1.  Monopoly:  Assume a monopoly has the following demand and total cost functions:
Q=8300-2.1P
TC=2200+480Q+20Q^2

A.  Derive an expression for the firms MR curve

Q=8300-2.1P
2.1P=8300-Q
P=(8300/2.1)-(Q/2.1)
P=3952.38-.476Q

TR=P*Q=(3952.38-.476Q)Q
TR=3952.38Q-.476Q^2

MR=dTR/dQ=3952.38-.952Q

B.  To max profits, what quantity should the firm produce and what quantity should they charge?

They will want to produce where MR=MC

MC=dTC/dQ=480+40Q
MR= from part A

480+40Q=3952.38-.952Q
Q=84.8

To find price plug quantity back into the demand function.  You find that P=3912.

C.  If the firm produces and sells like in part B, what are the firms total profits?

Profit=TR-TC=P*Q-TC
From earlier parts of the question you have all the parts in the profit function above.  Simply plug them in and solve.
You find that profit=145012.8

2.  Monopoly and perfect competition. 
TC=.28q  (individual firm cost curve)
Q=1000-1000P
A.  Assume the curves above are for a firm in a perfectly competitive industry.  What is the long run equilibrium price and quantity?

Equilibrium will occur at the Minimum of LRATC where LRATC=MC=MR=P
We need to find the LRATC

ATC=TC/q= .28q/q=.28  In this case the ATC per unit is constant so the minimum cost will be .28 (and the max will be .28)
Since ATC=P in equilibrium we know that P=.28 and we can sub. this back into demand to get q.
q=1000-1000(.28)=720

B.  What would change if this market suddenly became a monopoly?  Assume the original equations above were now for a monopolist.  Find LR equilibrium price and quantity.

In the LR, the monopoly will max profits where MR=MC
So first find MR and MC and then set them equal
Re-arrange Demand for p
P=1-.001Q   and use it to find TR
TR=P*Q=(1-.001Q)Q=Q-.001Q^2
MR=1-.002Q
MC=.28
1-.002Q=.28
Q=360
then plug Q back into the demand function to get P
P=.64
 

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