Question

3. Geena has the following utility function U = 2ln(QN ) + .5ln(QY ) where QN is the number of hours of Netflix Geena watches and QY is the number of hours of YouTube Geena watches. The only other activity Geena does is sleep for 8 hours a day. Geena’s Budget constraint is I = PN QN + PY QY where I is the number of hours a day she can watch Netflix or YouTube, PN is the time cost of watching Netflix relative to YouTube, and PY is the time cost of watching YouTube relative to Netflix. This means her Budget Constraint is 16 = QN + QY (i.e. every hour of Netflix watched is an hour that can not be spent watching YouTube and vice versa).

(a) Use the above information to determine the optimal quantity of Netflix and YouTube Geena watches. Draw Geena’s Budget Constraint and give a rough sketch of her indifference curves. Be sure to label the graph and all important points.

(b) Geena realizes that she can watch YouTube at double speed without any change to her enjoyment of the programs (in other words, Geena can watch an hour of YouTube in 1/2 an hour now). i. What is Geena’s new Budget Constraint and will she be better off or worse off (No math beyond the Budget Constraint is needed to answer this part of the question)? ii. What are the new values of QN and QY she selects? How does it compare with your answer in part a)?

(c) Bonus) This is for bonus marks only, you will not be penalized for not attempting this part of the problem. Assume Geena enjoys sleeping as much as she enjoys watching YouTube videos (but unlike YouTube videos, Geena can not sleep at double speed). How much sleep would Geena like to have if this were a choice variable? Does this change her choice of Netflix and YouTube? (Note: You will need to use the Lagrangian to solve this as there are 3 goods, QN , QY , and QZ where QZ is the quantity of sleep. Be careful about your Budget Constraint here.)

4. Suppose Tom has the following utility function U = ln(Qc − h) + ln(Qd) where Qc is the quantity of coffee Tom consumes, Qd is the quantity of doughnuts Tom consumes, and h is Tom’s coffee habit (i.e. Tom needs to consume at least h coffees before he gets any utility).

(a) Show the optimal choice of Qc and Qd that Tom would select using his budget constraint I = PcQc + PdQd (Hint: h will be in the optimal choice and Qc > h).

(b) Does Tom’s coffee habit influence his optimal quantity of doughnuts? Why would this be the case? (c) Find the own-price and cross-price elasticity for Qc and Qd. Let I = $20, Pc = $2, Pd = $3 and h = 2.

Answer 1

a) Since, the two options that Geena has are perfect substitutes. The ICs will be negatively sloped with slope equal to 1. The budget constraint is given by AB.

Since, the ICs are parallel to the budget constraint, IC3 is the highest curve that the consumer can reach subject to his budget constraint. Any point on this IC, which is also a point on the budget line, may be taken to be the consumer’s equilibrium point. Therefore, here, the consumer has not a unique equilibrium solution, he may have a large number of equilibrium points.

b) When Geena realizes that she can watch YouTube at double speed, in other words, Geena can watch an hour of YouTube in 1/2 an hour there will be a change in her budget constraint.

The budget constraint will become: 16 = QN + 2QY

The budget constraint will become flatter shown by AB. The ICs are negatively sloped straight lines and each of them is steeper than the budget line, then as the consumer moves downward towards right along his budget line, he would be reaching higher and higher ICs. Finally, at the corner point B of the budget line with the x-axis, the consumer would be in equilibrium be­cause, at this point on the budget line, he is on the highest possible IC, viz., IC4.

The equilibrium at a corner point is called a corner solution or a boundary solution. Here at the corner solution given by point B, the consumer would only watch youtube and no netflix. In this case, the consumer would have a unique equilibrium solution.