Question

Prove that $\Delta_w \Phi(p)=\Phi(p)-\sum_{i \in I} w_i$.

   Prove that $\Delta_w \Phi(p)=\Phi(p)-\sum_{i \in I} w_i$.

Competitive Equilibrium: Theory and Applications

Competitive Equilibrium: Theory and Applications

Bryan Ellickson 1st Edition

Chapter 6, Problem 1

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Step 1

$\Delta_w \Phi(p)$ is the discrete Laplacian of the function $\Phi(p)$ with respect to the weights $w_i$. In other words, it is the difference between the value of $\Phi(p)$ at a point $p$ and the weighted sum of the values of $\Phi(p)$ at its neighboring Show more…

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Prove that $\Delta_w \Phi(p)=\Phi(p)-\sum_{i \in I} w_i$.

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