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PROFESSOR: Today, we're going to be working on fall 2010, problem set number 2, and we're going to be doing problem number 4.

I'm going to start off by reading the problem. There's a lot of information here, so I've written up some of the key points on the board for you guys already. It is exactly 24 hours before Lauren's physics final. She has an economics final directly after her physics final, and has no time to study in between. Lauren wants to be a physicist, so she places more weight on her physics test score. Her utility function is given right here. Where p is the score of her physics final and e is the score of her economics final.

Although, she cares more about physics, she is better at economics. For each hour spent studying economics, she will increase her score by three points. But her physics score will only increase by two points for every hour spent studying physics. Studying zero hours results in a score of zero on both subjects.

Although natural log of zero is not defined, assume her utility for a score of zero is negative infinity. Now, we're going to go ahead and we're going to work through parts A, B, and C, A, B, C, and D and then we'll do part E, which is a new scenario afterwards.

Part A, we're going to find the constraints that Lauren faces in her test score maximization problem. And part B, we're going to find how many hours does Lauren optimally spend studying physics, how many hours does she spent studying economics. And hours are divisible, so we don't need whole number solutions for the hours spent studying physics and the hours spent studying economics.

So for part A, all we're looking for is we're looking for the constraints. And it sounds like the constraint that Lauren is really facing in this scenario is the amount of time that she has. It seems like Lauren's pretty intense about studying, so in the 24 hours before the test, she's not getting any sleep. She's going to spend all of this time studying. So we're going to have two hours variables to represent the hours she spends studying physics and the hours she spends studying economics.

And we know that the hours, when we add them together, they're going to have to be less than or equal to 24. Now, if this is our constraint, we really know that if she's trying to maximize her scores and if that's all she really cares about, she's not going to spend less than 24 hours studying. So we can actually just say that the sum of those two hours — her hours spent studying physics and her hours spent studying economics are going to be equal to 24. So that's the first constraint that Lauren is going to face.

The second constraint that she's going to face, and it's not necessarily an intuitive one — or it is actually really intuitive, but it's also a trivial one. It's just that she can't spend less than zero hours studying physics or studying economics. So we're going to add those constraints in as well. And the final constraints are actually production constraints. If the hours spent studying — these aren't actually what she's interested in. What she's interested in are the p and the e. That's how she's going to get her utilities — through the test scores.

So what we need is we need to find how she produces her physics score. If she gets two points for every hour she spent studying, her physics score is going to be p equals 2 times Hp — that's the production of her physics score.

I'm going to start off by reading the problem. There's a lot of information here, so I've written up some of the key points on the board for you guys already. It is exactly 24 hours before Lauren's physics final. She has an economics final directly after her physics final, and has no time to study in between. Lauren wants to be a physicist, so she places more weight on her physics test score. Her utility function is given right here. Where p is the score of her physics final and e is the score of her economics final.

Although, she cares more about physics, she is better at economics. For each hour spent studying economics, she will increase her score by three points. But her physics score will only increase by two points for every hour spent studying physics. Studying zero hours results in a score of zero on both subjects.

Although natural log of zero is not defined, assume her utility for a score of zero is negative infinity. Now, we're going to go ahead and we're going to work through parts A, B, and C, A, B, C, and D and then we'll do part E, which is a new scenario afterwards.

Part A, we're going to find the constraints that Lauren faces in her test score maximization problem. And part B, we're going to find how many hours does Lauren optimally spend studying physics, how many hours does she spent studying economics. And hours are divisible, so we don't need whole number solutions for the hours spent studying physics and the hours spent studying economics.

So for part A, all we're looking for is we're looking for the constraints. And it sounds like the constraint that Lauren is really facing in this scenario is the amount of time that she has. It seems like Lauren's pretty intense about studying, so in the 24 hours before the test, she's not getting any sleep. She's going to spend all of this time studying. So we're going to have two hours variables to represent the hours she spends studying physics and the hours she spends studying economics.

And we know that the hours, when we add them together, they're going to have to be less than or equal to 24. Now, if this is our constraint, we really know that if she's trying to maximize her scores and if that's all she really cares about, she's not going to spend less than 24 hours studying. So we can actually just say that the sum of those two hours — her hours spent studying physics and her hours spent studying economics are going to be equal to 24. So that's the first constraint that Lauren is going to face.

The second constraint that she's going to face, and it's not necessarily an intuitive one — or it is actually really intuitive, but it's also a trivial one. It's just that she can't spend less than zero hours studying physics or studying economics. So we're going to add those constraints in as well. And the final constraints are actually production constraints. If the hours spent studying — these aren't actually what she's interested in. What she's interested in are the p and the e. That's how she's going to get her utilities — through the test scores.

So what we need is we need to find how she produces her physics score. If she gets two points for every hour she spent studying, her physics score is going to be p equals 2 times Hp — that's the production of her physics score.

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