The discrete-time mean-reverting process can be modeled as: $\Delta S_t = S_t - S_{t-1} = a(b - S_{t-1})$

Where:

• $\Delta S_t$ is the change in the variable ($50 - 30 = 20$)
• $a$ is the mean reversion rate ($0.4$)
• $b$ is the long-run mean value
• $S_{t-1}$ is the previous value ($30$)

Plugging the given values into the equation: $50 - 30 = 0.4 \times (b - 30)$`$20` = 0.4 \times (b - 30)$$b - 30 = \frac{20}{0.4}$$b - 30 = 50$$b = 80$

The long-run mean value for the variable is 80.