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 $r$ are diﬀerent from each other, like $x$ and $y$ .

Furthermore, a version of a variable without a subscript should be interpreted as the population mean of that variable. Thus, if $R t+1$ is a stochastic variable, then $R$ denotes its mean value.

1 Utility Functions

1.1 [CRRALim]

Fact 1.

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

(1)

which follows from L’Hôpital’s rule¹ because for any $ρ ⁄= 1$ the derivative exists,

Thus, we can conclude that as $ρ → 1$ , the behavior of the consumer with $u (c) = c1− ρ∕ (1 − ρ )$ becomes identical to the behavior of a consumer with $u (c) = log c$ .²

1.2 [FinSum]

Fact 2.

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

(3)

1.3 [InfSum]

Fact 3. If $0 < γ < 1$ , then

∑∞ ( ) γi = --1---- 1 − γ i=0

(4)

1.4 [FinSumMult]

Fact 4.

∑T ( T+1 ) i γ-+--γ----(T-(γ-−--1)-−--1)- iγ = (1 − γ )2 i=0

(5)

1.5 [InfSumMult]

Fact 5. If $0 < γ < 1$ , then

∞ ( ) ∑ i --1---- γ = 1 − γ i=0

(6)

2 ‘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 useful for computational solutions; one of the purposes of the approximations is to show how the discrete-time solution becomes close to the corresponding continuous-time problem as the time interval shrinks).

2.1 [TaylorOne]

Fact 6. For $𝜖$ near zero (‘small’), a ﬁrst order Taylor expansion of $(1 + 𝜖)ζ$ around 1 yields

(1 + 𝜖)ζ ≈ 1 + 𝜖ζ

(8)

2.2 [TaylorTwo]

Fact 7. For $𝜖$ near zero (‘small’), a second order Taylor expansion of $ζ (1 + 𝜖)$ around 1 yields

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

(9)

2.3 [LogEps]

Fact 8. For $𝜖$ near zero (‘small’),

log (1 + 𝜖) ≈ 𝜖

(10)

2.4 [ExpEps]

Fact 9. For $𝜖$ near zero (‘small’),

𝜖 (1 + 𝜖) ≈ e

(11)

2.5 [OverPlus]

Fact 10. For $𝜖$ near zero (‘small’),

1∕(1 + 𝜖) ≈ 1 − 𝜖

(12)

2.6 [MultPlus]

Fact 11. For $𝜖$ and $ζ$ near zero (‘small’),

(1 + 𝜖 )(1 + ζ ) ≈ 1 + 𝜖 + ζ

(13)

2.7 [ExpPlus]

Fact 12. For real numbers $𝜖$ and $ζ$

exp (ζ )exp (𝜖 ) = exp (ζ + 𝜖)

(14)

2.8 [SmallSmallZero]

Fact 13. If $𝜖$ is small and $ζ$ is small then $𝜖ζ$ can be approximated by zero.

3 Statistics/Probability Facts

3.1 [SumNormsIsNorm]

Fact 14. If $2 rt+1 ∼ 𝒩 (r, σr)$ and $2 rt+1 ∼ 𝒩 (r,σ r)$ and $rt+1$ and $rt+1$ are independent (written $rt+1 ⊥ rt+1)$ then

r + r = 𝒩 (r + r,σ2 + σ2 ) t+1 t+1 r r

(15)

3.2 [ELogNorm]

Fact 15. If from the viewpoint of period $t$ the stochastic variable $Rt+1$ is lognormally distributed with mean $r$ and variance $2 σr$ , $2 rt+1 ∼ 𝒩 (r,σ r)$ , then

rt+1 r+ σ2∕2 𝔼t[e ] = e r

(16)

3.3 [ELogNormMeanOne]

Fact 16. If from the viewpoint of period $t$ the stochastic variable $Rt+1$ is lognormally distributed with mean $2 − σ ∕2$ and variance $2 σr$ , $log Rt+1 ∼ 𝒩 ( − σ2∕2, σ2 )$ , then