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MRS of the consumer at an arbitrary point

(10 points) Annie and David are painting their apartment. At the paint store, David says he prefers
Canary Yellow to Bumblebee Yellow, Lime Yellow, and Crayola Yellow. Annie finds new paint samples
and asks David to compare Canary Yellow to School Bus Yellow and to Sunrise Yellow. David prefers
Sunrise Yellow to Canary Yellow, and prefers School Bus Yellow to Canary Yellow. He also prefers
Sunrise Yellow to School Bus Yellow. The store is out of Sunrise Yellow, so they buy School Bus Yellow
and paint their apartment with it. David then insists that they go back, buy Lime Yellow, and repaint
the apartment.
True/False/Uncertain: David has rational preferences (as we define them).
Problem 1 courtesy of William Wheaton. Used with permission.
(20 points) In each of the following examples, a consumer purchases just two goods: x and y. Based on
the information in each of the following parts, sketch a plausible set of indifference curves (that is, draw
at least two curves on a set of labeled axes, and indicate the direction of higher utility). Also, write
down a utility function u(x, y) consistent with your graph. Note that although all these preferences
should be assumed to be complete and transitive (as required for utility representation), not all will be
monotone.
(a) (4 points) Jessica enjoys bagels x and coffee y, and consuming more of one makes consuming the
other more enjoyable.
(b) (4 points) Plamen loves mocha swirl ice cream x, but he hates mushrooms y.
(c) (4 points) Jennifer likes Cheerios x, and neither likes nor dislikes Frosted Flakes y.
(d) (4 points) Edward always buys three white tank tops x for every pair of jeans y.
(e) (4 points) Nancy likes both peanut butter x and jelly y, and always gets the same additional
satisfaction from an ounce of peanut butter as she does from two ounces of jelly.
(20 points) A consumer’s preferences are representable by the following utility function:
1
u(x, y) = x 2 + y.
(a) (10 points) Obtain the MRS of the consumer at an arbitrary point (¯x, y¯), where x >¯ 0 and y >¯ 0.
(b) (10 points) Suppose the price of the second good (y) is 1, and the price of the first good (x) is
denoted by p > 0. If the consumer’s income is m > 0, obtain the optimal consumption bundle of
the consumer (in terms of m and p). [Caution: make sure you cover cases in which m is relatively
low, as well as cases in which m is relatively high.]
(25 points) It is exactly 24 hours before Lauren’s physics final. She has an economics final directly after
the 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 by
u(p, e) = 0.6 ln(p) + 0.4 ln(e)
where p is the score on the physics final and e is the score on the 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 3 points, but her physics score will only increase by 2 points for every hour spent studying
physics. Studying zero hours results in a score of zero on both subjects (although ln(0) is not defined,
assume her utility for a score of zero is negative infinity).
(a) (5 points) What constraints does Lauren face in her test score maximization problem?
(b) (5 points) How many hours should Lauren optimally spend studying physics? How many hours
studying economics? (hours are divisible)
∗ (c) (5 points) What economics and physics test scores will she achieve (i.e. what are e and p∗)?
(d) (5 points) What utility level will she achieve?

Full Answer Section

Problem 2:

For each sub-part (a-e):

Identify the goods: Understand what x and y represent in each scenario.
Determine the nature of the preference: Based on the description, assess whether the consumer enjoys both goods, hates one, or has specific substitution patterns.
Sketch indifference curves: Draw two or more curves on a graph, considering the direction of higher utility (the further from the origin, the higher the utility).
Formulate a utility function: Based on the graph and understanding of the preference, write a function (u(x, y)) that reflects the consumer's satisfaction with different combinations of goods.

Problem 3:

(a) To find the Marginal Rate of Substitution (MRS) at point (x̄, ȳ), differentiate the given utility function (u(x, y)) partially with respect to x and then with respect to y. Set the equation to -ȳ/x̄ to solve for the MRS at that specific point.

(b) Analyze the consumer's budget constraint, considering income (m) and prices (p for x and 1 for y). Set up an optimization problem to maximize utility subject to the budget constraint. This may involve using Lagrange multipliers or substitution to solve for the optimal consumption bundle (x* and y*) in terms of m and p. Remember to consider both high and low income scenarios to see how budget limitations affect optimal choices.

Problem 4:

(a) Identify Lauren's constraints: limited time (24 hours) and the trade-off between studying physics and economics.

(b) Formulate an optimization problem: Maximize Lauren's utility (using the given function) subject to the time constraint and considering the different returns on investment (study hours versus score improvement) for each subject. This may involve setting up equations representing study hours allocated to each subject and relating them to final scores and utility.

(c) Solve for optimal study hours and final scores: Use the optimization problem you set up in (b) to find the values of study hours for physics and economics that maximize Lauren's utility. Plug these values back into the score equations to determine her predicted scores in both subjects.

(d) Calculate the final utility level: Substitute the optimal scores (p* and e*) into Lauren's utility function to determine the level of satisfaction she achieves with her chosen allocation of study time.

Remember, solving each problem effectively requires careful analysis of the context, applying relevant economic principles, and utilizing appropriate mathematical tools. Don't hesitate to consult additional resources like textbooks, lecture notes, or online explanations for further guidance on specific concepts or techniques

Sample Answer

Problem 1:

This problem requires analyzing David's preference order for yellow paint colors. Use the principle of transitivity to evaluate whether his choices violate any logical consistency. If his choices lead to contradictions (e.g., preferring A to B, B to C, and C to A), then his preferences could be considered irrational.