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I have been reading an Economics working paper and trying to derive the first-order conditions of a seemingly complicated optimization problem.

The optimization problem with choice variables $P_{t}^{R}$, $P_{t+i}^{S}$, $H_{t+i}$ is as follows.

$$\max E_{t} \sum_{i=0}^{\infty}(1-\delta)^{i}Q_{t,t+i}\pi_{t+i} = \pi_{t} + E_{t} \sum_{i=1}^{\infty}(1-\delta)^{i}Q_{t,t+i}\pi_{t+i} = \pi_{t} + V_{t}$$

$$\pi_{t+i} = \int_{H_{t+i}}^{1}\sigma^{F}(q_{t+i}^{R})\frac{P_{t+i}^{R}-W_{t+i}}{P_{t+i}}dt + \int_{0}^{H_{t+i}}\sigma^{F}(q_{t+i}^{S})\frac{P_{t+i}^{S}-W_{t+i}}{P_{t+i}}dt - \frac{\Psi }{2} \left ( \frac{P_{t+i}^{R}}{P_{t+i-1}^{R}} - 1\right )^{2}$$

s.t. $$\sigma^{H}(q_{t+i}^{R})\left ( U-\lambda_{t+i} P_{t+i}^{R}\right ) \geq 0,$$ $$\sigma^{H}(q_{t+i}^{S})\left ( U-\lambda_{t+i} P_{t+i}^{S}\right ) \geq J_{t+i},$$ where $Q_{t,t+i}$ is the stochastic discount factor defined $Q_{t,t+i}\pi_{t+i} = \frac{\beta^{i}\lambda_{t}}{\lambda_{t+i}}$, $\sigma$'s are the probabilities.

The first-order conditions are:

$$P_{t+i}^{S} = \frac{U}{\lambda_{t+i}} + \frac{q_{t+i}^{S}e^{-q_{t+i}^{S}}}{1-e^{-q_{t+i}^{S}}} \left ( \frac{U}{\lambda_{t+i}} - W_{t+i}\right ),$$

$$\sigma^{F}(q_{t+i}^{S})\left ( P_{t+i}^{S}-W_{t+i} \right ) = \sigma^{F}(q_{t+i}^{R})\left ( P_{t+i}^{R}-W_{t+i} \right ),$$

$$\mu_{t+i}\sigma^{H}(q_{t+i}^{R})\lambda_{t+i} = Q_{t,t+i}\left [ (1-H_{t+i})\frac{\sigma^{F}(q_{t+i}^{R})}{P_{t+i}} - \frac{\Psi }{P_{t+i}^{R}-1}\left ( \frac{P_{t+i}^{R}}{P_{t+i-1}^{R}} - 1\right ) \right ] + Q_{t,t+i+1} \Psi\left ( \frac{P_{t+i+1}^{R}}{P_{t+i}^{R}} - 1\right ) \frac{P_{t+i+1}^{R}}{(P_{t+i}^{R})^{2}} = \Phi (P_{t+i}^{R}),$$ where $\mu_{t+i}$ is the Lagrange multiplier w.r.t the constraint of $P_{t+i}^{R}$.

I am not sure how the first-order conditions were derived in the paper. I think the authors used the method of Lagrange multipliers, but I am not sure how to go about it. Can anyone help me with the process of getting the above first-order conditions?

EDIT: For more information, $P_{t+i}$ is aggregate price level, $\pi_{t+i}$ is firm's real profit in period $t+i$, $V_{t}$ represents the real net present value of an existing firm at the end of the period $t$, $\sigma^{F}$ is the firm's probability of getting at least one customer facing different queue length attracted by a posted price, $W_{t+i}$ is nominal wage, $\Psi$ is price adjustment cost in real term, $\sigma^{H}$ is the customer's probability of purchasing the good at the firm visited different queue length, $\lambda_{t+i}$ is the Lagrangian multiplier in the household's problem, and $J_{t+i}$ is the sale hunter's expected utility from visiting any other firm. $q^{r}$ and $q^{s}$ are queue lengths at the regular and sale prices.

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  • $\begingroup$ Can you share the link of the paper? There are many notations unexplained in the question. And maybe you can try to simplify the question by removing all the model features that are not core. This helps you and others understand the question better. $\endgroup$ Mar 16 at 11:53
  • $\begingroup$ @Alalalalaki The working paper cannot be publicly shared right now, unfortunately. I don't think other notations are that relevant for the FOC derivations. $\endgroup$
    – OGC
    Mar 17 at 6:31
  • $\begingroup$ @Alalalalaki I added more information about the notation if that helps. $\endgroup$
    – OGC
    May 12 at 8:23

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