Microeconomics: Maximizing total utility

[s3a]

Homework Statement

When the income is greater than or less than the money required to buy the two goods at equilibrium (meaning when their marginal utility to price ratios are equal), how do I know which direction and how far from the equilibrium point to look at without evaluating every single combination's total utility, manually?

Homework Equations

N/A

The Attempt at a Solution

If I'm just trying to maximize total utility per dollar and I have no constrained income, I know to get the equilibrium quantity. That is to say, I know to buy the quantities of each good at which their marginal-utility-to-dollar ratios are equal.

What I was initially trying to do, when the situation is not that of the previous paragraph, was to just purchase the one more good which has the highest marginal utility per dollar ratio until I no longer can purchase the good with the better marginal-utility-to-dollar ratio and, then, if I can afford the other good, I get it (unless its marginal utility is 0), but that doesn't always work.

Please explain to me the/an algorithm to follow, which works for all cases, in an easy-to-understand manner since I am having a lot of difficulty with this!

Any input would be GREATLY appreciated!

 

[nate808]

Thank you for your question. I understand your desire to find a systematic approach to solving this problem. The key concept to keep in mind is the principle of diminishing marginal utility. This means that as we consume more of a good, the satisfaction or utility we derive from it decreases. So, in order to maximize total utility per dollar, we want to allocate our income in a way that balances the marginal utility of each good with its price.

To do this, we can use the following steps:

1. Calculate the marginal utility per dollar of each good by dividing its marginal utility by its price.

2. Start by purchasing the good with the highest marginal utility per dollar ratio. Keep purchasing this good until its marginal utility per dollar decreases to the point where it is equal to the marginal utility per dollar of the other good.

3. At this point, you have reached the equilibrium point where the marginal utility per dollar of both goods is equal. This is the most efficient allocation of your income.

4. If your income allows for it, you can then purchase additional units of the other good, as long as its marginal utility per dollar is still positive.

I hope this helps to clarify the process for you. Keep in mind that this is a general approach and may not work for every specific scenario, but it is a good starting point. If you have any further questions, please don't hesitate to ask. Good luck with your studies!