Two firms (A and B) are planning to produce a new soft drink for the summer. The
ID: 1169811 • Letter: T
Question
Two firms (A and B) are planning to produce a new soft drink for the summer. The soft drinks produced by the two firms can differ only in the level of sugar, aside from that they will be exactly equal. Suppose firm A chooses to produce a soft drink with 0g of sugar and firm B chooses to produce a soft drink with 50g of sugar. The marginal cost of producing soft drinks is $1. There are 1000 consumers in this market. Consumers differ in their preference for sugar and are uniformly distributed according to their preference for sugar. Consumers with the lowest valuation for sugar prefer a soft drink with 0g of sugar, whereas consumers with the highest valuation for sugar prefer a soft drink with 50g. So, preferences for sugar are between 0g and 50g. Consumers get a disutility (in monetary value) of $0.10 for each gram of sugar different from their preferred level. Each consumer reservation value for the soft drink with the most preferred level of sugar is $10. The two firms compete for consumers by setting prices. Firm A sets the price PA first and then firm B sets the price PB after observing PA. Consumers buy the soft drink that provides the highest consumer surplus.
Suppose firm A invests in a new technology that reduces its marginal cost to $0.80. The marginal cost of firm B is still $1. What prices will firms set in equilibrium in this case?
Explanation / Answer
There are 1000 consumers. They are uniformly distributed in their preference of grams of sugar in the soft drink they consume, from 0 grams of sugar to 50 grams of sugar. Therefore, the uniform distribution is shown by curve AB in the following figure.
The vertical incept is 20 (=1000/50). For any level of preference y, the number of consumers who have a preference of less than y grams of sugar is given by the shaded area below the curve AB, and the number of consumers who have a preference of more than y grams of sugar is given by the unshaded area below the curve.
No. of consumers who have a preference of less than y grams of sugar
= shaded area below the curve
= Length × breadth
= 20y
No. of consumers who have a preference of more than y grams of sugar
= unshaded area below the curve
= Length × breadth
= 20(50 – y)
Each consumer gets disutility of $0.10 for every gram of sugar consumed different from his/her preference. The gross surplus from consuming his preference of soft drink is $10.
Consider a consumer with a preference of y grams of sugar in his/her soft drink. The consumer’s net surplus from buying from firm A is
Consumer’s net surplus from buying from A
= $10 – pA – $0.10(grams of sugar preferred – grams of sugar in the soft drink)
= $10 – pA – $0.10(y – 0)
= 10 – pA – 0.10y
Consumer’s net surplus from buying from B
= $10 – pB – $0.10(grams of sugar in the soft drink grams of sugar preferred)
= $10 – pB – $0.10(50 – y)
= 10 – pB – 5 + 0.10y
If this consumer in indifferent between buying from A and B then
10 – pA – 0.10y = 10 – pB – 5 + 0.10y
– pA – 0.10y = – pB – 5 + 0.10y
pA + 0.10y = pB + 5 – 0.10y
0.20y = pB – pA + 5
y = 5pB – 5pA + 25
and all consumers with preference below y will buy from A (because their net surplus from A will be more than their surplus from B) and all consumers with preference above y will buy from B (because their surplus from B will be more than their surplus from A).
Suppose each consumer buys only one soft drink and proceed.
Firm A’s profit is
A = revenue of firm A – cost of firm A
= (pA×no. of consumers served by A) – (per unit cost × no. of consumers served by A)
= pA(20y) – 1(20y)
= pA(100pB – 100pA + 500) (100pB – 100pA + 500)
= 100 pApB – 100p2A + 500pA 100pB + 100pA 500
To maximize profit, differentiate the above function with respect to pA and equate to 0.
A/pA = 0
100pB – 200pA + 500 + 100 = 0
100pB + 600 = 200pA
pA = (pB+6)/2 …… (1)
Firm B’s profit is
B = revenue of firm B – cost of firm B
= (pB×no. of consumers served by B) – (per unit cost × no. of consumers served by B)
= pB[20(50y)] – 1[20(50y)]
= pB(1000 100pB + 100pA – 500) – (1000 100pB + 100pA – 500)
= 1000 pB 100p2B + 100pA pB – 500 pB – 1000 + 100pB – 100pA + 500
= 600pB 100p2B + 100pA pB – 500 – 100pA
To maximize profit, differentiate the above function with respect to pB and equate to 0.
B/pB = 0
600 200pB + 100pA = 0
100pA + 600 = 200pB
pB = (pA+6)/2 …… (2)
Solving equation (2) in equation (1), we get
pA = pB = 6
These are the prices the firms will set if each has a marginal cost of $1.
Suppose the marginal cost of firm 1 is $0.80 and that of firm B is $1.
Then,
Firm A’s profit is
A = revenue of firm A – cost of firm A
= (pA×no. of consumers served by A) – (per unit cost × no. of consumers served by A)
= pA(20y) – 0.80(20y)
= pA(100pB – 100pA + 500) – 0.80(100pB – 100pA + 500)
= 100 pApB – 100p2A + 500pA 80pB + 80pA 400
To maximize profit, differentiate the above function with respect to pA and equate to 0.
A/pA = 0
100pB – 200pA + 500 + 80 = 0
100pB + 580 = 200pA
pA = (pB+5.8)/2 …… (1a)
Solving equation (2) in equation (1a), we get
pA = 5.87
pB = 5.93
These are the prices the firms will set if the marginal cost of firm 1 is $0.80 and that of firm B is $1.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.