Business, 01.01.2021 17:30 robert7248

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.

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Answer from: Meowkitty1894

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

Q = $q_{1}$ + $q_{2}$

Q = 100 + 200

Q = 300

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.

Answer from: Quest

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Answer from: Quest

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Suppose that the market demand curve for bean sprouts is given by P = 1,660 - 4Q, where P is the pri...

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