Calculate JointJuice’s optimal output level and profits if the market price for the product stays the same as for part a. What will the firm do in the long run? Is it reasonable to assume the market price prevailing today will remain the same in the long run? If so why? If not, why not? How about the number of firms in the market?
I need to second half answered.
JointJuice produces a prepackaged joint support supplement for relief of joint pain with 180 tablets per bottle and operates in a
C(q) = 250 + 6q + 0.1q^2
The
Q = 1848 - 2P
- If there are 20 identical firms in this industry, find the
market equilibrium price for the prepackaged supplements. - Calculate JointJuice’s optimal output level and profits given the market price for the product.
- If JointJuice is typical of the firms in this industry calculate the firm’s long-run equilibrium output, price, and profit level.
- answers for first half
JointJuice is a prepackages supplement-producing firm.
Suppose the firm produces prepackaged supplements and operates in a perfectly competitive market.
Suppose JointJuice’s total cost function is given by the following where q is JointJuice’s quantity of packages per day:
C(q) = 250 + 6q + 0.1q2.
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The market demand function for the output in this market is given by:
Q = 1848 - 2P
(1)
A perfectly competitive firm produces at P = MC.
P = MC is an individual firm supply function.
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C(q) = 250 + 6q + 0.1q2.
Differentiate C w.r.t q to get MC
=> MC = dC / dq
=> MC = 6 + 0.2q
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Set P = MC
=> P = 6 + 0.2q
=> q = (P - 6)/0.2
=> q = 5P - 30 -----------> individual firm supply curve equation.
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There are 20 identical firms in the industry.
Market supply = Number of firms * Individual firm supply
=> Q = 20 * q
=> Q = 20 * (5P - 30)
=> Q = 100P - 600 -----------------------> Market supply equation.
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The equilibrium price in the market is determined at the intersection of market demand and the market supply curve.
Q = 1848 - 2P ----> Market demand
Q = 100P - 600 ---------> Market supply
=> Set market demand = market supply
=> 1848 - 2P = 100P - 600
=> 1848 + 600 = 100P + 2p
=> 2448 = 102P
=> P = (2448 / 102)
=> P = 24
Hence, the equilibrium market price is 24.
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(2)
A perfectly competitive firm produces at P = MC
=> P =MC
=> 24 = 6 + 0.2q
=> q = (24 -6) / 0.2
=> q = 18 / 0.2
=> q = 90
Hence, the output produced by JointJuice is 90 units.
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Profit = TR - TC
=> Profit = (P*q) - (250 + 6q + 0.1q2)
=> Profit = (24 * 90) - (250 + (6*90) + 0.1*(90)2)
=> Profit = 2160 - (250 + 540 + 810)
=> Profit = 2160 - 1600
=> Profit = 560
Hence, the profit of JointJuice is 560
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(3)
A perfectly competitive firm produces at P = MC =
C = 250 + 6q + 0.1q2
Divide C by q to get AC
=> AC = C / q
=> AC = (250 + 6q + 0.1q2) / q
=> AC = (250 /q) + 6 + 0.1q
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Set MC = AC
=> 6 + 0.2q = (250 /q) + 6 + 0.1q
=> 0.2q - 0.1q = (250/q) + 6 - 6
=> 0.1q = (250 / q)
=> q2 = (250 / 0.1)
=> q2 = 2500
=> q = (2500)0.5
=> q = 50
Hence, in long run, JointJuice will produce 50 units in long run.
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Set P = MC
=> P = 6 + 0.2q
=> P = 6 + 0.2 * 50
=> P = 6 + 10
=> P = 16
Hence, the long-run price is 16.
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Since P = AC in long run, the profit will be zero in long run.
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(1) The market equilibrium price is 24
(2) JointJuice’s optimal output level and profits are 90 units and 560 respectively.
(3) Long-run equilibrium price is 16, the firm's long-run equilibrium quantity is 50 units and the profit is zero.
Suppose the situation changes. JointJuice has its plant in Portland Oregon. The local government passes a new tax on businesses that raises JointJuice’s fixed cost by $25 per period. The cost function now becomes:
C(q) = 300 + 6q + 0.1q^2
All the other firms in the market are in other states/cities that are not subject to the laws or taxes of Portland.
- Calculate JointJuice’s optimal output level and profits if the market price for the product stays the same as for part a. What will the firm do in the long run?
- Is it reasonable to assume the market price prevailing today will remain the same in the long run? If so why? If not, why not? How about the number of firms in the market?
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