31  Formula sheet

Print this. Bring it.

31.1 Identities

Y \equiv C + S \qquad S \equiv Y - C \qquad MPC + MPS = 1 \qquad APC + APS = 1

31.2 Consumption

C = \overline{C_0} + c \cdot Y_d \qquad Y_d = Y - \overline{T}

31.3 Equilibrium income

Closed: Y^* = \frac{\overline{C_0} + \overline{I}}{1-c}

Open with government: Y^* = \tfrac{1}{1-c}\,\overline{C_0} - \tfrac{c}{1-c}\,\overline{T} + \tfrac{1}{1-c}\,\overline{I} + \tfrac{1}{1-c}\,\overline{G} + \tfrac{1}{1-c}\,\overline{X} - \tfrac{1}{1-c}\,\overline{M}

31.4 Multipliers

K_X \equiv \frac{\Delta Y^*}{\Delta\overline{X}}

variable	multiplier
\overline{I}, \overline{C_0}, \overline{G}, \overline{X}	\dfrac{1}{1-c}
\overline{T}	-\dfrac{c}{1-c}
\overline{M}	-\dfrac{1}{1-c}
Balanced budget (\overline{G} and \overline{T} together)	K_{BB} = 1

At c = 0.75: K_G = 4, K_T = -3, K_{BB} = 1.

31.5 Inflation, CPI, unemployment

\text{CPI}_t = \frac{\sum P_t \cdot Q_{base}}{\sum P_{base} \cdot Q_{base}} \cdot 100 \qquad \pi_t = \frac{\text{CPI}_t - \text{CPI}_{t-1}}{\text{CPI}_{t-1}} \cdot 100

u = \frac{U}{E + U} \qquad LF = E + U \qquad \text{LFP} = \frac{LF}{\text{Pop}}

31.6 Money & banking

RR = rrr \cdot DD_p \qquad ER = TR - RR K_S = \frac{1}{rrr} \qquad \Delta M^S = K_S \cdot \Delta R

31.7 Bonds

PV = \frac{FV}{(1+r)^T} \qquad r = \left(\frac{FV}{PV}\right)^{1/T} - 1

31.8 PPP

E_{\$/£} = \frac{P_{US}}{P_{UK}}

31.9 Direction shortcuts

shock	effect
\uparrow M^S	\downarrow r^*
OMO purchase	\uparrow M^S, \uparrow bond prices, \downarrow r^*
OMO sale	\downarrow M^S, \downarrow bond prices, \uparrow r^*
\uparrow Y	\uparrow M^D, \uparrow r^*
Higher inflation country	currency depreciates
Higher interest rate country	currency appreciates
Currency depreciation	\uparrow EX, \downarrow IM, \uparrow Y^*, \uparrow P
Monetary policy, floating ER	both r channel and FX channel work
Monetary policy, fixed ER	only r channel; FX channel dead