Assignment Details:

Part A

A.2 A student has to decide whether to go to a party on the night before an exam. If they go to the party, then they have the probability Π of writing a good exam and probability 1- Π of writing a bad exam. If they do not go to the party, then they have the probability Φ > Π of writing a good exam.

The student has the utility u = x1/2 + H if they go to the party and get the mark x on the exam. But, if they do not go to the party they have the utility u = x1/2. The professor gets to choose xb (the mark she gives to bad exams) and xg the mark she gives to good exams. Students also have the option of going to the party and then quitting the course. This gives them the utility U + H where U > 0.

• What is the student’s expected utility from going to the party and her expected utility from not going to the party?
• Assuming students do not quit the course, what marks (xg, xb) can the professor set to stop students partying? Draw a picture of this set of marks with yg = pxg on one axis and yb = pxb on the other. Explain how this set changes as H and Φ- Π change.
• What marks can the professor set so the students prefer partying and taking the exam to quitting the course? What marks can the professor set so the students prefer not partying and taking the exam to quitting the course? Show these sets on a new picture.
• Suppose that the professor wants to promote equity and thus aims to make the difference xg xb as small as possible while still stopping students from going to the party, but she ignores the possibility that students will quit the course. What marks (xg, xb) should she set to achieve her objective? Will this result in the outcome she planned?
• Suppose now the professor wants to minimize xg xb while still getting students to not party and attend the course. Plot the contours of her objective function on your (yg, yb) picture. What marks (xg, xb) should she set to achieve her objective now?
• The professor has other variables that she can control: she can make the exam harder, which decreases Φ and Π, and she can make the party fail by calling the campus authorities, which decreases H. How does each of these variables affect her ability to achieve the objectives described above?

A.3 A firm can employ two types of workers: A workers and B workers. The A workers have ability α that is uniformly distributed on the interval [0, 2], that is, 0<= α <=2. The B workers have ability β that is also uniformly distributed on the interval [0, 2]. All the workers know their own abilities.

There are equal numbers of A and B workers. The firm can see whether the workers are type A or B, but cannot observe the workers’ abilities.

The A workers can earn 2α² from setting up their own company whereas the B workers can only earn β² from setting up their own company. A firm earns pα from every A worker of ability α that they employ and pβ for every B worker of ability β they employ.

• If the firm o↵ers the wage v for A workers and w for B workers, then what workers come to the firm?
• What is the firm’s expected profit from the wages (v, w) and how should it choose wages to maximize profits? Or to make zero profits in perfectly competitive labor markets?
• Does the firm employ more A workers or B workers? Explain why this is so.
• A piece of anti-discrimination legislation is published. The firm is no longer able to pay different wages to A and B workers but must pay them all the same wage u. What is the firm’s profit now? How should it choose u to maximize profits? Or to make zero profits
• Does the firm now employ more A workers or B workers? Explain why this is so.
• The government also decides the firm must employ equal numbers of A and B workers. What workers would the firm in part (d) like to sack to achieve this additional objective?
• What can we learn about the economics of discrimination from the answers you have given to the above?

PART B

B.1 An individual consumes two goods, food q1, and clothing q2. Their total budget is y, the prices of food and clothing are p1 and p2 and their utility is.

Suppose their preferences are represented by expenditure function c(υ, p1, p2) = √p1p2 eˆp1υ/p2 .

Assume that prices and utilities take values such that this function has all the required properties of an expenditure function, including concavity in prices, and that consumers demand positive quantities of food and clothing. Note that it is allowed to be positive or negative.

• Use Shephard’s Lemma to show that compensated budget shares of food and clothing are 1/2 + υp1/p2 and 1/2- υp1/p2.

Suppose that prices in a base period are p1 = p2 = 1. In a later period, they have changed to p1 = 2, p2 = 1/2. You are asked to consider whether the overall cost of living should be regarded as having risen or fallen.

• Explain what a true or Kon¨us cost of living index is and show that a Kon¨us cost of living index evaluated at initial utility 0 for these preferences and these prices has the form K(υ0)=eˆ3υ0
• Explain what a Laspeyres cost of living index is and show that a Laspeyres cost of living index for these preferences and these prices has the form
  L(υ0) = 5/4 + 3/2υ0
• Discuss the facts that

i. according to either index, whether the cost of living has risen of fallen depends on the initial utility υ0 at which it is evaluated

ii. the cost of living has risen according to L(υ0) but fallen according to K(υ0) if the initial utility is such that 0 > υ0 > -1/6.