Question

6. The manufacturer of high-quality flatbed scanners is trying to decide what price to set for its product. The costs of production and the demand for the product are assumed to be as follows:

TC = 500,000 + 0.85Q + (0.015Q)(0.015Q)

Q = 14,166 - 16.6P

a. Determine the short-run profit-maximizing price.

b. Plot this information on a graph showing AC, AVC, MC, P, and MR.

Answer 1

TC = 500,000 + 0.85Q + (0.015Q)² = 500,000 + 0.85Q + 0.000225Q²

MC = dTC/dQ = 0.85 + 0.00045Q

Q = 14,166 - 16.6P, so

16.6P = 14,166 - Q

P = (14,166 - Q) / 16.6

(a)

Profit is maximized when MR = MC

Total revenue (TR) = P x Q = (14,166Q - Q²) / 16.6

MR = dTR/dQ = (14,166 - 2Q) / 16.6

(14,166 - 2Q) / 16.6 = 0.85 + 0.00045Q

14,166 - 2Q = 14.11 + 0.00747Q

2.00747Q = 14,151.89

Q = 7,050

P = (14,166 - 7,050) / 16.6 = 7,116 / 16.6 = 428.67

(b)

AC = TC/Q = (500,000/Q) + 0.85 + 0.000225Q

TVC = 0.85Q + 0.000225Q²

AVC = TVC/Q = 0.85 + 0.000225Q

Data table used:

Q	P	MR	AC	AVC	MC
0	853.37	853.37
1000	793.13	732.89	501.08	1.075	1.3
2000	732.89	612.41	251.30	1.3	1.75
3000	672.65	491.93	168.19	1.525	2.2
4000	612.41	371.45	126.75	1.75	2.65
5000	552.17	250.96	101.98	1.975	3.1
6000	491.93	130.48	85.53	2.2	3.55
7000	431.69	10.00	73.85	2.425	4
8000	371.45	-110.48	65.15	2.65	4.45
9000	311.20	-230.96	58.43	2.875	4.9
10000	250.96	-351.45	53.10	3.1	5.35

Graph: