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Sagot :
Answer:
In equilibrium, total output by the two firms will be option e= 300.
Q = [tex]q_{1}[/tex] + [tex]q_{2}[/tex]
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 = [tex]q_{1}[/tex] + [tex]q_{2}[/tex]
where [tex]q_{}[/tex] being the individual firm output.
Solution:
P = 1660-4Q
P = 1660- 4([tex]q_{1}[/tex] + [tex]q_{2}[/tex])
P = 1660 - 4[tex]q_{1}[/tex] - 4[tex]q_{2}[/tex]
Including the marginal cost of firm 1 and multiplying the whole equation by [tex]q_{1}[/tex]
Let's suppose new equation is X
X = 1660[tex]q_{1}[/tex] - 4[tex]q_{1} ^{2}[/tex] - 4[tex]q_{1}[/tex][tex]q_{2}[/tex] - 60[tex]q_{1}[/tex]
Taking the derivative w.r.t to [tex]q_{1}[/tex], we will get:
[tex]X^{'}[/tex] = 1660 - 8[tex]q_{1}[/tex] - 4[tex]q_{2}[/tex] - 60 = 0
Making rearrangements into the equation:
8[tex]q_{1}[/tex] + [tex]q_{2}[/tex] = 1660 - 60
8[tex]q_{1}[/tex] + [tex]q_{2}[/tex] = 1600
Dividing the whole equation by 4
2[tex]q_{1}[/tex] +[tex]q_{2}[/tex] = 400
Solving for [tex]q_{1}[/tex]
2[tex]q_{1}[/tex] = 400 - [tex]q_{2}[/tex]
[tex]q_{1}[/tex] = 200 - 0.5 [tex]q_{2}[/tex]
Including the marginal cost of firm 1 and multiplying the whole equation by [tex]q_{2}[/tex]
P = 1660 - 4[tex]q_{1}[/tex] - 4[tex]q_{2}[/tex]
Let's suppose new equation is Y
Y = 1660[tex]q_{2}[/tex] - 4[tex]q_{1}[/tex][tex]q_{2}[/tex] -4[tex]q_{2} ^{2}[/tex] - 60[tex]q_{2}[/tex]
Pugging in the value of [tex]q_{1}[/tex]
Y = 1660[tex]q_{2}[/tex] - 4[tex]q_{2}[/tex](200 - 0.5 [tex]q_{2}[/tex]) -4[tex]q_{2} ^{2}[/tex] - 60[tex]q_{2}[/tex]
Y = 1660[tex]q_{2}[/tex] - 800[tex]q_{2}[/tex] +2[tex]q_{2} ^{2}[/tex] -4[tex]q_{2} ^{2}[/tex] - 60[tex]q_{2}[/tex]
Y = 1600[tex]q_{2}[/tex] - 800[tex]q_{2}[/tex] -2[tex]q_{2} ^{2}[/tex]
Taking the derivative w.r.t [tex]q_{2}[/tex]
[tex]Y^{'}[/tex] = 1600 - 800 - 4[tex]q_{2}[/tex] = 0
Solving for [tex]q_{2}[/tex]
4[tex]q_{2}[/tex] = 800
[tex]q_{2}[/tex] = 200
[tex]q_{1}[/tex] = 200 - 0.5 [tex]q_{2}[/tex]
Plugging in the value of [tex]q_{2}[/tex] to get the value of [tex]q_{1}[/tex]
[tex]q_{1}[/tex] = 200 - 0.5 (200)
[tex]q_{1}[/tex] = 200 - 100
[tex]q_{1}[/tex] = 100
Q = [tex]q_{1}[/tex] + [tex]q_{2}[/tex]
Q = 100 + 200
Q = 300
Hence, in equilibrium, total output by the two firms will be option
e= 300.
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