November 20, 2017, Christopher D. Carroll MathFactsList
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 and are different from each other, like and .
Furthermore, a version of a variable without a subscript should be interpreted as the population mean of that variable. Thus, if is a stochastic variable, then denotes its mean value.
which follows from L’H˘pital’s rule1 because for any the derivative exists,
Thus, we can conclude that as , the behavior of the consumer with becomes identical to the behavior of a consumer with .
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 .
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 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 while in discrete time we would write it as .)
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 12. If and and and are independent (written then
This follows from substituting for in .
This follows from substituting for in and taking the log.
which follows from taking the log of (22).