微观经济学Microeconomic ，当商品价格受数量影响而变化，怎样求Optimal Consumption Bundle(最佳消费束)?

Question

Mr. Johnson has an income equal to 100 which he uses to buy X and Y. Suppose Px=2 for the first five units and Px=5 for each additional unit, while Py=5 for each of the first six units and Py=10 for each additional unit.
a.) Draw Mr. Johnson's budget line.
b.) If Mr. Johnson has utility function U(X, Y) = X^1/24Y^1/5, how much X and Y will he consume? Would he prefer that the quantity penalty for X be eliminated?

我大致翻译一下，Mr. J收入为100，他全部用来买商品X和商品Y，假设商品X前5件价格为2，第6件起价格为5，而商品Y前6件价格为5，第7件起价格为10。
a.)画出Mr.J的消费者预算线
b.)如果Mr.J的效用函数为U(X, Y) = X^1/24Y^1/5，他的最佳消费束是什么? 假设X没有了数量惩罚，换句话说，商品X的价格始终为2无论购买数量，这样的价格对于Mr.J会更好吗？

我已经画出Mr. J的消费者预算线了，从左到右依次是三条斜率为-1/5,-1/2和-1的直线，Part B问Mr.J的最佳消费束，一般情况下是让 X边际效用/Y边际效用 等于 X价格/Y价格（MUx/MUy = Px/Py)，MUx/MUy我求出来是5y/24x，可是这个Px和Py在这种情况下并不是固定的，而是都有Quantity Penalty（数量惩罚），那我应该怎么求Optimal Consumption Bundle（最佳消费束）呢？

还有最后一问，假设X没有了数量惩罚，换句话说，商品X的价格始终为2无论购买数量，这样的价格对于Mr.J会更好吗？这个我也不太会解。

英文好的朋友最好用英文回答，特别是经济术语，中文回答也可以，But English is Preferred...Thx!!!

分数可增加如果回答很满意!

Answer 1

This is actually not a microeconomic problem, but integer programming instead.
As you have plot the budget line, you should have noticed that the feasibility set is convex but only integer nodes are valid. So a brute yet simple way is to enumerate all the possible frontier
x y U=x^1/24*y^1/5
0 13 0
5 12 1.757761708(*)
7 11 1.75182595
9 10 1.736841168
11 9 1.714904199
13 8 1.686678479
15 7 1.652050849
17 6 1.610270939
19 4 1.491740904
21 2 1.30406269
23 0 0
so the optimal bundle is (5,12).
The alternative method is to use Ux/Px = Uy/Py with check of boundary, but I do not suggest the method as it is an integer programming.

As for the second problem, you should always bear in mind that utility function is a non-increasing w.r.t price. So the decrease in Px enlarges the budget set, will not decrease the utility.
Now (10,11) is feasible, and its corresponding utility is 1.778055053, which is of course greater. Then you can conclude that Mr. J benefits from the price decrease.追问

These are actually microeconomics problems since I' taking Micro Econ 11 at UCLA this quarter, there's another problem i don't know if you can help...

Suppose two people have different preferences for X and Y, but identical incomes. If at their current consumption levels, their marginal rates of substitution are equal, has one of them failed to find the optimum?

追答

Well, I m telling you the best framework to fit the problem. You can apply economic analysis, which will incur tons of trouble, you know that well. I m in a mathematical economic program but in NY.

As for the added problem, the two people have identical income, facing the same relative price Px/Py (they have the same budget set), so Ux_1/Uy_1 = Ux_2/Uy_2 = Px/Py does not make any contradiction with the fact that both of them reach optimum. They can have different optimal point tangent the budget line.
For example for
agent 1: 1/2ln(x) + 1/2ln(y)
agent 2: 2/3ln(x) + 1/3ln(y)
budget: 1x + 1y = 12

agent 1: 1/x / 1/y = 1/1, so x = y = 6
agent 2: 2/x / 1/y = 1/1, so x = 8, y =4
they are both on the optimal point, their MRS's are equal, they have diff~ pref~.

本回答被提问者采纳