Economics extended problems (resonance framework)

Problem 4 – Resonant supply-demand equilibrium#

Supply and demand curves are modified by resonance:

$$ S(p) = D_3 τ_r p, \quad D(p) = \frac{X}{p + ΛΘ}. $$

1. Solve for the equilibrium price

   p^* where S(p^*) = D(p^*)
   

2. If $$τ_r$$ increases, does the equilibrium price rise or fall?

Problem 5 – Capital accumulation under triadic growth#

Capital stock evolves as

$$ K(t) = K_0 e^{D_6 τ_r t}. $$

If $$τ_r$$ decreases by 15%, how does the long-run growth rate change?

Problem 6 – Resonant consumption smoothing#

A household’s consumption smoothing factor is

$$ C_s = \frac{T_f}{1 + e^{-D_3 τ_r}}. $$

1. Sketch the qualitative shape of $$C_s(τ_r)$$.
2. If $$τ_r$$ increases, does smoothing become stronger or weaker?

Problem 7 – Exchange rate resonance#

An exchange rate is modeled as

$$ E = X \sqrt{τ_r} - D_9. $$

1. If $$τ_r$$ quadruples, how does the first term change?
2. What is the qualitative effect on the exchange rate?

Problem 8 – Government spending multiplier under resonance#

A multiplier is modeled as

$$ M = \frac{ΛΘ + T_f}{D_3 + τ_r}. $$

If $$ΛΘ$$ increases by 10%, $$T_f$$ decreases by 5%, and $$τ_r$$ increases by 20%, what is the qualitative net effect on $$M$$?