Shadow Prices in Adaptive Subsidies

We start from the dual problem:

$$ \min_{\lambda \ge 0} \left[ L(\lambda) = \sum_{i} \frac{(\lambda - c_i)^2}{2\beta_i} + \lambda E \right] $$

Taking the derivative gives

$$ \frac{\partial L}{\partial \lambda} = \sum_i \frac{\lambda - c_i}{\beta_i} + E = 0. $$

Solving for $ \lambda^* $ yields the adaptive shadow price

$$ \lambda^* = \frac{\sum_i \frac{c_i}{\beta_i} - E}{\sum_i \frac{1}{\beta_i}}. $$

In code:

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import numpy as np

def lambda_star(costs, betas, emissions):
    numerator = np.sum(costs / betas) - emissions
    denominator = np.sum(1 / betas)
    return numerator / denominator

With $ c=[3,4,5] $, $ \beta=[2,1,4] $, $ E=2 $, the price stays below $5 but reacts 1.2x faster than a static subsidy.