This is for equation reference only, for example, when you’re lost as hell during lectures!
I will make real cheat sheets that explain *what* things are.

## Calculating GDP

\[Y = C_a + I_a + G_a + NX_a\]where

- \(Y\) is
**GDP** - \(C_a\) is
**actual consumption** - \(I_a\) is
**actual investment**, which is equal to gross investment less depreciation - \(G_a\) is
**actual government expenditure** - \(NX_a = X_a - IM_a\) is
**actual net exports**- \(X_a\) is
**actual exports** - \(IM_a\) is
**actual imports**

- \(X_a\) is

## Simple Short-Run Model

Equilibrium GDP \(Y_e\) occurs when \(Y_e = AE\), the **aggregate expenditure**.

We have a few basic metrics:

- The
**autonomous spending**\(A\) is \(AE\) when \(Y = 0\) - \(\frac{\Delta Y}{\Delta AE} = z\) is the
**marginal propensity to spend** - \(\frac{\Delta A}{\Delta Y_e} = \frac{1}{1-z}\) is the
**simple multiplier**

All our models will be *linear*, that is, \(AE = A + zY\).
This should make it easy to identify \(A\) and \(z\).
The term with no \(Y\)-dependency is called *autonomous* and with is called *induced*.

Assuming linearity also gives us the really nice equation \(Y_e = \frac{A}{1-z}\).

This also gives a relation for change in marginal propensity to spend: \(Y_{e_2} = \frac{1-z_1}{1-z_2}Y_{e_1}\).

### Super Simple Model

In our simple model, we **ignore government and trade**, so that we can say

\(\begin{align} AE & = C + I \\ & = \underbrace{(a + I_0)}_{A} + \underbrace{(b)}_zY \end{align}\)

where

- \(C = a + bY\) is
**desired consumption**- \(a\) is
**autonomous consumption** - \(b\) is the
**marginal propensity to consume**. The opposite (*marginal propensity to save*) is \(1-b\). The**savings function**is \(-A + (1-b)Y\).

- \(a\) is
- \(I = I_0\) is
**desired investment**, which we set to a constant

### Adding Government

We now add government expenditure.

\(\begin{align} AE & = C + I + G \\ & = \underbrace{(a + I_0 - bT_0 + G_0)}_{A} + \underbrace{b(1-t)}_{z}Y \end{align}\)

where the only changes from above are that

- \(C = a + bY_D\)
- \(Y_D = Y - T\) is
**disposable income**- \(T = T_0 + tY\) is
**net tax revenue**- \(T_0\) is
**autonomous tax**that doesn’t depend on GDP, including transfers like CERB - \(t\) is the
**net tax rate**of all taxes net of subsidies

- \(T_0\) is

- \(T = T_0 + tY\) is

- \(Y_D = Y - T\) is
- \(G = G_0\) is
**desired government expenditure**, which we set to a constant

### Adding Trade

Adding the most annoying thing of all: the rest of the world.

\(\begin{align} AE & = C + I + G + NX \\ & = \underbrace{(a + I_0 - bT_0 + G_0 + X_0)}_{A} + \underbrace{(b(1-t) - m)}_{z}Y \end{align}\)

where the variables are the same except

- \(NX = X - IM\) is
**desired net exports**- \(X = X_0\) is
**desired exports**, which we set to a constant - \(IM = mY\) is
**desired imports**- \(m\) is the
**marginal propensity to import**

- \(m\) is the

- \(X = X_0\) is

## Government Policy

The **budget balance** is \(T - G\) where \(T\) is net tax revenue and \(G\) is government expenditure.