Indifference curves for utility functions of form min(ax+by, cx+dy)

[siddharth]

Hi , Amit
This is sid. i emailed you couple of days back.
My question is how do you approach drawing indifference curves for preferences of types min(ax+by, cx+dy)
then go on and find the optimal choice for given budget constrains..
These type of problems come up very frequently even in general Eq type questions and i am getting stuck with them.
Can you pls help me out with a detailed approach to sketch such graphs...
Thanks a lot.

[s]

hi sid for these utility functions if u have read about perfect complements u'l know that consumer will consume x and y at a fixed proportion and from ax+by and cx+dy the proportion is found as ax+by=cx+dy or x=(d-b)y/(a-c) now put this in the budget constraint say M=pxX + pyY and solve for optimum values of x and y. to draw the ICs and for better understanding refer Hal varian Intermediate Microeconomics workbook

[vikram]

This is a brilliant thread. To continue the discussion, I have a query on similar line. On another thread, Amit Sir has explained that for a utility function like U(x,y)=min{x+y,2y} the demand function is as given below.
---------------
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)
------------------
But how does one arrive at such a distribution using the optimization technique (Max U(x,y) and budget constraint)? One only touches upon the first result i.e. x = M/px+py. Please explain!!

[s]

hey vikram you are right i have jus mentioned how to find the ratio in which goods are consumed but to find out the optimum bundle the price ratio matters and you follow this exact rule
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)

[vikram]

Thanks, s!

But can you also state what will be the corresponding relation between px and py (because there will be a,b,c,d involved as well). px=py works only in case utility function is the simple case U(x,y)=min(x,y).

[sid]

hi.. thanks for the reply
but the tricky part here is how do we find the price ratios.. for solving a gen eq problem.
this has been mentioned below...
"This is a brilliant thread. To continue the discussion, I have a query on similar line. On another thread, Amit Sir has explained that for a utility function like U(x,y)=min{x+y,2y} the demand function is as given below.
---------------
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)
------------------
But how does one arrive at such a distribution using the optimization technique (Max U(x,y) and budget constraint)? One only touches upon the first result i.e. x = M/px+py. Please explain!! "
I shall take the discussion to higher level now.
Indifference curves look like the ones for perfect substitutes, in this case 2 lines intersecting each other with a locus ax+by = cx+dy, As it says min (ax+by, cx+dy) we only take the portions of line above the locus of points...( am i right?) i think this is very imp to draw the right IC to derive demand curves and i am kinda worried about it..
i see the logic behind deriving the demand curve is similar to the one we use for perfect substitutes.
Lets take just one half of the utility...
ax+b , demand would be
x=M/px +py if px/py < a/b
x=0 if px/py>a/b
x=(0, M/px+py)

Let me know if my approach is right here?