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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

    Mark44

    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

    Mark44

    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

    Mark44

    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
    Pffft!

    That's what I thought, but then where did C2 come from? Didn't it get substituted out?
     
    Last edited: Jun 20, 2012
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