Ejercicio 4 - Section B: IS-LM-PC model · Introducció a la Macroeconomia UPF 2024

a) Illustrate graphically the initial equilibrium across the three relevant diagrams and c) Show graphically the resulting changes in the three diagrams

Solución: a. Initial Equilibrium Diagrams: - Labor Market: Real wage $W/P$ on the vertical axis, unemployment rate $u$ on the horizontal axis. Downward-sloping Wage-Setting relation $WS: W/P = F(u, z)$ and horizontal Price-Setting relation $PS: W/P = \frac{1}{1+m}$. The intersection defines the natural rate of unemployment $u_n$ and the equilibrium real wage $(W/P)^*$. - Goods & Financial Markets (IS-LM): Real interest rate $r$ on the vertical axis, output $Y$ on the horizontal axis. Downward-sloping $IS$ curve and horizontal $LM$ line at $r = \bar{r}$. Intersection occurs at $Y = Y_n = L(1 - u_n)$. - Phillips Curve: Change in inflation $\pi - \pi^e$ (or $\pi$) on the vertical axis, output $Y$ on the horizontal axis. Upward-sloping $PC$ crossing the horizontal axis (where $\pi = \pi^e = 2\%$) exactly at $Y = Y_n$. c. Impact of the Supply Shock: - In the Labor Market, the increase in mark-up from $m$ to $m'$ shifts the horizontal $PS$ curve downward to $\frac{1}{1+m'}$. The intersection with $WS$ moves to a higher natural rate of unemployment $u'_n > u_n$. - In the Goods & Financial Markets (IS-LM), potential output contracts to $Y'_n = L(1 - u'_n)$. The initial $IS$ and $LM$ curves remain unchanged. - In the Phillips Curve diagram, the $PC$ curve shifts leftward/upward because the output level consistent with stable inflation is now lower ($Y'_n$).

b) Write the generic mathematical equations represented in each of the three diagrams of part 1. (Labor market: WS and PS)

Solución: WS: W/P = F(u, z), PS: W/P = 1 / (1 + m)

b) Write the generic mathematical equations represented in each of the three diagrams of part 1. (IS-LM)

Solución: IS: Y = C(Y - T) + I(Y, r + x) + G, LM: r = \bar{r}

b) Write the generic mathematical equations represented in each of the three diagrams of part 1. (Phillips curve)

Solución: \pi_t - \pi_t^e = (\alpha / L)(Y_t - Y_n)

c) Explain their economic mechanisms.

Solución: An increase in mark-up from $m$ to $m'$ shifts the horizontal $PS$ curve downward to $\frac{1}{1+m'}$. Real wages fall, and the intersection with $WS$ occurs at a higher natural rate of unemployment $u'_n > u_n$. Since $Y'_n = L(1 - u'_n)$, potential output contracts. In the PC diagram, the curve shifts leftward/upward because the output level consistent with stable inflation is now lower ($Y'_n$). The IS and LM curves are initially unaffected by the supply-side shock.

d) If the central bank keeps the real interest rate unchanged at t+1, describe the new short-run equilibrium across all markets. What occurs with inflation? Is the economy in a medium-run equilibrium?

Solución: Because the ECB maintains $r = \bar{r}$ and fiscal variables are unchanged, output remains at the initial level $Y_t = Y_n > Y'_n$. The economy now operates with a positive output gap relative to the new lower capacity ($Y - Y'_n > 0$). As a consequence, inflation rises above previous expectations ($\pi_{t+1} > \pi_t$). The economy is not in medium-run equilibrium because production exceeds potential output and inflation is accelerating.

e.1. Describe the policy applied by the ECB and illustrate the new equilibrium on the IS-LM and PC graphs.

Solución: To stabilize inflation, the ECB must raise the real interest rate to $r'$, shifting the horizontal LM curve upward until output contracts from $Y$ to the new potential output $Y'_n$. In the PC space, the economy moves down along the new PC curve until the output gap is zero ($Y = Y'_n$).

e.2. What will the inflation rate be in period t+3 once the central bank sets output equal to the new potential output?

Solución: 8

f) What will the inflation rate be in period t+3 if the ECB applies the same policy bringing output to its new potential level?

Solución: 2

f) Contrast this result with that of question e.

Solución: With anchored expectations, stabilizing output at its natural level is sufficient to bring inflation immediately back to target without requiring an economic downturn. Under adaptive expectations, closing the output gap merely stabilizes inflation at its prevailing high rate (8%), meaning disinflation requires an engineered recession.