Part 1 - Question 2.c - IS-LM-PC model · Introducció a la Macroeconomia UPF 2018

a. State the three structural equations of the model and show analytically and graphically how a tax reduction stimulates output beyond potential and impacts inflation.

Solución: 1. IS relation: $Y = C(Y - T) + I(Y, r + x) + G$ 2. LM relation (monetary policy rule): $r = \bar{r}$ 3. PC relation: $\pi - \bar{\pi} = \frac{\alpha}{L}(Y - Y_n)$ A tax cut ($\Delta T < 0$) raises disposable income and consumption, shifting the IS curve rightward. At the policy rate $\bar{r}$, short-run output expands above potential ($Y > Y_n$), generating a positive output gap that drives inflation above target ($\pi - \bar{\pi} > 0$).

b. Use the IS-LM-PC framework to depict the policy response implemented by the Federal Reserve to avert an acceleration in inflation.

Solución: The Fed raises the policy rate from $\bar{r}$ to $\bar{r}'$, shifting the LM line upward. Higher borrowing costs decrease investment and consumption, moving the economy along $IS'$ back to potential output $Y = Y_n$, which stabilizes inflation at target $\bar{\pi}$.

Graphical depiction of the IS-LM-PC framework for the tax cut and central bank reaction.

Solución: Top panel (IS-LM): $Y$ on horizontal axis, $r$ on vertical axis. Initial equilibrium at $(Y_n, \bar{r})$. Tax cut shifts IS rightward to $IS'$, increasing output to $Y > Y_n$. Fed raises rate to $\bar{r}'$, shifting LM upward to $LM'$, restoring output to $Y_n$. Bottom panel (PC): $Y$ on horizontal axis, $\pi - \bar{\pi}$ on vertical axis. Upward-sloping PC passing through $(Y_n, 0)$. When $Y > Y_n$, $\pi - \bar{\pi} > 0$; when Fed contracts back to $Y_n$, $\pi - \bar{\pi} = 0$.