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.

Answer the following questions:

(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

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.

Answer the following questions:

(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.

Answer the following questions:

(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

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.

Answer the following questions:

(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

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

Can you upload a pic of your working? Upload it to a site like imgur and link it.

Original post by mqb2766

Can you upload a pic of your working? Upload it to a site like imgur and link it.

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??

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

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.

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