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

## 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 $$-A + (1-b)Y$$.
• $$I = I_0$$ is desired investment, which we set to a constant

\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
• $$G = G_0$$ is desired government expenditure, which we set to a constant

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

## Government Policy

The budget balance is $$T - G$$ where $$T$$ is net tax revenue and $$G$$ is government expenditure.