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.

Answer 1

Solution:-

a) From the table we can find the demand function:

y = mx + c

y = ((y2-y1)/(x2-x1))x + c

put any two points to find the value of m and c.

we have equation:

P = 1000 - 20Q

From the graph, we have a cost function:

TC = 108Q² + 322.6Q + 1500

MR = d(P*Q) / dQ

MR = d(1000Q - 20Q²)/dQ

MR = 1000 - 40Q

ATC = TC / Q

ATC = 108Q + 322,6 + 1500/Q

VC = TC - FC

VC = 108Q² + 322.6Q

AVC = VC/Q

AVC = 108Q + 322.6

MC = d(TC)/dQ

MC = 216Q + 322.6

b) For profit maximization,

MR = MC

1000 - 40Q = 216Q + 322.6

256Q = 677.4

Q = 2.64

or

Q = 2 (from table)

P = 960

c) profit = revenue - total cost

profit = P*Q - TC(Q=2)

profit = 960*2 - 2568

profit = - 1920 - 2568 = -648

If fixed cost is removed then,

profit = 1500-648 = 852

d) L = (P - MC) / P

P = 960

MC = 216Q + 322.6

MC = 216 * 2 + 322.6

MC = 754.6

L = (960 - 754.6) / 960

L = 0.214

As lerner index of a firm is small, shows lower market power and high competiveness in the market.