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[ak2, ak_aiyagari, two_computation] Add figure captions and lw=2 #1081

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[ak2, ak_aiyagari, two_computation] Add figure captions and lw=2
mmcky Oct 9, 2026
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Merge remote-tracking branch 'origin/main' into style-figures-1080
mmcky Oct 9, 2026
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102 changes: 75 additions & 27 deletions lectures/ak2.md
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Original file line number Diff line number Diff line change
Expand Up @@ -681,23 +681,23 @@ class ClosedFormTrans:
# quantities
for i, name in enumerate(['K', 'Y', 'Cy', 'Co']):
ax = axs[i//3, i%3]
ax.plot(range(T+1), quant_seq[:T+1, i], label=name)
ax.plot(range(T+1), quant_seq[:T+1, i], label=name, lw=2)
ax.hlines(init_ss[i], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# prices
for i, name in enumerate(['W', 'r']):
ax = axs[(i+4)//3, (i+4)%3]
ax.plot(range(T+1), price_seq[:T+1, i], label=name)
ax.plot(range(T+1), price_seq[:T+1, i], label=name, lw=2)
ax.hlines(init_ss[i+4], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# policies
for i, name in enumerate(['τ', 'D', 'G']):
ax = axs[(i+6)//3, (i+6)%3]
ax.plot(range(T+1), policy_seq[:T+1, i], label=name)
ax.plot(range(T+1), policy_seq[:T+1, i], label=name, lw=2)
ax.hlines(init_ss[i+6], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')
Expand Down Expand Up @@ -756,6 +756,12 @@ Let's use the `simulate` method of `closed` to compute dynamic transitions.
Note that we leave `τ_pol` as `None`, since the tax rates need to be determined to satisfy the government budget constraint.

```{code-cell} ipython3
---
mystnb:
figure:
caption: Transition after a tax cut
name: ak2_exp1_tax_cut
---
quant_seq1, price_seq1, policy_seq1 = closed.simulate(T, init_ss,
D_pol=D_seq,
G_pol=G_seq)
Expand All @@ -779,31 +785,37 @@ quant_seq2, price_seq2, policy_seq2 = closed.simulate(T, init_ss,
```

```{code-cell} ipython3
---
mystnb:
figure:
caption: Tax cut transitions for two tax rates
name: ak2_exp1_two_rates
---
fig, axs = plt.subplots(3, 3, figsize=(14, 10))

# quantities
for i, name in enumerate(['K', 'Y', 'Cy', 'Co']):
ax = axs[i//3, i%3]
ax.plot(range(T+1), quant_seq1[:T+1, i], label=f'{name}, 1/3')
ax.plot(range(T+1), quant_seq2[:T+1, i], label=f'{name}, 0.2')
ax.plot(range(T+1), quant_seq1[:T+1, i], label=f'{name}, 1/3', lw=2)
ax.plot(range(T+1), quant_seq2[:T+1, i], label=f'{name}, 0.2', lw=2)
ax.hlines(init_ss[i], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# prices
for i, name in enumerate(['W', 'r']):
ax = axs[(i+4)//3, (i+4)%3]
ax.plot(range(T+1), price_seq1[:T+1, i], label=f'{name}, 1/3')
ax.plot(range(T+1), price_seq2[:T+1, i], label=f'{name}, 0.2')
ax.plot(range(T+1), price_seq1[:T+1, i], label=f'{name}, 1/3', lw=2)
ax.plot(range(T+1), price_seq2[:T+1, i], label=f'{name}, 0.2', lw=2)
ax.hlines(init_ss[i+4], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# policies
for i, name in enumerate(['τ', 'D', 'G']):
ax = axs[(i+6)//3, (i+6)%3]
ax.plot(range(T+1), policy_seq1[:T+1, i], label=f'{name}, 1/3')
ax.plot(range(T+1), policy_seq2[:T+1, i], label=f'{name}, 0.2')
ax.plot(range(T+1), policy_seq1[:T+1, i], label=f'{name}, 1/3', lw=2)
ax.plot(range(T+1), policy_seq2[:T+1, i], label=f'{name}, 0.2', lw=2)
ax.hlines(init_ss[i+6], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')
Expand All @@ -823,6 +835,12 @@ The government targets the same tax rate $\tau_t=\hat{\tau}$ and to accumulate
To conduct this experiment, we pass `τ_seq` and `G_seq` as inputs and let `D_pol` be determined along the path by satisfying the government budget constraint.

```{code-cell} ipython3
---
mystnb:
figure:
caption: Transition after a permanent expenditure cut
name: ak2_exp2_expenditure_cut
---
# government expenditure cut by a half
G_seq = τ_hat * 0.5 * Y_hat * np.ones(T+1)

Expand All @@ -844,7 +862,7 @@ mystnb:
caption: Ratio of government assets to capital
name: ak2_asset_capital_ratio
---
plt.plot(range(T+1), -closed.policy_seq[:-1, 1] / closed.quant_seq[:, 0])
plt.plot(range(T+1), -closed.policy_seq[:-1, 1] / closed.quant_seq[:, 0], lw=2)
plt.xlabel('t')
plt.ylabel('-D/K')
plt.show()
Expand Down Expand Up @@ -878,6 +896,12 @@ But now the cut lasts for one period only: the government sets $G_0 = 0$ and ret
From $t \geq 1$, government expenditures return to $\hat{G}$ and $\tau_t$ adjusts to maintain the asset level $-D_t = -D_1$.

```{code-cell} ipython3
---
mystnb:
figure:
caption: Transition after a temporary expenditure cut
name: ak2_exp3_temporary_cut
---
# sequence of government purchase
G_seq = τ_hat * Y_hat * np.ones(T+1)
G_seq[0] = 0
Expand Down Expand Up @@ -1059,11 +1083,11 @@ class AK2():
if verbose:
# plot current prices at ith iteration
for i, name in enumerate(['W', 'r']):
axs[i].plot(range(T+1), price_seq[:T+1, i])
axs[i].plot(range(T+1), price_seq[:T+1, i], lw=2)
axs[i].set_title(name)
axs[i].set_xlabel('t')
axs[2].plot(range(T+1), policy_seq[:T+1, 0],
label=f'{i_iter}th iteration')
label=f'{i_iter}th iteration', lw=2)
axs[2].legend(bbox_to_anchor=(1.05, 1), loc='upper left')
axs[2].set_title('τ')
axs[2].set_xlabel('t')
Expand Down Expand Up @@ -1141,23 +1165,23 @@ class AK2():
# quantities
for i, name in enumerate(['K', 'Y', 'Cy', 'Co']):
ax = axs[i//3, i%3]
ax.plot(range(T+1), quant_seq[:T+1, i], label=name)
ax.plot(range(T+1), quant_seq[:T+1, i], label=name, lw=2)
ax.hlines(init_ss[i], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# prices
for i, name in enumerate(['W', 'r']):
ax = axs[(i+4)//3, (i+4)%3]
ax.plot(range(T+1), price_seq[:T+1, i], label=name)
ax.plot(range(T+1), price_seq[:T+1, i], label=name, lw=2)
ax.hlines(init_ss[i+4], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# policies
for i, name in enumerate(['τ', 'D', 'G']):
ax = axs[(i+6)//3, (i+6)%3]
ax.plot(range(T+1), policy_seq[:T+1, i], label=name)
ax.plot(range(T+1), policy_seq[:T+1, i], label=name, lw=2)
ax.hlines(init_ss[i+6], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')
Expand Down Expand Up @@ -1186,13 +1210,25 @@ D_pol[1:] = D1
```

```{code-cell} ipython3
---
mystnb:
figure:
caption: Price iterations of the fixed point algorithm
name: ak2_fixed_point_iterations
---
quant_seq3, price_seq3, policy_seq3 = ak2.simulate(T, init_ss,
δy_seq, δo_seq,
D_pol=D_pol, G_pol=G_pol,
verbose=True)
```

```{code-cell} ipython3
---
mystnb:
figure:
caption: Tax cut transition computed with AK2
name: ak2_exp1_ak2
---
ak2.plot()
```

Expand All @@ -1215,31 +1251,37 @@ quant_seq4, price_seq4, policy_seq4 = ak2.simulate(T, init_ss,
Note how "crowding out" has been mitigated.

```{code-cell} ipython3
---
mystnb:
figure:
caption: Transitions with and without lump sum taxes
name: ak2_exp1_lump_sum
---
fig, axs = plt.subplots(3, 3, figsize=(14, 10))

# quantities
for i, name in enumerate(['K', 'Y', 'Cy', 'Co']):
ax = axs[i//3, i%3]
ax.plot(range(T+1), quant_seq3[:T+1, i], label=rf'{name}, $\delta$s=0')
ax.plot(range(T+1), quant_seq4[:T+1, i], label=rf'{name}, $\delta$s=0.005')
ax.plot(range(T+1), quant_seq3[:T+1, i], label=rf'{name}, $\delta$s=0', lw=2)
ax.plot(range(T+1), quant_seq4[:T+1, i], label=rf'{name}, $\delta$s=0.005', lw=2)
ax.hlines(init_ss[i], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# prices
for i, name in enumerate(['W', 'r']):
ax = axs[(i+4)//3, (i+4)%3]
ax.plot(range(T+1), price_seq3[:T+1, i], label=rf'{name}, $\delta$s=0')
ax.plot(range(T+1), price_seq4[:T+1, i], label=rf'{name}, $\delta$s=0.005')
ax.plot(range(T+1), price_seq3[:T+1, i], label=rf'{name}, $\delta$s=0', lw=2)
ax.plot(range(T+1), price_seq4[:T+1, i], label=rf'{name}, $\delta$s=0.005', lw=2)
ax.hlines(init_ss[i+4], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# policies
for i, name in enumerate(['τ', 'D', 'G']):
ax = axs[(i+6)//3, (i+6)%3]
ax.plot(range(T+1), policy_seq3[:T+1, i], label=rf'{name}, $\delta$s=0')
ax.plot(range(T+1), policy_seq4[:T+1, i], label=rf'{name}, $\delta$s=0.005')
ax.plot(range(T+1), policy_seq3[:T+1, i], label=rf'{name}, $\delta$s=0', lw=2)
ax.plot(range(T+1), policy_seq4[:T+1, i], label=rf'{name}, $\delta$s=0.005', lw=2)
ax.hlines(init_ss[i+6], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')
Expand Down Expand Up @@ -1281,31 +1323,37 @@ quant_seq5, price_seq5, policy_seq5 = ak2.simulate(T, init_ss,
```

```{code-cell} ipython3
---
mystnb:
figure:
caption: Tax cut versus social security transfer
name: ak2_exp4_social_security
---
fig, axs = plt.subplots(3, 3, figsize=(14, 10))

# quantities
for i, name in enumerate(['K', 'Y', 'Cy', 'Co']):
ax = axs[i//3, i%3]
ax.plot(range(T+1), quant_seq3[:T+1, i], label=f'{name}, tax cut')
ax.plot(range(T+1), quant_seq5[:T+1, i], label=f'{name}, transfer')
ax.plot(range(T+1), quant_seq3[:T+1, i], label=f'{name}, tax cut', lw=2)
ax.plot(range(T+1), quant_seq5[:T+1, i], label=f'{name}, transfer', lw=2)
ax.hlines(init_ss[i], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# prices
for i, name in enumerate(['W', 'r']):
ax = axs[(i+4)//3, (i+4)%3]
ax.plot(range(T+1), price_seq3[:T+1, i], label=f'{name}, tax cut')
ax.plot(range(T+1), price_seq5[:T+1, i], label=f'{name}, transfer')
ax.plot(range(T+1), price_seq3[:T+1, i], label=f'{name}, tax cut', lw=2)
ax.plot(range(T+1), price_seq5[:T+1, i], label=f'{name}, transfer', lw=2)
ax.hlines(init_ss[i+4], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')

# policies
for i, name in enumerate(['τ', 'D', 'G']):
ax = axs[(i+6)//3, (i+6)%3]
ax.plot(range(T+1), policy_seq3[:T+1, i], label=f'{name}, tax cut')
ax.plot(range(T+1), policy_seq5[:T+1, i], label=f'{name}, transfer')
ax.plot(range(T+1), policy_seq3[:T+1, i], label=f'{name}, tax cut', lw=2)
ax.plot(range(T+1), policy_seq5[:T+1, i], label=f'{name}, transfer', lw=2)
ax.hlines(init_ss[i+6], 0, T+1, color='r', linestyle='--')
ax.legend()
ax.set_xlabel('t')
Expand Down
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