# Model the uncertain impact of a proposed policy by expected utility or other probabilistic approach

The impact of a proposed policy is often uncertain and subjected to randomness. As such, it seems natural to use probabilistic models. How to model the policy impact using the expected utility approach or any other probabilistic approach?

My attempt:

Suppose a country has $$m$$ people and $$n$$ types of goods. In the current period, we know deterministically that the $$m$$ people have incomes $$w = (w_1, \dots, w_m) \in \mathbb{R}^m_{++}$$, and the $$n$$ types of goods have prices $$p = (p_1, \dots, p_n) \in \mathbb{R}^n_{++}$$.

A policy is being proposed and would affect the incomes and prices in the next period. Assume no person is dead or born and no new type of goods is created or obsolete in the next period. We model the change of incomes as an $$m$$-variate random vector $$Y = (Y_1, \dots, Y_m)$$ and the change of prices as an $$n$$-variate random vector $$R = (R_1, \dots, R_n)$$, so that the next-period incomes are $$(w_1 + Y_1, \dots, w_m + Y_m)$$ and the next-period prices are $$(p_1 + R_1, \dots, p_n + R_n)$$. We can interpret the randomness of $$Y$$ and $$R$$ as either due to uncertain exogenous influence (frequentist) or our belief of the possible impact (Bayesian).

Assume every person has the same consumer utility function $$u: \mathbb{R}^n_+ \to \mathbb{R}$$, where $$\mathbb{R}^n_+$$ is the commodity space. Also assume the utility function would not change in the next period. Everyone solves the utility maximization problem

$$\text{Maximize} \quad u(x) \quad \text{subjected to} \quad p \cdot x \leq w, x \geq 0$$

and obtains the same indirect utility function

$$v(p, w) = \max\{ u(x) : p \cdot x \leq w, x \geq 0 \} \quad \text{for} \quad p \in \mathbb{R}^n_{++}, w > 0$$

The expected utility approach means that we evaluate the policy impact by the change in total expected utility:

$$\text{Impact} = \sum_{i = 1}^m \mathbb{E}[v(p + R, w_i + Y_i)] - \sum_{i = 1}^m v(p, w_i)$$

The above formula is computationally intractable, so let's approximate it by Taylor's series. For each $$i$$, we have

$$v(p + R, w_i + Y_i) - v(p, w_i) \approx \nabla_p v(p, w_i) \cdot R + \frac{\partial v}{\partial w}(p, w_i) Y_i + \frac{1}{2} \frac{\partial^2 v}{\partial w^2}(p, w_i) Y_i^2$$

We only keep terms up to $$O(R)$$ and $$O(Y_i^2)$$ and drop the higher-order terms. We keep $$O(Y_i^2)$$ because we want to capture the risk aversion with respect to income: poor people have larger gains in utility than rich people when their incomes are increased by the same amount. On the other hand, the second-order effect of price change is not as important, so we drop it.

Taking expectation and summing over all $$i$$'s, we get

$$\text{Impact} \approx \sum_{i=1}^m \nabla_p v(p, w_i) \cdot \mathbb{E}[R] + \sum_{i=1}^m \frac{\partial v}{\partial w}(p, w_i) \mathbb{E}[Y_i] + \frac{1}{2} \sum_{i=1}^m \frac{\partial^2 v}{\partial w^2}(p, w_i) \mathbb{E}[Y_i^2]$$

If we assume $$Y_1, \dots, Y_m$$ are identically distributed as $$Y$$, then we have

$$\text{Impact} \approx \mathbb{E}[R] \cdot \sum_{i=1}^m \nabla_p v(p, w_i) + \mathbb{E}[Y] \sum_{i=1}^m \frac{\partial v}{\partial w}(p, w_i) + \frac{1}{2} \mathbb{E}[Y^2] \sum_{i=1}^m \frac{\partial^2 v}{\partial w^2}(p, w_i)$$