Question

Suppose John’s demand for a good is given by P=200-1/5 Q and that marginal revenue is MR=200-0.4Q. Marginal cost and average total cost are constant at $20 per unit.

Consider the following quantity discount.

The price for the first 400 units a customer purchases is $120 per unit, but any additional units have a price of $80 per unit. How many units will John purchase?

Answer 1

John will purchase where the marginal revenue is equal to the marginal cost.

Equating the marginal revenue to the marginal cost:

$$200 - 0.4Q = 20 \\[0.3cm] 0.4Q = 180\\[0.3cm] Q = 450$$

Without the quantity discount, John will purchase 450 units.

The price without quantity discount is:

$$P = 200 - \dfrac{1}{5}(450)\\[0.3cm] P = \$110$$

Without the quantity discount, John will spend:

$$\$110\times 450=\$49,500$$

The quantity discount is that John gets to pay $120 for the first 400 units. Thus, for 450 units, his spending will be:

$(\$120\times 400) + \$80(450 - 400) = \$52,000$

Therefore, John will buy just 450 units.