def compute_change_measure(u, c, θ, π):

"""

Compute the channge of measure given a utility function `u`, a consumption vector `c`,

Compute the change of measure given a utility function `u`, a consumption vector `c`,

a penalty parameter `θ` and a baseline probability vector `π`

"""

* Ex post Bayesian expected utility preferences

Expected utility, risk-sensitive, and ex post Bayesian prefernces are each cast in terms of a unique probability distribution, so they can express risk-aversion, but not model ambiguity aversion.

Expected utility, risk-sensitive, and ex post Bayesian preferences are each cast in terms of a unique probability distribution, so they can express risk-aversion, but not model ambiguity aversion.

Multiplier and constraint prefernces both express aversion to

Multiplier and constraint preferences both express aversion to

concerns about model misppecification, i.e., model uncertainty; both are cast in terms of a set or sets of probability distributions.

* The set of distributions expresses the decision maker's ambiguity about the probability model.

\sum_{i=1}^I \pi_i m_i = 1 .

$$ (tom4)

In {eq}`tom3`, $\eta \geq 0$ defines an entropy ball of probability distributions $\hat \pi = m \pi$ that surround a baseline distribution $\pi$.

In {eq}`tom3`, $\eta \geq 0$ determines an entropy ball of probability distributions $\hat \pi = m \pi$ that surround a baseline distribution $\pi$.

As noted earlier, $\sum_{i=1}^I m_i\pi_i u(c_i)$ is the expected value of $u(c)$ under a twisted probability distribution $\{\hat \pi_i\}_{i=1}^I = \{m_i \pi_i\}_{i=1}^I$.

The two panels in the next figure below can help us to visualize the extra adjustment for risk that the risk-sensitive operator entails.

This will help us understand how the $\mathbf{T}$ transformation works by envisioning what function is being averaged.

This will help us understand how the $\sf{T}$ transformation works by envisioning what function is being averaged.

```{code-cell} ipython3

:tags: [hide-input]

Constraint preferences express **concern about misspecification** or **ambiguity** for short with a positive

$\eta$ that circumscribes an entropy ball around an approximating

probability distribution $\pi$, and *aversion aversion to model misspecification* through

probability distribution $\pi$, and *aversion to model misspecification* through

minimization with respect to a likelihood ratio $m$.

Multiplier preferences express misspecification concerns with a parameter $\theta<+\infty$ that

η = 0.12

# Conustruct grid for c_1

# Constuct grid for c_1

c_1_grid_nb = 102

c_1_grid = np.linspace(0.5, 2., num=c_1_grid_nb)

**Tangent indifference curves off 45 degree line**

For a given $\eta$ and a given allocatin $(c_1, c_2)$ off the 45 degree line, by solving

For a given $\eta$ and a given allocation $(c_1, c_2)$ off the 45 degree line, by solving

equations {eq}`tom7` and {eq}`tom20`, we can find $\tilde \theta (\eta, c)$

and $\tilde \eta(\theta,c)$ that make indifference curves for

multiplier and constraint preferences be tangent to one another.