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kernelspec:

display_name: Python 3

display_name: Python 3 (ipykernel)

language: python

name: python3

---

First we plot the true finite stream present-value after computing it

below

```{code-cell} python3

```{code-cell} ipython3

# True present value of a finite lease

def finite_lease_pv_true(T, g, r, x_0):

G = (1 + g)

First we study the quality of our approximations

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Finite lease present value $T$ periods ahead"

name: finite_lease_present_value

---

def plot_function(axes, x_vals, func, args):

axes.plot(x_vals, func(*args), label=func.__name__)

funcs = [finite_lease_pv_true,

finite_lease_pv_approx_1,

finite_lease_pv_approx_2]

## the three functions we want to compare

# the three functions we want to compare

fig, ax = plt.subplots()

ax.set_title('Finite Lease Present Value $T$ Periods Ahead')

for f in funcs:

plot_function(ax, T, f, our_args)

ax.legend()

Next we compare the infinite and finite duration lease present values

over different lease lengths $T$.

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Infinite and finite lease present value $T$ periods ahead"

name: infinite_and_finite_lease_present_value

---

# Convergence of infinite and finite

T_max = 1000

T = np.arange(0, T_max+1)

fig, ax = plt.subplots()

ax.set_title('Infinite and Finite Lease Present Value $T$ Periods Ahead')

f_1 = finite_lease_pv_true(T, g, r, x_0)

f_2 = np.full(T_max+1, infinite_lease(g, r, x_0))

ax.plot(T, f_1, label='T-period lease PV')

Now we consider two different views of what happens as $r$ and

$g$ covary

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Value of lease of length $T$"

name: value_of_lease

---

# First view

# Changing r and g

fig, ax = plt.subplots()

ax.set_title('Value of lease of length $T$')

ax.set_ylabel('Present Value, $p_0$')

ax.set_xlabel('$T$ periods ahead')

T_max = 10

If you aren't enamored of 3-d graphs, feel free to skip the next

visualization!

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Three period lease PV with varying $g$ and $r$"

name: three_period_lease_PV

---

# Second view

fig = plt.figure()

fig = plt.figure(figsize = [16, 5])

T = 3

ax = plt.subplot(projection='3d')

r = np.arange(0.01, 0.99, 0.005)

ax.set_xlabel('$r$')

ax.set_ylabel('$g$')

ax.set_zlabel('Present Value, $p_0$')

ax.view_init(20, 10)

ax.set_title('Three Period Lease PV with Varying $g$ and $r$')

ax.view_init(20, 8)

plt.show()

```

After that, we'll use SymPy to compute derivatives

```{code-cell} python3

```{code-cell} ipython3

# Creates algebraic symbols that can be used in an algebraic expression

g, r, x0 = sym.symbols('g, r, x0')

G = (1 + g)

p0

```

```{code-cell} python3

```{code-cell} ipython3

print('dp0 / dg is:')

dp_dg = sym.diff(p0, g)

dp_dg

```

```{code-cell} python3

```{code-cell} ipython3

print('dp0 / dr is:')

dp_dr = sym.diff(p0, r)

dp_dr

time path of $y_t$, given that consumption is a constant fraction

of national income, and investment is fixed.

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Path of aggregate output tver time"

name: path_of_aggregate_output_over_time

---

# Function that calculates a path of y

def calculate_y(i, b, g, T, y_init):

y = np.zeros(T+1)

T = 100

fig, ax = plt.subplots()

ax.set_title('Path of Aggregate Output Over Time')

ax.set_xlabel('$t$')

ax.set_ylabel('$y_t$')

ax.plot(np.arange(0, T+1), calculate_y(i_0, b, g_0, T, y_init))

happen if we vary the so-called **marginal propensity to consume**,

i.e., the fraction of income that is consumed

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Changing consumption as a fraction of income"

name: changing_consumption_as_fraction_of_income

---

bs = (1/3, 2/3, 5/6, 0.9)

fig,ax = plt.subplots()

ax.set_title('Changing Consumption as a Fraction of Income')

ax.set_ylabel('$y_t$')

ax.set_xlabel('$t$')

x = np.arange(0, T+1)

Now we will compare the effects on output of increases in investment and government spending.

```{code-cell} python3

```{code-cell} ipython3

---

mystnb:

figure:

caption: "Different increase on output"

name: different_increase_on_output

---

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 10))

fig.subplots_adjust(hspace=0.3)