Marginal benefit AP free response question | APⓇ Microeconomics | Khan Academy

We're told Martha has a fixed budget of twenty dollars, and she spends it all on two goods: good X and good Y. The price of X is four dollars per unit, and the price of Y is two dollars per unit. The table below shows a total benefit measured in dollars Martha receives from the consumption of each good.

All right, we see that here this is total benefit, not marginal benefit. What is Martha's marginal benefit of the fifth unit of good X? So, just to answer this question, let's see; she has a total benefit of forty dollars when she has four of X, and then when she goes to the fifth, her total benefit is an incremental one dollar. So, she goes from forty dollars to forty-one dollars. The marginal benefit of that fifth one is that extra dollar. So, we added a dollar of total benefit, so that's the marginal benefit. So, it is one dollar.

Calculate the total consumer surplus if Martha consumes five units of X. Show your work. Well, the consumer surplus is going to be the benefit minus the cost, which is going to be equal to... Well, when she has five units of X, her total benefit is 41. So I'll write that here: 41. And then what's her cost of 5 units of X? Well, X costs four dollars per unit, so five times four is twenty dollars. Her cost is going to be twenty dollars. So, her consumer surplus is going to be equal to twenty-one dollars.

Martha is currently consuming four units of X and two units of Y. Use marginal analysis to explain why this combination is not optimal for Martha. So pause this video and see if you can answer that. All right, well, let's just think about what the marginal benefit is from every incremental unit of X or Y, and then let's think about the marginal benefit per dollar.

So, I'm going to make an extra column here. Let's call this marginal benefit of X, and then let's call this marginal benefit of Y. I'm doing it over this table just for the sake of space. And so the marginal benefit of this first one is going to be 16. We went from 0 to 16. The second one, we go from 16 to 28, so it's 12, and then to go from 28 to 36 is eight. To go from 36 to 40 is four extra dollars of benefit, and to go from 40 to 41, we already talked about that; that's one dollar of marginal benefit.

If we talk about why... Well, the first unit you get 10 of benefit; the next one is the total benefit increase by eight dollars, so that's the marginal benefit of the next unit. The next one to go from 18 to 24 is 6, to go from 24 to 28 is four more, and then 28 to 30 is two more.

Then we could use this information to think about marginal benefit of X per price of X, and we could also think about marginal benefit of Y per price of Y. And so, let's see... We're going to start with the first units. For this first unit, if you take the marginal benefit of X divided by the cost of unit of X, 16 divided by four dollars is going to get four. 12 divided by four is three. Eight divided by 4 is 2. Four divided by 4 is 1, and 1 divided by 4 is 0.25.

For Y, the cost of Y is 2 per unit, so the marginal benefit per dollar of this first unit right over here is ten dollars divided by two dollars, which is five; eight divided by two is four, six divided by two is three, four divided by two is two, and then 2 divided by 2 is 1.

And so, let's just think about what would be an optimal combination for Martha. When she's thinking about spending her first few dollars, she'll get a higher marginal benefit per dollar by going with Y, so she's going to start here. After that, her second unit of Y has the same marginal benefit per dollar as her first unit of X, so she's indifferent. She would do these in some order, so maybe she could do this one and then move on to that one or though it could go in the other order.

So far, she's only spent... Let's see: two dollars; two dollars is four dollars plus another four dollars. She's only spent eight dollars and has a budget of 20. After that, her next incremental unit of either X or Y, the marginal benefit per dollar is the same.

Just thinking about whether this could be an optimal combination, she's already bought two units of Y. Let's just give the benefit of the doubt here; let's say that she goes for the X, so she buys this one here. But once she has two units of both and she hasn't spent all of her money, she's spent eight dollars here plus another four dollars. She has eight dollars to spend.

The next rational thing for her to do, her marginal benefit per dollar for that third incremental Y is higher than the third incremental X. So it would be optimal for her to buy a third Y, but here we see that she only has two units of Y. That's why we know it's not an optimal combination.

So we could say once she has two of each, the marginal benefit of Y per price of Y is greater than the marginal benefit of X per price of X for the third unit. So, she will buy more than two Y's. Let me write... Let me scroll down a little bit: buy more than two Y's.

All right, the next they say is what is Martha's optimal combination of goods X and Y? Well, we've already started that conversation up here. She would buy this Y, and so far she spent eight dollars plus six dollars, so she has another six dollars to spend.

Now, her next incremental unit, she's indifferent, so maybe she buys another Y: eight dollars plus eight dollars, this is sixteen dollars, so she has four dollars left. Then she would buy this, and she has spent all of her money: she's twelve dollars here, eight dollars here, so she would buy three X's and four Y's. So, I would say three X's and four Y's.

All right, part E: indicate whether each of the following will cause the optimal quantity of good Y to increase, decrease, or stay the same. So look at these and pause this video and see if you can answer those.

So, the price of good Y doubles. Well, if the price of good Y doubles, then the marginal benefit per price of Y will go down, so she will buy less of Y. So, it would decrease; she would get less bang for her buck on Y, so she would buy less of Y.

Martha's income falls to ten dollars with no price changes. Well, if we go through the exercise we just did, her budget would run out much faster, and so she would definitely decrease the number of Y's she would buy. So, the Y's would decrease.

Martha's income doubles and the price of both goods doubles. Well, in that case, things would stay the same. Stay the same, because once again she could buy that exact same combination; it would just cost twice as much, but then her budget is now twice as much. So, things would stay the same; she would buy the same quantities of both X's and Y's, and they're just asking about Y's. And we're done.