in the cost function, which are not provided in the question. If you have those values, you can substitute them into the equations for a more concrete solution.
The profit-maximizing (or loss-minimizing) output occurs where Marginal Cost (MC) equals Marginal Revenue (MR). In perfect competition, MR equals the price (P). Set MC equal to P to find the equilibrium quantity.
\[MR = MC\]
The given Total Revenue function is \(TR = 6Q\), and the Total Cost function is \(TC = Q^3 - 2Q^2 + 50Q + 150\). First, find Marginal Revenue (\(MR\)) by taking the derivative of the Total Revenue with respect to quantity (\(Q\)).
\[MR = \frac{d(TR)}{dQ}\]
\[MR = 6\]
Now, find Marginal Cost (\(MC\)) by taking the derivative of the Total Cost with respect to quantity (\(Q\)).
\[MC = \frac{d(TC)}{dQ}\]
\[MC = 3Q^2 - 4Q + 50\]
Set \(MR\) equal to \(MC\) and solve for \(Q\):
\[6 = 3Q^2 - 4Q + 50\]
This is a quadratic equation that can be solved to find the equilibrium quantity.
**B. Compute the level of profit or loss at the above equilibrium quantity and comment on the decision of the firm:**
Once you find the equilibrium quantity (\(Q\)), substitute it back into the Total Revenue and Total Cost functions to calculate Total Revenue (\(TR\)), Total Cost (\(TC\)), and profit (\(\pi\)):
\[TR = 6Q\]
\[TC = Q^3 - 2Q^2 + 50Q + 150\]
\(\pi = TR - TC\)
**C. Mathematically and graphically derive the supply function of the firm:**
The supply function in a perfectly competitive market is determined by the portion of the MC curve above the Average Variable Cost (AVC) curve. Mathematically, it is given by:
\[Q_s = \frac{MC}{P} \text{ if } MC > AVC\]
Graphically, the supply curve is the portion of the MC curve that lies above the AVC curve.
Note: Solving the equations and graphing them would require specific numerical values for the constants in the cost function, which are not provided in the question. If you have those values, you can substitute them into the equations for a more concrete solution.
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