范里安《微观经济学(高级教程)》(第3版)课后习题-竞争市场(圣才出品)

第13章竞争市场

Let ()v p m +be the indirect utility function of a representative consumer,and let ()p πbe the profit function of a representative firm.Let welfare as a function of price be given by ()()v p p π+.Show that the competitive price minimizes this function.Can you explain why the equilibrium price minimizes this welfare measure rather than maximizes it?

答：令福利函数的一阶导数为零，则有：()()0v p p π''+=。根据罗伊尔定理和霍特林引理，一阶导表达式可写为：()()0x p y p -+=，这完全是需求等于供给的条件，也就是说，均衡时的价格能够满足福利函数的一阶导为零的条件。

同时，这一福利函数的二阶导数表达式为：()()x p y p ''-+，由需求曲线向下倾斜与供给曲线向上倾斜的性质可知它显然是正值。因此，该福利函数在均衡价格p 处具有极小值而不是极大值。

在任何不同于均衡价格的水平上，厂商供给的数量不同于消费者需求的数量。因此其他价格无法使得福利达到最大化。

2．证明：当价格从0p 变至1p 时，供给函数在0p 与1p 间积分所给出的利润变化。Show that the integral of the supply function between 0p and 1p gives the change in profits when price changes from 0p to 1p .

证明：由霍特林引理：()()/p w p y p π? ?=，，可以得到利润变化为：

3．一个有大量厂商的行业，每一厂商都取如下的成本函数形式

（a）找出厂商的平均成本曲线，描述它如何随要素价格

w w的变化而移动。

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（b）找出单个厂商的短期供给曲线。

（c）找出行业的长期供给曲线。

（d）描述单个厂商的投入要求集合。

An industry consists of a large number of firms,each of which has a cost function of the form

（a）Find the minimum of average cost curve of a firm and describe how it shifts as the factor price

w w changes.

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（b）Find the short-run supply curve of an individual firm.

（c）Find the long-run industry supply curve.

（d）Describe an input requirement set for an individual firm.

答：（a）平均成本AC为：

该曲线是凸的，将平均成本对y求导，得出平均成本最小时产量

y为：

m

并且该最低点是唯一的极小值。将m y 对12/w w 求导，得出导数为负值，即当12/w w 增加（减少）时，平均成本的极小值左（右）移。实际上，当这一比例趋于∞时，m y 收敛于1，当这一比例趋于0时，它收敛于2（b）边际成本为，所以短期供给函数（在平均可变成本线之上的部分）为：

（c）长期供给曲线为：

（d）从成本函数中可以看出两种要素是互补的，因此生产函数为：

从成本函数中可以得到投入需求为：

。并且，在任何程度上1x 和

2x 都不是替代的。所以，单个厂商的投入要素集是：

4．农场主生产棉花需土地和劳动。生产y 蒲式耳棉花需以货币计的劳动成本为()2c y y =。存在100个相同的农场主，其行为互相竞争。

（a）单个农场主的棉花供给曲线是什么？

（b）棉花市场供给曲线是什么？

（c）假定棉花的需求曲线为()20050D p p =-，售出的均衡价格和数量是多少？

（d）土地的均衡租金是多少？

Farmers produce corn from land and labor.The labor cost in dollars to produce y bushels of corn is ()2c y y =.There are 100identical farms which all behave competitively.

（a）What is the individual farmer’s supply curve of corn?

（b）What is the market supply of corn?

（c ）Suppose the demand curve of corn is ()20050D p p =-.What is the equilibrium price and quantity sold?

（d）What is the equilibrium rent on the land?解：（a）利润最大化条件为：

一阶条件为：

2y /2

p y p =?=（b）存在100个相互竞争的农场主，则市场供给函数为：

()()10050Y p y p p

=?＝（c）令()()D p Y p =，即：2005050p p -=，得：2100/1001p Y y Y ****= = ==，，。（d）在竞争条件下，土地的均衡租金必须等于均衡时农场主所得的收益和劳动成本之

间的差额，即全部的利润均为土地的租金：()22111r py c y =-=?-=。

（d）假设生产要素表现出常数规模报酬，这如何使得关于()t s '的公式变得简单？（e）在雨伞制造中，假定资本为一劣等生产要素，抵补英国资本补贴效应的关税有何不寻常？

Consider a model where the U.S.and England engage in trade in umbrellas.The representative firm in England produces the export model umbrella according to a production function ()f K L ，where K and L are the amounts of capital and labor used in production.Let r and w be the price of capital and the price of labor respectively in England,and let ()c w r y ，，be the cost function associated with the production function ()f K L ，.Suppose that initially the equilibrium price of umbrellas is p *and the equilibrium output is y *.Assume for simplicity that all of the export model umbrellas are exported,that there is no production of umbrellas in the U.S.,and that all markets are competitive.

（a）England decides to subsidize the production and export of umbrellas by

imposing an export subsidy s on each umbrella,so that each umbrella exported earns the exporter p s +.What size import tax ()t s should the U.S.choose so as to offset the imposition of this subsidy;i.e.,to keep the production and export of umbrellas constant at y *?（Hint:This is the easy part;don’t get too subtle.）

（b）Since it is so easy for the U.S.to offset the effects of this export subsidy,England decides instead to use a capital subsidy.In particular,they decide to subsidize capital purchase with a specific subsidy of s so that the price of capital to English umbrella makers is r s -.The U.S.decides to retaliate by putting a tax ()t s on imported umbrellas that will be sufficient to keep the number of umbrellas produced constant at y *.What must be the relationship between the price paid by the consumers,p,the tax,()t s ,and the cost function.

（c）Calculate an expression for ()t s 'involving the conditional factor demand function for capital,

()K w r y ，，.

（d）Suppose that the production function exhibits constant returns to scale.What simplification does this make to your formula for ()t s ?

（e）Suppose that capital is an inferior factor of production in umbrella making.What is unusual about the tariff

()t s that will offset the capital subsidy in England?

答：（a）为了抵消产出补贴，美国应选择与补贴同样大小的税收，即选择()t s s =。（b）如果英国政府对雨伞生产商进行资本补贴，那么生产商的成本函数为(),,c w r s y -，假设美国政府为了抵消英国政府的补贴政策的影响而对每把雨伞征收t 美元的关税，则有以