Economics core problem solutions

Solution to Problem 1 – Market resonance cycles#

We have

$$ ω = \frac{T_f}{D_3}. $$

To keep $$ω$$ constant when $$D_3$$ increases to $$D_3'$$, we need

$$ \frac{T_f'}{D_3'} = \frac{T_f}{D_3}. $$

Thus,

$$ T_f' = T_f \frac{D_3'}{D_3}. $$

Answer: Increase $$T_f$$ in direct proportion to the increase in $$D_3$$.

Solution to Problem 2 – Utility resonance#

Utility is

$$ U = X \ln(1 + τ_r). $$

If $$τ_r$$ increases by 25%:

$$ τ_r' = 1.25 τ_r. $$

Then

$$ U' = X \ln(1 + 1.25 τ_r). $$

The ratio is

$$ \frac{U'}{U} = \frac{\ln(1 + 1.25 τ_r)}{\ln(1 + τ_r)}. $$

Answer: Utility increases, but only logarithmically — less than 25% in most cases.

Solution to Problem 3 – Inflation drift#

Inflation drift is

$$ I = ΛΘ D_9. $$

• $$Λ$$ decreases by 10% → becomes $$0.9Λ$$
• $$D_9$$ increases by 5% → becomes $$1.05D_9$$

Thus,

$$ I' = (0.9Λ)Θ(1.05D_9) = 0.945 I. $$

Answer: Inflation decreases by 5.5%.