How was this equation differentiated?

[amaya244]

Hi guys, this is my first post, a friend of mine said I should try the site out. Here it goes.I have a function: Max (c1) = u(c1) + 1/1+p * u(c2)

(c2) is equal to this: (1+r)^2 * A0 + (1+r) (Y1-c1) + Y2

Substituting it into the max gives: u(c1) + 1/1+p * u [(1+r)^2 * A0 + (1+r) (Y1-c1) + Y2]FOC, I got: u'(c1) = 1+r/1+p * u'(c2)

However, I think I'm wrong. Can someone please check this for me?

(please show working)

Amy xx

 

[Mark44]

Hi guys, this is my first post, a friend of mine said I should try the site out. Here it goes.


I have an function: Max (c1) = u(c2) + 1/1+p * u(c2)

(c2) is equal to this: (1+r)^2 * A0 + (1+r) (Y1-C1) + Y2

I have no idea what you're asking. In the equation above, you have Max (c1) = u(c2) + 1/1+p * u(c2).

The way you wrote this suggests that Max is a function of variable named c1, but the other side of the equation involves c2, not c1.

Another point of confusion is trying to determine what you mean by u(c2) + 1/1+p * u(c2). Is u a number or a function? Also, what you wrote does not mean what you think it means. Apparently there is a fraction somewhere, but without parentheses, I can't tell what's in the numerator and what's in the denominator.

For example if you write (a + b)/(c + d) without parentheses, it would be a + b/c + d. Should this mean a + [b/(c + d)], [(a + b)/c] + d, or just plain a + [b/c] + d?

Substituting it into the max gives:


u(c2) + 1/1+p * u [(1+r)^2 * A0 + (1+r) (Y1-C1) + Y2]


FOC, I got: u'(c1) = 1+r/1+p * u'(c2)

What does FOC mean?

However, I think I'm wrong. Can someone please check this for me?

(please show working)

Amy xx

 

[amaya244]

Sorry Mark44, I totally understand what you mean. Let me explain a little.

Firstly, I made an error. c1 should indeed be in place of c2 (in the 1st utility function).

Secondly, 1/1+P is a 'discount factor' that can be simply written as a 'B':

u(c1) + B u(c2)

When B is close to 1, the individual is impatient and chooses to consume quite a lot of his future income.

Thirdly, let me explain the variables:

u(.) is an instantaneous utility function
ct is period t consumption
Yt is income at period t
At is wealth at period t

The problem is thus dynamic! The individual maximises consumption subject to a budget constraint.

u(c1) + B u(c2) s.t. c2 = (1+r)^2A0 + (1+r)(Y1-c1) + Y2

This is usually called Milton Friedman's Permanent Income Hypothesis. FOC is short for the 'first order condition'.

This first order condition is known as a Euler equation.
u'(c1) = (1+r) + B u'(c2)

I just want to know how c1 is maximised/differentiated.

 

[Mark44]

Sorry Mark44, I totally understand what you mean. Let me explain a little.

Firstly, I made an error. c1 should indeed be in place of c2 (in the 1st utility function).

Secondly, 1/1+P is a 'discount factor' that can be simply written as a 'B':

That would be 1/(1 + P).

u(c1) + B u(c2)

When B is close to 1, the individual is impatient and chooses to consume quite a lot of his future income.

Thirdly, let me explain the variables:

u(.) is an instantaneous utility function
ct is period t consumption
Yt is income at period t
At is wealth at period t

The problem is thus dynamic! The individual maximises consumption subject to a budget constraint.

u(c1) + B u(c2) s.t. c2 = (1+r)^2A0 + (1+r)(Y1-c1) + Y2

This is usually called Milton Friedman's Permanent Income Hypothesis. FOC is short for the 'first order condition'.

This first order condition is known as a Euler equation.
u'(c1) = (1+r) + B u'(c2)

I just want to know how c1 is maximised/differentiated.

I think this is what you're trying to say:

Maximize u(c₁) + B*u(c₂), where c₂ =A₀(1 + r)² + (1 + r)(Y₁ - c₁) + Y₂.

If you substitute for c₂ in the expression you want to maximize, you'll get a function that has one variable: c₁.

Let's call this function F.
F(c₁) = u(c₁) + B*u(A₀(1 + r)² + (1 + r)(Y₁ - c₁) + Y₂)).

The first part is easy enough to differentiate. For the second part you need to use the chain rule.

 

[amaya244]

That would be 1/(1 + P).

No, just 1/1 + P actually :-)

I think this is what you're trying to say:

Maximize u(c₁) + B*u(c₂), where c₂ =A₀(1 + r)² + (1 + r)(Y₁ - c₁) + Y₂.

If you substitute for c₂ in the expression you want to maximize, you'll get a function that has one variable: c₁.

Let's call this function F.
F(c₁) = u(c₁) + B*u(A₀(1 + r)² + (1 + r)(Y₁ - c₁) + Y₂)).

The first part is easy enough to differentiate. For the second part you need to use the chain rule.

Thank you Mark44,

Can I just ask, once you differentiated the first and second parts of F, with respect to c1, did you arrive at the following equation?

u'(c₁) = (1 + r) β*u'(c₂)

Amy xx

 

[Mark44]

No, just 1/1 + P actually :-)

Yeah, right...

Thank you Mark44,

Can I just ask, once you differentiated the first and second parts of F, with respect to c1, did you arrive at the following equation?

u'(c₁) = (1 + r) β*u'(c₂)

Well, I didn't do the differentiation.

The equation you show here is the result of setting F'(c₁) = 0, and then solving for u'(c₁).

 

[amaya244]

Yeah, right...

Pffft!

Well, I didn't do the differentiation.

The equation you show here is the result of setting F'(c₁) = 0, and then solving for u'(c₁).

That's what I thought, but then where did C₂ come from? Didn't it get substituted out?