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Hi I am working on a problem set in my macroeconomics course. I have a hard time figuring out how to get started on one of the problems.

The set-up is the following. A price-taking firm maximizes:

$\max_{L_{t},K_{t},I_{t}} E_{0} [\sum_{t=0}^{\infty}\prod_{s=0}^{t}(1+r_{s})^{-1}\{(1-\tau)K_{t}^{\alpha}L_{t}^{1-\alpha}-w_{t}L_{t}-I_{t}(1+a(I_{t}/K_{t}-\delta)) \}]$

subject to:

$K_{t} = (1-\delta)K_{t-1}+I_{t-1}$.

$K_{0}$ is given, $w_{t},r_{t}$ are stochastic, $\delta$ is the rate of capital depriciation, $I$ is the gross investment.

The question is the following: Let $q$ be the Lagrange multiplier on the constraint (it’s also Tobin’s Q). Derive the first order conditions for optimization with respect to $L, I$ and $K$. Given $q$, what information is needed to establish optimal level of investment? Log-linearize the FOC for investment. Suppose we observe $I, K$, and $q$. How would you estimate the parameter $a$ using this equation? How would you interpret the error term in this regression? What are the properties of the error term given your interpretation (e.g., serial correlation, correlation with other variables, etc.)?

I have set up the Lagrangian setting $\prod_{s=0}^{t}(1+r_{s})^{-1}=R_{t}$:

$ \mathcal{L} = E_{0} \big\{ \sum_{t=0}^{\infty} R_{t} \left[ (1-\tau)K_{t}^{\alpha}L_{t}^{1-\alpha} -w_{t}L_{t} -I_{t}(1+a(\frac{I_{t}}{K_{t}}-\delta)) + q_{t}(-K_{t}+(1-\delta)K_{t-1}+I_{t-1}) \right]\big\}$

and derived the following FOCs:

$ \frac{\partial \mathcal{L}}{\partial L_{t}} = R_{t} \left[ (1-\tau)(1-\alpha)K_{t}^{\alpha}L_{t}^{-\alpha}-w_{t} \right] = 0 \Leftrightarrow W_{t} = MPL_{t},\\ \frac{\partial \mathcal{L}}{\partial I_{t}} = R_{t} \left[ -1 -2a \frac{I_{t}}{K_{t}} + a\delta \right] +E_{t}[R_{t+1}q_{t+1}] = 0 \Leftrightarrow 1 +2a \frac{I_{t}}{K_{t}} - a\delta = E_{t}[(1+r_{t+1})^{-1} q_{t+1}],\\ \frac{\partial \mathcal{L}}{\partial K_{t}} = R_{t} \left[ (1-\tau) \alpha K_{t}^{\alpha-1}L_{t}^{1-\alpha} + a \left( \frac{I_{t}}{K_{t}} \right)^{2} - q_{t} \right] +E_{t}[R_{t+1}q_{t+1}(1-\delta)] = 0, \\ \Leftrightarrow (1-\delta)^{-1}(q_{t}-(1-\tau) \alpha K_{t}^{\alpha-1}L_{t}^{1-\alpha} - a \left( \frac{I_{t}}{K_{t}} \right)^{2}) = E_{t}[(1+r_{t+1})^{-1} q_{t+1}].$

But I have a hard time seeing how I would log-linearize $1 +2a \frac{I_{t}}{K_{t}} - a\delta = E_{t}[(1+r_{t+1})^{-1} q_{t+1}]$ to something useful in terms of a regression. I hope somebody can help explain how I should go from here? or if I have made a misstake in my derivations.

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