In: Economics
4) Suppose a product can be produced using virgin ore at a marginal cost given by MC1 = 0.5q1 and with recycled materials at a marginal cost of MC2 = 5 + 0.1q2.
a. If the inverse demand curve were given by P = 10 ? 0.5(q1 + q2), how many units of the product would be produced with virgin ore and how many with recycled units?
b. How would this change if the inverse demand function were P = 17 ? 0.5(q1 + q2)?
Answer : 4) Given , MC1 = 0.5q1 and MC2 = 5 + 0.1q2
a) Inverse demand function means there is a negative (-) relationship between price and quantity demanded. So, the given inverse demand function becomes, P = 10 - 0.5(q1 + q2)
=> P = 10 - 0.5q1 - 0.5q2
TR1 (Total Revenue) = P*q1 = (10 - 0.5q1 - 0.5q2)*q1 = 10q1 - 0.5q1^2 - 0.5q1q2
MR1 (Marginal Revenue) = TR1 / q1 = 10 - q1 - 0.5q2
At equilibrium condition for virgin ore, MR1 = MC1
=> 10 - q1 - 0.5q2 = 0.5q1
=> 10 - 0.5q2 = 0.5q1 + q1
=> 10 - 0.5q2 = 1.5q1
=> q1 = (10 - 0.5q2) / 1.5
=> q1 = 6.667 - 0.333q2 ...........................(i)
Now, TR2 = P*q2 = (10 - 0.5q1 - 0.5q2)*q2 = 10q2 - 0.5q1q2 - 0.5q2^2
MR2 = TR2 / q2 = 10 - 0.5q1 - q2
At equilibrium condition for recycled materials, MR2 = MC2
=> 10 - 0.5q1 - q2 = 5 + 0.1q2
=> 10 - 5 - 0.5q1 = 0.1q2 + q2
=> 5 - 0.5q1 = 1.1q2
=> q2 = (5 - 0.5q1) / 1.1
=> q2 = 4.545 - 0.455q1 ......................................(ii)
Now, by putting the value of q2 in equation (i), we have,
q1 = 6.667 - 0.333 ( 4.545 - 0.455q1)
=> q1 = 6.667 - 1.513 + 0.152q1
=> q1 - 0.152q1 = 5.154
=> 0.848q1 = 5.154
=> q1 = 5.154 / 0.848
=> q1 = 6.078
From equation (ii) we get,
q2 = 4.545 - 0.455 * 6.078
=> q2 = 1.78
Therefore, virgin ore produces , q1 = 6.078 units and recycled materials produces, q2 = 1.78 units.
b) In case of inverse demand function, P = 17 - 0.5(q1 + q2)
=> P = 17 - 0.5q1 - 0.5q2
TR1 = P*q1 = (17 - 0.5q1 - 0.5q2) *q1 = 17q1 - 0.5q1^2 - 0.5q1q2
MR1 = TR1 / q1 = 17 - q1 - 0.5q2
At equilibrium condition for virgin ore, MR1 = MC1
=> 17 - q1 - 0.5q2 = 0.5q1
=> 17 - 0.5q2 = 0.5q1 + q1
=> 17 - 0.5q2 = 1.5q1
=> q1 = (17 - 0.5q2) / 1.5
=> q1 = 11.333 - 0.333q2 ............................(iii)
Now, TR2 = P*q2 = (17 - 0.5q1 - 0.5q2)*q2 = 17q2 - 0.5q1q2 - 0.5q2^2
MR2 = TR2 / q2 = 17 - 0.5q1 - q2
At equilibrium condition for recycled materials, MR2 = MC2
=> 17 - 0.5q1 - q2 = 5 + 0.1q2
=> 17 - 5 - 0.5q1 = 0.1q2 + q2
=> 12 - 0.5q1 = 1.1q2
=> q2 = (12 - 0.5q1) / 1.1
=> q2 = 10.909 - 0.455q1 ................................(iv)
Now by putting the value of q2 in equation (iii), we have,
q1 = 11.333 - 0.333 (10.909 - 0.455q1)
=> q1 = 11.333 - 3.633 + 0.152q1
=> q1 - 0.152q1 = 7.7
=> 0.848q1 = 7.7
=> q1 = 7.7 / 0.848
=> q1 = 9.08
From equation (iv) we get,
q2 = 10.909 - 0.455 * 9.08
=> q2 = 6.78
Therefore, virgin ore produces, q1 = 9.08 units and recycled materials produces, q2 = 6.78 units.
By comparing the values of q1 and q2 for given question's a and b's demand functions, it is clear that in case of b's demand function both the virgin ore and recycled materials produces more units than a's demand function's production levels.