problem sets

[Akshay Jain]

have a look at the problem set given below.....its good and easy...similar to the varian workbook....
i will be uploading more in future.....
problem_set_1.pdf

[Mehak]

Thanks Akshay :)

[Akshay Jain]

consider a function f(x) equal to x(x-10)(x-20)(x-30)......find the possible ranges in wich the critical points of this function can lie.....

[X]

According to Wikipedia,"In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0."

So, f(x)=x(x-10)(x-20)(x-30)=x^4-60x^3+1100x^2-6000x
and it is differentiable everywhere w.r.t. x. and df/dx=x^3-45x^2+550x-1500 and to find out critical point, we have to find x such that df/dx=0.

now Akshay, can you solve df/dx=0 for x?

or any trick do you know to find critical point?

PLEASE HELP!!

[X]

Akshay, http://www.numberempire.com/equationsolver.php says,
the sol. for df/dx=0 are x1=15-5^(3/2), x2=5^(3/2)+15 and x3=15.
Out of which x3 can be found out manually by trial and error.

Do you know any better manual trick to solve eq.s?

Thanks.

[Mehak]

Hi
I got the same equation for df/dx. And yes we can find x=15 as one of the critical points by trial and error.
But other than that, I could just narrow down my answer to 0<x<30. What's the actual answer Akshay?

[Mehak]

Hii
I got the same equation for df/dx. And yes we can calculate x=15 as the solution by trial and error.
But other than that, I could just narrow down my solution to 0<x<30. What's the actual answer Akshay?

[Akshay Jain]

its a 1 marks question asked this year in DSE entrance.......
yes the function is everywhere differentiable....so critical points of this func is where dy/dx is zero.....
dy/dx = x(x-10)(x-20) + (x-10)(x-20)(x-30) + x(x-10)(x-30) + x(x-20)(x-30)....use product rule
now analyze the above derivative
just check the sign of the derivative
at x<0, dy/dx=-ve
at x=0, dy/dx=-ve
at x=10, dy/dx=+ve
at x=20, dy/dx=-ve
at x=30, dy/dx=+ve and vil remain +ve for greater dn 30
between 0 and 10 dy/dx changes its sign, hence there exist x such that dy/dx=0 where x belongs to (0,10)
similarly for the others
so ranges are (0-10), (10-20) and (20-30)

[Akshay Jain]

problem set 1 solutionsolution_1.pdf
problem set 2 problem_set_2.pdf

[Akshay Jain]

question..
Jaspreet, a college student, generally spends his summers working on the university maintenance crew at a wage rate of Rs.60/- per hour for a 40-hour week. Overtime work is always available at an hourly rate of 1.5 times the regular wage rate.
For the coming summer, he has been offered the pizza stand at the college canteen building, which would have to be open 10 hours per day, six days a week. Jaspreet estimates that he can sell 100 pizzas a week at Rs.60/- each. The production cost of each pizza is Rs.20/- and the rent on the stand is Rs.1500/- per week.
Should Jaspreet take the pizza stand? Why or why not?

[Ditro Raychaudhuri]

Case1 he works 40 hrs a week an earns           2400Rs / week
Case 2 he works 60 hrs a week and earns         6000-2000-1500 = 2500 Rs/ week i.e 41.6 Rs/hr

In case 2 he works 20hrs extra to earn 100 rs extra........which he could earn by working 1 hour and7mins( approx) more at the university and still save 18 hours and 53 minutes of his time.
  He should not take the stand

[Akshay Jain]

[Akshay Jain]

question
utility function is U = [(Min(x,y))^2 + (max(x,y))^2]^1/2
budget constraint is 10x + 10y = 100
find the utility max consumption bundle...

[X]

We should consider these three cases:

x>y,
x<y,
x=y.

In each of these cases,
[[Min(x,y)]^2+[Max(x,y)]^2]^1/2=(x^2+y^2)^1/2.

and, by Lagrangian or other method, x=y=5 is the optimum level of consumption to maximize utility (max U=50^1/2).

[Amit Goyal]

(10, 0) and (0, 10) are better than (5, 5) and are both affordable. So, (5, 5) cannot be a utility maximizing choice. Check that (10, 0) and (0, 10) both solves this problem.

[Akshay Jain]

@X.......this is a most common mistake dat everybody does.....notice dat the utility function is concave to the origin....in in dat case we have corner solution....think about it

[X]

Thank you.

[X]

Thanks a lot. Can we take monotonic transformations to make utility function easier?

[Amit Goyal]

Hi Akshay,

Let me correct you here. Please note that the utility function is not concave. Its the Indifference curves that are concave or we can say, in other words, utility function is quasi convex. And you are right when you said that in this case we will have a cornor solution.

By the way, you are doing a good work in helping aspirants prepare for masters.
Keep it up.

Amit

[Amit Goyal]

The answer to your question is yes.

Solution to the following two problems will be the same:
Max u(x)
s.t.
p.x \leq M

And

Max f(u(x))
s.t.
p.x \leq M

where f(u) is the monotonic transformation of u.