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Today, s-star.org would like to introduce to you Consumer Utility, Marginal Utility, and Marginal Rate of Substitution. Following along are instructions in the video below:
“This video. We re gonna look at a number of utility functions derive marginal utility utility from those functions and then calculate the marginal rate of substitution. Let s start. A utility function for consumer.
Given by x. Squared. Times. Y cubed.
Let s get the martian utility. For good x. Which is just going to be the change in utility from consuming one more unit of good x. It s given by the partial derivative of the utility function with respect to good acts.
So taking a partial derivative of the above equation you can get 2x bring down that 2 and front 2 1. Just leaves us axe here. I ll multiply by y raised to the third power. The martian utility of good y and over here.
It s going to be the partial derivative of the utility function with respect to good y here. We get 3x squared times. Y squared. Now from this.
Let s calculate the marginal rate of substitution just a martian utility of good x divided by the marginal utility of good y. We can simplify this slightly okay and ups and the three so that s our marginal rate of substitution. Why do we even care about the marginal rate of substitution well it is the absolute value of the slope of the indifference curve okay the marginal rate of substitution is the absolute value of the slope of the indifference curve and when a consumer is maximizing utility. The slope of the indifference curve will equal the slope of the budget constraint.
So that s one reason. Why we care about the marshall rate of substitution..
The definition of the marginal rate of substitution besides of being the absolute value of the slope of the indifference curve. It tells us how many units of good y. The consumer is willing to give up for one more unit of good x. Holding.
Utility. Constant. Let s do another. Example.
Here. Utility. Equals. X.
Raised to the 1 3. Power. And y. Raised.
To the 2 3. Power. Good. Accent.
Good. Y. Okay. X.
Units of good. X. Y. Represents.
Units. Of. Good. Y phi.
Did state that earlier. Margined utility of good x. In this case. Looks like this bring down the 1 3.
In front one third minus one leaves us minus two thirds okay and the marginal utility of good y bring down the two thirds in front two thirds minus one leaves us. Y raise to the minus 1 3. Power. The marginal rate of substitution.
Large utility of good x divided by the marginal utility. Good y. Using rules of exponents here. Bringing bringing down this x.
To the minus 2 3. Power. Into the denominator and then moving this y to the minus 1 3. In the denominator into the numerator.
I m gonna be left with this. Result..
Y. Over 2 x. Ok. This 1 3.
Divided by 2 3. Just leaves us with 1 2. So that s where the twos coming from ok. Let s do another.
Example. Here s a kind of interesting. Utility. Function.
Utility. Equals minus x. Raised to the 1 power. Minus.
Y. Raised to the power of minus. 1. Marsh.
Utility of good x. In this example looks something like this bring down the minus 1 in front. That s just gonna get minus minus. So just 1 and then minus 1 minus.
1 leaves us with minus 2. And doing a similar thing for good why bringing down the minus one in front of the wide leaves us just with plus y..
And then we re gonna go one. One y will then be raised to the minus 2. Power marginal rate of substitution. We get y squared divided by x.
Squared. Just using your rules of exponents here alright let s do another example. Let s use a utility function with natural logs. The margin utility of good x here is going to be 2 divided by x.
The marsh utility of good y just going to simply equal. 1 over y. The marginal rate of substitution large utility godox divided by the marginal utility of good y will simplify down to this expression okay just simplifying it by taking the denominator and multiplying. The denominator numerator by the reciprocal.
We ll have 1 over y or just y. And let s do one. Other example. Here s a perfect substitutes utility.
Function marja utility of good acts is the derivative of. 05. X plus. 2y which is 05.
The marsh utility a good y just going to equal. 2 so the marginal rate of substitution is 05. Divided by 2 and this is just 1 4. All right so that s a look at utility martian utility and the ” .
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