# Investment and probability

Being a mathematician, I am familiar with probability calculations, but I need to ask a question related to investments and probability, and how this is handled seen from an economics view.

Given a project which involves two tasks, task 1 and task 2. Developing task 1 cost X dollars, while developing task 2 takes Y dollars if developed at the same time as task 1 and Z dollars if developed at a later stage.

The probability that we actually need Task 2 i 70%

What is the the "probable cost" of implementing Task 1 and 2 at the same time. vs. Task 1 and then Task 2 with a certain probability. And what is this called in economics? Probable cost? Estimated cost??

The simplest thing is to calculate the expected cost in the two scenarios:

Scenario 1: Develop both at the same time $$E[cost]= X + Y$$

Scenario 2: Wait and see if task two is needed $$E[cost]= X + 0.7Z$$

However I would not say expected value calculation represents the "economics view". The general view rather depends on your preferences, often expressed as utility functions. It could be your utility is a function of expected value, but it could also not be the case. What is your risk tolerance? Some people seem risk-seeking. Are you ok with a a small possibility of ruin (bankruptcy)? Preferences can also depend on reference points and more...

For other simplistic approaches you could start with value at risk.

Regardless of period if we know the cost 100%, it is simply cost. If there is a degree of uncertainty on what the cost will be (such as in the future) we regard it as expected cost.

Regarding your question specifically about how the cost of a project can change in a future period, this falls under an area of study known as cost-benefit analysis. Your specific example is a topic within that domain known as irreversibility.

If we are uncertain if the project is needed(such as in the case of building a dam for a potential flood), this does not affect the expected cost. Instead it affects the expected benefits. Regardless of whether or not the dam is needed, we have still expended the funds. Our uncertainty is in fact, changes our expected benefit. It is important to note that no matter what side of the equation we handle this on (benefits or costs), the expected value will have the same result.

When making these kind of evaluations we calculate their expected value(expected benefits-expected cost) and then adjust for irreversibility.

• "If we are uncertain if the project is needed, this does not affect the expected cost."? Suppose you buy a used car for \\$1000 and you may need to spend the same amount to get it properly fixed it up. I would say the probability of needing the tune up affects the expected cost. Otherwise I am in full agreement with your answer. – Giskard Nov 8 '16 at 18:58
• I agree that example would fall under expected cost. What I mean by the necessity of a project is let's say in the future you don't even need a car, or it becomes illegal to drive cars. In that case, I would consider the uncertainty to be on the expected benefit not the expected cost. It's all the same however you slice it though since it piles into expected value the same way. – Lee Sin Nov 8 '16 at 19:02
• I see. That makes sense, though perhaps you could edit your answer to make this more clear. – Giskard Nov 8 '16 at 19:27

If I understand you correctly, there are two possible courses of action:

1. Develop only Task 1 now, and proceed with Task 2 if needed.
2. Develop both Tasks concurrently.

Each course of action has its own expected cost. I would have no problem calling this expected cost.