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Math Help - budget constraint and intertemporal substitution

  1. #1
    Member garymarkhov's Avatar
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    budget constraint and intertemporal substitution

    When forming the budget constraint for the intertemporal utility function U_t=lnc_t + \frac{1}{1+\rho}lnc_{t+1}, given that there is labour income in period 1 in the amount w_t and interest on savings from the first period at rate 1+r, should the interest income be discounted? That is, should the budget constraint be

    <br />
c_t + \frac{c_{t+1}}{1+r} \leq(1-s)w_t + sw_t(1+r)<br />

    (where s is savings, c is consumption)

    OR:

    <br />
 c_t + \frac{c_{t+1}}{1+r} \leq (1-s)w_t + sw_t(1+r)/(1+r)<br />

    The second makes better theoretical sense (both sides of the equation are in PV terms), but my professor claims the first one is correct. I'm at a loss.
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  2. #2
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    I'm with you on this one.

    Left side of equation = what you keep in 2nd periode, should be bigger than the consumption in the second period:

    (w_1 - C_1) (1+r) >= c_2

    this gives:

    w_1 (1+r) - C_1 (1+r) >= C_2

    C_2 + C_1 (1+r) <= W_1 (1+r)

     C_2 / (1+r) + C_1 <= W_1 (1)

    C_2 / (1+r) + C_1 <= W_1 (1-s) + s W_1 (1+r) / (1+r)

    More essential than me (only a Masterstudent of economics) agreeing with you is the fact that the result in (1) is the intertemporal budget constraint given in my coursnotes on Diamond's model of overlapping generations which utility function wise is exactly the same model you are using...

    So I guess your professor is wrong, which can be very annoying when they don't want to admit it...

    btw F.O.C. give then that

    C_2 / C_1 = (1+r) / (1+\rho)

    Studying for an exam on macro or growth economics?
    good luck, mine is due in 2 days !
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  3. #3
    Member garymarkhov's Avatar
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    Thanks for responding! I'm studying for macro, which I have on the 14th. Good luck with your exams!
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  4. #4
    Member garymarkhov's Avatar
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    beanbag, I'm guessing this will come too late for your exam but if anyone comes along this post in the future here is the no fail technique for getting the budget constraint right:

    1. Write c_t= all the present value stuff (current income minus taxes, pension contributions, etc.). Example: c_t= y_t(1-t) - \frac{b_t}{1+n} - s_t (where y is first period income, t is taxes, b is pension contributions, and s is savings).

    2. Write c_{t+1}= all stuff that you'll get or pay in the second period (but don't discount it yet!). Example: c_{t+1}= (1+r)s_t + y_{t+1} + b_{t+1} (where y_{t+1} is second period income, s is savings, r is the interest rate on savings, and b_{t+1} is the pension you get in the second period).

    3. Solve the second equation for s_t. Then plug that into your equation for c_t.

    4. Place c_t and c_{t+1} on the LHS and put everything else on the RHS.

    Using all the junk I included in my examples, the budget constraint would be:

    c_t + \frac{c_{t+1}}{1+r} = y_t(1-t) - \frac{b_t}{1+n} + \frac{y_{t+1}}{1+r} +\frac{b_{t+1}}{1+r}

    That mostly seems obvious, but I think it's easy to go wrong if you try to wing it. Anyway, hope your exams are going well beanbag.
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