Lucas Diversification

Background

See Microeconomic Origins of Aggregate Fluctuations.

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The Model

Consider the following economy: the economics is composed of a set $K$ production entities, note that these entities include factories, firms, sectors… (not necessarily to be firms). Factors are supplied inelastically, only influenced by productivity shocks. And entity $k\in K$ is subject to idiosyncratic (i.i.d.) output $a_{kt}$ . All these sum up, we have the GDP.

$$Y_t=\sum _{k\in K}{a}_{kt}$$

The GDP volatility in this economy

Since we already assume, the only supply volatility comes from productivity shocks.

$\frac{\Delta a_{k,t}}{a_{k,t-1}}=\frac{a_{k,t}-a_{k,t-1}}{a_{k,t-1}}={\sigma }_k{\epsilon }_{k,t}$

where ${ϵ}_{kt}$ is mean 0, var 1. GDP growth:

$\frac{\Delta Y_t}{Y_{t-1}}=\frac{1}{Y_{t-1}}{\sum }_{k\in K}\Delta a_{k,t}={\sum }_{k\in K}\frac{a_{k,t-1}}{Y_{t-1}}{\sigma }_k{\epsilon }_{k,t}$

Here, $\sigma$ is entity $k$ 的波动率参数，而 $ϵ$ 是标准化的冲击

Shocks are independent/uncorrelated, hence $\begin{array}{r}{\sigma }_Y=\sqrt{\mathsf{var}\frac{\Delta Y_t}{Y_{t-1}}}\end{array}$ is:

${\sigma }_Y={\left({\sum }_{k\in K}{\sigma }_k^2{\left[\frac{a_{k,t-1}}{Y_{t-1}}\right]}^2\right)}^{0.5}$

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Reference