Question

Assume P=$500-$4Qd and our fixed cost is $1000 and the variable cost is $12 per unit. At what unit is total revenue maximized? How much revenue is generated?

Determine the firm's breakeven points(s) and show graphically showing total revenue and total cost and labeling the breakeven points.

Answer 1

Given

P=500-4Q

Total Revenue=TR=P*Q=(500-4Q)*Q=500Q-4Q²

Marginal Revenue=MR=dTR/dQ=500-8Q

Set MR=0 for revenue maximization

500-8Q=0

Q=500/8=62.50

Revenue is maximized at Q=62.50

P=500-4Q=500-4*62.50=$250

Maximum Revenue=P*Q=250*62.50=$15625

Total Cost=TC=Fixed Cost+Variable Cost=1000+12Q

Set TC=TR to determine break even quantity

1000+12Q=500Q-4Q²

4Q²-488Q+1000=0

Q²-122Q+250=0

-6 + Q= = 62 - 4ac 2a

Q = 122 (-122)2 - 4 * 1 * 250 2*1

Q=(122+117.8304)/2=119.92

and Q=(122-117.8304)/2=2.08

Break even points are Q=2.08 and Q=119.92

Q	P=500-4Q	TR=P*Q	TC=1000+12*Q
0.00	500	0	1000
2.08	492	1025	1025
2.50	490	1225	1030
5.00	480	2400	1060
7.50	470	3525	1090
10.00	460	4600	1120
12.50	450	5625	1150
15.00	440	6600	1180
17.50	430	7525	1210
20.00	420	8400	1240
22.50	410	9225	1270
25.00	400	10000	1300
27.50	390	10725	1330
30.00	380	11400	1360
32.50	370	12025	1390
35.00	360	12600	1420
37.50	350	13125	1450
40.00	340	13600	1480
42.50	330	14025	1510
45.00	320	14400	1540
47.50	310	14725	1570
50.00	300	15000	1600
52.50	290	15225	1630
55.00	280	15400	1660
57.50	270	15525	1690
60.00	260	15600	1720
62.50	250	15625	1750
65.00	240	15600	1780
67.50	230	15525	1810
70.00	220	15400	1840
72.50	210	15225	1870
75.00	200	15000	1900
77.50	190	14725	1930
80.00	180	14400	1960
82.50	170	14025	1990
85.00	160	13600	2020
87.50	150	13125	2050
90.00	140	12600	2080
92.50	130	12025	2110
95.00	120	11400	2140
97.50	110	10725	2170
100.00	100	10000	2200
102.50	90	9225	2230
105.00	80	8400	2260
107.50	70	7525	2290
110.00	60	6600	2320
112.50	50	5625	2350
115.00	40	4600	2380
117.50	30	3525	2410
119.92	20	2439	2439
120.00	20	2400	2440
122.50	10	1225	2470
125.00	0	0	2500

18000 16000 14000 12000 10000 8000 6000 4000 2000 F119.92 72.08 10.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00