专业课资料 练习题 真题 北大CCER经典中级微观经济20次作业及考题 keyto05fhw2

Homework 2 Sep 15 2005 Chapters 4 and 5 1 写出下列情形的效用函数 画出无差异曲线 并在给定价格 p1 p2 和收入 m 的 情形下求最优消费选择 1 x1 一元纸币 x2 五元纸币 2 x1 一杯咖啡 x2 一勺糖 消费者喜欢在每杯咖啡加两勺糖 Solution 1 1 x is definitely the substitution of 2 x and five units of 1 x can bring the same utility as that one unit of 2 xcan do With the most simple form of the utility function 1212 1 5 u x xxx and assume that the prices of those two goods are 1 p and 2 prespectively and the total wealth of the consumer is m the problem can be written as 12 1212 1 122 1 max 5 x x u x xxx st p xp xm As for the solutions there may be three possible cases 1 x 12 1 5 pp 12 1 5 uxx 12 1 5 pp if 1 1 5 2 pp then all wealth will be spent on 2 x so 2 x 2 m p and 1 x 0 if 1 1 5 2 pp then any bundle 12 x x which satisfies the budget constraint 1 122 p xp xm is the solution of such problem 2 A cup of coffee is absolutely the complement of two spoons of sugar Let 1 x and 2 xrepresent these two kinds of goods then we can write the utility function as 1212 1 min 2 u x xxx 12 1 2 xx u u 1 x 2 x The problem of the consumer is 12 1212 1 122 1 max min 2 x x u x xxx st p xp xm Any solution should satisfies the rule that 1 1 2 2 xx and the budget constraint So replace 1 xwith 2 1 2 x in the budget constraint and we can get 2 12 2 2 m x pp and then 1 12 2 m x pp 2 给定下列效用函数 在价格 p1 p2 和收入 m 的情形下求解消费者的最优选择 1 2 1212 u x xxx 2 12 uxx Solution 1 the problem is 12 2 1212 1 122 max x x u x xxx st p xp xm i set up the Lagrange Function as 2 121 122 Lx xmp xp x where is the Lagrange Multiplier Since is concave to u i 12 x x and the budget constraint is linear the first order conditions are sufficient and necessary conditions for the maximum problem The F O C s are 1 0Lx 121 2x xp0 1 2 0Lx 2 11 0 xp 2 0L 1 122 0mp xp x 3 Form 1 and 2 we can get 12 21 2 xp xp or 22 1 1 2 p x x p replace it into 3 and then get 2 2 3 m x p and then 1 1 2 3 m x p 2 The problem is 12 12 1 122 max x x uxx st p xp xm and establish the Lagrange Function as 121 122 Lxxmp xp x Also is concave to u i 12 x x and the budget constraint is linear the first order conditions are sufficient and necessary conditions for the maximum problem The F O C s are 1 0Lx 1 1 1 0 2 p x 4 2 0Lx 2 1p0 5 0L 1 122 0mp xp x 6 From 4 and 5 we obtain 2 1 1 2 p x p or 2 12 4 2 1 xpp then replace it into 6 and get 2 2 1 2 1 2 22 4 4 p mp 2 1 ppm x ppp However the solution 12 22 2 21 21 4 4 pm xxpp pp is not the final answer Because of the nonnegative restraints of the comsumption bundles the above solution holds only if 12 x x 0 2 21 4mpp If 2 2 4mp