Question

Monopolistic firm has the inverse demand function p = 250 – 6Q. It’s total cost of production is C = 1250 + 10Q + 8Q² :

1. Create a spreadsheet for Q = 1 to Q = 20 in increments of 1. Determine the profit-maximizing output and price for the firm and the consequent level of profit.

2. Will the firm continue the production at the profit-maximizing level of output? Show why or why not?

3. Calculate the Lerner Index of monopoly power for each output level and verify its relationship with the value of the price elasticity of demand at the profit-maximizing level of output.

4. Suppose that a specific tax of 10 per unit is imposed on the monopoly. What is the effect on the monopoly’s profit-maximizing price?

Answer 1

Q	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
P	244	238	232	226	220	214	208	202	196	190	184	178	172	166	160	154	148	142	136	130
C	1268	1302	1352	1418	1500	1598	1712	1842	1988	2150	2328	2522	2732	2958	3200	3458	3732	4022	4328	4650
Profit = PQ - C	-1024	-826	-656	-514	-400	-314	-256	-226	-224	-250	-304	-386	-496	-634	-800	-994	-1216	-1466	-1744	-2050

Loss is minimized at Q=9, i.e. loss of 224. Hence, profit is maximized at Q=9.

2. The firm will not produce at this level because it is ultimately a loss making proposition, and hence the firm will have to shut down.

3. To calculate L, we have the following excel:

Q	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
P	244	238	232	226	220	214	208	202	196	190	184	178	172	166	160	154	148	142	136	130
C	1268	1302	1352	1418	1500	1598	1712	1842	1988	2150	2328	2522	2732	2958	3200	3458	3732	4022	4328	4650
Profit = PQ - C	-1024	-826	-656	-514	-400	-314	-256	-226	-224	-250	-304	-386	-496	-634	-800	-994	-1216	-1466	-1744	-2050
MC = CQ+1 - CQ		34	50	66	82	98	114	130	146	162	178	194	210	226	242	258	274	290	306	322
L = (P-MC)/P		0.857143	0.784483	0.707964602	0.627272727	0.542056	0.451923	0.35643564	0.255102	0.147368	0.032609	-0.08989	-0.22093	-0.36145	-0.5125	-0.67532	-0.85135	-1.04225	-1.25	-1.47692

At Q=9, (Change in Quantity/Quantity) / (Change in Price/Price) = 1/8 / -6/196 = -4.1

But,

So, 1/ |E| = .25 = L calculated

4. After imposing a specific tax, price increases by 10, and hence following excel is reached:

Q	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
P	254	248	242	236	230	224	218	212	206	200	194	188	182	176	170	164	158	152	146	140
C	1268	1302	1352	1418	1500	1598	1712	1842	1988	2150	2328	2522	2732	2958	3200	3458	3732	4022	4328	4650
Profit = PQ - C	-1014	-806	-626	-474	-350	-254	-186	-146	-134	-150	-194	-266	-366	-494	-650	-834	-1046	-1286	-1554	-1850

Here too, profit maximizing Quantity is same, and hence price increases to 206 apiece.