Examples of Optimization Problems in Microeconomics

Question 1: Speed Racer and His Car

1. Speed Racer likes to get where he is going fast, but he hates having accidents. Unfortunately, the faster he goes, the higher the probability that he will crash his car (the powerful Mach 5).

a. Draw Speed Racer's budget line between speed and probability of a fatal accident (PFA) putting speed on the y-axis Now draw Speed Racer's indifference curves assuming his preferences are convex. Show the optimal combination of speed and PFA. 

b. Suppose that Speed Racer's mother is worried about Speed's safety. She forces him to install a driver's side air bag, and other safety equipment in his car. However, she pays all the costs of these modifications. Show how Speed's budget line and optimal choice change. 

c. Can you say for certain whether the Speed Racer is more or less likely to die in a car crash after the equipment is installed? Be sure to illustrate your answer.

2. Midterm exams are usually offered within a week or so of each other. This means that you must decide how to allocate time to each of your courses. Suppose that you have both a Microeconomics, and an Econometrics exam this week. Suppose also that you have convex preferences
over the scores you receive on these exams (that is, you would rather get two B's than an A, and a C). Suppose you have 150 hours of time to study in total and it takes two hours of study to get each point on the Microeconomics midterm, but only one hour to get each point on the Econometrics exam. 

a. Write the equation of the budget constraint between points on the Microeconomics exam (M) and points on the Econometrics exam (E). Now sketch this budget constraint in a picture. Be sure to label you diagram. Draw in convex indifference curves and show your optimal choice.

b. In economics, we sometimes like to make ridiculous assumptions in order to illustrate a point. Here is one. Suppose that your microeconomics instructor suddenly becomes fantastically lucid, and the whole beautiful structure of economic reasoning suddenly becomes clear to you. Now you judge that it will only take you one hour of study to get each point on the Microeconomics exam. Write the equation for the new budget constraint. Now sketch the new budget constraint in same picture you used for part (a). show your new optimal choice.

c. True false or uncertain: if Econometrics grades are an inferior good, then you will end up getting a higher grade in Microeconomics? 

d. True false or uncertain: if Microeconomics grades are an inferior good, then you will end up getting a higher grade in Microeconomics?

3. Val Halla is a Norwegian farmer who owns 40 chicken and 20 sacks of wheat. As it happens, chicken and wheat are all he consumes as well. He goes to market one day and finds the price of chickens is $ 4 and the price of wheat is $ 2.

a. At these prices he chooses to consume 20 chickens and 60 sacks of wheat. In a diagram, show Val's endowment and budget constraint. Also, sketch in his indifference curves. Assume his preferences are convex and monotonic, and of course be sure to draw them such that 20 chickens and 60 sacks of wheat is his optimal choice.

b. Suppose the price of chickens drops to two dollars, and suppose that wheat is a normal good. Can you say for certain whether wheat consumption goes up? Can you say for certain whether Val is better off or worse off after the change in prices?