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 section is offered as an aide memoire and for reference purposes.

Throughout this document, typographical distinctions should be interpreted as meaningful; for example, the variables $\risky$ and $\rport$ 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 $\Risky_{t+1}$ is a stochastic variable, then $\Risky$ denotes its mean value.

1Utility Functions¶

This follows from L’Hopital’s rule^[1] because for any $\CRRA \neq 1$ the derivative exists,

and $\lim_{\CRRA \rightarrow 1} \cLev^{-\CRRA} = 1/\cLev$ but $\int (1/\cLev) = \log c$.

Thus, we can conclude that as $\CRRA \rightarrow 1$, the behavior of the consumer with $\uFunc(\cLev)=\cLev^{1-\CRRA}/(1-\CRRA)$ becomes identical to the behavior of a consumer with $\uFunc(\cLev)=\log c$.^[2]

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 $\divGro$.

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 $\divGro$ 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 $e^{0.1} \approx 1.10517$ while in discrete time we would write it as $\DivGro=1.1$.)

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).

3Statistics/Probability Facts¶

This follows from substituting $-\sigma_{\risky}^{2}/2$ for $\risky$ in ELogNorm.

This follows from substituting $\risky - \sigma_{\risky}^{2}/2$ for $\risky$ in ELogNorm and taking the log.

You should remember from your econometrics courses that the least-squares slope is the covariance of $c$ and $y$ divided by the variance of $y$. Substituting $c=k\,y^{P}$ and $y=y^{P}+y^{T}$, and using $\operatorname{cov}(y^{P},y^{T})=0$,

Their ratio is $\alpha_{1}$. The fitted line passes through the means, $\Ex[c]=\alpha_{0}+\alpha_{1}\Ex[y]$, and since $\Ex[y^{T}]=0$, both $\Ex[c]=k\,\Ex[y^{P}]$ and $\Ex[y]=\Ex[y^{P}]$; so $\alpha_{0}=(k-\alpha_{1})\Ex[y^{P}]$.

The intuition: households with high measured income are, on average, partly households with high permanent income and partly households that drew a positive transitory shock. Only the permanent part raises consumption, so consumption rises with measured income by $k$ times the fraction of the differences in income that are permanent. If consumption also has a transitory part uncorrelated with income, the slope is unchanged (AttenuationByNoise).

Since $p$, $\xi$ and $u$ are uncorrelated, $\operatorname{cov}(z,y)=\operatorname{cov}(k\,p+u,\,p+\xi)=k\operatorname{var}(p)$ and $\operatorname{var}(y)=\operatorname{var}(p)+\operatorname{var}(\xi)$. It generalizes FriedmanPIHRegression: $z$ need only depend on $p$, and may carry noise $u$ of its own.

Over $d$ periods the permanent component accumulates $d$ permanent shocks, while the transitory shocks of the first and last periods both appear:

These $d+2$ shocks are independent, so the variance of their sum is the sum of their variances, $d\,\sigma_{\psi}^{2}+2\,\sigma_{\xi}^{2}$.

With $0<\tau<1$, $p$ has the same variance in every period, and since $\psi_{t}$ is independent of $p_{t-1}$, $\operatorname{var}(p)=\tau^{2}\operatorname{var}(p)+\sigma_{\psi}^{2}$, which gives $\operatorname{var}(p)=\sigma_{\psi}^{2}/(1-\tau^{2})$. Since $p_{t+d}=\tau^{d}p_{t}$ plus shocks after $t$, $\operatorname{cov}(p_{t+d},p_{t})=\tau^{d}\operatorname{var}(p)$, so $\operatorname{var}(p_{t+d}-p_{t})=2\operatorname{var}(p)(1-\tau^{d})$, and the transitory shocks add $2\sigma_{\xi}^{2}$ as before. As $\tau\to1$, $(1-\tau^{d})/(1-\tau^{2})\to d/2$, which gives back $d\,\sigma_{\psi}^{2}$.

In the regression, $c_{t+d}-c_{t}=(p_{t+d}-p_{t})+(u_{t+d}-u_{t})$ and $y_{t+d}-y_{t}=(p_{t+d}-p_{t})+(\xi_{t+d}-\xi_{t})$. So AttenuationByNoise applies with $k=1$, a common part of variance $d\,\sigma_{\psi}^{2}$ and noise of variance $2\,\sigma_{\xi}^{2}$. The longer the difference, the larger the permanent part’s share of the change in income: that is why Friedman’s model predicts a larger coefficient in long differences than in short ones.

Measured transitory income is not white noise: Abowd & Card (1989) find that annual changes in earnings are correlated with the changes one and two years away, and not beyond. If $\xi_{t}$ is correlated with its own values up to $q$ periods apart, then for $d>q$, $\operatorname{var}(\xi_{t+d}-\xi_{t})=2\operatorname{var}(\xi)$, and the formulas hold with $\operatorname{var}(\xi)$ in place of $\sigma_{\xi}^{2}$: only the intercept changes. Carroll (1992) used only differences of three years or more for this reason.

The sum of squared deviations from the mean equals the sum of all the squared pairwise differences, divided by $2n$:

(Expanding the right side gives $\frac{1}{2n}\bigl[2n\sum_{s}x_{s}^{2}-2(\sum_{s}x_{s})^{2}\bigr]$, which is the left side.) By VarianceOfIncomeDifferences with no transitory shock, $\Ex[(x_{s}-x_{u})^{2}]=|s-u|\,\sigma^{2}$. Of the $n^{2}$ pairs, $2(n-d)$ are $d$ periods apart, for $d=1,\ldots,n-1$, so

This year’s income is $y_{t}=p_{t-1}+(\psi_{t}+\xi_{t})$: last year’s permanent income with the coefficient one, plus shocks uncorrelated with last year’s income $y_{t-1}=p_{t-1}+\xi_{t-1}$. So AttenuationByNoise applies with $k=1$, with the variation within the history in place of the variances: about $\sigma_{\psi}^{2}T/6$ for the random walk (RandomWalkSpread) and $\sigma_{\xi}^{2}$ for the white noise. The coefficient is

The random walk’s variation within the history grows with the history’s length and the white noise’s does not, so the transitory part’s weight, $6\sigma_{\xi}^{2}/(T\sigma_{\psi}^{2}+6\sigma_{\xi}^{2})$, falls as the history lengthens.

Now $y_{t}=\tau\,p_{t-1}+(\psi_{t}+\xi_{t})$ and $y_{t-1}=p_{t-1}+\xi_{t-1}$, so AttenuationByNoise applies with $k=\tau$. Since $\psi_{t}$ is independent of $p_{t-1}$, $\operatorname{var}(p)=\tau^{2}\operatorname{var}(p)+\sigma_{\psi}^{2}$, which gives ${\operatorname{var}(p)=\sigma_{\psi}^{2}/(1-\tau^{2})}$. In a long history the sample variances and covariances approach these population values. Substituting it and multiplying numerator and denominator by $1-\tau^{2}$ gives the result.

4Other Facts¶