[olg] rephrase or clarify

[shlff]

Based on the reading group's work thanks @HengchengZhang , @orectique , @DiaPorntipa .

[shlff]

Rephrase or clarify

• lecture-python-intro/lectures/olg.md

  Line 106 in 94272e4

  First let's consider the household side.

• lecture-python-intro/lectures/olg.md

  Line 150 in 94272e4

  Since $u$ is strictly increasing, both of these constraints will hold as equalities at the maximum.

• lecture-python-intro/lectures/olg.md

  Line 159 in 94272e4

  This leads to the **Euler equation** of the OLG model, which is

• lecture-python-intro/lectures/olg.md

  Lines 213 to 214 in 94272e4

  	Since the population size is normalized to 1, $s_t$ is also total savings in
  	the economy at time $t$.

• lecture-python-intro/lectures/olg.md

  Lines 281 to 282 in 94272e4

  	Rearranging [](interest_rate_one) gives the aggregate demand for capital
  	at time $t+1$

• lecture-python-intro/lectures/olg.md

  Line 350 in 94272e4

  the equilibrium quantity $k_{t+1}$.

• explain more about the relationship between interest rate and MPK

  lecture-python-intro/lectures/olg.md

  Line 560 in 94272e4

  The interest rate reflects the marginal product of capital, which is high when capital stock is low.

• lecture-python-intro/lectures/olg.md

  Line 573 in 94272e4

  This function is called the CRRA utility function.

• lecture-python-intro/lectures/olg.md

  Lines 834 to 835 in 94272e4

  	Generate three time paths for capital, from
  	three distinct initial conditions, under the parameterization listed above.

• lecture-python-intro/lectures/olg.md

  Lines 138 to 142 in 94272e4

  	$$
  	c_t + s_t \le w_t
  	\quad \text{and} \quad
  	c_{t+1} \le R_{t+1} s_t
  	$$

• lecture-python-intro/lectures/olg.md

  Lines 284 to 288 in 94272e4

  	```{math}
  	:label: aggregate_demand_capital_olg
  	k^d (R_{t+1})
  	:= \left (\frac{\alpha}{R_{t+1}} \right )^{1/(1-\alpha)}
  	```