micro

[aditi]

Sir , there is a ques. in 2008 dse paper that says :- Suppose a consumer has a utility function U = min. (x+y, 2y). He maximizes his utility subject to his budget constraint and consumes (x*,y*) = (3,3). Which one of the following statements must be true?
(a) price of good x is necessarily equal to price of good y
(b) price of good x is double the price of good y
(c) price of good x is less than or equal to price of good y
(d) none of the above

In the solutions you have stated that option (c) is correct. Can u please explain how do we arrive at this solution?

[Amit Goyal]

Hi Aditi, I generally explain this using graph but since i cannot do so here. Let me give you the demand correspondence for x.
x = M/(p(x)+p(y))  if p(x) < p(y)
   = [0, M/(p(x)+p(y))]  if p(x) = p(y)
   = 0 if p(x) > p(y)

The only possibilities where you consume equal amount of x and y are when either p(x) = p(y) or p(x) < p(y)

M ---> Income
p(i) --> price of good i ∈ {x, y}

[aditi]

thnx sir

[shashi]

utility funtion as well as additional information related to dd  of the consumer is also provided......so ,U = min. (x+y, 2y)=x+y....now we have the case of perfect substitute....since no information is given related to the availability of both goods...by this behaviour of consumer......it seems that px must=py

[shashi]

utility funtion as well as additional information related to dd  of the consumer is also provided......so ,U = min. (x+y, 2y)=x+y....now we have the case of perfect substitute....since no information is given related to the availability of both goods...by this behaviour of consumer......it seems that px must=py

[nidhi]

@ Amit sir
 i cud not understand this demand function. wil not the consumer in equilibrium always consume at x=y which is the locus of kinked points (x+y=2y)

and u have mentioned that X = M/ Px+Py when Px<Py but then will the demand for y not be zero

[Amit Goyal]

Hi nidhi,

Try to find out the optimal bundles in these three scenarios:
1) Use the above utility function, let prices be (2, 1) and income  M= 9
2) Use the above utility function, let prices be (1, 2) and income M= 9
3) Use the above utility function, let prices be (1.5, 1.5) and income M= 9

[nidhi]

@amit sir
if I use the normal method of taking locus of kink points x+y=2y i.e. x=y +M/(px+py)then i get x*=y*= 3 in each case

but in case 1 :P(2,1) px>py  if i take x=0 and so y=9 then U=9> U=6 when x=y=3 so this means x=0

case2: P(1,2) px<py.. U is maximum at x=y=3

case3: P(1.5,1.5) px=py...U is same at x=0, y=6 and x=y=3

thanx sir i understood the demand fn given by u using this.

but i have a question that does this mean  that the general way of taking the locus of kinked points and using it to solve for equilibrium in case of min function wrong because if we use it here i'll always get x=y=3 but that does not maximise utility in all scenarios.so shud that method not be used.
please explain

[Amit Goyal]

I have presented you the case where the method you use may not work
And The method you say will only work for case: min{ax, by} where a, b >0
and i guess this place is not suitable to teach the general method.
But still i will try to direct you, draw the indifference curve and look for the equilibrium.

[nidhi]

thanx a lot sir