# Economics: find when utility maximized

• 939
Then, I replaced x with that, giving me a y. Then, I plugged in the y to the budget constraint, giving me 80 = 7y + 2y. Then I solved for y, giving me a y value. Then I plugged the y value back into x.
939

## Homework Statement

I must "maximize utility". This is done when the slope of -px/py = -MUx/MUy.

MUx = (1/4x^-2/3)(y^2/5)
MUy = (2/4x^1/3)(y^-1/5)
Budget line: 80 = 7x + 2y

## Homework Equations

MUx = (1/4x^-2/3)(y^2/5)
MUy = (2/4x^1/3)(y^-1/5)
Budget line: 80 = 7x + 2y

## The Attempt at a Solution

-px/py = -7/2

Thus, I must find when MUx/MUy = -7/2. I don't need the actual answer, just wondering how I would do this?

939 said:

## Homework Statement

I must "maximize utility". This is done when the slope of -px/py = -MUx/MUy.

MUx = (1/4x^-2/3)(y^2/5)
MUy = (2/4x^1/3)(y^-1/5)
Budget line: 80 = 7x + 2y

## Homework Equations

MUx = (1/4x^-2/3)(y^2/5)
MUy = (2/4x^1/3)(y^-1/5)
Budget line: 80 = 7x + 2y

## The Attempt at a Solution

-px/py = -7/2

Thus, I must find when MUx/MUy = -7/2. I don't need the actual answer, just wondering how I would do this?

Do you mean
$$MU_x = \frac{1}{4} x^{-2/3} y^{2/5},$$
or do you mean
$$MU_x = \frac{1}{4 x^{-2/3}} y^{2/5}?$$ If you mean the former, use parentheses, like this: MUx = (1/4) x^(-2/3) y^(2/5), but if you mean the latter, use parenthese also, but in different places, like this: MUx = 1/(4 x^(-2/3)) y^(2/5), etc.

I will assume you want MUx = (1/4) x^(-2/3) y^(2/5) and MUy = (2/4) x^(1/3) y^(-1/5). So, just compute MUx/MUy using elementary algebra rules. Equating that to px/py gives one equation connecting x and y. The budget constraint gives another equation.

If the expressions you give above are correct, the final set of 2 equations in 2 unknowns does not gave any simple "closed-form" solution; it must be tackled numerically.

939 said:

## Homework Statement

I must "maximize utility". This is done when the slope of -px/py = -MUx/MUy.

MUx = (1/4x^-2/3)(y^2/5)
MUy = (2/4x^1/3)(y^-1/5)
Budget line: 80 = 7x + 2y

## Homework Equations

MUx = (1/4x^-2/3)(y^2/5)
MUy = (2/4x^1/3)(y^-1/5)
Budget line: 80 = 7x + 2y

## The Attempt at a Solution

-px/py = -7/2

Thus, I must find when MUx/MUy = -7/2. I don't need the actual answer, just wondering how I would do this?

What do MUx and MUy mean? Each one is a function of both x and y, so what is the purpose of x in MUx and the purpose of y in MUy?

I see where the -7/2 number comes from, which is solving the budget line equation for y (i.e., writing the line equation in slope-intercept form):
7x + 2y = 80 ##\Rightarrow## y = (-7/2)x + 80.

Is all that is asked for as simple as dividing MUx by MUy?

Ray Vickson said:
Do you mean
$$MU_x = \frac{1}{4} x^{-2/3} y^{2/5},$$
or do you mean
$$MU_x = \frac{1}{4 x^{-2/3}} y^{2/5}?$$ If you mean the former, use parentheses, like this: MUx = (1/4) x^(-2/3) y^(2/5), but if you mean the latter, use parenthese also, but in different places, like this: MUx = 1/(4 x^(-2/3)) y^(2/5), etc.

I will assume you want MUx = (1/4) x^(-2/3) y^(2/5) and MUy = (2/4) x^(1/3) y^(-1/5). So, just compute MUx/MUy using elementary algebra rules. Equating that to px/py gives one equation connecting x and y. The budget constraint gives another equation.

If the expressions you give above are correct, the final set of 2 equations in 2 unknowns does not gave any simple "closed-form" solution; it must be tackled numerically.

Thanks, and I wanted the first one.

Mark44 said:
What do MUx and MUy mean? Each one is a function of both x and y, so what is the purpose of x in MUx and the purpose of y in MUy?

They are different combinations of goods you can get within your budget constraint, i.e. in this case the budget constraint is 80, so you could buy so much of good x and so much of good y to remain in it, buy more of x, must buy less y, etc.

...

I'm just wondering if it would be possible to solve this by mapping out all the budget constraints, and then substituting the x and y values in for MUx/MUy and see which one = -7/2?

939 said:
Thanks, and I wanted the first one.

They are different combinations of goods you can get within your budget constraint, i.e. in this case the budget constraint is 80, so you could buy so much of good x and so much of good y to remain in it, buy more of x, must buy less y, etc.

...

I'm just wondering if it would be possible to solve this by mapping out all the budget constraints, and then substituting the x and y values in for MUx/MUy and see which one = -7/2?

Yes, but that is doing it the hard way. However, go ahead and try it for yourself. I already suggested another method; use whichever one you like best.

Ray Vickson said:
I already suggested another method; use whichever one you like best.

I know, but I'm really bad at algebra :(... Can you give some advice here?

7/2 = ((1/4) x^(-2/3) y^(2/5))/((2/4) x^(1/3) y^(-1/5))
7/2 = 1/2x^-1 y^3/5
...

939 said:
I know, but I'm really bad at algebra :(... Can you give some advice here?

7/2 = ((1/4) x^(-2/3) y^(2/5))/((2/4) x^(1/3) y^(-1/5))
7/2 = 1/2x^-1 y^3/5
...

So, you are saying
$$\frac{7}{2} = \frac{1}{2} \frac{y^{3/5}}{x}.$$
What is stopping you from solving for x in terms of y? After doing that, you can plug in that expression for x into the budget constraint, and you will have a single equation containing y alone.

Ray Vickson said:
So, you are saying
$$\frac{7}{2} = \frac{1}{2} \frac{y^{3/5}}{x}.$$
What is stopping you from solving for x in terms of y? After doing that, you can plug in that expression for x into the budget constraint, and you will have a single equation containing y alone.

Thanks, got it. I got it by solving x in terms of y.

## 1. What is utility maximization in economics?

Utility maximization is a concept in economics that refers to the idea that individuals or businesses will make choices that will bring them the most satisfaction or benefit. This is often measured in terms of utility, which can be thought of as the level of happiness or satisfaction derived from consuming a good or service.

## 2. How is utility maximization achieved?

There are several theories and models that explain how utility maximization is achieved. One of the most well-known is the theory of consumer behavior, which states that individuals will maximize their utility by allocating their limited income towards the goods and services that provide them with the most satisfaction.

## 3. What factors influence utility maximization?

There are a variety of factors that can influence utility maximization, such as income, preferences, and prices. For example, a person with a higher income may be able to afford more goods and services, thus increasing their overall utility. Preferences, on the other hand, can vary among individuals and may affect their choices and level of satisfaction.

## 4. How is utility maximization related to demand?

Demand is closely related to utility maximization, as individuals will demand more of the goods and services that provide them with the most satisfaction. As a result, when utility is maximized, demand for a particular good or service will also be at its highest.

## 5. Can utility maximization be measured?

While utility itself cannot be measured directly, there are ways to indirectly measure the level of utility that an individual or business receives from consuming a good or service. One method is through surveys or experiments that ask individuals to rate their level of satisfaction or happiness with different goods and services. Economists can use this data to estimate the level of utility and make predictions about consumer behavior.

Replies
2
Views
823
Replies
9
Views
1K
Replies
6
Views
986
Replies
6
Views
956
Replies
1
Views
8K
Replies
2
Views
2K
Replies
2
Views
719
Replies
5
Views
1K
Replies
9
Views
2K
Replies
2
Views
791