# Slow going at the pretzel factory

A chocolate covered, peanut butter filled, pretzel factory has three worker pods, and that each pod produces one important part of the snack: pod 1 prepares the chocolate, pod 2 the peanut butter, and pod 3 the pretzel shell.

The net production coming from each pod is balanced so that any given pod produces $$\lambda = N_i s_i$$ snacks worth of their part per unit time, where $$N_i$$ is the number of workers in pod $$i$$, and $$s_i(0)$$ is the speed at which a worker in pod $$i$$ can produce things.

When everything is working smoothly the factory can output 100 chocolate covered, peanut butter filled pretzels per unit time. One night the workers in pod 1 have a party late into the night and mistakenly dump a thickening agent into the chocolate supply. This makes the chocolate harder to handle, so that the rate at which workers can prepare chocolate drops to $$s_i(T) < s_i(0)$$. However, the speed of production in the other pods remains constant. The drop in $$s_1$$ causes the overall rate of pretzel production to fall to $$\lambda_T = 60$$ per unit time.

What is the new speed of chocolate production per worker, $$s_1(T)$$ (in snack equivalents per unit time)?

Assumptions and details

• The $$N_T = \sum_i N_i = 500$$ factory workers always distribute themselves across the pods such that the rate of snack production is maximal, given their current working conditions.
• When everything is working smoothly, a worker in pod 1 can produce chocolate at the rate $$s_i(0) = 10$$ snack equivalents per unit time.
• The company has a fixed amount of money each day $$M$$ to pay workers, and each worker gets paid fixed daily wage $$p$$.
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