Economics 704 Problem Set 3 Due Monday, Sept. 28 in lecture

THEORY 1. John von Neumann wrote a letter to the US government about the use of nuclear weapons during the Cold War. He argued that given the nuclear capabilities of Russia, a preemptive strike by the US was the optimal choice.

One way of looking at this is as a minimax argument:

the US

should take the action that minimizes the chance of the maximum loss possible (a Russian strike). Interpret Von Neumann's recommendation in the context of the lecture notes on decision theory.

In particular, what

happens if you change the supports in a formal analysis of his argument?

K ∼ N (µ, 1) (. You don't know exactly what µ is, but 0 with probability 12 . If you observe one your prior belief is that µ = 1 with probability 12 realization of K and it is equal to 1.96, what is your posterior belief of the value of µ? How does this change if your prior puts full weight of 1 on µ = 0?

2. You know that

3. Consider the following choices tax rates, state of the world is either

θ1

or

θ2 ,

τ1 < τ2 < τ3 .

You know the

which will yield the following levels

of revenue loss. Compute the max loss and max regret for each choice of tax rate, and identify the minimax and minimax regret choice of tax rate. Are they they same?

Table 1: Revenue loss

τ1 τ2 τ3

θ1

θ2

15

30

18

16

20

0

Max loss

Max regret

EMPIRICS 4. Empirically demonstrate the CLT using draws from an independent distribution of your choice. (a) Pick a distribution (say, the binomial distribution), and report the

µ and variance σ 2 you choose. Take n = 10 and generate Xn for 100 independent samples, each
√ sample having n draws. Plot a histogram of n Xn − µ (where Xn is the sample mean, n is the number of trials) with an overlaid true mean

(b)

normal distribution of the same mean and variance. (c) Repeat this for

n = 100

and

n = 10, 000.

1