Question

Suppose that the market demand curve for bean sprouts is given by P = 1,660 - 4Q, where P is the price and Q is total industry output. Suppose that the industry has two firms, a Stackleberg leader and a follower. Each firm has a constant marginal cost of $60 per unit of output. In equilibrium, total output by the two firms will be:_______.a. 200. b. 400. c. 50. d. 100. e. 300.

Answer 1

In equilibrium, total output by the two firms will be option e= 300.

Q =
`q_(1)` +
`q_(2)`

Q = 100 + 200

Q = 300

Step-by-step explanation:

Data Given:

Market Demand Curve = P = 1660-4Q

where, P = price and Q = total industry output

Each firm's marginal cost = $60 per unit of output

So, we know that Q =
`q_(1)` +
`q_(2)`

where
`q_{}` being the individual firm output.

Solution:

P = 1660-4Q

P = 1660- 4(
`q_(1)` +
`q_(2)`)

P = 1660 - 4
`q_(1)` - 4
`q_(2)`

Including the marginal cost of firm 1 and multiplying the whole equation by
`q_(1)`

Let's suppose new equation is X

X = 1660
`q_(1)` - 4
`q_(1) ^(2)` - 4
`q_(1)`
`q_(2)` - 60
`q_(1)`

Taking the derivative w.r.t to
`q_(1)`, we will get:

`X^(')` = 1660 - 8
`q_(1)` - 4
`q_(2)` - 60 = 0

Making rearrangements into the equation:

8
`q_(1)` +
`q_(2)` = 1660 - 60

8
`q_(1)` +
`q_(2)` = 1600

Dividing the whole equation by 4

2
`q_(1)` +
`q_(2)` = 400

Solving for
`q_(1)`

2
`q_(1)` = 400 -
`q_(2)`

`q_(1)` = 200 - 0.5
`q_(2)`

Including the marginal cost of firm 1 and multiplying the whole equation by
`q_(2)`

P = 1660 - 4
`q_(1)` - 4
`q_(2)`

Let's suppose new equation is Y

Y = 1660
`q_(2)` - 4
`q_(1)`
`q_(2)` -4
`q_(2) ^(2)` - 60
`q_(2)`

Pugging in the value of
`q_(1)`

Y = 1660
`q_(2)` - 4
`q_(2)`(200 - 0.5
`q_(2)`) -4
`q_(2) ^(2)` - 60
`q_(2)`

Y = 1660
`q_(2)` - 800
`q_(2)` +2
`q_(2) ^(2)` -4
`q_(2) ^(2)` - 60
`q_(2)`

Y = 1600
`q_(2)` - 800
`q_(2)` -2
`q_(2) ^(2)`

Taking the derivative w.r.t
`q_(2)`

`Y^(')` = 1600 - 800 - 4
`q_(2)` = 0

Solving for
`q_(2)`

4
`q_(2)` = 800

`q_(2)` = 200

`q_(1)` = 200 - 0.5
`q_(2)`

Plugging in the value of
`q_(2)` to get the value of
`q_(1)`

`q_(1)` = 200 - 0.5 (200)

`q_(1)` = 200 - 100

`q_(1)` = 100

Q =
`q_(1)` +
`q_(2)`

Q = 100 + 200

Q = 300

Hence, in equilibrium, total output by the two firms will be option

e= 300.