Question

3. Restricted maximization of three activities Bma = Bmb = Bmc

                                                                                                                    Pa Pb Pc

Conditions: that the marginal profit per dollar is the same for the three activities and that the assigned budget is not exceeded.

. A firm wants to maximize its benefits with three activities, whose costs are x = 2, y = 4, z = 6. The budget allocated is $ 46.00 Determine the optimal levels of x, y,z

Act.	Bmx	BMx/px	Costo x	Bmy	Bmy/py	Costo y	Bmz	Bmz/pz	costoz
1	16			40			54
2	14			36			48
3	12			32			42
4	10			24			36
5	8			20			24
6	6			16			18
7	4			12			12
8	2			8			6

Determine the optimal levels for the three activities

Activities	of x	Cost of x
	Activities of y	Cost of y
	Activities of z	Costo of Z
		Total

Answer 1

The equi marginal principle (production version) states that at the producers will maximize their output by allocating their cost such a way that the marginal product per dollar for the last unit of each input employed will be same for both the inputs. In production the minimization of cost requires:

The table is completed as below

ACT	BMx	BMx/Px	Costto x	Bmy	Bmy/Py	Costto y	BMz	BMz/Pz	Costto z
1	16	8	2	40	10	4	54	9	6
2	14	7	4	36	9	8	48	8	12
3	12	6	6	32	8	12	42	7	18
4	10	5	8	24	6	16	36	6	24
5	8	4	10	20	5	20	24	4	30
6	6	3	12	16	4	24	18	3	36
7	4	2	14	12	3	28	12	2	42
8	2	1	16	8	2	32	6	1	48

From the figure above, it can be seen that the marginal profit is same at 4 levels. For marginal profit level 8,6,4, and 2. The corresponding level of act is summarize in the table below

Bmi/Pi	x	y	z	Cost
8	1	3	2	26
6	3	4	4	46
4	7	6	5	68
2	8	8	7	90

From the table above, the cost constraint is satisfied for marginal profit level 6. Then the activity level and the cost is given as

Activities	of x 3	Cost of x	6
	Activities of y 4	Cost of y	16
	Activities of z 4	Costo of Z	24
		Total	46