# budget constraint and intertemporal substitution

• Jan 9th 2010, 04:13 AM
garymarkhov
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

$
c_t + \frac{c_{t+1}}{1+r} \leq(1-s)w_t + sw_t(1+r)
$

(where s is savings, c is consumption)

OR:

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

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. (Headbang)
• Jan 9th 2010, 03:02 PM
beanbag
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 !
• Jan 10th 2010, 02:08 AM
garymarkhov
Thanks for responding! I'm studying for macro, which I have on the 14th. Good luck with your exams!
• Jan 14th 2010, 03:06 PM
garymarkhov
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.