# Is this dynamic optimization?

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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Can you identify variables and assign typical values to constants in the above relation? – Narasimham Apr 9 at 16:05
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! – Luigi Apr 9 at 16:16