Β©April 2, 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 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.

     ( c1βˆ’ ρ βˆ’ 1)
lim    ----------  =  log c
ρ→1     1 βˆ’  ρ
(1)

which follows from L’HΓ΄pital’s rule1 because for any ρ≠1 the derivative exists,

 β€²          βˆ’ ρ
u (c)  =   c   ,                               (2)
and lim ρ→1cβˆ’Ο = 1βˆ•c but ∫ (1βˆ•c) = log c.

Thus, we can conclude that as ρ β†’ 1, the behavior of the consumer with u(c) = c1βˆ’Οβˆ•(1 βˆ’ ρ) becomes identical to the behavior of a consumer with u(c) = log c.2

1.2 [FinSum]

Fact 2.

βˆ‘ T        (       T+1 )
     Ξ³i =    1-βˆ’--Ξ³-----
               1 βˆ’ Ξ³
 i=0
(3)

1.3 [InfSum]

Fact 3. If 0 < Ξ³ < 1, then

 ∞         (       )
βˆ‘     i      --1----
     Ξ³  =    1 βˆ’ Ξ³
 i=0
(4)

2 β€˜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 β€˜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 e0.1 β‰ˆ 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 πœ– near zero (β€˜small’), a first order Taylor expansion of (1 + πœ–)ΞΆ around 1 yields

(1 +  πœ–)ΞΆ β‰ˆ 1 +  πœ–ΞΆ
(6)

2.2 [TaylorTwo]

Fact 5. For πœ– near zero (β€˜small’), a second order Taylor expansion of (1 + πœ–)ΞΆ around 1 yields

        ΞΆ                 2
(1 +  πœ–)   β‰ˆ   1 +  ΞΆπœ– + πœ–  ΞΆ(ΞΆ βˆ’  1)βˆ•2                      (7)
                    (     (  ΞΆ βˆ’ 1 )  )
           =   1 +    1 +    ------- πœ–   ΞΆπœ–                  (8)
                               2

2.3 [LogEps]

Fact 6. For πœ– near zero (β€˜small’),

log (1 +  πœ–) β‰ˆ  πœ–
(9)

2.4 [ExpEps]

Fact 7. For πœ– near zero (β€˜small’),

             πœ–
(1 +  πœ–) β‰ˆ e
(10)

2.5 [OverPlus]

Fact 8. For πœ– near zero (β€˜small’),

1βˆ•(1 +  πœ–) β‰ˆ  1 βˆ’  πœ–
(11)

2.6 [MultPlus]

Fact 9. For πœ– and ΞΆ near zero (β€˜small’),

(1 + πœ– )(1 + ΞΆ ) β‰ˆ 1 +  πœ– + ΞΆ
(12)

2.7 [ExpPlus]

Fact 10. For real numbers πœ– and ΞΆ

exp (ΞΆ )exp (πœ– ) = exp (ΞΆ +  πœ–)
(13)

2.8 [SmallSmallZero]

Fact 11. If πœ– is small and ΞΆ is small then πœ–ΞΆ can be approximated by zero.

3 Statistics/Probability Facts

3.1 [SumNormsIsNorm]

Fact 12. If 𝔯𝔯𝔯t+1 βˆΌπ’©(𝔯𝔯𝔯,σ𝔯𝔯𝔯2) and rt+1 βˆΌπ’©(r,Οƒr2) and 𝔯𝔯𝔯t+1 and rt+1 are independent (written 𝔯𝔯𝔯t+1 βŠ₯ rt+1) then

                            2     2
𝔯𝔯𝔯t+1 + rt+1  = 𝒩  (𝔯𝔯𝔯 +  r,σ𝔯𝔯𝔯 + Οƒ 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 Οƒr2, rt+1 βˆΌπ’©(r,Οƒr2), then

    rt+1     r+ Οƒ2βˆ•2
𝔼t[e    ] = e    r
(15)

3.3 [ELogNormMeanOne]

Fact 14. If from the viewpoint of period t the stochastic variable Rt+1 is lognormally distributed with mean βˆ’Οƒ2βˆ•2 and variance Οƒr2, log Rt+1 βˆΌπ’©(βˆ’Οƒ2βˆ•2,Οƒ2), then

     rt+1      βˆ’ Οƒ2βˆ•2+ Οƒ2βˆ•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

log 𝔼t[Rt+1  ] =   𝔼t [log  Rt+1 ] + Οƒ2rβˆ•2                  (17)
                          2
               =   r +  Οƒr βˆ•2                              (18)
which follows from taking the log of both sides of (15).

3.5 [NormTimes]

Fact 16. If rt+1 βˆΌπ’©(r,Οƒr2), then

                   2 2
Ξ³rt+1  ∼  𝒩 (Ξ³r, Ξ³  Οƒr )
(19)

3.6 [MeanOne]

Fact 17. If log Rt+1 βˆΌπ’©(βˆ’Οƒr2βˆ•2,Οƒr2), then

𝔼  [R    ]  =   1                              (20)
  t   t+1
for any value of Οƒr2 β‰₯ 0.

This follows from substituting βˆ’Οƒr2βˆ•2 for r in [ELogNorm].

3.7 [LogMeanMPS]

Fact 18. If log Rt+1 βˆΌπ’©(r βˆ’ Οƒr2βˆ•2,Οƒr2), then

log 𝔼  [R    ]  =   r                            (21)
      t   t+1
for any value of Οƒr2 β‰₯ 0.

This follows from substituting r βˆ’ Οƒr2βˆ•2 for r in [ELogNorm]and taking the log.

3.8 [ELogNormTimes]

Fact 19. If log Rt+1 = Ξ³ log Rt+1 where log Rt+1 βˆΌπ’©(r,Οƒr2), then

                   2 2
𝔼t[RΛ†t+1  ] = eΞ³r+ Ξ³ Οƒrβˆ•2
(22)

3.9 [LogELogNormTimes]

Fact 20. If log Rt+1 = Ξ³ log Rt+1 where log Rt+1 βˆΌπ’©(z,Οƒr2), then

        Λ†                2  2
log 𝔼t [Rt+1  ] = Ξ³r  + Ξ³  Οƒ rβˆ•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.