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I would like to know what I should know to understand this IMF paper. What kind of optimization is used to maximize the utility function on page 9 (number 1) subject to constraints (2) and (3)?

The function I must maximize is

$U_0^i=E_0\sum_{t=0}^{\infty} \beta_i^t \left[ \dfrac{(c_t^i - c_{min}^i)^{(1-\frac{1}{\sigma_i})}}{\left(1-\frac{1}{\sigma_i}\right)} + \xi_d log (d_t (1-(1-\gamma_\ell)\pi_t)) + \xi_k log (\overline{k} + k_t(1-(1-\gamma_k)\pi_t))\right] $

The constraints are

$k_t=(1-\delta)\Delta_{k_t} k_{t-1}+I_t$

and

$d_t q_t = \Delta_{\ell_t} d_{t-1}+r_t^k \Delta_{k_t}k_{t-1}-c_t^i-I_t$

The optimality conditions for $c$, $d$ and $k$ are:

$(c_t^i-c_{min}^i)^{-\frac{1}{\sigma_i}}=\lambda_t^i$

$1=\beta_i E_t \left( \dfrac{\lambda_{t+1}^i}{\lambda_t^i} \right) \dfrac{1-(1-\gamma_\ell)\pi_t}{q_t} + \dfrac{\xi_d}{\lambda_t^i d_t q_t}$

and

$1=\beta_i E_t \left( \dfrac{\lambda_{t+1}^i}{\lambda_t^i} \right) (r_{t+1}^k+1-\delta)(1-(1-\gamma_k)\pi_t)+\dfrac{\xi_k(1-(1-\gamma_k)\pi_t)}{\lambda_t^i(\overline{k}+k_t (1-(1-\gamma_k)\pi_t))}$

What should I study to understand this maximization? Is this dynamic optimization?

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  • $\begingroup$ Can you identify variables and assign typical values to constants in the above relation? $\endgroup$ – Narasimham Apr 9 '15 at 16:05
  • $\begingroup$ This question is very old and I barely know something about dynamic optimization (at that time I did not even know if this was dynamic optimization), so I cannot say more. Sorry! $\endgroup$ – Luigi Apr 9 '15 at 16:16
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This is dynamic optimization problem in infinite horizon, the standard solution method is dynamic programming.

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It is dynamic but it is not that much challenging to understand it. You already given the utility function. Just need to set up Lagrangian with given constraint which is something also we do in simple optimization. Only difference here is that flipping time index from $t$ to ${t+1}$ and taking two FOC for the variables $k_t$ and $d_t$ since you are given different period indexes for those variables.

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