Pasting Subhayu's earlier post below for convenience. I guess the paper series is different.
This post was updated on May 26, 2014; 10:42am.
Q41) The consumer will be indifferent between the two bundles if the utility derived under optimum situation is equal in bothe the cases.
Now consider case 1: the optimum demand bundles are x1=x2=M/(p1+p2)=M/3.
In case 2: M=px*X+py*Y,
or, M=1*X+(2-θ)*Y+T.
or, M-T=1*X+(2-θ)*Y.
The optimum bundles in this case is given by x1=x2=(M-T)/(1+(2-θ))=(M-T)/(3-θ).
Utilities will be equal implies, M/3=(M-T)/(3-θ)......................(a).
from (a) we can derive, θ=3T/M (option b).
"for the second part: the number of people who wud consume in period A is intergration from 1/2 to 1/1+r of 2..(uniform distri)...which comes out as N(1-r/1+r)..so if the optimal choice period 1 is 1/2..the demand will be N/2(1-r/1+r)"
Hi Akshay.. why are we not using the basic definition of PO..that if someone is made better off the other is made worse off which happens in the interval [0.2,0.7]
For question 1 , dont u think that if the person will run at 5 m/s once, he will have to attain 4 m/s at least once, cos speed is a continuous function in time . one can not fly to 5 m/s from 0. He will have to pass through every value of speed in the interval [0,5]. Moreover if we restrict the speed to just below 6 and have the value of acceleration (in going from 0 to 6-e m/s) modulated so as to cover 100 m in 20 sec, the answer comes out to be 'a'.
Pr(β<=(1/1+r)) would give us the fraction of the N people who would have the budget line flatter than the IC, implying that they would demand only good 1. Thus, N*(1-r)/(1+r) people demand only good 1. Each of them, however demands 1/(1+r), making the total demand N*(1-r)/(1+r)^2, making none of the options correct in my opinion.
This is my understanding of the question. Do let me know any flaw in my reasoning.