Estimating the demand equation

Question

Question 1) A monopoly firm in a small island country produces and supplies to the country. It faces a market demand curve for its products. The demand equation is linear, so is the firm’s total cost (TC) function. The following table exhibits the data for Quantity, Price, and TC.
 
(1) Estimate the demand equation.
[Show the estimated equation.]

(2) Estimate the TC function.
[Show the estimated TC equation.]

(3) What is the profit-maximizing quantity?

[Profit-max quantity is determined at the level where MC = MR. Hence, Derive MC function from the estimated TC function. Derive MR from TR function which is PQ, where P = P(Q). In other words, P is the estimated demand equation expressed by Q. But, since the demand function is linear, the MR function is linear with a twice steeper slope than the demand equation and with the same intercept.]

(4)  What is the profit-maximizing price?

[Once the quantity is determined, find the corresponding price at the demand curve.]

(5) What is the monopoly profit?  [Profit = TR – TC.]
 
Q       P    TC
1     100    11
2      97    19
3      98    31
4      95    39
5      96    51
6      93    59
7      94    71
8      91    79
9      92    91
10    89    99
11    90    111
12    87    119
13    88    131
14    85    139
15    86    151
16    83    159
17    84    171
18    81    179
19    82    191
20    79    199
21    80    211
22    77    219
23    78    231
24    75    239
25    76    251
 
Question 2) A manufacturing firm operating in a perfectly-competitive market has the following cost function:

TC = a + b Q + c Q² + d Q³

The TC data with respect to Q are provided in the table below.
 
(1) Estimate the TC equation and show the estimated equation.
[a = round up to a whole number (no decimal), b = round up and no decimal, c = round up to the first decimal (one decimal), and d = round up to the first decimal (one decimal)]

(2) What are the equations for AC and MC?

(3) The market price is $637.5. What is the firm’s profit-maximizing level of output?

(4) Does the firm make profit or loss? Show either the profit or loss?

(5) The market price changes. At which price will the firm shut down the operation?
 
TC          Q
1780       1
2021       2
2238       3
2435       4
2626       5
2814       6
3013       7
3227       8
3472       9
3750      10
4074      11
4451      12
4893      13
5405      14
6001      15
6684      16
7468      17
8357      18
9366      19
10500    20
11770    21
13182    22
14749    23
16475    24
18375    25