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PROFESSOR: Hi, and welcome back to the 14.01 problem solving videos. Today, we're going to do Fall 2010, Problem Set 3, Problem Number 5. And we're going to go ahead and we're going to work through parts A, B, C, D, and E, and then we're going to finish up parts F and G.

Problem 5 says that Xiao spends all her income on statistical softwares and clothes. Her preferences can be represented by the utility function where her utility equals 4 times the natural log of S plus 6 times the natural log of C, where S is software and C is clothes.

Part A asks us to compute the marginal rate of substitution of software for clothes, asks us if the MRS is increasing or decreasing in S, and also asks us how we interpret the MRS.

So before we start with this, we should really think about, conceptually, what the marginal rate of substitution of software for clothes looks like on our graph. So on this graph, I have clothes on the y-axis — the quantity of clothes — and the quantity of software on the x-axis here. Looking at this graph, this line that I've drawn is one utility level. So at this place, she might have a utility equal to 1. So she's indifferent on this indifference curve between being at point here or at a point here.

And what the marginal rate of substitution is really asking us, it's asking us how much clothing is she willing to give up to get one more unit of software? So she's going to have to give up a certain amount of clothes to get one more unit of software. And the marginal rate of substitution tells us exactly how much clothing she's willing to give up.

To calculate this algebraically, all we're going to do is we're going to take the marginal utility of software and divide it by the marginal utility of clothes.

So we're going to take the derivative with respect to software and the derivative with respect to clothing and divide. When we do this, we find that the MRS is going to be equal to 4 over S, which is our marginal utility of software, all over 6 divided by C, which is our marginal utility of clothes. Solving through, we find that our MRS is 4C over 6S.

Now, we have to think about, conceptually, what happens when software increases? When we have S increase, since it's in the denominator, we're also going to have the MRS decrease.

So what this means is as software is increasing, or as she has more software, she's going to be willing to give up fewer clothing, or less clothing, to get another unit of software.

So looking at our graph, when she's at this point, she's more willing to give up clothing to get more software. But when she has more software down here, she's less willing to give up the clothing.

Let's go ahead and move on to Part B. Part B, find Xiao's demand functions for software and clothes — so we're going to call those QS and QC — in terms of the price of software PS, the price of clothes PC, and Xiao's income.

Now, before we move on with this, what we want to do is we want to solve for one of the variables C or S in terms of the prices and the other variable.

So to do this, we're going to set the MRS equal to the price of the software over the price of the clothes.

From here, we can solve through for C, and we find that C is going to be equal to 3/2 times PS over PC times S.

Now, since we have two variables — we have a variable for clothes and a variable for software — we're going to have to introduce another constraint into this problem.

Problem 5 says that Xiao spends all her income on statistical softwares and clothes. Her preferences can be represented by the utility function where her utility equals 4 times the natural log of S plus 6 times the natural log of C, where S is software and C is clothes.

Part A asks us to compute the marginal rate of substitution of software for clothes, asks us if the MRS is increasing or decreasing in S, and also asks us how we interpret the MRS.

So before we start with this, we should really think about, conceptually, what the marginal rate of substitution of software for clothes looks like on our graph. So on this graph, I have clothes on the y-axis — the quantity of clothes — and the quantity of software on the x-axis here. Looking at this graph, this line that I've drawn is one utility level. So at this place, she might have a utility equal to 1. So she's indifferent on this indifference curve between being at point here or at a point here.

And what the marginal rate of substitution is really asking us, it's asking us how much clothing is she willing to give up to get one more unit of software? So she's going to have to give up a certain amount of clothes to get one more unit of software. And the marginal rate of substitution tells us exactly how much clothing she's willing to give up.

To calculate this algebraically, all we're going to do is we're going to take the marginal utility of software and divide it by the marginal utility of clothes.

So we're going to take the derivative with respect to software and the derivative with respect to clothing and divide. When we do this, we find that the MRS is going to be equal to 4 over S, which is our marginal utility of software, all over 6 divided by C, which is our marginal utility of clothes. Solving through, we find that our MRS is 4C over 6S.

Now, we have to think about, conceptually, what happens when software increases? When we have S increase, since it's in the denominator, we're also going to have the MRS decrease.

So what this means is as software is increasing, or as she has more software, she's going to be willing to give up fewer clothing, or less clothing, to get another unit of software.

So looking at our graph, when she's at this point, she's more willing to give up clothing to get more software. But when she has more software down here, she's less willing to give up the clothing.

Let's go ahead and move on to Part B. Part B, find Xiao's demand functions for software and clothes — so we're going to call those QS and QC — in terms of the price of software PS, the price of clothes PC, and Xiao's income.

Now, before we move on with this, what we want to do is we want to solve for one of the variables C or S in terms of the prices and the other variable.

So to do this, we're going to set the MRS equal to the price of the software over the price of the clothes.

From here, we can solve through for C, and we find that C is going to be equal to 3/2 times PS over PC times S.

Now, since we have two variables — we have a variable for clothes and a variable for software — we're going to have to introduce another constraint into this problem.

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