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考虑一个竞争激烈、成本不变的行业。在这个市场中运营的公司是相同的,每个公司的总成本由 TC = 180 + 43q + 5q 2给出。如果市场需求曲线为 P = 331 - 2Q,从长远来看,该市场上的企业数量是多少?

Question

考虑一个竞争激烈、成本不变的行业。在这个市场中运营的公司是相同的,每个公司的总成本由

TC = 180 + 43q + 5q² 给出。如果市场需求曲线为
P = 331 - 2Q,从长远来看,该市场上的企业数量是多少?

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Solution

1. Break Down the Problem

To determine the number of firms in a competitive market in the long run, we need to find:

  1. The marginal cost (MC) of the firms.
  2. The average total cost (ATC) of the firms.
  3. The equilibrium price (P) at which the firms break even: P=ATC P = ATC .
  4. The market demand to find the total quantity supplied (Q) and then the number of firms.

2. Relevant Concepts

  1. Calculate Marginal Cost (MC) from Total Cost (TC): MC=d(TC)dq MC = \frac{d(TC)}{dq}

  2. Calculate Average Total Cost (ATC): ATC=TCq ATC = \frac{TC}{q}

  3. Equilibrium in a perfectly competitive market occurs where P=MC=ATC P = MC = ATC .

3. Analysis and Detail

Step 1: Calculate MC

Given: TC=180+43q+5q2 TC = 180 + 43q + 5q^2

Differentiate TC with respect to q q : MC=d(TC)dq=43+10q MC = \frac{d(TC)}{dq} = 43 + 10q

Step 2: Calculate ATC

ATC=TCq=180+43q+5q2q=180q+43+5q ATC = \frac{TC}{q} = \frac{180 + 43q + 5q^2}{q} = \frac{180}{q} + 43 + 5q

Step 3: Set P = MC = ATC

From the market demand curve: P=3312Q P = 331 - 2Q

In the long run equilibrium: P=MCandP=ATC P = MC \quad \text{and} \quad P = ATC

Set MC=ATC MC = ATC : 43+10q=180q+43+5q 43 + 10q = \frac{180}{q} + 43 + 5q

By simplifying, we eliminate 43 43 from both sides: 10q=180q+5q 10q = \frac{180}{q} + 5q 5q=180q 5q = \frac{180}{q} Multiplying both sides by q q : 5q2=180 5q^2 = 180 q2=36 q^2 = 36 q=6 q = 6

Step 4: Find equilibrium price (P)

Substituting q=6 q = 6 back into the MC: MC=43+10(6)=43+60=103 MC = 43 + 10(6) = 43 + 60 = 103 Thus, P=103 P = 103 .

Step 5: Find total quantity demand (Q)

Set P=3312Q P = 331 - 2Q and solve for Q Q : 103=3312Q 103 = 331 - 2Q 2Q=331103=228 2Q = 331 - 103 = 228 Q=114 Q = 114

Step 6: Calculate number of firms (n)

If each firm produces q=6 q = 6 : n=Qq=1146=19 n = \frac{Q}{q} = \frac{114}{6} = 19

4. Verify and Summarize

We have verified the calculations through differentiation and substitution. The equilibrium conditions and total quantity produced have been confirmed.

Final Answer

The number of firms in the market in the long run is 19.

This problem has been solved

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