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 September 29th, 2013, 02:22 PM #1 Newbie   Joined: Sep 2013 Posts: 5 Thanks: 0 Lagrangian Multiplier Hi everyone, I am trying to solve the following maximization problem: \begin{align*} \max_{\theta_i,k_i,p_i,a_i} \pi =\iint_S [p_i(b,y)\alpha_i(b,y)f(b,y)dbdy]-V(k_i)\\ s.t.\\ k_i= \iint_S \alpha_i(b,y)f(b,y)dbdy\\ \theta_i = \frac{1}{k_i}\iint_S b\alpha_i(b,y)f(b,y)dbdy \end{align*} where S is the support of b and y, and f() is the joint distribution of b and y. p is the price for a combination b,y and a is the proportion of b,y (so a()*f() is the total number for each b,y combination). The solution states that assuming we have an optimal $p_i^*$, we can pointwise maximize over $\alpha_i$ forming a Lagrangian taking both constraints into account and obtain: \begin{align*} p_i^*(b,y,\theta_i)=V'(k_i)+\eta_i(\theta_i-b)\\ \eta_i =\frac{1}{k_i}\iint_S \frac{\partial p_i^*(b,y,\theta_i) }{\partial \theta_i} \alpha_i(b,y)f(b,y)dbdy \end{align*} where $\eta_i$apparently is the Lagrange multiplier of the second constraint. I have not provided too many details on the meaning of the variables, please let me know if you need additional information. I am thankful for any ideas, this should be pretty straightforward but I could not figure this out for quite a while now. Thanks, Tartaglia

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