ę November 20, 2017, 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 𝔯𝔯𝔯  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 Rt+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’H˘pital’s rule1 because for any ρ ⁄=  1  the derivative exists,

u′(c)  =   c- ρ,                               (2)
and          - ρ
lim ρ→1  c   =  1 ∕c  but ∫
  (1∕c ) = log c  .

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

1.2 [FinSum]

Fact 2.

∑ T        (       T+1 )
      i      1----γ-----
     γ  =      1 - γ
 i=0
(3)

1.3 [InfSum]

Fact 3. If 0 <  γ <  1  , then

∑∞         (   1   )
     γi =    -------
             1 - γ
 i=0
(4)

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  .

 gt                             t     t
e    corresponds   to  (1 +  g ) ≡  G  .                   (5)

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

(1 +  ϵ)ζ ≈ 1 +  ϵζ
(6)

2.2 [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

2.3 [LogEps]

Fact 6. For ϵ  near zero (‘small’),

log (1 +  ϵ) ≈  ϵ
(9)

2.4 [ExpEps]

Fact 7. For ϵ  near zero (‘small’),

             ϵ
(1 +  ϵ) ≈ e
(10)

2.5 [OverPlus]

Fact 8. For ϵ  near zero (‘small’),

1∕(1 +  ϵ) ≈  1 -  ϵ
(11)

2.6 [MultPlus]

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

(1 + ϵ )(1 + ζ ) ≈ 1 +  ϵ + ζ
(12)

2.7 [ExpPlus]

Fact 10. For real numbers ϵ  and ζ

exp (ζ )exp (ϵ ) = exp (ζ +  ϵ)
(13)

2.8 [SmallSmallZero]

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

3 Statistics/Probability Facts

3.1 [SumNormsIsNorm]

Fact 12. If                 2
𝔯𝔯𝔯t+1  ~ N  (𝔯𝔯𝔯,σ 𝔯𝔯𝔯)  and                 2
rt+1 ~  N  (r,σ r)  and 𝔯𝔯𝔯t+1   and rt+1   are independent (written 𝔯𝔯𝔯t+1 ⊥  rt+1)  then

                            2     2
𝔯𝔯𝔯t+1 + rt+1  = N  (𝔯𝔯𝔯 +  r,σ𝔯𝔯𝔯 + σ r)
(14)

3.2 [ELogNorm]

Fact 13. If from the viewpoint of period t  the stochastic variable Rt+1   is lognormally distributed with mean r  and variance σ2
 r   , rt+1 ~  N  (r,σ2 )
                r  , then

                 2
Et[ert+1] = er+ σr∕2
(15)

3.3 [ELogNormMeanOne]

Fact 14. If from the viewpoint of period t  the stochastic variable R
  t+1   is lognormally distributed with mean     2
-  σ ∕2  and variance  2
σr   , log  R     ~  N ( - σ2∕2, σ2 )
      t+1  , then

     rt+1      - σ2∕2+ σ2∕2    0
Et [e    ] =  e  r     r  =  e  =  1
(16)

3.4 [LogELogNorm]

Fact 15. If Rt+1   is lognormally distributed as in [ELogNormMeanOne      ]  , then

                                      2
log Et[Rt+1  ] =   Et [log  Rt+1 ] + σ r∕2                  (17)
               =   r +  σ2 ∕2                              (18)
                         r
which follows from taking the log of both sides of (15).

3.5 [NormTimes]

Fact 16. If                 2
rt+1  ~  N (r, σr)  , then

γrt+1  ~  N (γr, γ2 σ2 )
                     r
(19)

3.6 [MeanOne]

Fact 17. If log R     ~  N  (- σ2 ∕2,σ2 )
      t+1           r      r  , then

Et [Rt+1 ]  =   1                              (20)
for any value of σ2  ≥ 0.
  r

This follows from substituting -  σ2r ∕2  for r  in [ELogNorm    ]  .

3.7 [LogMeanMPS]

Fact 18. If                        2      2
log Rt+1  ~  N  (r - σ r∕2, σr )  , then

log E  [R    ]  =   r                            (21)
      t   t+1
for any value of σ2r ≥ 0.

This follows from substituting        2
r -  σr ∕2  for r  in [ELogNorm    ]  and taking the log.

3.8 [ELogNormTimes]

Fact 19. If log ˆR     =  γ log R
      t+1            t+1   where log R     ~  N  (r,σ2 )
      t+1            r  , then

              γr+ γ2σ2∕2
Et[Rˆt+1  ] = e       r
(22)

3.9 [LogELogNormTimes]

Fact 20. If     ˆ
log Rt+1  =  γ log Rt+1   where                      2
log Rt+1  ~  N  (z, σr)  , then

                         2  2
log Et [Rˆt+1  ] = γr  + γ  σ r∕2
(23)

which follows from taking the log of (22).

4 Other Facts

4.1 [EulersTheorem]

Fact 21. If Y  =  F (K,  L)  is a constant returns to scale production function, then

Y   =   FK  K  +  FLL,                            (24)
and if this production function characterizes output in a perfectly competitive economy then FK   is the interest factor and FL   is the wage rate.