Question

company is producing 2 items A and B joint]y in fixed proportions of 1 :1 .

The following indicate their market demand functions: QA : 700 - 1 00 PA and QB : 500 - 200 P B. The MC of the joint operations is constant at Rs- 2 per unit

What quantities of A and B will company produce for optimum profits? (b) Will the company need to store in inventory at least one of the items or both items?
Support your answer with calculations.

Answer 1

To find the profit maximizing quantity and price for good A and B we need to use the profit maximizing condition, marginal revenue = marginal cost.

So let's calculate the profit maximizing quantity and price for good A.

First we need to inverse demand function for good A,

QA = 700 - 100PA

100PA = 700 - QA

PA = 7 - QA/100

Total revenue = PA × QA

Total revenue = (7 - QA/100) × QA

Differentiating both the sides with respect to QA to calculate marginal revenue,

Marginal revenue = (7 - QA/100)×1 + QA(-1/100)

Marginal revenue = 7 - 2QA/100

And marginal cost of production = 2

At profit maximizing quantity, MR=MC

7 - 2QA/100 = 2

700 - 2QA = 200

2QA = 700 - 200

QA = 500/2

QA* = 250

Putting QA in demand function we get,

PA = 7- QA/100

PA = 7- 250/100

PA = 7 - 2.5

PA* = 4.5

So the profit maximizing price PA* and quantity QA* for good A is 4.5 and 250 respectively.

Now let's calculate the profit maximizing price and quantity for good B,

QB = 500 - 200PB

200PB = 500 - QB

PB = (500 - QB)/200

PB = 5/2 - QB/200

Total revenue = PB×QB

Total revenue = (5/2 - QB/200)×QB

Differentiating both the sides with respect to QB to calculate the marginal revenue,

Marginal revenue = (5/2 - QB/200) × 1 + QB(-1/200)

Marginal revenue = 5/2 - 2QB/200

Marginal revenue = (500 - 2QB)/200

And putting MR=MC,

(500 - 2QB)/200 = 2

500 - 2QB = 400

2QB = 500 - 400

QB* = 100/2 = 50

Putting QB in the demand function we get,

PB = 5/2 - QB/200

PB = 5/2 - 50/200

PB = 5/2 - 1/4

PB* = (10 - 1)/4

PB* = 9/4 = 2.25

So the profit maximizing price PB* and quantity QB* for good B is 2.25 and 50 respectively.

But note that the company produces good A and B jointly in 1:1, that is to say they are producing good A and good B in equal numbers.

So the company needs to produce 250 units of good A to maximize profit from Good A but for producing 250 units of good A they'll have to produce 250 units of good B also. But as we have already calculated that the profit maximizing quantity of good B is only 50. So out of 250 units of good B company is going to sell only 50 units of good B, and rest will stored as inventory. So the amount of inventory for good B will be,

Inventory = total production - total sell

Inventory = 250 - 50

Inventory = 200.

So the company will need to store inventory of good B which is equal to 200 units.