Bob and Ray are thinking of buying a sofa. Bob's utility function is $U_B(S,M_B)=(1+S)M_B$ and Ray's utility function is $U_R(S,M_R)=(3+S)M_R$ where $S=0$ where S=0 if they do not get the sofa and S=1 if they do and where $M_B$ and $M_R$ are the amounts of money they have respectiely to spend on their private consumptions. Bob has a total of 2000 dollars to spend on the sofa and other stuff Ray has a total of 1600 to spend on the sofa and other stuff. The maximum amount that they could pay for the sofa and still arrange to both be better off that without it is:
a) 2100
b) 533.33
c)750
d)1400
e)2800
How to solve this question? I have utterly no idea whastoever.
Bob's budget: $B_B = 2000$
Price of sofa: $X$
Bob's money to be spent on private consumptions: $M_B = B_B - X$ if buying the sofa or $M_B = B_B$ if not buying.
So, if Bob buys the sofa, $U_B = (1+1)(B_B-X) $. If he does not, $U_B = (1+0)(B_B)$
Bob is better off buying the sofa, when $(1+1)(B_B-X) > (1)(B_B)$. That would be $4000-2X > 2000$ and that means $X<1000$.
For Bob, the max. price he can pay is <1000 dollars.
Similarly for Ray, $4(1600-X)>3(1600)$. That is $6400-4X>4800$ , and that gives $X<400$.
If Ray can only spend less than 400 and Bob less than 1000, that means together they can spend max. anything less than 1400 on the sofa and still their utility functions will be higher than without the sofa, in other words, will be better off than without it.