MicroEcon: 竞争下的 Cost 与 Profit Maximization

大前提

在完全竞争的市场下，任何一家企业都无法操控市场价格，只能根据市场价格调整自身的产量。即市场价格决定每一家企业的定价。

逻辑

由于企业无法操控市场价格，如果企业的定价高于市场价格，那么消费者必然转向其替代品；如果企业定价低于市场价格

企业运行的驱动动力：利益

企业必然追求利益。当产量为 $Q$ 时，利润 Profit $\pi(Q)$ 定义为

\boxed{\pi(Q)=TR(Q)-TC(Q)}

这里的 Total Cost $TC(Q)$ 包含 explicit cost 和 implicit cost.

Economic Profit 与 Accounting Profit

这两个概念会在分析长期行为时提到，简而言之

Economic Profit 需要包含 Implicit Cost
Accounting Profit 则不需要包含

后文提到的 Profit 若无特殊说明都是指 Economic Profit.

利益最大化

根据 Cost-Benefit Analysis，利润最大化的时候对应的产量 $Q^\ast$ 必然有

MR(Q^\ast)=MC(Q^\ast)

而且，此处的 $MR$ 恒等于市场价格，即 $MR(Q^\ast)=P$ 。因此，在完全竞争的市场下，总是有

\boxed{P=MC(Q^\ast)}

企业的成本

分为两种：

固定成本 (Fixed Cost)：短期内无法改变 Quantity，例如生产机器、办公楼等等
可变成本 (Variable Cost)：短期内可以改变 QUantity，例如劳动力、生产原料等等

在后面分析企业行为的时候也会用到这两个概念。简单来说，如果考察企业的短期行为，由于固定成本可以看作是已经产生费用，因此应当看作 Sunk Cost，不应参与短期行为决策；但对于长期行为而言，也应将固定成本考虑进去。

企业的市场行为

既然要赚取利益，那么企业是否继续参与市场必然与 Profit 有关。当 $\pi(Q^\ast)\lt 0$ 时，企业的利润小于零，企业会选择退出市场；否则就有利可图，会继续参与市场。

也可以理解为 $\pi(Q=Q^\ast)$ 与 $\pi(Q=0)$ 之间进行比较

短期行为

从短期来看，企业的 Fixed Cost 不应计入决策过程，企业是否退出市场取决于利润

\begin{aligned} \pi(Q^\ast)&\ge \pi(0)\\ TR(Q^\ast)-TC(Q^\ast)&\ge TR(0)-TC(0)\\ PQ^\ast-TVC(Q^\ast)-TFC&\ge 0-TVC(0)-TFC\\ PQ^\ast&\ge TVC(Q^\ast)\\ \end{aligned}

所以有

\boxed{P\ge AVC(Q^\ast)}

因此从短期来看，企业是否会退出市场，取决于市场价格（会影响收入）和自身生产的平均可变成本 (Average Variable Cost, AVC)。此时的利润为

\boxed{\pi(Q^\ast)=\Big( P-\textcolor{red}{AC} \Big)\times Q^\ast}

注意！

计算利润时要注意包含 Fixed Cost。只有在做决策时才不计算 FC.

曲线的性质

MC 曲线总是和 AVC 曲线交于 AVC 曲线的最低点。

证明

略。

进一步的，我们可以推断，企业短期内的生产曲线 Supply Curve 由 MC Curve 和 Shutdown Decision 共同决定。

Supply Curve

长期行为

对于长期而言，Fixed Cost 此时也应该算入（例如办公楼续约费可以视为支出）。类似的，应有

\boxed{P\ge AC(Q^\ast)}

当且仅当市场价格高于平均成本（含固定成本），企业才会考虑进入市场（不然无法收回固定成本）。

合并来看，Supply Curve 差不多长这样：

Aggregate Supply Curve

例题

2020 Spring Final

题目

There are many identical firms (having the same production costs and producing the same product) in a competitive market. Their product is assumed to be perfectly divisible. When one firm produces $q$ units, its marginal cost is

MC(q) = q

and the corresponding total variable cost is

TVC(q) = 0.5q^2

The demand curve in the market is

Q = 9000 - 90P

Moreover, assume that the industry is constant cost. The market is currently in a long-run equilibrium, and the equilibrium price is $\$45$ per unit.

Using the information, we conclude that there are [ Answer36 ] firms in the market.
Using this information, we conclude that the fixed cost of an individual firm is [ Answer37 ] dollars.
Using this information, we conclude that the short-run supply curve of this industry is of the form

Q = A + B \times P

where $A$ is equal to [ Answer38A ], and $B$ is equal to [ Answer38B ]

Suppose now the market demand has become

Q = 15300 - 153P

Using the information, we can calculate that in the long run, an individual firm will produce [ Answer39 ] units.

解答

Question Bank

Determine whether the following statements about a perfectly competitive market are TRUE OR FALSE.
Statement 1:
The demand curve facing the market is horizontal.
Statement 2:
Entry of new firms will drive the accounting profits of the existing firms to zero in the long run.

【解答】False, False
【解说】对于 Statement 1，完全竞争市场的 Demand Curve 应该是 Downward Sloping 的。陈述错误。
对于 Statement 2，长期来看，完全竞争市场下，企业的 Economic profit 会趋于 $0$ ，但是 Accounting profit 相比 Economic profit 无须考虑 implicit cost，因此即使 economic profit 为 $0$ ，accounting profit 依旧可能 $\gt 0$ 。因此陈述错误。

Determine whether the following statements are TRUE OR FALSE.
Statement 1:
Competitive firms can still earn positive accounting profits in the long run.
Statement 2:
A higher total cost to a firm must imply a lower level of output.

【解答】True, False
对 Statement 1 的解说见 Q1 statement 2
对于 statement 2，因为

\text{Total cost}=\text{Variable Cost}+\text{Fixed Cost}

如果 Fixed cost 增加，那么短期内企业仍然会维持产量。因此陈述错误。

Suppose the government reduces a commercial registration fee for all firms in a competitive industry. Evaluate whether the following statements are TRUE or FALSE.
Statement 1: A typical firm of the industry will raise its output in the short run.
Statement 2: A typical firm of the industry will see an increase in producer surplus in the short run.

题目减少的是 fixed cost，对于 short run 没有影响。

\text{生产者剩余}=\text{总收益}-\text{可变成本}

The government invested $10 million in an infrastructure project two years ago. The amount of money was spent on wages and material costs, which cannot be recovered. This year, the government realizes that she needs to invest another $4 million in the project because of construction delays, while the $11 million expected revenue remains unchanged and can only be generated upon the completion of the project. The government should ________ the additional budget because ________.

Select one:
a. approve; the project can generate positive revenue.
b. approve; marginal benefit of the extra payment is greater than its marginal cost.
c. disapprove; the cost is higher than expected
d. disapprove; the government will suffer from a $3 million loss.

【解答】B
【解说】A 为什么错误呢？分析应该基于边际成本与收益。

There are $100$ firms in a perfectly competitive decreasing cost industry at the long-run equilibrium. A typical firm faces the following cost curves ( $q$ is firm quantity, $n$ is number of firms):

Average Variable Cost ($): $AVC=q+50$
Marginal Cost ($): $MC=2q+50$
Fixed Cost ($): $FC=810000/n$
What are the market price and market quantity?

Select one:
a. $\$230$ , $9000$ units
b. $\$15.5$ , $8100$ units
c. $\$155$ , $81$ units
d. $\$230$ , $90$ units

长期平衡的话，考虑 $TC=TVC+TFC$ ， $AC=AVC+AFC$ 。这里，单个企业的 Total Fixed Cost 为 $\frac{8.1\times 10^5}{100}=8100$ ，因此平均固定成本为 $AFC=\frac{8100}{q}$ ，因此平均总成本为

AC=AVC+AFC=q+50+\frac{8100}{q}

考虑市场价格，对于单个企业有 $MC=AC$ ，因此 $q=90,p=230$ 。

这里要求市场上总的产量，因此 $Q=nq=9000$ units.

(2020 Spring Final Exam Q31-34) The market of fortune cookies is perfectly competitive. John’s Fortune Cookies is one of the many perfectly competitive firms. Assume that the only variable input required for the production is workers. The following table shows the relationship between the number of workers hired and the total output (packs of fortune cookies produced).

# of workers Total output (packs)

1 98

2 183

3 264

4 330

5 391

6 441

7 482

8 517

9 536

10 552

John pays a fixed cost of $380 per day, and to each employee a wage of $54 per day. The price of fortune cookies is $1.14 per pack.

Given such information, we can calculate the marginal product due to the 3-rd worker is [Answer1].

Given such information, in the short run, John should hire [Answer2] workers.

Given such information, in the short run, John should [Answer3] packs of fortune cookies.

Consequently, in the short run, John will make a profit of [Answer4] dollars.

Suppose the fixed cost has become $570 per day. In the short run, John should hire [Answer5] workers.

# of workers	Total output (packs)
1	98
2	183
3	264
4	330
5	391
6	441
7	482
8	517
9	536
10	552

【解答】
决定产量的时候不需要考虑 fixed cost，但是计算利润的时候还是要计算固定成本的。

(2020 Spring Final Exam Q36-39) There are many identical firms (having the same production costs and producing the same product) in a competitive market. Their product is assumed to be perfectly divisible. When one firm produces $q$ units, its marginal cost is

$MC(q) = q$
and the corresponding total variable cost is

$TVC(q) = 0.5q^2$
The demand curve in the market is

$Q = 9180 - 90P$
Moreover, assume that the industry is constant cost. The market is currently in a long-run equilibrium, and the equilibrium price is $45 per unit.

Using the information, we conclude that there are [Answer1] firms in the market.

Using this information, we conclude that the fixed cost of an individual firm is [Answer2] dollars.

Using this information, we conclude that the short-run supply curve of this industry is of the form

$Q = A + B \times P$

$A$ is equal to [Answer3].

$B$ is equal to [Answer4].

Suppose now the market demand has become

$Q = 15300 - 150P$

Using the information, we can calculate that in the long run, an individual firm will produce [Answer5] units.

【解答】
对于单个企业而言，其 supply curve 就是 MC curve，也就是单个企业的 $P-Q$ 曲线为 $q=0+1P$ ，但是这里题目要的是社会总产量 $Q=nq$ ，因此 $Q=0+114P$

In a perfectly competitive constant cost industry, all firms are identical. A typical firm possesses the cost curves:
$TC=q^2+80q+100$

$MC=2q+80$
(q is firm quantity). The market demand is
$P=1100-Q$
(Q is market quantity). The industry is initially at the long-run equilibrium.

(a) A typical firm will shut-down and produce zero quantity in the short run if the price is lower than [Answer1] dollars. (Hint: P_shutdown = min AVC)

A typical firm will exit from the industry in the long run if the price is lower than [Answer2] dollars. (Hint: P_exit = min AC, where MC=AC).

(b) At initial long-run equilibrium, the price is [Answer3] dollars. (Hint: What is the relationship between long-run equilibrium price and firm exit price?)

At initial long-run equilibrium, each firm produces [Answer4] units.

© At initial long-run equilibrium, there are [Answer5] firms in the market. (Hint: number of firms = market quantity / firm quantity.)

(d) Because of change in environmental regulation that affects production cost, the marginal cost is reduced by 17 dollars but the fixed cost is increased by 300 dollars permanently. The post-regulation short-run supply curve of this industry is:

$P = A + B \times Q$
where A is equal to [Answer6]. (Hint: What will be the new MC curve?)

Continue above, B is equal to [Answer7].

(e) At the post-regulation short-run equilibrium, the price is [Answer8] dollars.

At the post-regulation short-run equilibrium, each hotel produces [Answer9] units.

At the post-regulation short-run equilibrium, each firm earns a profit of [Answer10] dollars. (Hint: What will be the new TC curve?)

(f) With free entry and exit of firms, at the post-regulation long-run equilibrium, the price will be [Answer11] dollars.

At the post-regulation long-run equilibrium, there will be [Answer12] firms in the market.

(h) Suppose the government wants to induce the market to generate a post-regulation long-run equilibrium quantity the same as the pre-regulation long-run equilibrium quantity. The government should impose per-unit subsidy of [Answer13] dollars.

【解答】

分析 $MC$ 的表达式，发现当 shutdown 即 $q=0$ 时，对应的价格为 $80$ ，也就是说当价格低于 $80$ 企业就会选择不生产。因此短期 shutdown price 为 $80$ .
exit price 的话只需要列出方程即可，即 $AVC=MC$
$\frac{q^2+80q+100}{q}=2q+80$
解得 $q=10$ ，所以 exit price 为 $2q+80=100$ .
长期平衡价格等于企业 exit price
上面刚刚解出来的 $q=10$
代入 Demand curve 可知，市场总产量为 $Q=1000$ ，因此，总共有 $n=\frac{Q}{q}=100$ 家企业
(7.) Marginal Cost 降低 $17$ 而 fixed cost 增加 $300$ ，于是， $MC$ 和 $TC$ 变成了
$MC=2q+63\\ TC=q^2+63q+400$
从 $P=MC=2q+63$ 和 $q=\frac{Q}{n}$ ，可知 $P=0.02Q+63$ ，所以 $A=63$
$B=0.02$
联立 $P=0.02Q+63$ 和 $P=1100-Q$ ，可知 $Q=1017.67,P=83.33$
由于 $Q=1017.67$ ，由于市场刚刚变化，因此此时还不会有企业退出，所以 $q=Q/n=10.17$
把 $q=10.17$ 代入， $\text{Profit}=Pq-TC=-296.67$

h. 最后可以分析出来原来的长期市场价格为 $100$ 而当前为 $103$ ，在 Demand Curve 上我们可以发现，要想让 $Q'$ 恢复到 $Q$ 的水平，长期均衡的价格必须是 $100$ ，但是企业现在的 $AC$ 最低点是 $103$ ，需要降到 $100$ . 因此需要补贴 $103-100=3$ 块钱。