©February 21, 2019, Christopher D. Carroll MathFactsList

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 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.

which follows from L’Hôpital’s rule^{1}
because for any ρ≠1 the derivative exists,

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

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 e^{0.1} ≈ 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).

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

| (6) |

Fact 12. If _{t+1} ∼N(,σ_{}^{2}) and r_{t+1} ∼N(r,σ_{r}^{2}) and _{t+1} and r_{t+1} are
independent (written _{t+1} ⊥ r_{t+1}) then

| (14) |

Fact 13. If from the viewpoint of period t the stochastic variable R_{t+1} is
lognormally distributed with mean r and variance σ_{r}^{2}, r_{t+1} ∼N(r,σ_{r}^{2}),
then

| (15) |

Fact 14. If from the viewpoint of period t the stochastic variable
R_{t+1} is lognormally distributed with mean −σ^{2}∕2 and variance σ_{r}^{2},
log R_{t+1} ∼N(−σ^{2}∕2,σ^{2}), then

| (16) |

Fact 15. If R_{t+1} is lognormally distributed as in [ELogNormMeanOne], then

This follows from substituting −σ_{r}^{2}∕2 for r in [ELogNorm].

This follows from substituting r − σ_{r}^{2}∕2 for r in [ELogNorm]and taking the
log.

Fact 20. If log _{t+1} = γ log R_{t+1} where log R_{t+1} ∼N(z,σ_{r}^{2}), then

| (23) |

which follows from taking the log of (22).