Question

Can you explain the paper Good and Bad Uncertainty: Macroeconomic and Financial Market Implications (Gill Segal et. al) with math examples?

Answer 1

Consider an economy with a continuum of firms with productivity $\theta^i \backsim N(\bar{\theta},\sigma^2_\theta).$

Assume linear technologies: $\frac{dB^i_t}{B^i_t}=\theta^idt+\sigma dW_t+\sigma_1dW_{i,t}$

Aggregate capital $B_t=\int B^i_tdt$

The law of large numbers $\implies\frac{dB_t}{B_t}=(\bar{\theta}+\frac{1}{2}\sigma^2_\theta t)dt+\sigma dW_{t}$

 higher dispersion of productivity increase the drift rate of aggregate capital (and hence, investments, consumption etc).

Moreover, aggregate price is increasing in $σ_θ$ . Given a log-normal SDF and assuming only one dividend

$D_T = B_T$ at T we have

$\frac{M_t}{B_t}=e^{(\bar{\theta}-r-\sigma \sigma_M)(T-t)+\frac{1}{2}\sigma^2_\theta(T-t)^2}$

In this model:

– $σ_θ =$ Good uncertainty

–$σ, σ_M =$ Bad uncertainty