Expectations in Gali's classical monetary model

Question

In Gali chapter 2 we have the following constraint to the classical monetary model

$P_t C_t + Q_t B_t \leq B_{t-1} + W_t N_t-T_t$.

Then it seems that this is treated as an equality. Therefore my question is: are we assumig that $\partial U/\partial B_t >0$? So households would achieve the optimun at the equality.

Another question is, in general if I have the problem

$Max E_0 \sum_{t=0}^{\infty}\beta^t U(C_t,N_t)$

Should I solve the OP

$Max \sum_{t=0}^{\infty}\beta^t U(C_t,N_t)$

And then take expectation to the relations I obtain by optimizing?. For example if we add the constraint

$P_t C_t + Q_t B_t \leq B_{t-1} + W_t N_t-T_t$ to the OPs, in the problem without expectation I get $Q_t=\beta \frac{\partial U/\partial C_{t+1}}{\partial U/\partial C_{t}} \frac{P_t}{P_{t+1}}$ and taking $E_t$ I obtain Euler's equation. Is the procedure of solving the problem without expectation and then take expectation correct?

EDIT: If it is incorrect to forget the expectation and optimize, then I am confused about this: How to relate real rate of return on capital to bond interest rate: Lagrangian . As there the procedure to solve the problem seems to forget the expectation.

Second, if I do $\frac{\partial }{\partial C_t}E[U]=E\left[\frac{\partial U}{\partial C_t}\right]$. I obtain $Q_t=\beta\frac{ E_0\left[\partial U/\partial C_{t+1} \right]}{E_0\left[\partial U/\partial C_{t} \right]}\frac{P_t}{P_{t+1}}$ because I can not get rid off $E_0$. Nevertheless this would be equivalent to $Q_t=\beta E_0 \left[\frac{\partial U/\partial C_{t+1} }{\partial U/\partial C_{t} }\right]\frac{P_t}{P_{t+1}}$ if $E_0 \left[ \frac{\partial U/\partial C_{t+1} }{\partial U/\partial C_{t} }\right]=\frac{ E_0\left[\partial U/\partial C_{t+1} \right]}{E_0\left[\partial U/\partial C_{t} \right]}$. Is this because $C_t$ is independent of $C_{t+1}$?.

Third, I realized that if I solve $Max \sum_{t=0}^{\infty}\beta^t E_tU(C_t,N_t)$ s.t the constraint as before and proceed with $\frac{\partial }{\partial C_t}E[U]=E\left[\frac{\partial U}{\partial C_t}\right]$ I obtain the correct Euler's equation. Is this the correct problem to optimize? but what about Gali's propose?

Answer 1

As for your first question, we are not just assuming $\partial U/\partial B > 0$. We simply assume monotonicity of preferences, which for the model here is a plausible condition. In simple terms this means that "more is better". Hence, the consumer will always choose to spend all of his budget for current or future consumption, given by $C$ and $B$ respectively. Hence, the constraint holds with equality. In these types of models we pretty much always assume equality of the budget constraint.

As for your second question, solving without expectations and then taking the expectation is strictly speaking incorrect. It will give you the correct result here though. What you can do here is to take the derivative in the expectation. For these models that usually works in the non-stochastic setting as here. If you have for example stochastic income with log-normal distributions you would have to be more careful. What you can do then, depending on the case, is solve the expectation before taking the derivative or use the Leibniz Rule.

Hence for your problem you don't forget the expectation in the first place, which corresponds to $\partial U / \partial C_{t+1}$. Instead for your model you can take $\partial E_0[U] / \partial C_{t+1} = E_0[\partial U / \partial C_{t+1}]$. As you can see, it will yield the same result as simply taking the expectation later on $\partial U / \partial C_{t+1}$. However, that is the wrong way and you'll probably lose points in an exam if you write it down that way.