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[LakeModel] Simplify computation of the steady state #355

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Merged
merged 11 commits into from
Dec 20, 2023
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[LakeModel] Simplify computation of the steady state #355

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63d0c92
[LakeModel] Simplify computation of the steady state
mnshkw Jul 10, 2023
60e7b02
fix
shlff Dec 19, 2023
4de55c7
rephrase
shlff Dec 19, 2023
2473f00
Update lectures/lake_model.md
mmcky Dec 19, 2023
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Update lectures/lake_model.md
mmcky Dec 19, 2023
9f5259c
Update lectures/lake_model.md
mmcky Dec 19, 2023
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Revert "Update lectures/lake_model.md"
mmcky Dec 19, 2023
726e17c
Revert "Update lectures/lake_model.md"
mmcky Dec 19, 2023
6d67088
incorporate text changes from @jstac
mmcky Dec 19, 2023
1cb72ac
clarification
shlff Dec 20, 2023
c53b86a
Update lectures/lake_model.md
mmcky Dec 20, 2023
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19 changes: 6 additions & 13 deletions lectures/lake_model.md
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Original file line number Diff line number Diff line change
Expand Up @@ -218,7 +218,7 @@ This class will
1. store the primitives $\alpha, \lambda, b, d$
1. compute and store the implied objects $g, A, \hat A$
1. provide methods to simulate dynamics of the stocks and rates
1. provide a method to compute the steady state of the rate
1. provide a method to compute the steady state of the rate (We will discuss it in section {ref}`dynamics_workers`)

Please be careful because the implied objects $g, A, \hat A$ will not change
if you only change the primitives.
Expand Down Expand Up @@ -265,12 +265,8 @@ class LakeModel:
--------
xbar : steady state vector of employment and unemployment rates
"""
x = np.full(2, 0.5)
error = tol + 1
while error > tol:
new_x = self.A_hat @ x
error = np.max(np.abs(new_x - x))
x = new_x
x = np.array([self.A_hat[0, 1], self.A_hat[1, 0]])
x /= x.sum()
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Suggest

        x /= x.sum()
        return x

to

        return x / x.sum()

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The former is in principle strictly better: x / x.sum() creates an intermediate array while x /= x.sum() is an in-place operation. (Of course, the difference is negligible for this size of the data.)

return x

def simulate_stock_path(self, X0, T):
Expand Down Expand Up @@ -415,6 +411,7 @@ plt.tight_layout()
plt.show()
```

(dynamics_workers)=
## Dynamics of an Individual Worker

An individual worker's employment dynamics are governed by a {doc}`finite state Markov process <finite_markov>`.
Expand Down Expand Up @@ -1016,12 +1013,8 @@ class LakeModelModified:
--------
xbar : steady state vector of employment and unemployment rates
"""
x = np.full(2, 0.5)
error = tol + 1
while error > tol:
new_x = self.A_hat @ x
error = np.max(np.abs(new_x - x))
x = new_x
x = np.array([self.A_hat[0, 1], self.A_hat[1, 0]])
x /= x.sum()
return x

def simulate_stock_path(self, X0, T):
Expand Down