# Consumer's maximization problem

I am an undergrad trying to learn the basics of NK models (on my own) and I am having some difficulties understanding the derivation of certain equations.

So we have the representative household seeking to maximize $E_{0}\sum _{0}^{\infty}\beta^{t}U(C_{t}, N_{t}; Z_{t})$ subject to $P_{t}C_{t} + Q_{t}B_{t} \leq B_{t-1} + W_{t}N_{t} + D_{t}$

Where $B_{t}$ is the amount of one-period nominally riskless bonds, maturing in period $t+1$ and $Q_{t}$ is the value of said bond.

I am able to derive the optimal consumption-work hour pair by finding the total derivative of the utility function subject to the budget constraint in $C_{t},N_{t}$ space.

However, next the authors wants to examine the effects of reallocating consumption between period $t$ and $t+1$ on expected utility (holding all else equal). He shows this is;

$U_{c,t}dC_{t} + \beta E_{t}[U_{c,t+1}dC_{t+1}] = 0$ (1)

for any pair $(dC_{t},dC_{t+1}$)

satisfying $P_{t+1}dC_{t+1}= -\frac{P_{t}}{Q_{t}}dC_{t}$ (2)

I kind of understand where equation (1) is coming from (although if someone could derive this mathematically I would be very grateful). But equation (2) I don't understand at all. Is this some sort of budget constraint on consumption?I have tried finding the total differential of the budget while considering changes in $C_{t}$ and $B_{t}$. $B_{t}$, because I assume since we're considering consumption between two periods, savings is going to matter. However, because it is consumption between two periods and the budget constraint in just for a given period, I am becoming confused.

I was hoping someone could explain how to think about this intuitively and the mathematical derivation behind it.

• Is this Gali? You should think of this more as a heuristic variational argument. (2) is a budget-balance condition -- you are only considering reallocations between $C_t$ and $C_{t+1}$ whose net cost is $0$. – Theoretical Economist Jan 20 '17 at 0:43
• Gali indeed. Net cost being zero makes sense, thanks for that. I am still a little confused as to why I am dividing by $Q_{t}$. Is there a mathematical derivation? Thanks for the help! – BenBernke Jan 20 '17 at 0:51
• $Q_t$ is how much I have to pay, at period $t$ (today), for a dollar of consumption at period $t+1$ (tomorrow). (Notice this is just a rephrasing of what a bond is.) Hence paying $1 / Q_t$ today buys me one dollar tomorrow. – Theoretical Economist Jan 20 '17 at 1:00
• Ugh, I'm stupid. Thank you so much for clarifying. – BenBernke Jan 20 '17 at 1:08
• You aren't -- this stuff just takes getting used to. – Theoretical Economist Jan 20 '17 at 1:09