Question

Please answer the following question. Please show all your working/solutions.

In dealing with macroeconomic data, it is often informative to express GDP- per-capita in logs. To see the convenience, consider a variable X_t over time. The growth rate of a variable X_t from period t − 1 to period t is given by (X_t − _Xt−1)/X_t−1. If we let ∆(X_t) denote the growth rate of the variable X_t from t − 1 to t, we can say that ∆(X_t) is well approximated by log(X_t/X_t−1). Let’s see why.

(a) Show that for small values of z, we must have log(1 + z) ≈ z. (Hint: Use Taylor expansion!). Use this result in the following parts.

(b) Show that log(X_t/X_t−1) ≈ (X_t − X_t−1)/X_t−1 = ∆(X_t). (Hint: add and subtract 1 to X_t/X_t−1 inside the log function.) For the rest of the question, assume ∆(X_t) = log(X_t/X_t−1).

(c) Show ∆(X_tY_t) = ∆(X_t) + ∆(Y_t)

(d) Show ∆(X_t/Y_t) = ∆(X_t) − ∆(Y_t)

(e) Show ∆(X_t²) = 2∆(X_t)

Answer 1

Ans A)

Log(1+z) with Taylor expansion can be written as follows

Log(1+z)=z+z^2/2+z^3/3+...

As z tends to 0

Higher order of z tends to 0

Hence Log(1+z)=z (approximately)

Ans B)

Log(1+((Xt/Xt-1)-1))=(Xt/Xt-1)-1 (approx.) As per proof above

Now

Log(Xt/Xt-1)=LogXt-LogXt-1=(Log(1+Xt-1)-Log(1+Xt-1-1))=

Xt-1+(Xt-1)^2/2+(Xt-1)^3/3+...-(Xt-1-1)-(Xt-1-1)^2/2-...=Xt-Xt-1

If we ignore higher order terms of Xt and Xt-1

Log(Xt/Xt-1)=∆Xt

Ans C)

∆(Xt)+∆(Yt)=log(Xt/Xt-1)+log(Yt/Yt-1)=log(XtYt/(Xt-1)(Yt-1))=∆(XtYt)

Ans D)

∆(Xt)-∆(Yt)=log(Xt/Xt-1)-log(Yt/Yt-1)=log(Xt/(Xt-1)/Yt/(Yt-1))=log((Xt/Yt)/(Xt-1/Yt-1))=∆(Xt/Yt)

Ans E)

∆(Xt^2)=log(Xt^2/Xt-1^2)=Log((Xt/Xt-1)^2)=2Log((Xt/Xt-1))=2∆Xt