# Maths for economists - constrained maximisation questoin

Hiya, I'm a bit stuck on this maximisation problem. I've done all the steps but seem to be getting too large an answer. Any help would be greatly appreciated!

A student has current income y1 and expects future income y2. She plans
current consumption c1 and future consumption c2 in order to maximise
utility
U = 2√c1 +2β√c2
c1, c2 > 0
β > 0 is her discount rate.
If she borrows now, c1 > y1, then future
consumption, after repaying the loan c1 −y1 with interest r, will be
c2 = y2 −(1+r)(c1 −y1).
Alternatively, if she saves now, c1 < y1, future consumption will be
c2 = y2 + (1+r)(y1 −c1)
after receiving interest r on her savings.
(a) [10 marks] Find the optimal consumption plan c1, c2. Show your
workings.
(b) [15 marks] Show how an increase in the interest rate affects the level
of borrowing or saving. Show your workings
Original post by ankita.sahax
Hiya, I'm a bit stuck on this maximisation problem. I've done all the steps but seem to be getting too large an answer. Any help would be greatly appreciated!

A student has current income y1 and expects future income y2. She plans
current consumption c1 and future consumption c2 in order to maximise
utility
U = 2√c1 +2β√c2
c1, c2 > 0
β > 0 is her discount rate.
If she borrows now, c1 > y1, then future
consumption, after repaying the loan c1 −y1 with interest r, will be
c2 = y2 −(1+r)(c1 −y1).
Alternatively, if she saves now, c1 < y1, future consumption will be
c2 = y2 + (1+r)(y1 −c1)
after receiving interest r on her savings.
(a) [10 marks] Find the optimal consumption plan c1, c2. Show your
workings.
(b) [15 marks] Show how an increase in the interest rate affects the level
of borrowing or saving. Show your workings

What did you do?
Original post by mqb2766
What did you do?

the regular steps for maximisation:
find the lagrangean equation
partial differentiation in terms of c1 and then c2
then rearranging to make lamda the subject

but there are so many variables like beta and r in my answers that i am a bit confused whether it is correct or not
Original post by ankita.sahax
the regular steps for maximisation:
find the lagrangean equation
partial differentiation in terms of c1 and then c2
then rearranging to make lamda the subject

but there are so many variables like beta and r in my answers that i am a bit confused whether it is correct or not

Original post by mqb2766

https://imgur.com/h90LJtQ

does this work??
Original post by ankita.sahax

It worked. I may be missing something, but you have c2 = f(c1) so why cant you write it as a single variable maximisation problem in terms of just c1?
Original post by mqb2766
It worked. I may be missing something, but you have c2 = f(c1) so why cant you write it as a single variable maximisation problem in terms of just c1?

so you mean, replace the utility functions c2, with the equation provided for c2.
and then treat the whole question as a unconstrained maximisation problem??
and then upon working out c1, i can then substitute the value of c1 into the c2 equation and work that out??
Original post by ankita.sahax
so you mean, replace the utility functions c2, with the equation provided for c2.
and then treat the whole question as a unconstrained maximisation problem??
and then upon working out c1, i can then substitute the value of c1 into the c2 equation and work that out??

Yes. Seems the simplest thing to do. The two equations for c2 are really the same as you have effectively -1*-1.
Original post by mqb2766
Yes. Seems the simplest thing to do. The two equations for c2 are really the same as you have effectively -1*-1.

perfect! i will try this now and let you know
Original post by mqb2766
Yes. Seems the simplest thing to do. The two equations for c2 are really the same as you have effectively -1*-1.

I can't lie, its a bit more complicated doing it like that, a lot of c1 with different coefficients or powers. A bit lost now
Original post by mqb2766
Yes. Seems the simplest thing to do. The two equations for c2 are really the same as you have effectively -1*-1.

https://imgur.com/a/tm2Lhga

i think ive done it?? looks a bit odd still, but not sure what else i can do
Original post by ankita.sahax
https://imgur.com/a/tm2Lhga

i think ive done it?? looks a bit odd still, but not sure what else i can do

Ive not worked it through, but the coefficients in the utillity/constraints must be reflected in the "general" solution. Tbh, Id have written
c2 = a - b*c1
as its a linear relationship where a=.... and b=... Then sub the relationships in at the end. Using that, the utility is
sqrt(c1) + 2beta*sqrt(a-b*c1)
and its fairly obvious you have some sort of tradeoff between c1=0 and c1=a/b. It probably makes the optimal value of c1 more intuitive. In addition you culd take the 2beta inside the square root to absorb it into the a and b coefficients which makes it slightly simpler to derive.
(edited 3 months ago)