update calvo addon

lectures/calvo_addon.md

@@ -58,10 +58,6 @@ This function will be an input into solving for a Ramsey plan using a version of
 
 I'll move on to describe that function implicitly as a system of constraints.
 
-```{code-cell} ipython3
-!pip install jaxopt
-```
-
 ```{code-cell} ipython3
 from quantecon import LQ
 import numpy as np
@@ -324,11 +320,6 @@ def objective(x, m0, model):
     m, p = solve_mp(x, m0, model)
     return -jnp.sum(β**jnp.arange(T+1) * (u(f(x)) + j(m - p)))
 
-def penality(x, m0, model):
-    β, T = model.β, model.T
-    m, p = solve_mp(x, m0, model)
-    return -jnp.sum(β**jnp.arange(T+1) * (u(f(x)) + j(m - p)))
-
 grad_objective = grad(lambda x: objective(x, m0=jnp.log(100), model=model))
 x_init = jnp.ones(model.T+1)# Initial guess for x
 optimized_x = adam_optimizer(grad_objective, x_init)
@@ -341,6 +332,10 @@ print("Optimized m_t:", optimized_m)
 print("Optimized p_t:", optimized_p)
 ```
 
+```{code-cell} ipython3
+xs = np.linespace(0, 100, 1)
+```
+
 $$
 P(\vec x, \vec m, \vec p) = \lambda_1 \sum_{t=0}^\infty \beta^t \left( m_{t} - p_t + \alpha (p_{t+1} - p_t) \right)^2 
 + \lambda_2 \sum_{t=0}^\infty \beta^t \left( \exp(m_{t+1}) - \exp(m_t) + x_t \exp(p_t) \right)^2