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⑸ ltc derivative
1.LMC = dLTC/dQ = 3Q2 – 24Q + 40 = 100
Q2 – 8Q – 20 = 0
Q = 10
Yield = 10
Average Cost = LTC/Q = Q2 – 12Q + 40 = 20
Profit = output * (price-average cost) = 10*(100-20) = 800
2. LTC reaches a minimum value in long-run equilibrium. Derivation of LTC is
2Q – 12 = 0
Q = 6
Yield = 6
Price = LMC = 3Q2 – 24Q + 40 = 4
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⑺ The long-term total cost function of a firm in a perfectly competitive industry is LTC=0.1Q3-10Q2+300Q, where Q is the monthly output, find the firm’s long-term equilibrium output, price and
The conditions for the long-term equilibrium of a perfectly competitive firm are: LAC=MC=P when the profit is zero
The lowest point of LAC That is, the equilibrium output, derivation of LAC to get 0.2Q-10=0 to get Q=50 Substitute into LAC to get P=50
or to find MC=0.3Q^2-20Q+300 by LTC Substitute Q=50 into P =MC=50
⑻ Regarding the problem of “Western Economics”, it is known that the long-term cost function of a single firm in a perfectly competitive market is LTC=Q3-20Q2+200Q
1. Profit = 600Q -LTC = 600Q – (Q3-20Q2+200Q) = -Q3 + 20Q2 + 400Q
Derivation from Q, we get
-3Q2 + 40Q + 400 = 0
Take the positive root and get
Q = 20
Maximum profit output = 20
Average cost = LTC/Q = Q2 – 20Q + 200 = 200
Profit = -Q3 + 20Q2 + 400Q = 8000
2. The average cost reaches a minimum value in long-run equilibrium
Average cost = LTC/Q = Q2 – 20Q + 200
Find Q Derive, get
2Q – 20 = 0
Q = 10
Production = 10
Marginal cost = dLTC/dQ = 3Q2 – 40Q + 200 = 100
Market Price = Marginal Cost = 100
Average Cost = LTC/Q = Q2 – 20Q + 200 = 100
Profit = Output*(Market Price – Average Cost) = 0
⑼ The long-term cost function of a single firm is LTC=Q3-20Q2+200Q, and the market price is P=600. Find out whether the industry is in long-term equilibrium
(1) LMC=30Q^2-40Q+200 and it is known that P=600
According to the profit maximization principle of mining current competitors, LMC=P, there are
Solution to 3Q^2-40Q-400=0, Q=20 (negative rounding)
From the known conditions, it can be obtained: LAC=Q=Q^2-20Q+200
Substituting Q=20 into the LAC function, the long-term average cost when the profit is maximized is
In addition, the profit value when the profit is maximized is: P·Q-LTC
So, the output Q= 20, the average cost LAC=200, and the profit is 8000.
(2) Set dQ=0, that is, dQ=2Q-20, so Q=10
So, when Q=10, the LAC curve reaches the minimum value.
Substituting Q=10 into the LAC function, we can get:
Combining the calculation results of (1) and (2), we can judge that the industry has not achieved long-term equilibrium. Because, it can be seen from (2) that when the industry achieves long-term equilibrium, the equilibrium price of the market should be equal to the height of the lowest point of the LAC curve of a single firm, that is, there should be a long-term equilibrium price P=100, and the long-term equilibrium output of a single firm should be is Q=10, and there should also be profit л=0 for each firm.
In fact, it can be seen from (1) that the price P=600, the output Q=20, and π=8000 when the firm maximizes profits. Obviously, the price, output, and profit of the manufacturer to maximize profits are greater than the requirements for a single manufacturer in the long-term equilibrium of the industry, that is, price 600>100, output 20>10, and profit 8000>0.
Therefore, the industry is not in a long-run equilibrium. (3) It is known from (2) that when the industry is in long-term equilibrium, the output of a single firm is Q=10, and the price is equal to the lowest long-term average cost, that is, there is P=minimum LAC=100, and profit л=0. (4) From the above analysis, it can be judged that: (1) The manufacturers instage. The reason is: in (1), the output of a single firm is Q=20 and the price is P=600, which are respectively greater than the output of a single firm at the lowest point of the LAC curve when the industry is in long-term equilibrium Q=10 and P=100. In other words, the profit-maximizing output and price combination of a single firm in (1) occurs on the right side of the lowest point of the LAC curve
, that is, the LAC curve is in the rising segment, so a single firm is in diseconomies of scale Stage.