Math Facts Useful for Graduate Macroeconomics

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

Throughout this document, typographical distinctions should be interpreted as meaningful; for example, the variables $r$ and $𝔯𝔯𝔯$ are different from each other, like $x$ and $y$ .

1 Utility Functions

1.1 [CRRALim]

Fact 1.

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

(1)

2 Geometric Series

2.1 [FinSum]

Fact 2.

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

(3)

2.2 [InfSum]

Fact 3. If $0 < γ < 1$ , then

∑∞ ( ) i --1---- γ = 1 - γ i=0

(4)

3 ‘Small’ Number Approximations

Sometimes economic models are written in continuous time and sometimes in discrete time. Generically, there is a close correspondence between the two approaches, which is captured (for example) by the future value of a series that is growing at rate $g$ .

The words ‘corresponds to’ are not meant to imply that these objects are mathematically identical, but rather that these are the corresponding ways in which constant growth is treated in continuous and in discrete time; while for small values of $g$ they will be numerically very close, continuous-time compounding does yield slightly different values after any given time interval than does discrete growth (for example, continuous growth at a 10 percent rate after 1 year yields $0.1 e ≈ 1.10517$ while in discrete time we would write it as $G = 1.1$ .)

Many of the following facts can be interpreted as manifestations of the limiting relationships between continuous and discrete time approaches to economic problems. (The continuous time formulations often yield simpler expressions, while the discrete formulations are needed for computational solutions; one of approximations is to show how the discrete-time solution becomes close to the corresponding continuous-time problem as the time interval shrinks).

3.1 [TaylorOne]

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

ζ (1 + ϵ) ≈ 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

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

3.8 [SmallSmallZero]

Fact 11. If $ϵ$ is small and $ζ$ is small then $ϵζ$ can be approximated by zero.

4 Statistics/Probability Facts

4.1 [SumNormsIsNorm]

Fact 12. If $𝔯𝔯𝔯t+1 ~ N (𝔯𝔯𝔯,σ2𝔯𝔯𝔯)$ and $rt+1 ~ N (r,σ2r)$ then

2 2 𝔯𝔯𝔯t+1 + rt+1 = N (𝔯𝔯𝔯 + r,σ𝔯𝔯𝔯 + σ r)

(14)

4.2 [ELogNorm]

Fact 13. If from the viewpoint of period $t$ the stochastic variable $Zt+1$ is lognormally distributed with mean $z$ and variance $2 σ z$ (Defining $zt+1 = log Zt+1$ , write this as $zt+1 ~ N (z,σ2z)$ ), then