1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: How was this equation differentiated?

  1. Jun 19, 2012 #1
    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
    Last edited: Jun 19, 2012
  2. jcsd
  3. Jun 19, 2012 #2


    Staff: Mentor

    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?
    What does FOC mean?
  4. Jun 19, 2012 #3
    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 alot 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.
    Last edited: Jun 19, 2012
  5. Jun 19, 2012 #4


    Staff: Mentor

    That would be 1/(1 + P).
    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.
  6. Jun 19, 2012 #5
    No, just 1/1 + P actually :-)

    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
  7. Jun 19, 2012 #6


    Staff: Mentor

    Yeah, right...
    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).
  8. Jun 20, 2012 #7

    That's what I thought, but then where did C2 come from? Didn't it get substituted out?
    Last edited: Jun 20, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook