垄断寡头模型分析

作成日: 2024年12月11日

使用モデル O1 by Chat01

質問

1．已知垄断者成本函数为TC=6Q+0.05Q2，产品需求函数为Q=360-20P，求：
(1)利润最大的销售价格、产量和利润。
(2)如果政府试图对该垄断企业采取规定产量措施使其达到完全竞争行业所能达到的产量水平。求解这个产量水平和此时的价格，以及垄断者的利润。
(3)如果政府试图对垄断企业采取限价措施使其只能获得生产经营的正常利润。求解这个限价水平以及垄断企业的产量。

2．假设有两个寡头厂商行为遵循古诺模型，其成本函数分别为：
TCl=O.1Q21+20Q1+10000
TC2=O.4Q22+32Q2+20000
这两个厂商生产同一质量产品，其市场需求函数为：
Q=4000-10P
根据古诺模型，试求：(1)厂商l和厂商2的反应函数；
(2)均衡价格和厂商1和厂商2的均衡产量；
(3)厂商1和厂商2的利润。

思考

Analyzing monopoly dynamics

I’m examining how monopoly power shapes market outcomes, considering pricing strategies and whether regulatory actions are justified.

Calculating profit

I’m defining total cost and revenue, finding profit by subtracting total cost from total revenue. By maximizing profit through differentiation, I determine the profit-maximizing quantity and price.

Verifying calculations

I’m confirming profit by checking total cost, total revenue, and profit calculations. Now, I’m working through the competitive equilibrium, where price equals marginal cost, to determine the optimal output.

Confirming the equilibrium

Interestingly enough, setting price equal to marginal cost yields a different equilibrium. I'm securing the equilibrium price and quantity, and checking the resulting profit level.

Confirming the approach

I’m analyzing the cost structure to ensure there are no fixed costs. This leads to the conclusion that setting production quantity to match perfect competition is the proper course of action.

Setting the price cap

OK, let's analyze the scenario where the government imposes a price cap, leading to zero economic profit. I'll determine the capped price that aligns marginal revenue with marginal cost.

Verifying the outcome

OK, let's check: zero economic profit at $P=12$ and $Q=120$ . Total cost and total revenue align perfectly, confirming the demand function's accuracy at this equilibrium.

Pinpointing optimal output

I'm concluding that the firm can't reach a positive profit ceiling with $P=12$ and $Q=120$ . It should aim for $P=MC$ to maximize profit at this price and quantity.

Reassessing strategy

Setting $P=MC$ at $P=12$ and finding $Q=60$ indicates the firm earns $180$ profit at this price and quantity, contradicting the aim for normal profit.

Adjusting my strategy

I’m thinking about how the firm should adjust its production to align with the price ceiling, considering linear demand and quadratic cost constraints, while tracking potential profit implications.

Balancing production

I'm noting the firm needs to balance maximizing profit with production constraints. Ensuring the quantity produced is feasible given the market demand and price ceiling is key.

Balancing production and market demand

I’m thinking about how the firm should match production quantity with market demand, ensuring total revenue aligns with total cost, while keeping profit at zero.

Ensuring market alignment

I’m noting that the firm should produce up to the market demand at a fixed price, balancing production quantity with market constraints to achieve zero profit.

Balancing price and quantity

I’m working through the equilibrium point, ensuring the chosen quantity maximizes profit while adhering to the market demand and price constraints. This approach maintains zero profit as the target.

Aligning conditions

I’m piecing together the optimal quantity given a price cap, ensuring the profit-maximizing quantity aligns with market demand constraints. This avoids conflicting conditions and maintains zero profit.

Confirming equilibrium

I’m verifying that the equilibrium quantity is determined by the smaller quantity between supply and demand constraints, ensuring zero profit condition is satisfied without conflicts.

Balancing constraints

I’m gauging that the firm must produce the maximum quantity the market demands at the given price cap to keep zero profit, without creating discrepancies.

Evaluating price feasibility

I’m working through the potential prices $P=18$ and $P=12$ , ensuring they align with market constraints. This helps confirm the firm can achieve zero profit under these circumstances.

Piecing together price constraints

OK, I’m confirming which price cap fits zero profit conditions without contradictions. It’s about aligning supply and demand effectively to achieve equilibrium.

Achieving zero profit

Trying to align the firm's production with market demand at a fixed price, aiming to ensure zero economic profit while balancing supply and demand constraints effectively.

Pushing for zero profit

I’m working to match the firm’s production to market demand at a fixed price, aiming for zero economic profit while balancing supply and demand constraints effectively.

Aligning equilibrium dynamics

I’m showing that with a price ceiling, equilibrium ensures the firm produces where $P_c = MC$ , which surprisingly leads to a trivial zero profit scenario.

Aligning costs and profits

I’m figuring out that setting the price ceiling equal to the average cost maximizes profit while keeping it zero, aligning quantity with the optimal production level.

Balancing production

OK, let me see. Aligning the price ceiling with total cost maximizes profit while maintaining zero economic profit, keeping production and market demands in sync.

Balancing profit and market constraints

I’m thinking about how the firm should set production to zero when price is capped at $12$ , aligning with market demand to achieve zero profit dynamically.

Ensuring equilibrium production

I’m figuring out that the firm should adjust production to match demand at a fixed price of $12. This approach maintains zero profit, balancing supply and market constraints effectively.

Munking through equilibrium

OK, so under a price ceiling, equilibrium depends on supply curve intersections with the price cap, ensuring zero economic profit and balancing production with market constraints effectively.

Examining market demands

I’m analyzing the scenario where the firm must meet all market demand at the price ceiling. This leads to a possible shortage since the firm can't produce enough to match the increased demand.

Balancing production and demand

OK, let's see. Setting a price ceiling at $12$ ensures zero profit by making the firm produce exactly $120$ units, matching market demand perfectly.

Assessing duopoly behavior

I’m evaluating the Cournot duopoly, focusing on firm 1’s profit by combining like terms for an accurate total profit calculation. This helps determine each firm’s optimal production.

Determining equilibrium quantities

I’m optimizing each firm’s production with the combined profit functions, considering the interdependence in this Cournot duopoly setup. Progress is being made in finding an efficient equilibrium.

Adjusting production plans

OK, let’s see. Each firm is changing its production levels to boost profits. This leads to a sharper equilibrium under Cournot competition, with both firms maximizing their profits.

Thought about monopolistic and oligopolistic problems，持续 2m 4s

回答

下面为详细解答步骤。

第一题

已知条件：

垄断者总成本函数： $TC = 6Q + 0.05Q^2$
市场需求函数： $Q = 360 - 20P$ 由需求函数可得价格函数：
$P = \frac{360 - Q}{20} = 18 - 0.05Q.$

(1) 求利润最大时的销售价格、产量和利润。

垄断者利润 $\pi = TR - TC$ 。
总收益（TR）： $TR = P \cdot Q = (18 - 0.05Q)Q = 18Q -0.05Q^2.$

总成本（TC）： $TC = 6Q +0.05Q^2.$

利润：
$\pi = TR - TC = (18Q - 0.05Q^2) - (6Q + 0.05Q^2) = 18Q -0.05Q^2 -6Q -0.05Q^2.$

合并项：
$\pi = (18Q -6Q) - (0.05Q^2 +0.05Q^2) = 12Q -0.1Q^2.$

求最大利润的产量：
对 $\pi$ 关于 Q 求导：
$\frac{d\pi}{dQ}=12 -0.2Q =0 \implies 0.2Q=12 \implies Q=60.$

将 $Q=60$ 代入价格函数：
$P = 18 -0.05 \times 60 =18 -3=15.$

利润：
$\pi(60)=12 \times 60 -0.1 \times 60^2 =720 -0.1 \times 3600 =720 -360=360.$

故利润最大时：

最优产量 $Q^* =60$
最优价格 $P^* =15$
最大利润 $\pi^* =360$ .

(2) 如果政府要求该垄断企业生产达到完全竞争产量水平。

在完全竞争市场中，长期均衡下价格等于边际成本 (P=MC)。先求边际成本：
$MC=\frac{dTC}{dQ}=6 +0.1Q.$

完全竞争产量由市场需求和 $P=MC$ 条件决定：
$P=18 -0.05Q, \quad MC=6+0.1Q.$

设 $P=MC$ ：
$18 -0.05Q =6 +0.1Q$ $18 -6=0.1Q +0.05Q$ $12=0.15Q \implies Q=80.$

价格：
$P=18 -0.05 \times 80 =18 -4=14.$

利润：
总收益： $TR = P \cdot Q =14 \times 80=1120.$
总成本： $TC=6 \times 80 +0.05 \times 80^2 =480 +0.05 \times 6400=480+320=800.$

利润： $\pi=1120 -800=320.$

故在完全竞争产量下(即政府强制达到此产量)：

产量 $Q=80$
价格 $P=14$
利润 $\pi=320$ .

(3) 如果政府对垄断企业实行限价使其获得正常利润（即零经济利润）。

零经济利润条件： $\pi=0$ 。利润函数为：
$\pi(Q)=TR -TC=(18Q -0.05Q^2) - (6Q+0.05Q^2)=12Q -0.1Q^2.$

零利润条件：
$12Q -0.1Q^2=0$ $0.1Q^2=12Q$ $Q=0 \text{ 或 } Q=120.$

$Q=120$ 时得到零利润。求此时价格：
$P=18 -0.05 \times 120=18-6=12.$

在价格 $P=12$ 时市场需求量为 $Q_d=360-20 \times 12=360-240=120.$

要实现零利润，政府若将价格上限设定为12，那么市场需求为120，此时若垄断者必须满足市场需求(即以该价格出售120单位)，则利润为零。这里的关键在于政府限价并迫使企业满足市场需求（不出现供给不足的情况）。在此假设下：

限价 $P=12$
垄断者产量 $Q=120$
利润 $\pi=0$ .

第二题

有两个寡头厂商，遵循古诺(Cournot)模型。

已知：
厂商1的成本函数： $TC_1=0.1Q_1^2+20Q_1+10000$
厂商2的成本函数： $TC_2=0.4Q_2^2+32Q_2+20000$

市场需求函数： $Q=Q_1+Q_2=4000 -10P$
则价格函数： $P = 400 -0.1(Q_1+Q_2).$

(1) 厂商1和厂商2的反应函数

厂商1的利润：
$\pi_1 = P Q_1 - TC_1 = (400 -0.1(Q_1+Q_2))Q_1 - (0.1Q_1^2 +20Q_1+10000).$

展开：
$\pi_1 = 400Q_1 -0.1Q_1(Q_1+Q_2) -0.1Q_1^2 -20Q_1 -10000.$

整理二次项：
收入中有： $400Q_1 -0.1Q_1^2 -0.1Q_1Q_2$
减成本 $0.1Q_1^2+20Q_1+10000$ 后，合并同类项：

\pi_1 = 400Q_1 -0.1Q_1Q_2 - (0.1Q_1^2+0.1Q_1^2) -20Q_1 -10000

\pi_1 = 400Q_1 -0.1Q_1Q_2 -0.2Q_1^2 -20Q_1 -10000

\pi_1 = (400-20)Q_1 -0.1Q_1Q_2 -0.2Q_1^2 -10000

\pi_1 = 380Q_1 -0.1Q_1Q_2 -0.2Q_1^2 -10000.

对 $Q_1$ 一阶条件：
$\frac{\partial \pi_1}{\partial Q_1}=380 -0.1Q_2 -0.4Q_1=0.$

得：
$-0.4Q_1 = -380 +0.1Q_2$ $Q_1 = \frac{380 -0.1Q_2}{0.4}=950 -0.25Q_2.$

这就是厂商1的反应函数：

Q_1 = 950 -0.25Q_2.

同理，厂商2的利润：
$\pi_2 = P Q_2 - TC_2 = (400 -0.1(Q_1+Q_2))Q_2 - (0.4Q_2^2+32Q_2+20000).$

展开：

\pi_2 = 400Q_2 -0.1Q_2(Q_1+Q_2) -0.4Q_2^2 -32Q_2 -20000.

\pi_2 = 400Q_2 -0.1Q_1Q_2 -0.1Q_2^2 -0.4Q_2^2 -32Q_2 -20000

合并二次项： $-0.1Q_2^2 -0.4Q_2^2 = -0.5Q_2^2$ ，线性项 $400Q_2-32Q_2=368Q_2$ ：

\pi_2 = 368Q_2 -0.1Q_1Q_2 -0.5Q_2^2 -20000.

对 $Q_2$ 求导：

\frac{\partial \pi_2}{\partial Q_2} = 368 -0.1Q_1 -1.0Q_2 =0.

-1.0Q_2 = -368 +0.1Q_1

Q_2 = 368 -0.1Q_1.

这是厂商2的反应函数：

Q_2=368 -0.1Q_1.

(2) 均衡产量与价格

将厂商2的反应函数代入厂商1反应函数求解库诺均衡：

厂商1反应函数：
$Q_1 =950 -0.25Q_2.$

厂商2反应函数：
$Q_2 =368 -0.1Q_1.$

代入：
从第二个方程中： $Q_2=368 -0.1Q_1$

将 $Q_2$ 代入第一个：
$Q_1=950 -0.25(368 -0.1Q_1)$ $Q_1=950 -0.25 \times 368 +0.025Q_1$ $0.25 \times 368=92$ : $Q_1=950 -92 +0.025Q_1$ $Q_1=858 +0.025Q_1$ $Q_1 -0.025Q_1=858$ $0.975Q_1=858$ $Q_1=\frac{858}{0.975}=880.$

有 $Q_1=880$ 后，用 $Q_2=368 -0.1 \times 880=368-88=280.$

故均衡产量：

Q_1^*=880, \quad Q_2^*=280.

市场总产量 $Q=Q_1+Q_2=880+280=1160.$

价格： $P=400 -0.1 \times 1160=400 -116=284.$

(3) 厂商1和厂商2的利润

厂商1利润：

\pi_1 = P Q_1 -TC_1 =284 \times 880 - (0.1 \times 880^2+20 \times 880 +10000).

计算成本项：
$880^2=774400$
$0.1 \times 774400=77440$
$20 \times 880=17600$
总成本： $77440+17600+10000=77440+27600=105040.$

总收益：
$TR_1=284 \times 880=249920.$

利润：
$\pi_1=249920 -105040=144880.$

厂商2利润：

\pi_2 =284 \times 280 - (0.4 \times 280^2 +32 \times 280 +20000).

$280^2=78400$ $0.4 \times 78400=31360$ $32 \times 280=8960$ 总成本： $31360+8960+20000=31360+(8960+20000)=31360+28960=60320.$

总收益：
$TR_2=284 \times 280=79680.$ （请校对一下：284×280）计算：284×200=56800, 284×80=22720, 和=56800+22720=79520（之前为79680需更正）
正确计算：284×280=284×(200+80)=284×200+284×80=56800+22720=79520。

利润： $\pi_2=79520 -60320=19200.$

最终答案汇总

第一题：
(1) 垄断均衡：

最优产量 $Q=60$
最优价格 $P=15$
最大利润 $\pi=360.$

(2) 完全竞争产量水平下：

产量 $Q=80$
价格 $P=14$
利润 $\pi=320.$

(3) 正常利润限价：

限价 $P=12$
产量 $Q=120$ (假设在该限价下垄断者需满足全部需求)
利润 $\pi=0.$

第二题：
(1) 反应函数：
$Q_1=950 -0.25Q_2,$ $Q_2=368 -0.1Q_1.$

(2) 古诺均衡：
$Q_1^*=880,\quad Q_2^*=280,\quad P^*=284.$

(3) 各自利润：
$\pi_1=144880,\quad \pi_2=19200.$

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