Unit 10 Probability 102 Homework ((EXCLUSIVE))

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Prerequisites: PHYS 4C. This course introduces computational techniques in Physics. Basic programming in Python will be presented, with focus on designing computer simulations and applying them to solve problems in statistical physics, quantum mechanics and modeling of complex systems. Topics in probability theory and statistics will be discussed and a modern introduction to artificial intelligence will be presented. (3 lecture hours).

The term wealth is also sometimes used in a broader sense to include immaterial aspects such as your health, skills, and ability to earn an income (your human capital). But we will use the narrower definition of material wealth in this unit.

Since it is measured over a period of time (such weekly or yearly), it is a flow variable. Wealth is a stock variable, meaning that it has no time dimension. At any moment of time it is just there. In this unit we only consider after-tax income, also known as disposable income.

Here you will see that the same feasible set and indifference curve analysis applies to choosing between having something now, and having something later. In earlier units we saw that giving up free time is a way of getting more goods, or grades, or grain. Now we see that giving up some goods to be enjoyed now will sometimes allow us to have more goods later. The opportunity cost of having more goods now is having fewer goods later.

The fact that Julia can borrow means that she does not have to consume only in the later period. She can borrow now and choose any combination on her feasible frontier. But the more she consumes now, the less she can consume later. With an interest rate of r = 10%, the opportunity cost of spending one dollar now is that Julia will have to spend 1.10 = 1 + r dollars less later.

One plus the interest rate (1 + r) is the marginal rate of transformation of goods from the future to the present, because to have one unit of the good now you have to give up 1 + r goods in the future. This is the same concept as the marginal rate of transformation of goods, grain, or grades into free time that you encountered in Units 3 and 5.

Given the opportunities for bringing forward consumption shown by the feasible set, what will Julia choose to do? How much consumption she will bring forward depends on how impatient she is. She could be impatient for two reasons:

More generally, the value to the individual of an additional unit of consumption in a given period declines the more that is consumed. This is called diminishing marginal returns to consumption. You have already encountered something similar in Unit 3, in which Alexei experienced diminishing marginal returns to free time. Holding his grade constant, the more free time he had, the less each additional unit was worth to him, relative to how important the grade would be.

To see what pure impatience means, we compare two points on the same indifference curve in Figure 10.3b. At point A she has $50 now and $50 later. We ask how much extra consumption she would need to have later in order to compensate her for losing $1 now. Point B on the same indifference curve gives us the answer. If she had only $49 now, she would need $51.50 later in order to stay on the same indifference curve and be equally happy. So she needed $1.50 later to compensate for losing $1 now. Julia has pure impatience because rather than preferring to perfectly smooth her consumption, she places more value on an additional unit of consumption today than in the future.

The slope of the indifference curve of 1.5 (in absolute value) at point A in Figure 10.3b means that she values an extra unit of consumption now 1.5 times as much as an extra unit of consumption later.

A better plan, if Marco could find a trustworthy borrower, would be to lend the money. If he did this and could be assured of repayment of $(1 + r) for every $1 lent, then he could have feasible consumption of 100 × (1 + r) later, or any of the combinations along his new feasible consumption line. The light line in Figure 10.6 shows the feasible frontier when Marco lends at 10%. As you can see from the figure, compared to storage or putting the money under his mattress, his feasible set is now expanded by the opportunity to lend money at interest. Marco is able to reach a higher indifference curve.

Long before there were the employers, employees, and the unemployed that we studied in the previous unit, there were lenders and borrowers. Some of the first written records of any kind were records of debts. Differences in income between those who lend (people like Marco) and those who borrow (people like Julia) remain an important source of economic inequality today.

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In contrast to prevalence, incidence is a measure of the occurrence of new cases of disease (or some other outcome) during a span of time. There are two related measures that are used in this regard: incidence proportion (cumulative incidence) and incidence rate. A useful way to think about cumulative incidence (incidence proportion) is that it is the probability of developing disease over a stated period of time; as such, it is an estimate of risk. Ken Rothman uses the example of a newspaper article that states that women who are 60 years of age have a 2% risk of dying from cardiovascular disease. As written this statement is impossible to interpret, because it doesn't specify a time period. In order to interpret risk it is necessary to know the length of time that applies. A 2% risk has a very different meaning if it is over the next 12 months vs. the next 10 years. Therefore, the incidence proportion (cumulative incidence) must specify a time period. For example, the incidence proportion of neonatal mortality is the number of deaths divided by the number of births over the first 30 days after birth.

The concept of risk is fairly intuitive - if a group of disease-free people were followed for a period of time, one could determine the proportion of people who developed the disease at some point during the observation period in order to arrive at an estimate of the probability of developing that disease, i.e. the risk. However appealing this is for its simplicity, there are some drawbacks to this approach to assessing the occurrence of health outcomes, because an accurate assessment of probability relies on observing all subjects for the entire observation period. This is particularly a problem when assessing long term risk.

Remember that a rate almost always contains a dimension of time. Therefore, the incidence rate is a measure of the number of new cases ("incidence") per unit of time ("rate"). Compare this to the cumulative incidence (incidence proportion), which measures the number of new cases per person in the population over a defined period of time. Because studies of incidence in epidemiology are conducted among groups of people as they move through time, the denominator is actually a combination of the number of people and the amount of time. This is expressed as person-time. The time units can be expressed in days, months, or years, but should be tied to the length of the study and aid interpretation of the results. The most frequently encountered expression is "person-years". The characteristics of cumulative incidence and incidence rate are illustrated in the examples below.

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