It is exactly 24 hours before Jennifer's exams and she is going to spend the entire time studying. She has an Economics exam followed directly by a Philosophy exam, with no time to study in between. Jennifer's utility is represented by

U(e, b) = e² b

where e is the score on the Economics exam and bis the score on the Philosophy exam. Jennifer wants to be an economist, so she places more weight on her Economics exam score - it has a higher exponent in her utility function, 2 instead of 1.

Although Jennifer cares more about Economics, she is better at Philosophy. Specifically, for each hour spent studying Philosophy, she will increase her score by 5 points, but her Economics score will only increase by 3 points for every hour spent studying Economics. Before studying, she starts at a score of O for each test.

(a) We often write budget equations in the form

Px^x+ PY^Y M

with Px, py indicating prices of goods X and Y; x, y indicating quantities of those goods; and M indicating the total budget to be spent. Let's write a budget equation for this problem.

Say Jennifer will spend h_E hours studying economics and h_B hours studying philosophy out of her 24 total hours.

1. Write out the budget equation describing the possible values of h_Eand h_Bthat Jennifer can choose.

2. Explain what it means that the "prices" of h_Eand h_B - the coefficients in front of them - are both equal to 1 in this equation.

(b) We can also write Jennifer's budget equation in terms of test scores rather than hours.

1. Rewrite your budget equation from the previous question in terms of e and b (instead of h_E and h_B ) by plugging in the relationships e = 3h_E and b = 5h_B .

Once again, we can think of the coefficients in front of e and b as representing "prices", PE and PB Reinterpret each price in economic terms: it "costs" Jennfier [some amount] of [something] in order to get [some other amount] of [something else].

( c) Given some number of points e on Economics and b on Biology, calculate Jennifer's marginal utility from additional points on each exam. That is, calculate MU_E= (e, b) and MUB = (e, b), expressing both as a general function of e and b.

Next, put these two formulas together to calculate the Marginal Rate of Substitution of Economics points for Philosphy points (e for b, i.e., treating e as the x variable and b as the y variable).

What would Jennifer's MRS be if she were scoring e = 30 and b = 30? What if she were scoring e = 30 and b = 10? Give meaningful interpretations of both of these numbers.

(d) Put together the prices from part (b) and the Marginal Utility numbers from part (c) into a bang-for-the-buck equation of the form Mux px = Muy py. (Simplify the PX PY equation as much as you can.)

In words, what would it mean if Jennifer were choosing an outcome withe and b such that the marginal utility divided by the "price" of Economics points were higher than that for Biology? What mistake would she be making?

(e) Now combine the budget equation of part (b) with the bang-for-the-buck equation of part ( d) to figure out what test scores Jennifer will get based on her preferred allocation of studying. That is, what e and b will she choose?

(f) Based on the answer to the previous part, how many hours will Daryl spend studying

economics? How many hours will she study biology?