# New-Keynesian Model: Log-linearizing the firm's FOC

In Gali's book (chapter 3), the FOC of a firm is given by:

$$(\sum_{k=0}^\infty \theta^k E_k(Q_{t,t+k} Y_{t+k|t} (P_t^*/P_{t-1} - \alpha MC_{t,t+k} \beta_{t-1,t+k}))) = 0$$

Basically, the first order condition for maximizing a firm's profit. $0< \theta <=1$, $\alpha$ is markup over competitive price. MC is marginal cost. And $beta = P_{t+k}/P_t$.

At the zero inflation steady state (the point we're supposed to linearize this around),

$P_t^*/P_{t-1} = 1$ $beta = P_{t+k}/P_t = 1$ $P^*=P_{t+k}$ Thus Y is constant. $Y_{t+k|t} = Y$ and so is MC. Q is stocastic discount factor and equals $B^k$ around steady. Similarly, Marginal cost (MC) is the reciprocal of the markup. i.e. $MC = 1/\alpha$

I need to expand this lovely expression. The problem I'm having is that if I do an expansion around the steady state, everything seems to be cancelling out, leaving me without anything that looks like the intended solution to the problem.

$$(\sum_{k=0}^\infty \theta^k E_k(Q_{t,t+k} Y_{t+k|t} (P_t^*/P_{t-1} - \alpha MC_{t,t+k} \beta_{t-1,t+k}))) = 0$$

Log linearize around the zero inflation steady state.

$$p_t^* - p_{t-1} = (1 - \beta\theta) \sum_{k=0}^\infty (\beta\theta)^k E_t(\widehat{mc}_{t+k|t} + p_{t+k} - p_{t-1})$$

Where $$\widehat{mc}_{t+k|t} \equiv {mc}_{t+k|t} - mc$$ In other words, $$p_t^* = \gamma + (1 - \beta\theta) \sum_{k=0}^\infty (\beta\theta)^k E_t(\widehat{mc}_{t+k|t} + p_{t+k})$$ where $\gamma \equiv log \frac{\epsilon}{\epsilon + 1}$

If I remember correctly, this model uses Calvo pricing, so inflation is just from wage stickiness. So if you set $\theta = 0$ (no price stickiness) $$p_t^* = \gamma + mc_t + p_t$$

See what you can work with from there.

Edit: If you're wondering how this is done, then "An Old Man in the Sea"'s resource in the comments is helpful, in sections 3.2 and 3.3. It will show how to do the First-Order Taylor Expansion to derive the result.