Problem Set 2

**Q1**

1 Point

What is the expected value of the lottery L*L* where

L=[24:frac{1}{2}, 12:frac{1}{6}, 48:frac{1}{6}, 6:frac{1}{6}]*L*=[24:21,12:61,48:61,6:61]

Choice 1 of 4:12Choice 2 of 4:14Choice 3 of 4:23Choice 4 of 4:85.5

**Answer:**

**Q2**

1 Point

Consider two lotteries A*A* and B*B*. For lottery A*A*, there is a 20-percent chance that you will receive $80, and 50-percent chance that you will receive $40, and a 30-percent chance that you will receive $10. For lottery B*B*, there is a 40-percent chance that you will receive $30, a 30-percent chance that you will receive $40 and a 30-percent chance that you will receive $50.

Suppose that the decision maker’s utility for money is given by U($m)=sqrt{m}*U*($*m*)=*m* (i.e., the utility for money is the square root of the monetary value). Which of the following is true?

The following checkbox options contain math elements, so you may need to read them in your screen reader’s “reading” or “browse” mode instead of “forms” or “focus” mode.

Choice 1 of 3:The decision maker is indifferent between A*A* and B*B*.

Choice 2 of 3:The decision maker strictly prefers A*A* over B*B*.

Choice 3 of 3:The decision maker strictly prefers B*B* over A*A*.

**Answer:**

** **

** **

**Q3**

1 Point

Suppose that Ann is offered a choice between the following two lotteries:

A=[$4,000: 0.8, $0: 0.2]*A*=[$4,000:0.8,$0:0.2] and B=[$3,000:1]*B*=[$3,000:1].

Ann says she strictly prefers B*B* to A*A*.

Is Ann maximizes expected monetary value?

The following multiple-choice options contain math elements, so you may need to read them in your screen reader’s “reading” or “browse” mode instead of “forms” or “focus” mode.

Choice 1 of 2:Yes, B*B* has a greater expected monetary value.Choice 2 of 2:No, B*B* does not have a greater expected monetary value.

**Answer:**

** **

**Q4**

1 Point

Suppose that Ann is offered a choice between the following two lotteries:

A=[$4,000: 0.8, $0: 0.2]*A*=[$4,000:0.8,$0:0.2] and B=[$3,000:1]*B*=[$3,000:1].

Ann says she strictly prefers B*B* to A*A*.

Suppose that Ann is an expected **utility** maximizer. Which of the following two lotteries will Ann choose?

C=[$4,000: 0.2, $0: 0.8]*C*=[$4,000:0.2,$0:0.8] or D=[$3,000: 0.25, $0: 0.75]*D*=[$3,000:0.25,$0:0.75].

The following multiple-choice options contain math elements, so you may need to read them in your screen reader’s “reading” or “browse” mode instead of “forms” or “focus” mode.

Choice 1 of 3:Ann strictly prefers C*C* to D*D*Choice 2 of 3:Ann strictly prefers D*D* to C*C*Choice 3 of 3:Ann is indifferent between C*C* and D*D*

**Answer:**

** **

**Q5**

2 Points

Explain your answer to question 4.

**Answer:**

** **

**Q6**

2 Points

Suppose that we have four different foods: ice cream, chocolate sauce, french fries, and ketchup. These foods can be mixed together into a bowl in different proportions. Let (a,b,c,d)(*a*,*b*,*c*,*d*) represent the amount of ice cream, chocolate sauce, french fries, and ketchup (respectively) that gets mixed together. For example, (1,1,0,0)(1,1,0,0) represents 11 oz of ice cream mixed with 11 oz of chocolate sauce (and no french fries or ketchup). Or (1,0,2,0)(1,0,2,0) represents 11 oz of ice cream mixed with 22 oz of french fries (and no chocolate sauce or ketchup). Now, suppose a concatenation operation oplus⊕ which works like this: (a,b,c,d)(*a*,*b*,*c*,*d*) mixed with (w,x,y,z)(*w*,*x*,*y*,*z*), written (a,b,c,d) oplus (w,x,y,z)(*a*,*b*,*c*,*d*)⊕(*w*,*x*,*y*,*z*) produces a new mixture (a+w, b+x, c+y, d+z)(*a*+*w*,*b*+*x*,*c*+*y*,*d*+*z*). So, for example, (1,1,0,0)oplus (1,0,2,0)(1,1,0,0)⊕(1,0,2,0) results in a food mixture consisting of 2 oz ice cream, 1 oz chocolate sauce, 2 oz french fries, and no ketchup. Consider your run-of-the-mill preferences over all objects of the form (a,b,c,d)(*a*,*b*,*c*,*d*). Do they satisfy this axiom?

**Definition (Independence)** For all food bowls f*f* and g*g*, if f*f* is strictly preferred to g*g* then for any food bowl h*h*, foplus h*f*⊕*h* is strictly preferred to goplus h*g*⊕*h*.

Explain why or why not.

**Answer:**

** **

**Q7**

2 Points

You are playing a game for money. There are two envelopes on a table. You know that one contains $X$*X* and the other $2X$2*X*, but you do not know which envelope is which or what the number X is. Initially you are allowed to pick one of the envelopes, to open it, and see that it contains $Y$*Y*. You then have a choice: walk away with the $Y$*Y* or return the envelope to the table and walk away with whatever is in the other envelope. What should you do?

Since the envelop you are holding contains Y*Y* dollars, you know that the other envelop contains either frac{1}{2}Y21*Y* dollars or 2Y2*Y* dollars, with equal probability. So, you must compare keeping Y*Y* dollars with the expected value of switching envelops. The expected value of switching is:

frac{1}{2}(frac{1}{2}Y) + frac{1}{2}2Y=frac{1}{4}Y + Y = frac{5}{4}Y21(21*Y*)+212*Y*=41*Y*+*Y*=45*Y*

Since frac{5}{4}Y > Y45*Y*>*Y*, you should switch envelops. But, of course, the same argument applies to the envelop you are now holding. So, expected utility theory seems to suggest that you should keep switching. But this is absurd. What is wrong with this reasoning? This is problem is known as the **two-envelop paradox**. There are many explanations of this problem found on the internet. Summarize how you would answer this question. Make sure to cite your source (i.e., provide a link to the online article or video that you used to come up with your answer).

**Answer:**