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Fix headiing Capitalizations
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lectures/lake_model.md

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@@ -42,7 +42,7 @@ import matplotlib.pyplot as plt
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from graphviz import Digraph
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```
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## The Model
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## The Lake model
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This model is sometimes called the **lake model** because there are two pools of workers:
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Do long-run outcomes depend on the initial values $(u_0, e_o)$?
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### Visualising the Long-Run Outcomes
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### Visualising the long-run outcomes
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Let us first plot the time series of unemployment $u_t$, employment $e_t$, and labor force $n_t$.
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Moreover, the times series of unemployment and employment seems to grow at some stable rates in the long run.
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### The Application of Perron-Frobenius Theorem
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### The application of erron-Frobenius theorem
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Since by intuition if we consider unemployment pool and employment pool as a closed system, the growth should be similar to the labor force.
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This theorem helps characterise the dominant eigenvalue $r(A)$ which
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determines the behavior of this iterative process.
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#### Dominant Eigenvector
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#### Dominant eigenvector
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We now illustrate the power of the Perron-Frobenius theorem by showing how it
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helps us to analyze the lake model.
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This suggests that all such sequences share strong similarities in the long run, determined by the dominant eigenvector $\bar{x}$.
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#### Negative Growth Rate
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#### Negative growth rate
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In the example illustrated above we considered parameters such that overall growth rate of the labor force $g>0$.
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