Math Facts Useful for Graduate Macroeconomics

The following collection of facts is useful in many macroeconomic models. No proof is offered for some facts because the derivations are standard elements of prerequisite mathematics or microeconomics classes; this handout is offered as an aide memoire and for reference purposes.

1 Utility Functions

1.1 [CRRALim]

Fact 1

( ) c1- ρ lim ------- = log c ρ→1 1 - ρ

(1)

2 Geometric Series

2.1 [FinSum]

Fact 2

( ) ∑T 1 - γT +1 γi = ----------- i=0 1 - γ

(4)

2.2 [InfSum]

Fact 3 If 0 < γ < 1, then

∑∞ ( ) γi = ---1--- 1 - γ i=0

(5)

3 ‘Small’ Number Approximations

3.1 [TaylorOne]

Fact 4 For ε near zero (‘small’), a first order Taylor expansion of (1 + ε)^ζ around 1 yields

(1 + ε)ζ ≈ 1 + εζ

(6)

3.2 [TaylorTwo]

Fact 5 For ε near zero (‘small’), a second order Taylor expansion of (1 + ε)^ζ around 1 yields

(1 + ε)ζ ≈ 1 + ζε + ε2ζ(ζ - 1 )∕2 (7) ( ( ) ) = 1 + 1 + ζ----1- ε ζ ε (8) 2

3.3 [LogEps]

Fact 6 For ε near zero (‘small’),

log (1 + ε) ≈ ε

(9)

3.4 [ExpEps]

Fact 7 For ε near zero (‘small’),

ε (1 + ε) ≈ e

(10)

3.5 [OverPlus]

Fact 8 For ε near zero (‘small’),

1 ∕(1 + ε) ≈ 1 - ε

(11)

3.6 [MultPlus]

Fact 9 For ε and ζ near zero (‘small’),

(1 + ε)(1 + ζ) ≈ 1 + ε + ζ

(12)

3.7 [ExpPlus]

Fact 10 For real numbers ε and ζ

exp (ζ) exp (ε) = exp (ζ + ε)

(13)

4 Statistical Facts

4.1 [ELogNorm]

Fact 11 If from the viewpoint of period t the stochastic variable Z_t+1 is lognormally distributed with mean and variance σ_z² (Defining z_t+1 = log Z_t+1, write this as z_t+1 ~( ,σ_z²)), then