1. You are given the following data for your firm, which sells high-capacity vid
ID: 1117547 • Letter: 1
Question
1. You are given the following data for your firm, which sells high-capacity video MP3 players.
Q
P
TC
0
$1,000
$1,500
2
$960
$2,568
4
$920
$3,660
6
$880
$4,824
8
$840
$6,108
10
$800
$7,560
12
$760
$9,228
14
$720
$11,160
16
$680
$13,404
18
$640
$16,008
20
$600
$19,020
a. Determine equations for P=f(Q), MR=f(Q), ATC=f(Q, Q2), AVC=f(Q, Q2), MC=f(Q, Q2). Recall that your marginal equations should be derivatives of your totals!
b. Determine the profit-maximizing price and quantity. (Since MC is in terms of Q2, solving with calculus and algebra can be messy. Your table should give an exact answer.)
c. How much total profit would your firm earn if you set P and Q according to part b?
d. Describe the competitiveness of the market by calculating the Lerner index.
Q
P
TC
0
$1,000
$1,500
2
$960
$2,568
4
$920
$3,660
6
$880
$4,824
8
$840
$6,108
10
$800
$7,560
12
$760
$9,228
14
$720
$11,160
16
$680
$13,404
18
$640
$16,008
20
$600
$19,020
Explanation / Answer
a. Determine equations for P=f(Q), MR=f(Q), ATC=f(Q, Q2), AVC=f(Q, Q2), MC=f(Q, Q2). Recall that your marginal equations should be derivatives of your totals!
From the table, we figure out that fixed cost is $1500. Total cost TC = Q^3 - 3*Q^2 + 536Q + 1500.
Total variable cost = Q^3 - 3*Q^2 + 536Q. MC = 3Q^2 - 6Q + 536. AVC = TVC/Q = Q^2 - 3Q + 536. ATC = TC/Q = Q^2 - 3Q + 536 + 1500.
Price P = 1000 - 20Q. MR = 1000 - 40Q.
b. Determine the profit-maximizing price and quantity.
Note that MR = 1000 - 40Q and MC = 3Q^2 - 6Q + 536. We have
3Q^2 - 6Q + 536 = 1000 - 40Q
3Q^2 + 34Q - 464 = 0
This is a quadratic equation which solves for Q = 8
Price = 1000 - 8*20 = $840
c. How much total profit would your firm earn if you set P and Q according to part b?
Profit = 840*8 - (8^3 - 3*8^2 + 536*8 + 1500) = 612
d. Describe the competitiveness of the market by calculating the Lerner index
Price = 840, MC = 3*8^2 - 6*8 + 536 = 680
Lerner index = (840 - 680)/840 = 0.19
Market is highly competitive as lerner index is close to zero.
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