<p>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.</p>

<p>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 $P_{t}C_{t} + Q_{t}B_{t} \leq B_{t-1} + W_{t}N_{t} + D_{t}$</p>

<p>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.</p>

<p>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. </p>

<p>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;</p>

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

<p>for any pair $(dC_{t},dC_{t+1}$) </p>

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

<p>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. </p>

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

CC BY-SA 3.0
2017-01-20T00:43:21.877Z
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
CC BY-SA 3.0
2017-01-20T00:51:45.047Z
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!
user11767
CC BY-SA 3.0
2017-01-20T01:00:50.170Z
$Q_t$ 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
CC BY-SA 3.0
2017-01-20T01:08:33.450Z
Ugh, I'm stupid. Thank you so much for clarifying.
user11767
CC BY-SA 3.0
2017-01-20T01:09:43.627Z
You aren't -- this stuff just takes getting used to.
Theoretical Economist
CC BY-SA 3.0
2017-01-21T19:44:15.840Z
@TheoreticalEconomist Perhaps you could transform your comments into an (acceptable) answer?
FooBar
CC BY-SA 3.0
2017-01-21T19:47:57.030Z
@FooBar I know I ought to do that -- I am unfortunately drowning in work at the moment, so I don't have the time to write a proper answer. Apologies. I'm happy for anyone else (perhaps even the OP?) to take whatever I've said above and place it in an answer. (Hopefully one slightly expanded.) I'll even give it an upvote.
Theoretical Economist
CC BY-SA 3.0
2017-04-10T20:51:26.643Z
Isn't $Q_t$ the price of the bond in period t? I think the value of the bond is 1 paid in period $t+1$.
123
<utility><self-study><consumption><new-keynesian-economics>
Consumer's maximization problem
user11767
# Consumer's maximization problem

(Asked by: username_0 on 2017-01-19)

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.

**Tags:** utility, self-study, consumption, new-keynesian-economics

> **Comments:**
> **username_1**: 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 $0$.  
> **username_0**: 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!  
> **username_1**: $Q_t$ 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.  
> **username_0**: Ugh, I'm stupid. Thank you so much for clarifying.  
> **username_1**: You aren't -- this stuff just takes getting used to.  
> **username_2**: @TheoreticalEconomist Perhaps you could transform your comments into an (acceptable) answer?  
> **username_1**: @username_2 I know I ought to do that -- I am unfortunately drowning in work at the moment, so I don't have the time to write a proper answer. Apologies. I'm happy for anyone else (perhaps even the OP?) to take whatever I've said above and place it in an answer. (Hopefully one slightly expanded.) I'll even give it an upvote.  
> **username_3**: Isn't $Q_t$ the price of the bond in period t? I think the value of the bond is 1 paid in period $t+1$.  

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## 0 answers

