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

Answer:

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.