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12.3 SECOND-DEGREE PRICE DISCRIMINATION: QUANTITY DISCOUNTS 497
with this part of the demand curve is the segment BN. Since the second
block will be sold at a single, uniform price, the optimal quantity for
this second block will correspond to the intersection of the marginal
revenue curve and the marginal cost curve MC, at Q 2 . Since the de-
mand curve is linear, the marginal revenue curve has twice the slope of
the demand curve, and Q must lie halfway between Q and 18 (as we
1
2
showed when deriving the monopoly midpoint rule in Chapter 11—see
Learning-By-Doing Exercise 11.5). That is, Q (Q 18)/2.
2
1
Step 2. Producer surplus is total revenue minus total variable cost. The revenue
from the first block is P Q , the revenue from the second block is
1 1
P (Q Q ), and total variable cost is 2Q . Thus, producer surplus
1
2
2
2
PS P Q P (Q Q ) 2Q . The demand equation tells us
2
1 1
2
2
1
that P 20 Q and that P 20 Q , which means that
1
1
2
2
PS (20 Q )Q (20 Q )(Q Q ) 2Q , which reduces
1
1
2
2
1
2
2
to PS (3/4)(Q 6) 108.
1
2
Step 3. Since the expression (3/4)(Q 6) is negative for any value of Q other
1
1
than 6, PS is maximized (at 108) when this expression equals zero, or
when Q 6. Thus, the optimal quantity for the first block Q 6
1
1
units of electricity, with an optimal price P 20 6 $14 per unit;
1
the optimal quantity for the second block is then Q (6 18)/2 12
2
units, with an optimal price P 20 12 $8 per unit; and the
2
maximum producer surplus is $108. 8
In this example, second-degree price discrimination with the optimal block tariff
(assuming just two blocks) increased producer surplus by $27 over producer surplus
with uniform pricing ($108 versus $81).
LEARNING-BY-DOING EXERCISE 12.3
S
D
E
Increasing Profits with a Block Tariff
Softco is a software company that sells a Softco were to sell the first block at the price you deter-
patented computer program to businesses. Each busi- mined in (a), and that the quantity for that block is the
ness it serves has the demand for Softco’s product: P quantity you determined in (a). Find the profit-maximizing
70 0.5Q. The marginal cost for each program is $10. quantity and price per unit for the second block. How
Assume there are no fixed costs. much extra profit would Softco earn from each of its
business customers?
Problem
(c) Do you think Softco could earn even more profits
(a) If Softco sells its program at a uniform price, what with a set of prices and quantities for the two blocks dif-
price would maximize profit? How many units would it ferent from those in part (b)? Explain.
sell to each business customer? How much profit would
it earn from each business customer? Solution
(b) Softco would like to know if it is possible to improve (a) The marginal revenue for each customer is MR
its profit by implementing block pricing. Suppose that 70 Q. We can find the optimal quantity by setting
8 One can also find the optimal block tariffs using calculus. As above, PS (20 Q 1 )Q 1 (20 Q 2 )
(Q 2 Q 1 ) 2Q 2 . If we set the partial derivative of PS with respect to Q 1 equal to zero, we find that Q 2
2Q 1 . If we set the partial derivative of PS with respect to Q 2 equal to zero, we find that 18 2Q 2 Q 1 0.
Then we solve these two equations in two unknowns to find that Q 1 6 and Q 2 12, from which we
can calculate the block prices and the producer surplus. For more on the use of derivatives to find a maxi-
mum, see the Mathematical Appendix at the end of the book.