monetary econ

Monetary Economics Homework 2:Additional Guidance

Professor Sang-Sub Lee

Problem 3-1. Look up log normal distribution in the Wekipedia to find out:

ln(x) ∼ N(µx,σ2x) → E(x) =? 6= eµx

var(x) =?

From the mean and variance of x derived above, you can also derive the expressions:

variance(x) [E(x)]2

and standard deviation of x E(x)

,which is required for problem 3-3 below.

Problem 3-2 (see also lecture note)

Et(r̃ i t+1 − r

f t+1) = −

covt[β uct+1 uct

, r̃it+1 − r f t+1]

Et[β uct+1 uct

] = covt[mt,t+1, r̃

i t+1 − r

f t+1]

Et[mt,t+1]

ρxy = Cov(X,Y )

σxσy

Et(r̃t+1 − r f t+1)

σr̃ = −

ρσmt,t+1 Et(mt,t+1)

(1)

Note that covt[mt,t+1, r̃ i t+1 −r

f t+1] = covt[mt,t+1, r̃

i t+1] because r

f t+1 is known at t (i.e., risk

free).

Problem 3-3. Note that mt,t+1 = β( ct+1 ct

)−γ

⇒ ln(m) = ln(β) −γln(ct+1 ct

) ∼ N(?, ?) because ln(ct+1 ct

) is normal and a linear combination of normal is also a normal. You just need to find out what the mean and variance of the lin- ear combination are. From this, you can derive Et(mt,t+1), V ar(m), and

standard deviation of m E(m)

based on the results from Problem 3-1.

Problem 3-4. From 1-3, you derived Et(mt,t+1).(Again note that m is log normal and you can apply the formula from problem 1-1 to find out the expected vale of m). Then,just take

1

a log of 1 Et(mt,t+1)

2