this is part-1 of __that article__

## Abstract

Since the introduction of covid vaccine to the public in Israel, continuous debate on non-vaccination has arisen. While the main debate focuses on misconception and fake news that push some of the public to non-vaccination, the economical causes of vaccination and non-vaccination stay outside the room. some of the reason for that, may be the fact that governmental funding for the vaccine. In this article I will test the effect of social insurance and alike on vaccination rate across different cities in Israel.

## Theoretical framework

In the economic literature, a consumer is said to be rational. Thus, to allocate her consumption in such way she maximizes some subjective utility function which represent her preference. In the case of uncertainty, the consumer said to maximize the expected utility, in the statistical sense. Thus, to choose the option, that on average should give the user the maximal wellbeing with the greatest certainty. For analyzing and measuring uncertainty, a mathematical framework of probability theory will be used.

In our framework, the subjects need to choose whereases to vaccinate or not, while the cost of the vaccine as seeing by the consumer is zero, due to the government procurement of the vaccine and its social policy. Thus, the only remaining explanation of the subject’s decision to vaccinate or not is all the remaining socioeconomic factors. This setting allows as to estimate the effect of several different socioeconomic features on vaccination motivation, but firs, a theoretic explanation of rationality is needed.

since non-vaccination has negative effect on the effort to terminate the disease, the Israeli government decided to restrict some economic activity to non-vaxxers. This includes restriction on goods consumption.

Some of the goods can be restricted and some are not. I will denote these goods Cr and Cu respectively. The unrestricted goods are those who one can’t (or find it very hard) to live without. i.e., food. One can imagine that if the government will restrict the entrance to groceries base on vaccination status, it will be blamed in immoral action and lose its popularity. The goods that can be restricted are those who one can easily give up for (relatively). i.e., concerts, sports shows, and any activity that may cause the public to get infected.

To introduce this restriction into a rational model, we may see it as tax on the price of the restricted goods, which can be put only on some level of consumption. one may think of it as tax on the subject’s income. Whatever the most accurate way to introduce the restriction, the exact form is not important for our analysis. The only importance is that the non-vaxxers wellbeing from consumption has decrease, thus, the utility function has shift down.

Now our consumer needs to choose whether to vaccinate or not. In this kind of decision, we said the decision maker tries to maximize the expected utility.

Thus, Let’s the subject have utility function:

When the subjects stand in front of decision to vaccinate, they are considering the possible dangerous or the bad thigs that can happen due to the vaccine. we know that the utility of bad state of nature, denoted U2 is not greater than the utility in non-vaccination with world without covid-19.

Thus,

The expected utility when the consumer decides to vaccinate is,

And,

To put the equation into words, the participate in the vaccination project is a lottery, which the probability to get back into the original utility U0 is P, and (1-P) to get things worst toward U2.

If U2 is lay between U0 and U1, the subject evaluates that the worst state of nature with vaccination is not much worse from his current wellbeing, and thus decide to vaccinate. If so, the subjects revealed preference shows us that if one finds himself hesitate if to take the vaccine or not, he is probably not sure if U~ is bigger or smaller than U1.

Mathematically,

For estimating U~, one need to know the probability p, which is the probability for the vaccine to work with no harm.

A short algebra shows that if subjects preferred U~ over U1, thus,

their subjective p should be

Which means that if the probability for the good scenario is less then, the subjects decide to avoid vaccination.

Since the homogeneity of U, we can write the critical p as function of subject’s income:

Where k_i denote the difference between the different utility outcomes.

If the subjects know the real value of p for sure, as we assume, the difference between subjects’ decision is only due to Income.

The meaning is that poor consumers are more sensitive to the restriction, as their lower level of consumption indicate bigger marginal utility, thus, bigger incentive to participate in the vaccination project. Another explanation for in vaccination may be the subjective probability that consumers have duo to lack of scientific understanding. Or very small U2 relative to U1 and U0.

Our analysis shows that tax on non-vaccination will affect mostly on poor subjects, and less on rich one’s.

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