Question

A company is selling an item that has made the following assumptions:

• Sold over a 15 year period

• The item’s demand in year one will be 15,000 units.

• During years 2-5, the annual growth of demand will be 10%.

• During years 6-15, the annual growth of demand will be 5%.

• It costs $5, payable at the end of year 1, to build each unit of annual production capacity.

• During year one, the item will sell for $7 per unit and will incur a variable cost of $4 to produce.

• The cost of maintaining a unit capacity during year 1 is $1.

• The sales price, unit variable cost, and unit capacity maintenance cost will increase by 3% per year.

• All cash flows are assumed to occur at the end of each year, and the corporate discount rate is 8%.

I would like to know what is the capacity level that should be chosen for the production facility based on its fifteen year cash flows?

Answer 1

Let say Capacity Level is 15000

Year	Max. Production	Contribution per Unit	Earnings	Discounted Earnings @ 8%
1	15,000	2.00	30,000.00	27,777.78
2	15,000	2.06	30,900.00	26,491.77
3	15,000	2.12	31,827.00	25,265.30
4	15,000	2.19	32,781.81	24,095.61
5	15,000	2.25	33,765.26	22,980.07
6	15,000	2.32	34,778.22	21,916.18
7	15,000	2.39	35,821.57	20,901.54
8	15,000	2.46	36,896.22	19,933.88
9	15,000	2.53	38,003.10	19,011.01
10	15,000	2.61	39,143.20	18,130.87
11	15,000	2.69	40,317.49	17,291.48
12	15,000	2.77	41,527.02	16,490.95
13	15,000	2.85	42,772.83	15,727.48
14	15,000	2.94	44,056.01	14,999.36
15	15,000	3.03	45,377.69	14,304.94
				305,318.22
Cost of Building Capacity			15000*5	69,444.44
Net Profit				235,873.77

Now let say Capacity Level is maximum demand i.e at end of 15 years = 35773

Year	Max. Production	Contribution per Unit	Earnings	Discounted Earnings @ 8%
1	15,000	2.00	30,000.00	27,777.78
2	16,500	2.06	33,990.00	29,140.95
3	18,150	2.12	38,510.67	30,571.01
4	19,965	2.19	43,632.59	32,071.26
5	21,962	2.25	49,435.72	33,645.12
6	23,060	2.32	53,464.73	33,691.85
7	24,213	2.39	57,822.11	33,738.65
8	25,423	2.46	62,534.61	33,785.51
9	26,694	2.53	67,631.18	33,832.43
10	28,029	2.61	73,143.13	33,879.42
11	29,431	2.69	79,104.29	33,926.47
12	30,902	2.77	85,551.29	33,973.59
13	32,447	2.85	92,523.72	34,020.78
14	34,069	2.94	100,064.40	34,068.03
15	35,773	3.03	108,219.65	34,115.35
				492,238.19
Cost of Building Capacity			35773*5	165,615.60
Net Profit				326,622.59

Now lets take a Capacity level in between these 2 no. says 25000

Year	Max. Production	Contribution per Unit	Earnings	Discounted Earnings @ 8%
1	15,000	2.00	30,000.00	27,777.78
2	16,500	2.06	33,990.00	29,140.95
3	18,150	2.12	38,510.67	30,571.01
4	19,965	2.19	43,632.59	32,071.26
5	21,962	2.25	49,435.72	33,645.12
6	23,060	2.32	53,464.73	33,691.85
7	24,213	2.39	57,822.11	33,738.65
8	25,000	2.46	61,493.69	33,223.13
9	25,000	2.53	63,338.50	31,685.02
10	25,000	2.61	65,238.66	30,218.12
11	25,000	2.69	67,195.82	28,819.13
12	25,000	2.77	69,211.69	27,484.92
13	25,000	2.85	71,288.04	26,212.47
14	25,000	2.94	73,426.69	24,998.93
15	25,000	3.03	75,629.49	23,841.57
				447,119.90
Cost of Building Capacity			=25000*5	115,740.74
Net Profit				331,379.15

i.e Maximum demand is not the optimum level of capacity as it generates lesser Profit as compared to operating at 25000 capacity level.

By Hit & trial, we came to know 29,431 is the optimum level of capacity as it generates maximum profit.

Year	Max. Production	Contribution per Unit	Earnings	Discounted Earnings @ 8%
1	15,000	2.00	30,000.00	27,777.78
2	16,500	2.06	33,990.00	29,140.95
3	18,150	2.12	38,510.67	30,571.01
4	19,965	2.19	43,632.59	32,071.26
5	21,962	2.25	49,435.72	33,645.12
6	23,060	2.32	53,464.73	33,691.85
7	24,213	2.39	57,822.11	33,738.65
8	25,423	2.46	62,534.61	33,785.51
9	26,694	2.53	67,631.18	33,832.43
10	28,029	2.61	73,143.13	33,879.42
11	29,431	2.69	79,104.29	33,926.47
12	29,431	2.77	81,477.42	32,355.80
13	29,431	2.85	83,921.74	30,857.85
14	29,431	2.94	86,439.39	29,429.25
15	29,431	3.03	89,032.58	28,066.78
				476,770.12
Cost of Building Capacity				136,252.36
Net Profit				340,517.76

At any other level of Capacity, Net Profit won't be Higher than this amount of Net Profit.

Happy learning!